Literature

BTZ Solutions

1. • The BTZ black hole as a Lorentz-flat geometry
   • May 26, 2014 | Pedro D. Alvarez, Pablo Pais, Eduardo Rodriguez, Patricio Salgado-Rebolledo, Jorge Zanelli
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     It is shown that 2+1 dimensional anti-de Sitter spacetimes are Lorentz-flat. This means, in particular, that any simply-connected patch of the BTZ black hole solution can be endowed with a Lorentz connection that is locally pure gauge. The result can be naturally extended to a wider class of black hole geometries and point particles in three-dimensional spacetime.

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   • [ pdf | 1405.6657 [gr-qc] | Phys. Lett. B server | inspire | doi ]
2. • Phase transition and entropy spectrum of BTZ black hole in three-dimensional gravity with torsion
   • Oct 5, 2013 | Meng-Sen Ma, Ren Zhao
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     In this paper, we study the phase transition and the entropy spectrum of BTZ black hole obtained in a model of three-dimensional gravity with torsion. By calculating the heat capacity we find that the BTZ black hole we considered will experience phase transition at some critical point. This indicates that the critical behaviors of black holes do not only depend on the spacetime metric, but have to do with the theory of gravity under consideration. In addition we derived the entropy spectrum of the BTZ black hole according to the quasinormal modes(QNMs) and the adiabatic invariance. It shows that the area or entropy spectrum will also rely on the concrete gravitational action.

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     • torsion may be linked to thermodynamical properties of black holes
   • [ pdf | 1310.1491 [gr-qc] | Phys. Rev. D server | inspire | doi ]
3. • Exact vacuum solution of a (1 + 2)-dimensional Poincaré gauge theory: BTZ solution with torsion
   • Jun 20, 2003 | Alberto A. Garcia, Friedrich W. Hehl, Christian Heinicke, Alfredo Macias
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     In the framework of (1+2)-dimensional Poincare gauge gravity, we start from the Lagrangian of the Mielke-Baekler (MB) model that depends on torsion and curvature and includes translational and Lorentzian Chern-Simons terms. We find a general stationary circularly symmetric vacuum solution of the field equations. We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution with torsion. In this framework, we discuss Cartan's (3-dimensional) spiral staircase and find that it is not only a special case of our new vacuum solution, but can alternatively be understood as a solution of the 3-dimensional Einstein-Cartan theory with matter of constant pressure and constant torque.

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   • [ pdf | gr-qc/0302097v2 | Phys. Rev. D server | inspire | doi ]
4. • The (2+1)-Dimensional Black Hole
   • Jun 29, 1995 | Steven Carlip
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     I review the classical and quantum properties of the (2+1)-dimensional black hole of Banados, Teitelboim, and Zanelli. This solution of the Einstein field equations in three spacetime dimensions shares many of the characteristics of the Kerr black hole: it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse; it exhibits mass inflation; and it has a nonvanishing Hawking temperature and interesting thermodynamic properties. At the same time, its structure is simple enough to allow a number of exact computations, particularly in the quantum realm, that are impractical in 3+1 dimensions.

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     • p. 28: Einstein–Hilbert action is a Chern–Simons action already in (2+1) dimensions
     • p. 32: boundary degrees of freedom in quantum gravity provide a possible explanation for the thermodynamic properties of black holes
   • [ pdf | gr-qc/9506079v3 | Class. Quant. Grav. server ]
5. • Geometry of the 2 + 1 Black Hole
   • Feb 10, 1993 | Máximo Bañados, Marc Henneaux, Claudio Teitelboim, Jorge Zanelli
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     The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti-de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface r = 0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at r = 0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti-de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum . A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an Appendix.

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     • p. 14: Killing vectors induce isometries (clear), building the quotient space may induce closed timelike curves
   • [ pdf | gr-qc/9302012v1 | Phys. Rev. D server ]
6. • The Black Hole in Three Dimensional Space Time
   • Apr 29, 1992 | Máximo Bañados, Claudio Teitelboim, Jorge Zanelli
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     The standard Einstein-Maxwell equations in 2+1 spacetime dimensions, with a negative cosmological constant, admit a black hole solution. The 2+1 black hole -characterized by mass, angular momentum and charge, defined by flux integrals at infinity- is quite similar to its 3+1 counterpart. Anti-de Sitter space appears as a negative energy state separated by a mass gap from the continuous black hole spectrum. Evaluation of the partition function yields that the entropy is equal to twice the perimeter length of the horizon.

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   • [ pdf | hep-th/9204099v3 | Phys. Rev. Lett. server | inspire ]

Charged BTZ Solutions

1. • Five-dimensional Janis-Newman algorithm
   • Nov 7, 2014 | Harold Erbin, Lucien Heurtier
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     The Janis-Newman algorithm has been shown to be successful in finding new stationary solutions of four-dimensional gravity. Attempts for a generalization to higher dimensions have already been found for the restricted cases with only one angular momentum. In this paper we propose an extension of this algorithm to five dimensions with two angular momenta - using the prescription of G. Giampieri - through two specific examples, that are the Myers-Perry and BMPV black holes. We also discuss possible enlargements of our prescriptions to other dimensions and maximal number of angular momenta, and show how dimensions higher than six appear to be much more challenging to treat within this framework. Nonetheless this general algorithm provides a unification of the formulation in d = 3, 4, 5 of the Janis-Newman algorithm, from which which expose several examples including the BTZ black hole.

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   • [ pdf | 1411.2030 [gr-qc] ]
2. • Untangling the Newman-Janis algorithm
   • Nov 15, 2013 | Rafael Ferraro
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     Newman-Janis algorithm for Kerr-Newman geometry is reanalyzed in the light of Cartan calculus.

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   • [ pdf | 1311.3946 [gr-qc] ]
3. • Spinning BTZ Black Hole versus Kerr Black Hole : A Closer Look
   • Sept 16, 1998 | Hongsu Kim
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     By applying Newman's algorithm, the AdS_3 rotating black hole solution is ``derived'' from the nonrotating black hole solution of Banados, Teitelboim, and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in ``Boyer-Lindquist-type'' coordinates whereas the form of the solution originally given by BTZ is given in a kind of an ``unfamiliar'' coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two time coordinates is carefully studied. Since the Kerr-type and Boyer-Lindquist-type coordinates for rotating BTZ solution are newly found via Newman's algorithm, next, the transformation to Kerr-Schild-type coordinates is looked for. Indeed, such transformation is found to exist. And in this Kerr-Schild-type coordinates, truely maximal extension of its global structure by analytically continuing to ``antigravity universe'' region is carried out.

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   • [ pdf | gr-qc/9809047 ]
4. • Charged rotating black hole in three space-time dimensions
   • Dec 27, 1999 | Cristian Martinez, Claudio Teitelboim, Jorge Zanelli
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     The generalization of the black hole in three-dimensional spacetime to include an electric charge Q in addition to the mass M and the angular momentum J is given. The field equations are first solved explicitly when Q is small and the general form of the field at large distances is established. The total ``hairs'' M, J and Q are exhibited as boundary terms at infinity. It is found that the inner horizon of the rotating uncharged black hole is unstable under the addition of a small electric charge. Next it is shown that when Q=0 the spinning black hole may be obtained from the one with J=0 by a Lorentz boost in the ??t plane. This boost is an ``illegitimate coordinate transformation'' because it changes the physical parameters of the solution. The extreme black hole appears as the analog of a particle moving with the speed of light. The same boost may be used when Q?0 to generate a solution with angular momentum from that with J=0, although the geometrical meaning of the transformation is much less transparent since in the charged case the black holes are not obtained by identifying points in anti-de Sitter space. The metric is given explicitly in terms of three parameters, M~, Q~ and ? which are the ``rest mass'' and ``rest charge'' and the angular velocity of the boost. These parameters are related to M, J and Q through the solution of an algebraic cubic equation. Altogether, even without angular momentum, the electrically charged 2+1 black hole is somewhat pathological since (i) it exists for arbitrarily negative values of the mass, and (ii) there is no upper bound on the electric charge.

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   • [ pdf | hep-th/9912259 | Phys. Rev. D server | inspire ]
5. • On the rotating charged BTZ metric
   • Sep 15, 1999 | Alberto Garcia
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     It is shown that the charged non-diagonal BTZ (2+1)-spacetime is not a solution of the Einstein-Maxwell field equations with cosmological constant.

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   • [ pdf | hep-th/9909111 | inspire ]
6. • Rotating Charged Solutions to Einstein-Maxwell-Chern-Simons Theory in 2+1 Dimensions
   • May 9, 1997 | Sharmanthie Fernando, Freydoon Mansouri
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     We obtain a class of rotating charged stationary circularly symmetric solutions of Einstein-Maxwell theory coupled to a topological mass term for the Maxwell field. These solutions are regular, have finite mass and angular momentum, and are asymptotic to the uncharged extreme BTZ black hole.

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     • Chern–Simons part in action makes abelian gauge bosons become massive
   • [ pdf | gr-qc/9510025 | inspire ]
7. • 2+1 dimensional charged black hole with (anti-)self dual Maxwell fields
   • Jan 9, 1997 | Masaru Kamata, Takao Koikawa
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     We discuss the exact electrically charged BTZ black hole solutions to the Einstein-Maxwell equations with a negative cosmological constant in 2+1 spacetime dimensions assuming a (anti-)self dual condition between the electromagnetic fields. In a coordinate condition there appears a logarithmic divergence in the angular momentum at spatial infinity. We show how it is to be regularized by taking the contribution from the boundary into account. We show another coordinate condition which leads to a finite angular momentum though it brings about a peculiar spacetime topology.

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   • [ pdf | hep-th/9605114 | Phys. Lett. B server | inspire ]
8. • Spinning charged BTZ black holes and self-dual particle-like solutions
   • Jan 18, 1996 | Gérard Clément
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     We generate from the static charged BTZ black hole a family of spinning charged solutions to the Einstein-Maxwell equations in 2+1 dimensions. These solutions go over, in a suitable limit, to self-dual spinning charged solutions, which are horizonless and regular, with logarithmically divergent mass and spin. To cure this divergence, we add a topological Chern-Simons term to the gauge field action. The resulting self-dual solution is horizonless, regular, and asymptotic to the extreme BTZ black hole.

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   • [ pdf | gr-qc/9510025 | Phys. Lett. B server | inspire ]

Topological Gauge Theories of Gravity (Mielke-Baekler, massive gravity)

1. • Minimal Massive 3D Gravity
   • Apr 10, 2014 | Eric Bergshoeff, Olaf Hohm, Wout Merbis, Alasdair J. Routh, Paul K. Townsend
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     We present an alternative to Topologically Massive Gravity (TMG) with the same "minimal" bulk properties; i.e. a single local degree of freedom that is realized as a massive graviton in linearization about an anti-de Sitter (AdS) vacuum. However, in contrast to TMG, the new "minimal massive gravity" has both a positive energy graviton and positive central charges for the asymptotic AdS-boundary conformal algebra.

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   • [ pdf | 1404.2867 [gr-qc] | Class. Quant. Grav. server ]
2. • Dirac field in topologically massive gravity
   • Nov 2, 2011 | Özcan Sert, Muzaffer Adak
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     We consider a Dirac field coupled minimally to the Mielke-Baekler model of gravity and investigate cosmological solutions in three dimensions. We arrive at a family of solutions which exists even in the limit of vanishing cosmological constant.

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   • [ pdf | 1111.0395 [gr-qc] | Gen. Relat. Grav. server ]
3. • SU(2) gauge theory of gravity with topological invariants
   • Oct 19, 2011 | Sandipan Sengupta
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     The most general gravity Lagrangian in four dimensions contains three topological densities, namely Nieh-Yan, Pontryagin and Euler, in addition to the Hilbert-Palatini term. We set up a Hamiltonian formulation based on this Lagrangian. The resulting canonical theory depends on three parameters which are coefficients of these terms and is shown to admit a real SU(2) gauge theoretic interpretation with a set of seven first-class constraints. Thus, in addition to the Newton's constant, the theory of gravity contains three (topological) coupling constants, which might have non-trivial imports in the quantum theory.

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   • [ pdf | 1110.4185 [gr-qc] | J. Phys. server ]
4. • Conserved charges in 3D gravity
   • Mar 19, 2010 | M. Blagojevic, B. Cvetkovic
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     The covariant canonical expression for the conserved charges, proposed by Nester, is tested on several solutions in 3D gravity with or without torsion and topologically massive gravity. In each of these cases, the calculated values of energy-momentum and angular momentum are found to satisfy the first law of black hole thermodynamics.

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   • [ pdf | 1003.3782 [gr-qc] | inspire ]
5. • S. S. Chern and Chern-Simons Terms
   • Aug 26, 2004 | R. Jackiw
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     Some properties of Chern-Simons terms are presented and their physical utility is surveyed.

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   • [ pdf | math-ph/0408051 | inspire ]
6. • Chern-Simons Reduction and non-Abelian Fluid Mechanics
   • Apr 11, 2000 | R. Jackiw, V.P. Nair, So-Young Pi
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     We propose a non-Abelian generalization of the Clebsch parameterization for a vector in three dimensions. The construction is based on a group-theoretical reduction of the Chern-Simons form on a symmetric space. The formalism is then used to give a canonical (symplectic) discussion of non-Abelian fluid mechanics, analogous to the way the Abelian Clebsch parameterization allows a canonical description of conventional fluid mechanics.

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   • [ pdf | hep-th/0004084 | inspire | Phys. Rev. D server | doi ]
7. • Torsional Topological Invariants (and their relevance for real life)
   • Aug 26, 1997 | Osvaldo Chandia, Jorge Zanelli
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     The existence of topological invariants analogous to Chern/Pontryagin classes for a standard SO(D) or SU(N) connection, but constructed out of the torsion tensor, is discussed. These invariants exhibit many of the features of the Chern/Pontryagin invariants: they can be expressed as integrals over the manifold of local densities and take integer values on compact spaces without boundary; their spectrum is determined by the homotopy groups ?D?1(SO(D)) and ?D?1(SO(D+1)).

     These invariants are not solely determined by the connection bundle but depend also on the bundle of local orthonormal frames on the tangent space of the manifold. It is shown that in spacetimes with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature.

     Explicit examples of topologically stable configurations carrying nonvanishing instanton number in four and eight dimensions are given, and they can be conjectured to exist in dimension 4k. It is also shown that the chiral anomaly in a spacetime with torsion receives a contribution proportional to this instanton number and hence, chiral theories in 4k-dimensional spacetimes with torsion are potentially anomalous.

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   • [ pdf | hep-th/9708138 | inspire ]
8. • Topological Invariants, Instantons and Chiral Anomaly on Spaces with Torsion
   • Feb 3, 1997 | Osvaldo Chandia, Jorge Zanelli
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     In a spacetime with nonvanishing torsion there can occur topologically stable configurations associated with the frame bundle which are independent of the curvature. The relevant topological invariants are integrals of local scalar densities first discussed by Nieh and Yan (N-Y). In four dimensions, the N-Y form N=(Ta?Ta?Rab?ea?eb) is the only closed 4-form invariant under local Lorentz rotations associated with the torsion of the manifold. The integral of N over a compact D-dimensional (Euclidean) manifold is shown to be a topological invariant related to the Pontryagin classes of SO(D+1) and SO(D). An explicit example of a topologically nontrivial configuration carrying nonvanishing instanton number proportional to ?N is costructed. The chiral anomaly in a four-dimensional spacetime with torsion is also shown to contain a contribution proportional to N, besides the usual Pontryagin density related to the spacetime curvature. The violation of chiral symmetry can thus depend on the instanton number of the tangent frame bundle of the manifold. Similar invariants can be constructed in D>4 dimensions and the existence of the corresponding nontrivial excitations is also discussed.

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   • [ pdf | hep-th/9702025 | Phys. Rev. D server | inspire ]
9. • Dynamical Symmetries in Topological 3D Gravity with Torsion
   • Jul 22, 1991 | P. Baekler, E. W. Mielke, F. W. Hehl
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     Recently two of us generalized the topological massive gauge model of gravity of Deser, Jackiw, and Templeton (DJT) by liberating its translational gauge degrees of freedom. Consequently, the new 3?SO(1,2) gauge model «lives» in a 3-dimensional Riemann-Cartan space-time with torsion. The extended Lagrangian consists, of the familiar Einstein-Cartan term, the Chern-Simons 3-form for the curvature, and, in addition, of a new translational Chern-Simons term. In this article we uncover a «dynamical symmetry» of the new theory by inquiring how the two Noether identities, the two Bianchi identities, and the two field equations are interrelated to each other. This includes two important subcases in which the first Bianchi identity is mapped into the second one and the first (energy-momentum) Noether into the second (angular-momentum) Noether identity. As a furtherexact result, the topological gauge field equations imply a covariant Proca-type field equation, for the translational gauge potential,i.e. the coframe. Thus the theory encompasses massive gravitons, as in the DJT model.

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     • p. 93: EC field equations map the two Noether identities for total angular momentum and energy-momentum into the two Bianch identities. This can even be extended to spaces carrying nonmetricity.
   • [ pdf | Springer server | inspire ]
10. • Topological gauge model of gravity with torsion
    • Apr 30, 1991 | E. W. Mielke, P. Baekler
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      We generalize the topological massive gauge model of gravity to a three-dimensional Riemann-Cartan spacetime with torsion. Our Lagrangian consists of the familiar Einstein-Cartan term, the Chern-Simons three-form for the curvature and, in addition, of a new "translational" Chern-Simons term. An exact vacuum solution is derived with purely axial torsion and constant curvature.

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    • [ pdf | Phys. Lett. A server ]
11. • Three-Dimensional Massive Gauge Theories
    • Apr 30, 1991 | S. Deser, R. Jackiw, S. Templeton
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      Three-dimensional Yang-Mills and gravity theories augmented by gauge-invariant mass terms are analyzed. These topologically nontrivial additions profoundly alter the particle content of the models and lead to quantization of a dimensionless mass-coupling-constant ratio. The vector field excitations become massive, with spin 1 (rather than massless with spin 0), and the mass provides an infrared cutoff. The gravitation acquires mass, mediates finite-range interactions, and has spin 2 (rather than being absent altogether); although its mass term is of third derivative order, there are no ghosts or acausalities.

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    • [ pdf | Phys. Rev. Lett. server | inspire | doi ]
12. • An Identity in Riemann–Cartan Geometry
    • Feb 24, 1981 | H.T. Nieh, M.L. Yan
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      We derive a new Gauss-Bonnet type identity in Riemann-Cartan geometry: (?g)1/2????? (R ???? + (1/2) C ? ?? C ???) = ?? (?(?g)1/2????? C ????), where C ? ?? is the torsion tensor.

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13. • Introduction to the Yang-Mills quantum theory
    • Oct 1, 1980 | R. Jackiw
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      We propose a non-Abelian generalization of the Clebsch parameterization for a vector in three dimensions. The construction is based on a group-theoretical reduction of the Chern-Simons form on a symmetric space. The formalism is then used to give a canonical (symplectic) discussion of non-Abelian fluid mechanics, analogous to the way the Abelian Clebsch parameterization allows a canonical description of conventional fluid mechanics.

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      • vacuum angle and topological implications explained
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    • [ pdf | inspire | Rev. Mod. Phys. server | doi ]
14. • Vacuum Periodicity in a Yang-Mills Quantum Theory
    • Jul 19, 1976 | R. Jackiw, C. Rebbi
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      We propose a description of the vacuum in Yang-Mills theory and arrive at a physical interpretation of the pseudoparticle solution and the attendant violation of symmetries. The existence of topologically inequivalent classical gauge fields gives rise to a family of quantum mechanical vacua, parametrized by a CP-nonconserving angle. The requirement of vacuum stability against gauge transformations renders the vacua chirally noninvariant.

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      • vacuum angle
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    • [ pdf | inspire | Phys. Rev. Lett. server | doi ]

Riemann–Cartan Geometry, Gravitation as a Gauge Theory

1. • Nonlocal Gravity: The General Linear Approximation
   • Sep 17, 2014 | B. Mashhoon
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     The recent classical nonlocal generalization of Einstein's theory of gravitation is presented within the framework of general relativity via the introduction of a preferred frame field. The nonlocal generalization of Einstein's field equations is derived. The linear approximation of nonlocal gravity (NLG) is thoroughly examined and the solutions of the corresponding field equations are discussed. It is shown that nonlocality, with a characteristic length scale of order 1 kpc, simulates dark matter in the linear regime while preserving causality. Light deflection in linearized nonlocal gravity is studied in connection with gravitational lensing; in particular, the propagation of light in the weak gravitational field of a uniformly moving source is investigated. The astrophysical implications of the results are briefly mentioned.

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     • contains a concise summary of teleparallel gravity in terms of tensor calculus
   • [ pdf | 1409.4472 [gr-qc] ]
2. • Lagrangian analysis of `trivial' symmetries in models of gravity
   • Jun 18, 2014 | Debraj Roy
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     We study the differences between Poincare and canonical hamiltonian symmetries in models of gravity through the corresponding Noether identities and show that they are equivalent modulo trivial gauge symmetries.

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   • [ pdf | 1406.4284 [gr-qc] ]
3. • Effects of curvature and gravity from flat spacetime
   • Jun 18, 2014 | Debraj Roy
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     We study some aspects of gravity in relation to flat spacetime. At first, we study an accelerated observer in Minkowski space as a quantum tunnelling problem in Rindler space. Both Bosonic and Fermionic modes are calculated to construct a reduced density matrix of particles tunnelling out across the accelerated Rindler horizon, giving a thermal spectrum characterized by a temperature proportional to the local acceleration - Unruh temperature. So, we calculate both the spectrum and temperature from within the tunnelling framework.

     In another direction, following Utiyama-Sciama-Kibble, we localise the Poincare group to obtain a Poincare gauge theory (PGT) of gravity. It had been pointed before in the literature, that the Poincare symmetries seemed to be recoverable canonically only on-shell. This would however mean existence of two independent sets of symmetries, each by itself having number of gauge parameters equal to the Poincare group, implying an apparent doubling of symmetries. To resolve this, we first start by a constraint analysis and construct the first-class gauge generator through an explicitly off-shell algorithm. We investigate both the Mielke-Baekler 3d model and a PGT formulation of New-Massive gravity. Next, we carefully study the symmetries and corresponding Noether identities to show that the difference between the two sets is a `trivial symmetry.' These is a transformation of fields that is not a true gauge symmetry as it gives rise to no new gauge degeneracy in defining physical states.

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   • [ pdf | 1406.4303 [gr-qc] ]
4. • On Poincaré gauge theory of gravity, its equations of motion, and Gravity Probe B
   • Jun 7, 2013 | Friedrich W. Hehl, Yuri N. Obukhov, Dirk Puetzfeld
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     Ever since E.Cartan in the 1920s enriched the geometric framework of general relativity (GR) by introducing a {\it torsion} of spacetime, the question arose whether one could find a measurement technique for detecting the presence of a torsion field. Mao et al.(2007) claimed that the rotating quartz balls in the gyroscopes of the Gravity Probe B experiment, falling freely on an orbit around the Earth, should "feel" the torsion. Similarly, March et al.(2011) argue with the precession of the Moon and the Mercury and extend later their considerations to the Lageos satellite.--- A consistent theory of gravity with torsion emerged during the early 1960's as gauge theory of the Poincar\'e group. This Poincar\'e gauge theory of gravity incorporates as simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory (EC), the teleparallel equivalent GR|| of GR, and GR itself. So far, PG and, in particular, the existence of torsion have {\it not} been experimentally confirmed. However, PG is to be considered as the standard theory of gravity with torsion because of its very convincing gauge structure.--- Since the early 1970s up to today, different groups have shown more or less independently that torsion couples only to the {\it elementary particle spin} and under no circumstances to the orbital angular momentum of test particles. This is established knowledge and we reconfirm this conclusion by discussing the energy-momentum law of PG, which has same form for all versions of PG. Therefore, we conclude that, unfortunately, the investigations of Mao et al. and March et al. do not yield any information on torsion.

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   • [ pdf | 1304.2769 [gr-qc] ]
5. • Derivation of Einstein-Cartan theory from general relativity
   • Jan 8, 2013 | R. J. Petti
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     General relativity cannot describe exchange of intrinsic and orbital angular momentum. In 1922 E. Cartan proposed extending general relativity by including affine torsion, which resolves this problem. In 1986 the author published a derivation of Einstein Cartan theory from general relativity with classical spin, with no additional assumptions. This paper adds simpler explanations, more details of the proof of the main result, correction of a factor of 2, a summary of the evidence in support of Einstein Cartan theory, and a discussion of comments about the derivation.

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   • [ pdf | 1301.1588 [gr-qc] | inspire ]
6. • Gauge Theory of Gravity and Spacetime
   • Apr 17, 2012 | Friedrich W. Hehl
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     The advent of general relativity settled it once and for all that a theory of spacetime is inextricably linked to the theory of gravity. From the point of view of the gauge principle of Weyl and Yang-Mills-Utiyama, it became manifest around the 1960s (Sciama--Kibble) that gravity is closely related to the Poincare group acting in Minkowski space. The gauging of this external group induces a Riemann-Cartan geometry on spacetime. If one generalizes the gauge group of gravity, one finds still more involved spacetime geometries. If one specializes it to the translation group, one finds a specific Riemann-Cartan geometry with teleparallelism (Weitzenbock geometry).

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     • p. 7: Gauge principles are always linked to conserved currents.
     • p. 7: The translation group T(4), subgroup of the Poincare group P(1, 3), acting in Minkowski space, creates a gravitational potential: the coframe. Similarly, the Lorentz subgroup SO(1, 3) creates another gravitational potential: the connection
     • p. 7: In this sense: the theory starts out with global, rigid symmetries of Minkowski space. By turning these symmetries into local ones, and demanding the theory to remain locally Minkowskian, the geometry arises. If the theory is only gauged to T(4), GR and Riemannian geometry arise; additionally gauging it to SO(1, 3) results in a more general geometry: Riemann-Cartan geometry.
     • p. 13: The connection two-forms represent the n × n potentials associated with the group GL(n, R).
     • p. 14: The conformally invariant part of the metric gives us the light cone. It has been shown in premetric electroynamics, that a light cone arises naturally from electromagnetic considerations without the assumption of the metric. In this sense, the conformally invariant part of the metric is not fundamental.
     • p. 18: The vanishing of curvature (Riemann = 0) and the vanishing of torsion (T = 0) can be considered as constraints. Upon releasing the first one, GR arises, and upon releasing both, Einstein-Cartan theory emerges.
     • p. 25: P(1, 3) symmetry induces Riemann-Cartan geometry
     • p. 25: Demanding dilation invariance (11-parametric Weyl group) brings the non-metricity covector potential into existence.
   • [ pdf | 1204.3672 [gr-qc] ]
7. • Fibre Bundles, Connections, General Relativity, and Einstein-Cartan Theory
   • Oct 5, 2011 | M. Socolovsky
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     We present in the most natural way, that is, in the context of the theory of vector and principal bundles and connections in them, fundamental geometrical concepts related to General Relativity and one of its extensions, the Einstein-Cartan theory.

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   • [ pdf | 1110.1018 [gr-qc] ]
8. • On the Poincare Gauge Theory of Gravitation
   • Dec 16, 2009 | S. A. Ali, C. Cafaro, S. Capozziello, Ch. Corda
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     We present a compact, self-contained review of the conventional gauge theoretical approach to gravitation based on the local Poincare group of symmetry transformations. The covariant field equations, Bianchi identities and conservation laws for angular momentum and energy-momentum are obtained.

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   • [ pdf | 0907.0934 [gr-qc] | Int. J. Theor. Phys. server ]
9. • Space and Time 100 Years after Minkowski
   • Sep 5, 2008 | Friedrich W. Hehl, Yakov Itin, Yuri Obukhov
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     Our program: Taking the Minkowski space M of 1908 and reading off the gravitational properties of matter by going over to non-inertial frames of reference in M and studying the behavior of matter therein. For a review see Gronwald and H., gr-qc/9602013.

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   • [ pdf ]
10. • S-duality in 3D gravity with torsion
    • Jan 25, 2006 | Eckehard W. Mielke, Ali A. Rincon Maggiolo
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      The deformation of the connection in three spacetime dimensions by the kinematically equivalent coframe is shown to induce a duality between the (Lorentz-) rotational and translational field momenta, for which the coupling to the deformation parameter is inverted. This new kind of strong/weak duality, pertinent to 3D, is instrumental for studying exact solutions of the 3D Poincaré gauge field equations and the particle content of propagating modes on a background of constant curvature. For a topological Chern-Simons model of gravity, the propagating modes 'living' on an Anti-de Sitter (AdS) background correspond to real massive particles. Yang-Mills type generalizations and new cubic Lagrangians are found and completely classified in 3D. AdS or black hole type solutions with constant axial torsion emerge, also for these higher-order Lagrangians with new 'exotic' torsion-curvature couplings. Their pattern complies with our S-duality, with new repercussions for the field redefinition of the metric in 3D quantum gravity and the cosmological constant problem.

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    • [ Annals Phys. server | inspire ]
11. • Exact solutions in Einstein's theory and beyond
    • 2005 | Christian Heinicke
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12. • General Relativity with Torsion
    • Sep 7, 2004 | Tomoki Watanabe, Mitsuo J. Hayashi
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      General Relativity with nonvanishing torsion has been investigated in the first order formalism of Poincare gauge field theory. In the presence of torsion, either side of the Einstein equation has the nonvanishing covariant divergence. This fact turned out to be self-consistent in the framework under consideration. By using Noether's procedure with the definition of the Lie derivative where the general coordinate transformation and the local Lorentz rotation are combined, the revised covariant divergence of the energy momentum is consistently obtained. Subsequently we have definitely derived the spin correction to the energy momentum tensor for the Dirac field and the Rarita-Schwinger field in the Einstein equation. We conjecture that the accelerated expansion of the universe possibly arises due to the spin correction in our framework.

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    • [ pdf | gr-qc/0409029 ]
13. • Three lectures on Poincare gauge theory
    • Feb 11, 2003 | M. Blagojevic
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      In these lectures we review the basic structure of Poincare gauge theory of gravity, with emphasis on its fundamental principles and geometric interpretation. A specific limit of this theory, defined by the teleparallel geometry of spacetime, is discussed as a viable alternative for the description of macroscopic gravitational phenomena.

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      • caveat: appendix A implies that Einstein–Cartan theory is not locally translation-invariant
    • [ pdf | gr-qc/0302040 ]
14. • Black Holes in Two Dimensions
    • Jul 14, 1998 | Yuri N. Obukhov, Friedrich W. Hehl
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      Models of black holes in (1+1)-dimensions provide a theoretical laboratory for the study of semi-classical effects of realistic black holes in Einstein's theory. Important examples of two-dimensional models are given by string theory motivated dilaton gravity, by ordinary general relativity in the case of spherical symmetry, and by {\em Poincar\'e gauge gravity} in two spacetime dimensions. In this paper, we present an introductory overview of the exact solutions of two-dimensional classical Poincar\'e gauge gravity (PGG). A general method is described with the help of which the gravitational field equations are solved for an arbitrary Lagrangian. The specific choice of a torsion-related coframe plays a central role in this approach. Complete integrability of the general PGG model is demonstrated in vacuum, and the structure of the black hole type solutions of the quadratic models with and without matter is analyzed in detail. Finally, the integrability of the general dilaton gravity model is established by recasting it into an effective PGG model.

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    • [ pdf | hep-th/9807101 ]
15. • On the Gauge Aspects of Gravity
    • Feb 8, 1996 | F. Gronwald, F.W. Hehl
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      We give a short outline, in Sec. 2, of the historical development of the gauge idea as applied to internal (U(1), SU(2), ...) and external (R4, SO(1,3), ...)symmetries and stress the fundamental importance of the corresponding conserved currents. In Sec. 3, experimental results with neutron interferometers in the gravitational field of the earth, as interpreted by means of the equivalence principle, can be predicted by means of the Dirac equation in an accelerated and rotating reference frame. Using the Dirac equation in such a non-inertial frame, we describe how in a gauge-theoretical approach (see Table 1) the Einstein-Cartan theory, residing in a Riemann-Cartan spacetime encompassing torsion and curvature, arises as the simplest gravitational theory. This is set in contrast to the Einsteinian approach yielding general relativity in a Riemannian spacetime. In Secs. 4 and 5 we consider the conserved energy-momentum current of matter and gauge the associated translation subgroup. The Einsteinian teleparallelism theory which emerges is shown to be equivalent, for spinless matter and for electromagnetism, to general relativity. Having successfully gauged the translations, it is straightforward to gauge the four-dimensional affine group R4?×GL(4,R) or its Poincare subgroupR4?×SO(1,3). We briefly report on these results in Sec. 6 (metric-affine geometry) and in Sec. 7 (metric-affine field equations (111, 112, 113)). Finally, in Sec. 8, we collect some models, currently under discussion, which bring life into the metric-affine gauge framework developed.

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    • [ pdf | gr-qc/9602013 ]
16. • Consequences of Propagating Torsion in Connection-Dynamic Theories of Gravity
    • Jun 16, 1994 | Sean M. Carroll, George B. Field
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      We discuss the possibility of constraining theories of gravity in which the connection is a fundamental variable by searching for observational consequences of the torsion degrees of freedom. In a wide class of models, the only modes of the torsion tensor which interact with matter are either a massive scalar or a massive spin-1 boson. Focusing on the scalar version, we study constraints on the two-dimensional parameter space characterizing the theory. For reasonable choices of these parameters the torsion decays quickly into matter fields, and no long-range fields are generated which could be discovered by ground-based or astrophysical experiments.

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    • [ pdf | gr-qc/9403058 | Phys. Rev. D server ]
17. • Metric-Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation Invariance
    • Feb 4, 1994 | F.W. Hehl, J.D. McCrea, E.W. Mielke, Y. Ne'eman
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      In Einstein's gravitational theory, the spacetime is Riemannian, that is, it has vanishing torsion and vanishing nonmetricity (covariant derivative of the metric). In the gauging of the general affine group A(4,R) and of its subgroup GL(4,R) in four dimensions, energy--momentum and hypermomentum currents of matter are canonically coupled to the one--form basis and to the connection of a metric--affine spacetime with nonvanishing torsion and nonmetricity, respectively. Fermionic matter can be described in this framework by half--integer representations of the SL¯¯¯¯¯(4,R) covering subgroup. --- We set up a (first--order) Lagrangian formalism and build up the corresponding Noether machinery. For an arbitrary gauge Lagrangian, the three gauge field equations come out in a suggestive Yang-Mills like form. The conservation--type differential identities for energy--momentum and hypermomentum and the corresponding complexes and superpotentials are derived. Limiting cases such as the Einstein--Cartan theory are discussed. In particular we show, how the A(4,R) may ``break down'' to the Poincar\'e (inhomogeneous Lorentz) group. In this context, we present explicit models for a symmetry breakdown in the cases of the Weyl (or homothetic) group, the SL(4,R), or the GL(4,R).

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    • [ pdf | gr-qc/9402012 | Phys. Rept. server | inspire ]
18. • Irreducible decompositions of nonmetricity, torsion, curvature and Bianchi identities in metric-affine spacetimes
    • May 23, 1991 | J. Dermott McCrea
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      The irreducible components of the curvature under the Lorentz group are of direct physical relevance in the four-dimensional Riemannian geometry of general relativity. The same is true for both curvature and torsion in the four-dimensional Riemann-Cartan geometry of the Poincare gauge theory of gravitation. In the latter theory a knowledge of these irreducible components is also extremely useful when setting up the Lagrangian and searching for exact solutions. The author deals with an n-dimensional metric-affine spacetime of arbitrary signature. In such a spacetime the connection is no longer metric so that there is an additional geometric object-the nonmetricity. The irreducible decompositions of nonmetricity, torsion and curvature under the pseudo-orthogonal group as well as those of the corresponding Bianchi identities are derived. Because of the increasing use made of it in the literature, the exterior form notation is used throughout.

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    • [ pdf | Class. Quant. Grav. server | inspire ]
19. • On Hyperbolic U(4) Manifolds With Local Duality
    • Dec 1982 | R. P. Wallner
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      We use the decomposition of the Riemann/Cartan curvature 2-forms ω^ij in terms of their irreducible parts under the Lorentz group to examine the irreducible content of self- and anti-self double dual curvature forms Ω^+-ij and their further refinements involving left and right duals.

      In the case of local duality (i.e. Ω^ij = Ω^+-ij locally), some consequences to curvature and torsion are easily derived in this way.

      As Riemann/Cartan space-times (U₄ space-times) are subject to generalized gravity theories, some (vacuum) field equations proposed there are also taken into consideration. As an application of the various decompositions of curvature and torsion we point out their utility in the search of exact solutions of U₄-field equations. To simplify notations and calculations, the calculus of exterior forms is used throughout.

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    • [ pdf | inspire ]
20. • Geometric Techniques in Gauge Theories
    • 1981 | Editors: Rodolfo Martini, Eduardus M. de Jager
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      Proceedings of the Fifth Scheveningen Conference on Differential Equations, The Netherlands August 23-28, 1981

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    • [ pdf (Ch. 7) | Springer server ]
21. • Hidden gauge symmetry
    • Jan, 1979 | Lochlainn O'Raifeartaigh
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      The past 15 years has seen the gradual emergence of the idea that the fundamental physical interactions are determined by gauge symmetry or, more precisely, by hidden (spontaneously broken) gauge symmetry. The importance of gauge symmetry is that it reduces considerably the possible forms of interaction, gives the interactions a geometrical meaning, and introduces a certain degree of unification to the different known interactions (gravitational, weak, etc.). A description of the principles of hidden gauge symmetry and of its application ot the fundamental interactions is presented. The emphasis is on the structure, gauge symmetry and hidden symmetry are first treated as independent phenomena before being combined into a single (hidden gauge symmetry) theory. The main application of the theory is to the weak and electromagnetic interactions of the elementary particles, and although models are used for comparison with experiment and for illustration, emphasis is placed on those features of the application which are model-independent.

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    • [ pdf | Rep. Prog. Phys. server ]
22. • General relativity with spin and torsion: Foundations and prospects
    • Jul 1, 1976 | Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, James M. Nester
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      A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the U4 theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the U4 theory arises as a local gauge theory for the Poincaré group in space-time. The deviations of the U4 theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

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    • [ pdf | Rev. Mod. Phys. server | inspire ]
23. • Nonlinear Spinor Equation and Asymmetric Connection in General Relativity
    • 1971 | F. W. Hehl, K. Datta
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      In order to take full account of spin in general relativity, it is necessary to consider space-time as a metric space with torsion, as was shown elsewhere. We treat a Dirac particle in such a space. The generalized Dirac equation turns out to be of a Heisenberg-Pauli type. The nonlinear terms induced by torsion express a universal spin-spin interaction of range zero.

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    • [ pdf | J. Math. Phys. server | inspire | doi ]
24. • Lorentz Invariance and the Gravitational Field
    • 1961 | T. W. B. Kibble
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      An argument leading from the Lorentz invariance of the Lagrangian to the introduction of the gravitational field is presented. Utiyama's discussion is extended by considering the 10-parameter group of inhomogeneous Lorentz transformations, involving variation of the coordinates as well as the field variables. It is then unnecessary to introduce a priori curvilinear coordinates or a Riemannian metric, and the new field variables introduced as a consequence of the argument include the vierbein components hk ? as well as the ``local affine connection'' Ai j? . The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system. The free Lagrangian for the new fields is shown to be a function of two covariant quantities analogous to F?? for the electromagnetic field, and the simplest possible form is just the usual curvature scalar density expressed in terms of hk ? and Ai j? . This Lagrangian is of first order in the derivatives, and is the analog for the vierbein formalism of Palatini's Lagrangian. In the absence of matter, it yields the familiar equationsR?? =0 for empty space, but when matter is present there is a difference from the usual theory (first pointed out by Weyl) which arises from the fact that Ai j? appears in the matter field Lagrangian, so that the equation of motion relating Ai j? to hk ? is changed. In particular, this means that, although the covariant derivative of the metric vanishes, the affine connection ?? ?? is nonsymmetric. The theory may be reexpressed in terms of the Christoffel connection, and in that case additional terms quadratic in the ``spin density'' Sk ij appear in the Lagrangian. These terms are almost certainly too small to make any experimentally detectable difference to the predictions of the usual metric theory.

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    • [ pdf | J. Math. Phys. server ]

Torsion

1. • Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics
   • Mar 9, 2012 | Arash Yavari, Alain Goriely
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     We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold-where the body is stress free-is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan's moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.

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   • [ pdf | Arch. Rational Mech. Anal. server ]
2. • Cartan's spiral staircase in physics and, in particular, in the gauge theory of dislocations
   • Nov 11, 2009 | Markus Lazar, Friedrich W. Hehl
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     In 1922, Cartan introduced in differential geometry, besides the Riemannian curvature, the new concept of torsion. He visualized a homogeneous and isotropic distribution of torsion in three dimensions (3d) by the "helical staircase", which he constructed by starting from a 3d Euclidean space and by defining a new connection via helical motions. We describe this geometric procedure in detail and define the corresponding connection and the torsion. The interdisciplinary nature of this subject is already evident from Cartan's discussion, since he argued - but never proved - that the helical staircase should correspond to a continuum with constant pressure and constant internal torque. We discuss where in physics the helical staircase is realized: (i) In the continuum mechanics of Cosserat media, (ii) in (fairly speculative) 3d theories of gravity, namely a) in 3d Einstein-Cartan gravity - this is Cartan's case of constant pressure and constant intrinsic torque - and b) in 3d Poincare gauge theory with the Mielke-Baekler Lagrangian, and, eventually, (iii) in the gauge field theory of dislocations of Lazar et al., as we prove for the first time by arranging a suitable distribution of screw dislocations. Our main emphasis is on the discussion of dislocation field theory.

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   • [ pdf | 1403.3467 [cond-mat.mtrl-sci] | Found. Phys. server | inspire | doi ]
3. • Einstein's Apple: His First Principle of Equivalence
   • Mar 29, 2007 | Engelbert L. Schucking, Eugene J. Surowitz
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     After a historical discussion of Einstein's 1907 principle of equivalence, a homogeneous gravitational field in Minkowski spacetime is constructed. It is pointed out that the reference frames in gravitational theory can be understood as spaces with a flat connection and torsion defined through teleparallelism. This kind of torsion was introduced by Einstein in 1928. The concept of torsion is discussed through simple examples and some historical observations.

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     • p. 30: "If one takes a sphere, removes the North and South poles, and puts unit vectors pointing North along meridians and along latitude circles pointing East, one has created an orthonormal frame everywhere. If one now moves vectors tangent to the sphere by keeping their frame components constant, one has a connection with torsion."
     • p. 31, ch. 17: Poincar\'e's upper half plane carries torsion!
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   • [ pdf | gr-qc/0703149v2 ]
4. • Volterra distortions, spinning strings, and cosmic defects
   • May 5, 1997 | Roland A Puntigam, Harald H Soleng
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     Cosmic strings, as topological spacetime defects, show striking resemblance to defects in solid continua: distortions, which can be classified into disclinations and dislocations, are line-like defects characterized by a delta-function-valued curvature and torsion distribution giving rise to rotational and translational holonomy. We exploit this analogy and investigate how distortions can be adapted in a systematic manner from solid-state systems to Einstein - Cartan gravity. As distortions are efficiently described within the framework of an SO(3) \semidirect T(3) gauge theory of solid continua with line defects, we are led in a straightforward way to a Poincaré gauge approach to gravity which is a natural framework for introducing the notion of distorted spacetimes. Constructing all ten possible distorted spacetimes, we recover, inter alia, the well known exterior spacetime of a spin-polarized cosmic string as a special case of such a geometry. In a second step, we search for matter distributions which, in Einstein - Cartan gravity, act as sources of distorted spacetimes. The resulting solutions, appropriately matched to the distorted vacua, are cylindrically symmetric and are interpreted as spin-polarized cosmic strings and cosmic dislocations.

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     • Volterra distortions: order 1-3: dislocations (translations), order 4-6: disclination (rotations)
     • p. 6, Lorenz connection: SO(1,3)-algebra valued connection 1-form
     • p. 6, translation connection: R⁴-valued connection 1-form (this one also has value in the algebra of T(4), but since it is an Abelian group, its algebra structure is somewhat trivial and can hence be omitted)
     • p. 7, Burgers vector: translational holonomy \in T(4), defined by contour integral of vierbein around linelike defect
     • p. 8, Frank matrix: rotational holonomy \in SO(1,3), defined by contour integral of Lorenz connection around linelike defect
     • p. 9: these defects correspond to torsion and curvature singularities (delta "functions")
     • pp. 9, 11: all vacuum line elements are specified for linelike defects, they may correspond to something like cosmic strings, i.e. topological defects in spacetime related to the structure formation in the early universe
     • p. 13, junction conditions: demanding that energy-momentum-, spin, and contorsion tensor are continuous across the boundary of the matter distribution
     • p. 14f: a spinning particle in (1+2)D corresponds to a cosmic string in (1+3)D with unspecified four-veclocity. Depending on the (timelike, lightlike, or spacelike) direction of the four-velocity, different solutions arise.
     • p. 19: a Poincar\'e gauge theory is a natural four-dimensional generalization of the SO(3) \semidirect T(3) gauge theory of solid continua with line defects
     • ...
   • [ pdf | Class. Quant. Grav. server ]
5. • Torsion and related concepts: An introductory overview
   • Mar 8, 1980 | Borut Gogala
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     The aim of this paper is to provide an overview of all the basic aspects of the torsion of a manifold, with particular stress on the expressions in an anholonomic basis. After a brief review of anholonomic bases and Koszul covariant derivative, we show how the expressions for the torsion and the Riemann tensors in a general (anholonomic) basis arise from their expressions in a coordinate basis. We further derive the expression for the contortion tensor, which arises from the requirement that an affine connection with torsion be metric (preserving). The latter requirement is related to the equivalence principle, whose mathematical aspects in a manifold with torsion are discussed next. Finally, we derive the expression for the distortion tensor, which is an analog of the curvature tensor but arising from the torsion rather than the metric tensor.

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   • [ pdf | Int. J. Theor. Phys. server ]

Geometry - Specific Topics

1. • Black holes, hidden symmetries, and complete integrability
   • May 15, 2017 | Valeri P. Frolov, Pavel Krtous, David Kubiznak
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     The study of higher-dimensional black holes is a subject which has recently attracted a vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr-NUT-(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional black holes. We start with the overview of the Liouville theory of completely integrable systems and introduce Killing and Killing-Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a `seed object' which generates all these symmetries. It determines the form of the black hole geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.

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   • [ pdf | inspire ]
2. • Force-Free Foliations
   • Dec 15, 1977 | Geoffrey Compère, Samuel E. Gralla, Alexandru Lupsasca
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     Electromagnetic field configurations with vanishing Lorentz force density are known as force-free and appear in terrestrial, space, and astrophysical plasmas. We explore a general method for finding such configurations based on formulating equations for the field lines rather than the field itself. The basic object becomes a foliation of spacetime or, in the stationary axisymmetric case, of the half-plane. We use this approach to find some new stationary and axisymmetric solutions, one of which could represent a rotating plasma vortex near a magnetic null point.

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   • [ pdf | inspire | doi ]
3. • Killing Tensor Quantum Numbers and Conserved Currents in Curved Space
   • Dec 15, 1977 | Brandon Carter
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     The relationship between relativistic quantum current conservation laws in a curved-space background and the corresponding "good quantum numbers," i.e., operators that commute with the fundamental wave operator in a first-quantized field theory, is considered. It is shown that under favorable circumstances (such as vanishing Ricci curvature) the existence of such an operator for scalar fields is automatically implied by the existence of the corresponding constant for particle trajectories in the classical limit, that is to say, by the existence of a Killing vector or a "Killing tensor" in the first- and second-order cases, respectively. Thus the fourth constant of the motion for a scalar quantum field in the Kerr metric background arises automatically from the Killing tensor defining the fourth constant of the classical motion. Another application is to the Runge-Lenz constants in the nonrelativistic hydrogen atom problem. The "Schiff conjecture" concerning the relationship between classical mechanics and first-quantized field theory in connection with the equivalence principle is discussed in passing.

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   • [ inspire | doi ]
4. • Conserved charges, surface degrees of freedom, and black hole entropy
   • Mar 9, 2016 | Ali Seraj
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     In this thesis, we study the Hamiltonian and covariant phase space description of gravitational theories. The phase space represents the allowed field configurations and is accompanied by a closed nondegenerate 2 form- the symplectic form. We will show that local/gauge symmetries of the action fall into two different categories in the phase space formulation. Those corresponding to constraints in the phase space, and those associated with nontrivial conserved charges. We argue that while the former is related to redundant gauge degrees of freedom, the latter leads to physically distinct states of the system, known as surface degrees of freedom and can induce a lower dimensional dynamics on the system. These ideas are then implemented to build the phase space of specific gravitational systems: 1) asymptotically AdS3 spacetimes, and 2) near horizon geometries of extremal black holes (NHEG) in arbitrary dimension. In the AdS3 phase space, we show that Brown-Henneaux asymptotic symmetries can be extended inside the bulk of spacetime and hence become symplectic symmetries of the phase space. We will show that in the NHEG phase space, surface gravitons form a Virasoro algebra in four dimensions, and a novel generalization of Virasoro in higher dimensions. The central charge of the algebra is proportional to the entropy of the corresponding extremal black hole. We study the holographic description of NHEG phase space and show that the charges can be computed through a Liouville type stress tensor defined over a lower dimensional torus. We will discuss whether surface gravitons can serve as the microscopic origin of black hole entropy.

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     general remarks

     • Noether I: global symmetry leads to conservation law
     • Noether II: local symmetry leads to Bianchi identity
     • Hamiltonian picture: a charge generates a symmetry transformation (via the Poisson bracket); charge = on-shell value of generator; energy = on-shell value of Hamiltonian
     • can have unique charges for an asymptotic symmetry
     • (interesting fact regarding U₄ paper: electric part of Weyl tensor is related to charge in asymptotic AdS spacetimes, see Ashtekar et al. [24, 25])
     • idea: extend asymptotic symmetries into the bulk (thereby they become symplectic symmetries of phase space), and hence charges can be computed on any closed surface in the bulk

     Hamiltonian formalism

     • gauge symmetries of action = first class constraints in phase space
     • compact spacetimes: Hamiltonian is a collection of constraints and vanishes weakly
     • spacetimes with boundaries: bulk term vanishes on-shell, but there exists necessary non-vanishing boundary term that makes the Hamiltonian non-vanishing on phase space
     • first class constraints form an algebra, and that is why they correspond to gauge symmetries (which also form an algebra): gauge transformations are generated by first class constraints
   • [ pdf | 1603.02442 [hep-th] | inspire ]
5. • Local subsystems in gauge theory and gravity
   • Jan 18, 2016 | William Donnelly, Laurent Freidel
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     We consider the problem of defining localized subsystems in gauge theory and gravity. Such systems are associated to spacelike hypersurfaces with boundaries and provide the natural setting for studying entanglement entropy of regions of space. We present a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary. In Yang-Mills theory the new degrees of freedom are a choice of gauge on the boundary, transformations of which are generated by the normal component of the nonabelian electric field. In general relativity the new degrees of freedom are the location of a codimension-2 surface and a choice of conformal normal frame. These degrees of freedom transform under a group of surface symmetries, consisting of diffeomorphisms of the codimension-2 boundary, and position-dependent linear deformations of its normal plane. We find the observables which generate these symmetries, consisting of the conformal normal metric and curvature of the normal connection. We discuss the implications for the problem of defining entanglement entropy in quantum gravity. Our work suggests that the Bekenstein-Hawking entropy may arise from the different ways of gluing together two partial Cauchy surfaces at a cross-section of the horizon.

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   • [ pdf | 1601.04744 [hep-th] ]
6. • Gravitational Energy for GR and Poincare Gauge Theories: a Covariant Hamiltonian Approach
   • Jul 27, 2015 | Chiang-Mei Chen, James M. Nester, Roh-Suan Tung
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     Our topic concerns a long standing puzzle: the energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy-momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein's general relativity (GR), noting especially Noether's contribution. In the second part we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincar\'e gauge theories (GR is a special case). The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and, furthermore it determines the boundary conditions for the dynamical variables. Energy-momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincar\'e transformations. They are best described in a local Poincar\'e gauge theory. The type of spacetime that naturally has this symmetry is Riemann-Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy-momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to Poincar\'e gauge theories.

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   • [ pdf | 1507.07300 [gr-qc] | Int. J. Mod. Phys. server | inspire | doi ]
7. • Actions, topological terms and boundaries in first order gravity: A review
   • Apr 26, 2016 | Alejandro Corichi, Irais Rubalcava-Garcia, Tatjana Vukasinac
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     In this review we consider first order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad eIa and a SO(3,1) connection ?aIJ. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein-Hilbert-Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space ? is given by solutions to the equations of motion. For each of the possible terms contributing to the action we consider the well posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way we point out and clarify some issues that have not been clearly understood in the literature.

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   • [ pdf | 1604.07764 [gr-qc] ]
8. • Hamiltonian Analysis of non-chiral Plebanski Theory and its Generalizations
   • Sep 27, 2008 | Sergei Alexandrov, Kirill Krasnov
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     We consider non-chiral, full Lorentz group-based Plebanski formulation of general relativity in its version that utilizes the Lagrange multiplier field Phi with "internal" indices. The Hamiltonian analysis of this version of the theory turns out to be simpler than in the previously considered in the literature version with Phi carrying spacetime indices. We then extend the Hamiltonian analysis to a more general class of theories whose action contains scalars invariants constructed from Phi. Such theories have recently been considered in the context of unification of gravity with other forces. We show that these more general theories have six additional propagating degrees of freedom as compared to general relativity, something that has not been appreciated in the literature treating them as being not much different from GR.

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   • [ pdf | 0809.4763 [gr-qc] ]
9. • Local Invariants Vanishing on Stationary Horizons: A Diagnostic for Locating Black Holes
   • Jan 14, 2015 | Don N. Page, Andrey A. Shoom
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     Inspired by the example of Abdelqader and Lake for the Kerr metric, we construct local scalar polynomial curvature invariants that vanish on the horizon of any stationary black hole: the squared norms of the wedge products of n linearly independent gradients of scalar polynomial curvature invariants, where n is the local cohomogeneity of the spacetime.

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   • [ pdf | 1501.03510 [gr-qc] | inspire ]
10. • The Kummer tensor density in electrodynamics and in gravity
    • Mar 14, 2014 | Peter Baekler, Alberto Favaro, Yakov Itin, Friedrich W. Hehl
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      Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, Kijkl. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four Tijkl, which is antisymmetric in its first two and its last two indices: Tijkl=?Tjikl=?Tijlk. Thus, K?T3, see Eq.(46). (i) If T is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If T is identified with the {\it curvature} tensor Rijkl of a Riemann-Cartan spacetime, then K?R3 and, in the special case of general relativity, K reduces to the Kummer tensor of Zund (1969). This K is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose K irreducibly under the 4-dimensional linear group GL(4,R) and, subsequently, under the Lorentz group SO(1,3).

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    • [ pdf | 1403.3467 [gr-qc] | Annals of Phys. server | inspire | doi ]
11. • Normal frames for general connections
    • Nov 2, 2009 | James M. Nester
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      At any point on a manifold certain normal frames that are well adapted to general connections are identified. For manifolds with metrics the series expansion terms for the metric/frame (second order) and the connection (first order) are specified. Some special cases as well as mathematical and physical applications are noted.

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      • p. 49: in a Weizenbopeck space, the connection can be transformed to zero everywhere, cf. Eq. (43) and paragraph below.
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    • [ pdf | Annals Phys. server ]
12. • Conformal transformations and conformal invariance in gravitation
    • Jun 16, 2008 | Mariusz P. Dabrowski, Janusz Garecki, David B. Blaschke
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      Conformal transformations are frequently used tools in order to study relations between various theories of gravity and the Einstein relativity. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy-momentum tensor. We show the subtlety of the matter energy-momentum conservation law which refers to the fact that the conformal transformation "creates" an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is "created" due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants R2, RabRab, RabcdRabcd and the Gauss-Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.

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    • [ pdf | 0806.2683 [gr-qc] | Ann. Phys. server | inspire | doi ]
13. • Stationary spacetimes and the Simon tensor
    • Nov 10, 2004 | Donato Bini, Robert T. Jantzen
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      For stationary vacuum spacetimes the Bianchi identities of the second kind equate the Simon tensor to the Simon-Mars tensor, the latter having a clear geometrical interpretation. The equivalence of these two tensors is broken in the nonvacuum case by additional source energy-momentum terms, but absorbing these source terms into a redefinition of the Simon tensor restores the equality. Explicit examples are discussed for electrovacuum and rigidly rotating matter fields.

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    • [ pdf | gr-qc/0411051 | Italian Phys. Soc. server ]
14. • Second Order Scalar Invariants of the Riemann Tensor: Applications to Black Hole Spacetimes
    • Feb 23, 2003 | C. Cherubini, D. Bini, S. Capozziello, R. Ruffini
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      We discuss the Kretschmann, Chern-Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman-Penrose formalism and in the framework of gravitoelectromagnetism, using the Kerr-Newman geometry as an example. An analogy with electromagnetic invariants leads to the definition of regions of gravitoelectric or gravitomagnetic dominance.

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    • [ pdf | gr-qc/0302095 | Int. J of Mod. Phys. server ]
15. • The Cotton tensor in Riemannian spacetimes
    • Sep 1, 2003 | A. Garcia, F.W. Hehl, C. Heinicke, A. Macias
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      Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.

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      • p. 3: Cotton tensor plays a role in the Hamiltonian formulation of General Relativity
    • [ pdf | gr-qc/0309008 | Class. Quant. Grav. server | inspire | doi ]
16. • The Cotton, Simon-Mars and Cotton-York Tensors in Stationary Spacetimes
    • Oct 11, 2001 | Donato Bini, Robert T Jantzen, Giovanni Miniutti
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      The Cotton-York and Simon-Mars tensors in stationary vacuum spacetimes are studied in the language of the congruence approach pioneered by Hawking and Ellis. Their relationships with the Papapetrou field defined by the stationary Killing congruence and with a recent characterization of the Kerr spacetime in terms of the alignment between of the principal null directions of the Weyl tensor with those of the Papapetrou field are also investigated in this more transparent language.

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      • p. 4: The Weyl tensor is the part of curvature which is not directly determined by the energy-momentum tensor
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    • [ pdf | gr-qc/0110059 | Class. Quant. Grav. server ]
17. • The Einstein 3-form G_a and its equivalent 1-form L_a in Riemann-Cartan space
    • Dec 11, 2000 | Christian Heinicke
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      The definition of the Einstein 3-form G_a is motivated by means of the contracted 2nd Bianchi identity. This definition involves at first the complete curvature 2-form. The 1-form L_a is defined via G_a = L^b \wedge #(o_b \wedge o_a). Here # denotes the Hodge-star, o_a the coframe, and \wedge the exterior product. The L_a is equivalent to the Einstein 3-form and represents a certain contraction of the curvature 2-form. A variational formula of Salgado on quadratic invariants of the L_a 1-form is discussed, generalized, and put into proper perspective.

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    • [ pdf | gr-qc/0012037 | Gen. Rel. Grav. server | inspire ]
18. • A small guide to variations in teleparallel gauge theories of gravity and the Kaniel-Itin model
    • Jan 12, 1998 | Uwe Muench, Frank Gronwald, Friedrich W. Hehl
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      Recently Kaniel & Itin proposed a gravitational model with the wave type equation [◻ + λ(x)] ϑ^α = 0 as vacuum field equation, where ϑ^α denotes the coframe of spacetime. They found that the viable Yilmaz-Rosen metric is an exact solution of the tracefree part of their field equation. This model belongs to the teleparallelism class of gravitational gauge theories. Of decisive importance for the evaluation of the Kaniel-Itin model is the question whether the variation of the coframe commutes with the Hodge star. We find a master formula for this commutator and rectify some corresponding mistakes in the literature. Then we turn to a detailed discussion of the Kaniel-Itin model.

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      • variations of the Hodge star
    • [ pdf | gr-qc/9801036 | Gen. Rel. Grav. server | inspire | doi ]
19. • On the Relation Between Quadratic and Linear Curvature Lagrangians in Poincaré Gauge Gravity
    • Feb 8, 1996 | Yuri N. Obukhov, Friedrich W. Hehl
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      We discuss the choice of the Lagrangian in the Poincare gauge theory of gravity. Drawing analogies to earlier de Sitter gauge models, we point out the possibility of deriving the Einstein-Cartan Lagrangian without cosmological term from a modified quadratic curvature invariant of topological type.

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    • [ pdf | gr-qc/9602014 | inspire ]

Computer Algebra

1. • Computer algebra in spacetime embedding
   • May 28, 2014 | Waldir L. Roque, Renato P. dos Santos
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     In this paper we describe an algorithm to determine the vectors normal to a space-time V4 embedded in a pseudo-Euclidean manifold M4+n. An application of this algorithm is given considering the Schwarzchild space-time geometry embedded in a 6 dimensional pseudo-Euclidean manifold, using the algebraic computing system REDUCE.

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   • [ pdf | 1405.7404 [gr-qc] | J. Symbolic Computation server ]
2. • Computer algebra in gravity: Programs for (non-)Riemannian spacetimes. I
   • Apr 24, 1998 | Jose Socorro, Alfredo Macias, Friedrich W. Hehl
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     Computer algebra programs are presented for application in general relativity, in electrodynamics, and in gauge theories of gravity. The mathematical formalism used is the calculus of exterior differential forms, the computer algebra system applied Hearn's Reduce with Schruefer's exterior form package Excalc. As a non-trivial example we discuss a metric of Plebanski & Demianski (of Petrov type D) together with an electromagnetic potential and a triplet of post-Riemannian one-forms. This whole geometrical construct represents an exact solution of a metric-affine gauge theory of gravity. We describe a sample session and verify by computer that this exact solution fulfills the appropriate field equations.-- Computer programs are described for the irreducible decomposition of (non-Riemannian) curvature, torsion, and nonmetricity.

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   • [ pdf | gr-qc/9804068v1 | Comput. Phys. Commun. server ]
3. • Exterior calculus on the computer: The REDUCE-package EXCALC applied to general relativity and to the Poincare gauge theory
   • May 22, 1986 | Eberhard Schruefer, Friedrich W. Hehl, J. Dermott McCrea
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     The computer algebra system REDUCE has recently been enriched by a package on exterior calculus. Here we apply the EXCALC package to the calculation of quantities within the Poincare gauge theory of gravity, general relativity being included in this scheme as a special case. Thereby we simplify and streamline earlier results found by means of tensor-analytical REDUCE calculations.

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   • [ pdf | Gen. Rel. Grav. server ]

Quantized BTZ Solution

1. • Canonical Quantization of the BTZ Black Hole using Noether Symmetries
   • May 2, 2014 | T. Christodoulakis, N. Dimakis, Petros A. Terzis, G. Doulis
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     The well-known BTZ black hole solution of (2+1) Einstein's gravity, in the presence of a cosmological constant, is treated both at the classical and quantum level. Classically, the imposition of the two manifest local Killing fields of the BTZ geometry at the level of the full action results in a mini-superspace constraint action with the radial coordinate playing the role of the independent dynamical variable. The Noether symmetries of this reduced action are then shown to completely determine the classical solution space, without any further need to solve the dynamical equations of motion. At a quantum mechanical level, all the admissible sets of the quantum counterparts of the generators of the above mentioned symmetries are utilized as supplementary conditions acting on the wave-function. These additional restrictions, in conjunction with the Wheeler-DeWitt equation, help to determine (up to constants) the wave-function which is then treated semiclassically, in the sense of Bohm. The ensuing space-times are, either identical to the classical geometry, thus exhibiting a good correlation of the corresponding quantization to the classical theory, or are less symmetric but exhibit no Killing or event horizon and no curvature singularity, thus indicating a softening of the classical conical singularity of the BTZ geometry.

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   • [ pdf | 1405.0363 [gr-qc] ]
2. • Entropy of thin shells in a (2+1)-dimensional asymptotically AdS spacetime and the BTZ black hole limit
   • Mar 3, 2014 | José P. S. Lemos, Gonçalo M. Quinta
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     The thermodynamic equilibrium states of a static thin ring shell in a (2+1)-dimensional spacetime with a negative cosmological constant are analyzed. Inside the ring, the spacetime is pure anti-de Sitter (AdS), whereas outside it is a Ba\~nados-Teitelbom-Zanell$ (BTZ) spacetime and thus asymptotically AdS. The first law of thermodynamics applied to the thin shell, plus one equation of state for the shell's pressure and another for its temperature, leads to a shell's entropy, which is a function of its gravitational radius alone. A simple example for this gravitational entropy, namely, a power law in the gravitational radius, is given. The equations of thermodynamic stability are analyzed, resulting in certain allowed regions for the parameters entering the problem. When the Hawking temperature is set on the shell and the shell is pushed up to its own gravitational radius, there is a finite quantum backreaction that does not destroy the shell. One then finds that the entropy of the shell at the shell's gravitational radius is given by the Bekenstein-Hawking entropy.

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   • [ pdf | 1403.0579 [gr-qc] ]
3. • Mass Spectrum and Statistical Entropy of the BTZ black hole from Canonical Quantum Gravity
   • Feb 20, 2008 | Cenalo Vaz, Sashideep Gutti, Claus Kiefer, T. P. Singh, L.C.R. Wijewardhana
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     In a recent publication we developed a canonical quantization program describing the gravitational collapse of a spherical dust cloud in 2+1 dimensions with a negative cosmological constant [...]. In this paper we address the quantization of the Banados-Teitelboim-Zanelli (BTZ) black hole. We show that the mass function describing the black hole is made of two pieces, a constant non-vanishing boundary contribution and a discrete spectrum [...]. The discrete spectrum is obtained by applying the Wheeler--DeWitt equation with a particular choice of factor ordering and interpreted as giving the energy levels of the collapsed matter shells that form the black hole. Treating a black hole microstate as a particular distribution of shells among the levels, we determine the canonical entropy of the BTZ black hole. Comparison with the Bekenstein-Hawking entropy shows that the boundary energy is related to the central charge of the Virasoro algebra that generates the asymptotic symmetry group of the three-dimensional anti-de Sitter space AdS3. This gives a connection between the Wheeler-DeWitt approach and the conformal field theory approach.

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   • [ pdf | 0712.1998 [gr-qc] | Phys. Rev. D server ]

Quasinormal Modes

1. • On quasi-normal modes, area quantization and Bohr correspondence principle
   • Mar 18, 2015 | Christian Corda
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     In Int. Journ. Mod. Phys. D 14, 181 (2005) Khriplovich verbatim claims that "the correspondence principle does not dictate any relation between the asymptotics of quasinormal modes and the spectrum of quantized black holes" and that "this belief is in conflict with simple physical arguments". In this paper we analyze Khriplovich's criticisms and realize that they work only for the original proposal by Hod, while they do not work for the improvements suggested by Maggiore and recently finalized by the author and collaborators through a connection between Hawking radiation and black hole (BH) quasi-normal modes (QNMs). This is a model of quantum BH somewhat similar to the historical semi-classical model of the structure of a hydrogen atom introduced by Bohr in 1913. Thus, QNMs can be really interpreted as BH quantum levels (the "electrons" of the "Bohr-like BH model"). Our results have also important implications on the BH information puzzle.

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   • [ pdf | 1503.05551 [gr-qc] | inspire ]
2. • Bohr-like black holes
   • Mar 11, 2015 | Christian Corda
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     The idea that black holes (BHs) result in highly excited states representing both the "hydrogen atom" and the "quasi-thermal emission" in quantum gravity is today an intuitive but general conviction. In this paper it will be shown that such an intuitive picture is more than a picture. In fact, we will discuss a model of quantum BH somewhat similar to the historical semi-classical model of the structure of a hydrogen atom introduced by Bohr in 1913. The model is completely consistent with existing results in the literature, starting from the celebrated result of Bekenstein on the area quantization.

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   • [ pdf | 1503.03474 [gr-qc] | inspire | AIP server | doi ]
3. • Black hole quasi-normal modes: the "electrons" of quantum gravity? Implications for the black hole information puzzle
   • Feb 26, 2015 | Christian Corda
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     Some recent important results on black hole (BH) quantum physics concerning the BH effective state and the natural correspondence between Hawking radiation and BH quasi-normal modes (QNMs) are reviewed, clarified and refined. Such a correspondence permits to naturally interpret QNMs as quantum levels in a semi-classical model. This is a model of BH somewhat similar to the historical semi-classical model of the structure of a hydrogen atom introduced by Bohr in 1913. In a certain sense, QNMs represent the "electron" which jumps from a level to another one and the absolute values of the QNMs frequencies "triggered" by emissions (Hawking radiation) and absorption of particles represent the energy "shells" of the "gravitational hydrogen atom". Important consequences on the BH information puzzle are discussed. In fact, it is shown that the time evolution of this "Bohr-like BH model" obeys to a time dependent Schr\"odinger equation which permits the final BH state to be a pure quantum state instead of a mixed one. Thus, information comes out in BH evaporation, in agreement with the assumption by 't Hooft that Schr\"oedinger equations can be used universally for all dynamics in the universe. We also show that, in addition, our approach solves the entanglement problem connected with the information paradox. We emphasize that Bohr model is an approximated model of the hydrogen atom with respect to the valence shell atom model of full quantum mechanics. In the same way, we expect the Bohr-like BH model to be an approximated model with respect to the definitive, but at the present time unknown, BH model arising from a full quantum gravity theory. If the analogy between electron and QNMs is correct, this could be the first, important step for the realization of a new approach to quantum gravity that we could call "QNMs quantum gravity".

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   • [ pdf | 1503.00565 [gr-qc] | inspire ]
4. • Dirac Quasinormal modes of MSW black holes
   • Jan 15, 2014 | Saneesh Sebastian, V. C. Kuriakose
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     In this paper we study the Dirac quasinormal modes of an uncharged 2 + 1 black hole proposed by Mandal et. al and referred to as MSW black hole in this work. The quasi- normal mode is studied using WKB approximation method. The study shows that the imaginary part of quasinormal frequencies increases indicating that the oscillations are damping and hence the black hole is stable against Dirac perturbations.

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   • [ pdf | 1401.3496 [gr-qc] | inspire | Mod. Phys. Lett. A server | doi ]
5. • From bricks to quasinormal modes: A new perspective on black hole entropy
   • May 15, 2013 | Michele Arzano, Stefano Bianco, Olaf Dreyer
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     Calculations of black hole entropy based on the counting of modes of a quantum field propagating in a Schwarzschild background need to be regularized in the vicinity of the horizon. To obtain the Bekenstein-Hawking result the short distance cut-off needs to be fixed by hand. In this note we give an argument for obtaining this cut-off in a natural fashion. We do this by modelling the black hole by its set of quasinormal modes. The horizon then becomes a extended region: the quantum ergosphere. The interaction of the quantum ergosphere and the quantum field provides a natural regularization mechanism. The width of the quantum ergosphere provides the right cut-off for the entropy calculation. We arrive at a dual picture of black hole entropy. The entropy of the black hole is given both by the entropy of the quantum field in the bulk and the dynamical degrees of freedom on the horizon.

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   • [ pdf | 1305.3479 [gr-qc] | inspire | Int. J. Mod. Phys. server | doi ]
6. • A new approach to the study of quasi-normal modes of rotating stars
   • Sep 18, 2007 | V. Ferrari, L. Gualtieri, S. Marassi
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     We propose a new method to study the quasi-normal modes of rotating relativistic stars. Oscillations are treated as perturbations in the frequency domain of the stationary, axisymmetric background describing a rotating star. The perturbed quantities are expanded in circular harmonics, and the resulting 2D-equations they satisfy are integrated using spectral methods in the (r,theta)-plane. The asymptotic conditions at infinity, needed to find the mode frequencies, are implemented by generalizing the standing wave boundary condition commonly used in the non rotating case. As a test, the method is applied to find the quasi-normal mode frequencies of a slowly rotating star.

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   • [ pdf | 0709.2925 [gr-qc] | Phys. Rev. D server ]
7. • Hawking temperature from quasi-normal modes
   • Jun 24, 2004 | Claus Kiefer
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     A perturbed black hole has characteristic frequencies (quasi-normal modes). Here I apply a quantum measurement analysis of the quasi-normal mode frequency in the limit of high damping. It turns out that a measurement of this mode necessarily adds noise to it. For a Schwarzschild black hole, this corresponds exactly to the Hawking temperature. The situation for other black holes is briefly discussed.

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   • [ pdf | gr-qc/0406097 | Class. Quant. Grav. server ]
8. • Scalar, electromagnetic and Weyl perturbations of BTZ black holes: quasi normal modes
   • Jan 13, 2001 | Vitor Cardoso, Jose' P. S. Lemos
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     We calculate the quasinormal modes and associated frequencies of the Banados, Zanelli and Teitelboim (BTZ) non-rotating black hole. This black hole lives in 2+1-dimensions in an asymptotically anti-de Sitter spacetime. We obtain exact results for the wavefunction and quasi normal frequencies of scalar, electromagnetic and Weyl (neutrino) perturbations.

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   • [ pdf | gr-qc/0101052 | Phys. Rev. D server | inspire ]
9. • New approach to the quasinormal modes of a black hole
   • Jul 15, 1984 | Valeria Ferrari, Bahram Mashhoon
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     We describe a new analytic approach to the problem of black-hole oscillations, which has been investigated numerically thus far. Our treatment is based on a connection between the quasinormal modes and the bound states of the inverted black-hole effective potentials. Approximate analytic formulas for the quasinormal frequencies of Schwarzschild, Reissner-Nordström, and slowly rotating Kerr black holes are provided. We find that a real quasinormal frequency for an extreme Kerr black hole has vanishing amplitude in the ordinary (i.e., nonsuperradiant) regime; therefore, extreme Kerr black holes are not marginally unstable in this case. These results are significant for the question of the stability of a black hole as well as for the late-time behavior of radiation from gravitationally collapsing configurations.

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   • [ pdf | Phys. Rev. D server | inspire | doi ]
10. • Quantized Dirac field in curved Riemann-Cartan background. I. Symmetry properties, Green's function
    • May 14, 1981 | H. T Nieh, M. L Yan
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      In the present series of papers, we study the properties of quantized Dirac field in curved Riemann-Cartan space, with particular attention on the role played by torsion. In this paper, we give, in the spirit of the original work of Weyl, a systematic presentation of Dirac's theory in curved Riemann-Cartan space. We discuss symmetry properties of the system, and derive conservation laws as direct consequences of these symmetries. Also discussed is conformal gauge symmetry, with torsion effectively playing the role of a conformal gauge field. To obtain short-distance behavior, we calculate the spinor Green's function, in curved Riemann-Cartan background, using the Schwinger-DeWitt method of proper-time expansion. The calculation corresponds to a generalization of DeWitt's calculation for a Riemannian background.

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    • [ Annals of Phys. server | inspire ]

Thermodynamics

1. • Gravitational Black Hole Hair from Event Horizon Supertranslations
   • Jan 14, 2016 | Artem Averin, Gia Dvali, Cesar Gomez, Dieter Lust
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     We discuss BMS supertranslations both at null-infinity and on the horizon for the case of the Schwarzschild black hole. We show that both kinds of supertranslations lead to infinetly many gapless physical excitations. On this basis we construct a quotient algebra using suited superpositions of both kinds of transformations which cannot be compensated by an ordinary BMS-supertranslation and therefore are intrinsically due to the presence of an event horizon. We show that these quotient transformations are physical and generate gapless excitations on the horizon that can account for the gravitational hair as well as for the black hole entropy. We identify the physics of these modes as associated with Bogolioubov-Goldstone modes due to quantum criticality. Classically the number of these gapless modes is infinite. However, we show that due to quantum criticality the actual amount of information-carriers becomes finite and consistent with Bekenstein entropy. Although we only consider the case of Schwarzschild geometry, the arguments are extendable to arbitrary space-times containing event horizons.

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   • [ pdf | 1601.03725 [gr-qc] | inspire ]
2. • The entropy formula of black holes in Minimal Massive Gravity and its application for BTZ black holes
   • Jan 1, 2015 | M. R. Setare, H. Adami
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     In this paper we obtain the entropy formula of black hole solutions of Minimal Massive Gravity (MMG) by Tachikawa method \cite{a}. Then we apply this formula for BTZ black hole solution. We find that the usual Bekenstein-Hawking entropy is modified. The modification come from Chern-Simons (CS) term and new term in MMG. The contribution of CS term, which is proportional with inner radius of horizon is not new, but last term which is due to the new term of MMG, as usual is proportional to the outer radius of horizon and is new result. Then we show that the total entropy exactly can be reproduced by Cardy formula for entropy of the dual boundary CFT.

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   • [ pdf | 1501.00920 [gr-qc] | inspire ]

Quantum aspects of black holes

1. • Hawking radiation, the Stefan-Boltzmann law, and unitarization
   • Nov 25, 2015 | Steven B. Giddings
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     Where does Hawking radiation originate? A common picture is that it arises from excitations very near or at the horizon, and this viewpoint has supported the "firewall" argument and arguments for a key role for the UV-dependent entanglement entropy in describing the quantum mechanics of black holes. However, closer investigation of both the total emission rate and the stress tensor of Hawking radiation supports the statement that its source is a near-horizon quantum region, or "atmosphere," whose radial extent is set by the horizon radius scale. This is potentially important, since Hawking radiation needs to be modified to restore unitarity, and a natural assumption is that the scales relevant to such modifications are comparable to those governing the Hawking radiation. Moreover, related discussion suggests a resolution to questions regarding extra energy flux in "nonviolent" scenarios, that does not spoil black hole thermodynamics as governed by the Bekenstein-Hawking entropy.

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   • [ pdf | 1511.08221 [hep-th] ]
2. • Anomalies, Hawking Radiations and Regularity in Rotating Black Holes
   • Jun 2, 2006 | Satoshi Iso, Hiroshi Umetsu, Frank Wilczek
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     This is an extended version of our previous letter hep-th/0602146. In this paper we consider rotating black holes and show that the flux of Hawking radiation can be determined by anomaly cancellation conditions and regularity requirement at the horizon. By using a dimensional reduction technique, each partial wave of quantum fields in a d=4 rotating black hole background can be interpreted as a (1+1)-dimensional charged field with a charge proportional to the azimuthal angular momentum m. From this and the analysis gr-qc/0502074, hep-th/0602146 on Hawking radiation from charged black holes, we show that the total flux of Hawking radiation from rotating black holes can be universally determined in terms of the values of anomalies at the horizon by demanding gauge invariance and general coordinate covariance at the quantum level. We also clarify our choice of boundary conditions and show that our results are consistent with the effective action approach where regularity at the future horizon and vanishing of ingoing modes at r=\infty are imposed (i.e. Unruh vacuum).

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   • [ pdf | 0606018 [hep-th] ]
3. • Evanescent Black Holes
   • Nov 28, 2015 | C. Callan, S. Giddings, J. Harvey, A. Strominger
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     A renormalizable theory of quantum gravity coupled to a dilaton and conformal matter in two space-time dimensions is analyzed. The theory is shown to be exactly solvable classically. Included among the exact classical solutions are configurations describing the formation of a black hole by collapsing matter. The problem of Hawking radiation and backreaction of the metric is analyzed to leading order in a 1/N expansion, where N is the number of matter fields. The results suggest that the collapsing matter radiates away all of its energy before an event horizon has a chance to form, and black holes thereby disappear from the quantum mechanical spectrum. It is argued that the matter asymptotically approaches a zero-energy ``bound state'' which can carry global quantum numbers and that a unitary S-matrix including such states should exist.

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   • [ pdf | 9111056 [hep-th] ]
4. • Trace anomalies and the Hawking effect
   • Apr 15, 1977 | S. M. Christensen and S. A. Fulling
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     The general spherically symmetric, static solution of ??T?? = 0 in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of T??. One constant is the magnitude of Ttr at infinity, and the other is determined if the physically normalized components of T?? are finite on the future horizon. The trace of the stress tensor of a conformally invariant quantum field theory may be nonzero (anomalous), but must be proportional (here) to the Weyl scalar, 48M2r?6; we fix the coefficient for the scalar field by indirect arguments to be (2880?2)?1. In the two-dimensional analog, the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the anomalous trace (a multiple of the curvature scalar); a knowledge of either number completely determines the stress tensor outside a body in the final state of collapse. In four dimensions, one obtains instead a relation constraining the remaining undetermined function, which we choose as T???T??4. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of ?T???. Groundwork is laid for explicit calculations of ?T???.

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   • [ Phys. Rev. D server | doi | inspire ]
5. • Entanglement entropy production in gravitational collapse: covariant regularization and solvable models
   • Apr 30, 2014 | Eugenio Bianchi, Tommaso De Lorenzo, Matteo Smerlak
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     We study the dynamics of vacuum entanglement in the process of gravitational collapse and subsequent black hole evaporation. In the first part of the paper, we introduce a covariant regularization of entanglement entropy tailored to curved spacetimes; this regularization allows us to propose precise definitions for the concepts of black hole "exterior entropy" and "radiation entropy." For a Vaidya model of collapse we find results consistent with the standard thermodynamic properties of Hawking radiation. In the second part of the paper, we compute the vacuum entanglement entropy of various spherically-symmetric spacetimes of interest, including the nonsingular black hole model of Bardeen, Hayward, Frolov and Rovelli-Vidotto and the "black hole fireworks" model of Haggard-Rovelli. We discuss specifically the role of event and trapping horizons in connection with the behavior of the radiation entropy at future null infinity. We observe in particular that (i) in the presence of an event horizon the radiation entropy diverges at the end of the evaporation process, (ii) in models of nonsingular evaporation (with a trapped region but no event horizon) the generalized second law holds only at early times and is violated in the "purifying" phase, (iii) at late times the radiation entropy can become negative (i.e. the radiation can be less correlated than the vacuum) before going back to zero leading to an up-down-up behavior for the Page curve of a unitarily evaporating black hole.

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   • [ pdf | 1409.0144 [gr-qc] ]
6. • Entanglement entropy and negative energy in two dimensions
   • Apr 2, 2014 | Eugenio Bianchi, Matteo Smerlak
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     It is well known that quantum effects can produce negative energy densities, though for limited times. Here we show in the context of two-dimensional CFT that such negative energy densities are present in any non-trivial conformal vacuum and can be interpreted in terms of the entanglement structure of such states. We derive an exact identity relating the outgoing energy flux and the entanglement entropy in the in-vacuum. When applied to two-dimensional models of black hole evaporation, this identity implies that unitarity is incompatible with monotonic mass loss.

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   • [ pdf | 1404.0602 [gr-qc] ]
7. • Mining Energy from a Black Hole by Strings
   • Dec 29, 2000 | V. Frolov, D. Fursaev
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     We discuss how cosmic strings can be used to mine energy from black holes. A string attached to the black hole gives rise to an additional channel for the energy release. It is demonstrated that when a string crosses the event horizon, its transverse degrees of freedom are thermally excited and thermal string perturbations propagate along the string to infinity. The internal metric induced on the 2D worldsheet of the static string crossing the horizon describes a 2D black hole. For this reason thermal radiation of string excitations propagating along the string can be interpreted as Hawking radiation of the 2D black hole. It is shown that the rate of energy emission through the string channel is of the same order of magnitude as the bulk radiation of the black hole. Thus, for N strings attached to the black hole the efficiency of string channels is increased by factor N. We discuss restrictions on N which exist because of the finite thickness of strings, the gravitational backreaction and quantum fluctuations. Our conclusion is that the energy emission rate by strings can be increased as compared to the standard emission in the bulk by the factor 10^3 for GUT strings and up to the factor 10^{31} for electroweak strings.

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   • [ pdf | hep-th/0012260 | Phys. Rev. D server | inspire ]
8. • Black Hole Evaporation along Macroscopic Strings
   • Dec 14, 1993 | A. Lawrence, E. Martinec
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     We develop the quantization of a macroscopic string which extends radially from a Schwarzschild black hole. The Hawking process excites a thermal bath of string modes that causes the black hole to lose mass. The resulting typical string configuration is a random walk in the angular coordinates. We show that the energy flux in string excitations is approximately that of spacetime field modes.

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   • [ pdf | hep-th/9312127 ]
9. • Entanglement Entropy of BTZ Black Hole and Conformal Anomaly
   • Dec 22, 2014 | Shobhit Sachan, Dharm Veer Singh
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     We study the logarithmic divergence term of the entropy of scalar fields propagating in BTZ black hole space-time. The logarithmic divergence term is related to the conformal anomalies and its coefficient is proportional to the "a" and "c" type anomalies. The (2+1) dimensional massive theory is obtained from (3+1) dimensional free massless theory via dimensional reduction. We obtained the divergence term of (2+1) dimensional massive theory after integrating over the masses. The mass term does not affect the area law. The logarithmic coefficient of divergence term is directly related to the c(?1), which we calculate numerically using the entanglement entropy approach. The numerical results are in agreement with the analytic results.

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   • [ pdf | 1412.7170 [hep-th] ]
10. • Black Hole Thermodynamics
    • Oct 6, 2014 | S. Carlip
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      The discovery in the early 1970s that black holes radiate as black bodies has radically affected our understanding of general relativity, and offered us some early hints about the nature of quantum gravity. In this chapter I will review the discovery of black hole thermodynamics and summarize the many independent ways of obtaining the thermodynamic and (perhaps) statistical mechanical properties of black holes. I will then describe some of the remaining puzzles, including the nature of the quantum microstates, the problem of universality, and the information loss paradox.

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    • [ pdf | 1410.1486 [gr-qc] ]
11. • Quantizing models of (2+1)-dimensional gravity on non-orientable manifolds
    • Feb 14, 2014 | Si Chen, Donald M. Witt, Steven S. Plotkin
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      Motivated by the relation between the Chern-Simons gauge theory and (2+1)-dimensional gravity, we find a formulation of gauge theories which applies to both orientable and non-orientable manifolds, using orientation bundles and density-valued forms. We show that on a non-orientable manifold, (2+1)D gravity is equivalent to BF theory, which is still topological and can be mapped in turn to Chern-Simons theory on the orientable double cover. By quantizing U(1) BF theory on a non-orientable manifold, we find that non-orientability introduces additional constraints on the quantized BF theory beyond those present for an orientable manifold, such that the coupling constant can only adopt a small number of discrete values. Specifically, for both the Klein bottle of demigenus 2 (N₂) and the compact surface of non-orientable genus 3 (Dyck's surface or N₃), we find explicit representations for the holonomy, large gauge, and mapping class groups, as well as the Hilbert space; here the above symmetries along with the non-orientability of the surface constrain the coupling constant k to only take values 1/2, 1, or 2.

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    • [ pdf | Class. Quant. Grav. server ]
12. • Dirac, Majorana and Weyl fermions
    • Jun 9, 2010 | Palash B. Pal
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      This is a pedagogical article which discusses various kinds of fermion fields: Dirac, Majorana and Weyl. The definitions and motivations for introducing each kind of fields is discussed, along with the connections between them. It is pointed out that these definitions have to do with the proper Lorentz group, and not with respect to any discrete symmetry. The relationship of discrete symmetries like charge conjugation and CP, particularly important for Majorana fermions, has also been clarified.

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    • [ pdf | 1006.1718 [gr-qc] | Am. J. Phys. server ]
13. • Semiclassical approximation to supersymmetric quantum gravity
    • Aug 15, 2005 | Claus Kiefer, Tobias Lueck, Paulo Moniz
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      We develop a semiclassical approximation scheme for the constraint equations of supersymmetric canonical quantum gravity. This is achieved by a Born-Oppenheimer type of expansion, in analogy to the case of the usual Wheeler-DeWitt equation. The formalism is only consistent if the states at each order depend on the gravitino field. We recover at consecutive orders the Hamilton-Jacobi equation, the functional Schrodinger equation, and quantum gravitational correction terms to this Schrodinger equation. In particular, the following consequences are found:
      (i) the Hamilton-Jacobi equation and therefore the background spacetime must involve the gravitino, (ii) a (many fingered) local time parameter has to be present on SuperRiem? (the space of all possible tetrad and gravitino fields), (iii) quantum supersymmetric gravitational corrections affect the evolution of the very early universe. The physical meaning of these equations and results, in particular the similarities to and differences from the pure bosonic case, are discussed.

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    • [ pdf | gr-qc/0505158 | Phys. Rev. D server ]
14. • On the interaction of mesoscopic quantum systems with gravity
    • Nov 18, 2004 | Claus Kiefer, Carsten Weber
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      We review the different aspects of the interaction of mesoscopic quantum systems with gravitational fields. We first discuss briefly the foundations of general relativity and quantum mechanics. Then, we consider the non-relativistic expansions of the Klein-Gordon and Dirac equations in the post-Newtonian approximation. After a short overview of classical gravitational waves, we discuss two proposed interaction mechanisms: (i) the use of quantum fluids as generator and/or detector of gravitational waves in the laboratory, and (ii) the inclusion of gravitomagnetic fields in the study of the properties of rotating superconductors. The foundations of the proposed experiments are explained and evaluated.

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    • [ pdf | gr-qc/0408010 | Ann. Phys. server ]
15. • Hawking radiation from decoherence
    • Oct 15, 2001 | Claus Kiefer
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      It is argued that the thermal nature of Hawking radiation arises solely due to decoherence. Thereby any information-loss paradox is avoided because for closed systems pure states remain pure. The discussion is performed for a massless scalar field in the background of a Schwarzschild black hole, but the arguments should hold in general. The result is also compared to and contrasted with the situation in inflationary cosmology.

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    • [ pdf | gr-qc/0110070 | Class. Quant. Grav. server ]
16. • Physical Aspects of the Space-Time Torsion
    • Mar 13, 2001 | I. L. Shapiro
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      We review many quantum aspects of torsion theory and discuss the possibility of the space-time torsion to exist and to be detected. The paper starts, in Chapter 2, with an introduction to the classical gravity with torsion, that includes also interaction of torsion with matter fields. In Chapter 3, the renormalization of quantum theory of matter fields and related topics, like renormalization group, effective potential and anomalies, are considered. Chapter 4 is devoted to the action of particles in a space-time with torsion, and to possible physical effects generated by the background torsion. In particular, we review the upper bounds for the background torsion which are known from the literature. In Chapter 5, the comprehensive study of the possibility of a theory for the propagating completely antisymmetric torsion field is presented. We show, that the propagating torsion may be consistent with the principles of quantum theory only in the case when the torsion mass is much greater than the mass of the heaviest fermion coupled to torsion. Then, universality of the fermion-torsion interaction implies that torsion itself has a huge mass, and can not be observed in realistic experiments. In Chapter 6, we briefly discuss the string-induced torsion and the possibility to induce torsion action and torsion itself through the quantum effects of matter fields.

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    • [ pdf | hep-th/0103093 | Phys. Rept. server ]
17. • Lectures in (2+1)-Dimensional Gravity
    • Mar 18, 1995 | Steven Carlip
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      These lectures briefly review our current understanding of classical and quantum gravity in three spacetime dimensions, concentrating on the quantum mechanics of closed universes and the (2+1)-dimensional black hole. Three formulations of the classical theory and three approaches to quantization are discussed in some detail, and a number of other approaches are summarized. An extensive, although by no means complete, list of references is included. (Lectures given at the First Seoul Workshop on Gravity and Cosmology, February 24-25, 1995.)

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    • [ pdf | gr-qc/9503024 | Phys. Rept. server ]
18. • 2 + 1 dimensional gravity as an exactly soluble system
    • Dec 19, 1988 | Edward Witten
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      By disentangling the hamiltonian constraint equations, 2 + 1 dimensional gravity (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels. Indeed, it is closely related to Yang-Mills theory with purely the Chern-Simons action, which recently has turned out to define a soluble quantum field theory. 2 + 1 dimensional gravity has a straightforward renormalized perturbation expansion, with vanishing beta function. 2 + 1 dimensional quantum gravity may provide a testing ground for understanding the role of classical singularities in quantum mechanics, may be related to the discrete series of Virasoro representations in 1 + 1 dimensions, and may be a useful tool in studying three-dimensional geometry.

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    • [ pdf | Nucl. Phys. B server ]
19. • The Thermodynamic Theory of Black Holes
    • Apr 21, 1977 | P. C. W. Davies
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      The thermodynamic theory underlying black hole processes is developed in detail and applied to model systems. It is found that Kerr-Newman black holes undergo a phase transition at a = 0.68M or Q = 0.86M, where the heat capacity has an infinite discontinuity. Above the transition values the specific heat is positive, permitting isothermal equilibrium with a surrounding heat bath. Simple processes and stability criteria for various black hole situations are investigated. The limits for entropically favoured black hole formation are found.

      The Nernst conditions for the third law of thermodynamics are not satisfied fully for black holes. There is no obvious thermodynamic reason why a black hole may not be cooled down below absolute zero and converted into a naked singularity. Quantum energy-momentum tensor calculations for uncharged black holes are extended to the Reissner-Nordstrom case, and found to be fully consistent with the thermodynamic picture for Q < M. For Q > M the model predicts that 'naked' collapse also produces radiation, with such intensity that the collapsing matter is entirely evaporated away before a naked singularity can form.

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    • [ pdf | Proc. Phys. Soc. London, Sect. A server ]

Quantum field theory in curved spacetime

1. • Cosmological Non-Constant Problem: Cosmological bounds on TeV-scale physics and beyond
   • Mar 31, 2015 | Niayesh Afshordi, Elliot Nelson
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     We study the influence of the fluctuations of a Lorentz invariant and conserved vacuum on cosmological metric perturbations, and show that they generically blow up in the IR. We compute this effect using the K\"all\'en-Lehmann spectral representation of stress correlators in generic quantum field theories, as well as the holographic bound on their entanglement entropy, both leading to an IR cut-off that scales as the fifth power of the highest UV scale (in Planck units). One may view this as analogous to the Heisenberg uncertainty principle, which is imposed on the phase space of gravitational theories by the Einstein constraint equations. The leading effect on cosmological observables come from anisotropic vacuum stresses which imply: i) any extension of the standard model of particle physics can only have masses (or resonances) ? 35 TeV, and ii) perturbative quantum field theory or quantum gravity becomes strongly coupled beyond a cut-off scale of ??1 PeV. Such a low cut-off is independently motivated by the Higgs hierarchy problem. This result, which we dub the cosmological non-constant problem, can be viewed as an extension of the cosmological constant (CC) problem, demonstrating the non-trivial UV-IR coupling and (yet another) limitation of effective field theory in gravity. However, it is more severe than the old CC problem, as vacuum fluctuations cannot be tuned to cancel due to the positivity of spectral densities or entropy. We thus predict that future advances in cosmological observations and collider technology will sandwich from above and below, and eventually discover, new (non-perturbative) physics beyond the Standard Model within the TeV-PeV energy range.

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   • [ pdf | 1504.00012 [hep-th] ]

Quantum gravity

1. • Quantum gravity at the corner
   • Jul 9, 2015 | Laurent Freidel, Alejandro Perez
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     We investigate the quantum geometry of 2d surface S bounding the Cauchy slices of 4d gravitational system. We investigate in detail and for the first time the symplectic current that naturally arises boundary term in the first order formulation of general relativity in terms of the Ashtekar-Barbero connection. This current is proportional to the simplest quadratic form constructed out of the triad field, pulled back on S. We show that the would-be-gauge degrees of freedom---arising from SU(2) gauge transformations plus diffeomorphisms tangent to the boundary, are entirely described by the boundary 2-dimensional symplectic form and give rise to a representation at each point of S of SL(2,R)×SU(2). Independently of the connection with gravity, this system is very simple and rich at the quantum level with possible connections with conformal field theory in 2d. A direct application of the quantum theory is modelling of the black horizons in quantum gravity.

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   • [ pdf | 1507.02573 [gr-qc] ]
2. • Minimal Length Scale Scenarios for Quantum Gravity
   • Mar 28, 2012 | Sabine Hossenfelder
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     We review the question of whether the fundamental laws of nature limit our ability to probe arbitrarily short distances. First, we examine what insights can be gained from thought experiments for probes of shortest distances, and summarize what can be learned from different approaches to a theory of quantum gravity. Then we discuss some models that have been developed to implement a minimal length scale in quantum mechanics and quantum field theory. These models have entered the literature as the generalized uncertainty principle or the modified dispersion relation, and have allowed the study of the effects of a minimal length scale in quantum mechanics, quantum electrodynamics, thermodynamics, black-hole physics and cosmology. Finally, we touch upon the question of ways to circumvent the manifestation of a minimal length scale in short-distance physics.

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   • [ pdf | 1203.6191 [gr-qc] ]

Quantum field theory

1. • The Ubiquitous 'c': from the Stefan-Boltzmann Law to Quantum Information
   • Aug 13, 2010 | John Cardy
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     I discuss various aspects of the role of the conformal anomaly number c in 2- and 1+1-dimensional critical behaviour: its appearance as the analogue of Stefan's constant, its fundamental role in conformal field theory, in the classification of 2d universality classes, and as a measure of quantum entanglement, among other topics.

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   • [ pdf | 1008.2331 [cond-mat] ]

Miscellaneous

1. • Axion Cosmology
   • Oct 26, 2015 | David J. E. Marsh
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     Axions comprise a broad class of particles that can play a major role in explaining the unknown aspects of cosmology. They are also well-motivated within high energy physics, appearing in theories related to CP-violation in the standard model, super-symmetric theories, and theories with extra-dimensions, including string theory, and so axion cosmology offers us a unique view onto these theories. I review the motivation and models for axions in particle physics and string theory. I then present a compre- hensive and pedagogical view on the cosmology and astrophysics of axion-like particles, starting from inflation and progressing via BBN, the CMB, reionization and structure formation, up to the present-day Universe. Topics covered include: axion dark matter (DM); direct and indirect detection of axions, reviewing existing and future experi- ments; axions as dark radiation; axions and the cosmological constant problem; decays of heavy axions; axions and stellar astrophysics; black hole superradiance; axions and astrophysical magnetic fields; axion inflation, and axion DM as an indirect probe of inflation. A major focus is on the population of ultralight axions created via vacuum realignment, and its role as a DM candidate with distinctive phenomenology. Cosmo- logical observations place robust constraints on the axion mass and relic density in this scenario, and I review where such constraints come from. I next cover aspects of galaxy formation with axion DM, and ways this can be used to further search for evidence of axions. An absolute lower bound on DM particle mass is established. It is m_a>10^?24eV from linear observables, extending to m_a >= 10^?22 eV from non-linear observables, and has the potential to reach m_a >= 10^?18eV in the future. These bounds are weaker if the axion is not all of the DM, giving rise to limits on the relic density at low mass. This leads to the exciting possibility that the effects of axion DM on structure formation could one day be detected, and the axion mass and relic density measured from cosmological observables.

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   • [ pdf | 1510.07633 [astro-ph.CO] | Phys. Rept. server | inspire ]
2. • Emergent Photons and Gravitons: The Problem of Vacuum Structure
   • Jul 30, 2010 | James D. Bjorken
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     We discuss vacuum condensates associated with emergent QED and with torsion, as well as the possible role of the Kodama wave function in quantum cosmology.

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   • [ pdf | 1008.0033 [hep-ph] ]
3. • Quantum gravity in terms of topological observables
   • Jan 24, 2005 | Laurent Freidel, Artem Starodubtsev
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     We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. The coupling constant becomes dimensionless (G_{Newton} \Lambda) and extremely small 10^{-120}. We give an expression for the generating functional of perturbation theory. We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable. This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory.

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   • [ pdf | hep-th/0501191 ]
4. • Spin Connection and Renormalization of Teleparallel Action
   • Apr 30, 2015 | Martin Krssák, J. G. Pereira
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     In general relativity, inertia and gravitation are both included in the Levi-Civita connection. As a consequence, the gravitational action, as well as the corresponding energy-momentum density, are always contaminated by spurious contributions coming from the inertial effects. Since these contributions can be removed only quasi-locally, one usually ends up with a quasi-local notion of energy and momentum. In teleparallel gravity, on the other hand, because the spin connection represents inertial effects only, it is possible to separate inertia from gravitation. Relying on this property, it is shown that to each tetrad there is naturally associated a spin connection that locally removes the inertial effects from the action, being thus possible to obtain local notions of energy and momentum. The use of the appropriate spin connection can be viewed as a renormalization process in the sense that the computation of energy and momentum naturally yields the physically relevant values.

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   • [ pdf | 1504.07683 [gr-qc] ]
5. • Visualizing Interstellar's Wormhole
   • Feb 12, 2015 | Oliver James, Eugenie von Tunzelmann, Paul Franklin, Kip S. Thorne
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     Christopher Nolan's science fiction movie Interstellar offers a variety of opportunities for students in elementary courses on general relativity theory. This paper describes such opportunities, including: (i) At the motivational level, the manner in which elementary relativity concepts underlie the wormhole visualizations seen in the movie. (ii) At the briefest computational level, instructive calculations with simple but intriguing wormhole metrics, including, e.g., constructing embedding diagrams for the three-parameter wormhole that was used by our visual effects team and Christopher Nolan in scoping out possible wormhole geometries for the movie. (iii) Combining the proper reference frame of a camera with solutions of the geodesic equation, to construct a light-ray-tracing map backward in time from a camera's local sky to a wormhole's two celestial spheres. (iv) Implementing this map, for example in Mathematica, Maple or Matlab, and using that implementation to construct images of what a camera sees when near or inside a wormhole. (v) With the student's implementation, exploring how the wormhole's three parameters influence what the camera sees---which is precisely how Christopher Nolan, using our implementation, chose the parameters for \emph{Interstellar}'s wormhole. (vi) Using the student's implementation, exploring the wormhole's Einstein ring, and particularly the peculiar motions of star images near the ring; and exploring what it looks like to travel through a wormhole.

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   • [ pdf | 1502.03809 [gr-qc] ]
6. • The Kerr-Newman metric: A Review
   • Oct 24, 2014 | Tim Adamo, E.T. Newman
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     The Kerr-Newman metric describes a very special rotating, charged mass and is the most general of the asymptotically flat stationary 'black hole' solutions to the Einstein-Maxwell equations of general relativity. We review the derivation of this metric from the Reissner-Nordstrom solution by means of a complex transformation algorithm and provide a brief overview of its basic geometric properties. We also include some discussion of interpretive issues, related metrics, and higher-dimensional analogues.

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     • spin coefficients are explained
     • ...
   • [ pdf | 1410.6626 [gr-qc] ]
7. • Kaluza-Klein multidimensional models with Ricci-flat internal spaces: the absence of the KK particles
   • Jan 9, 2014 | Alexey Chopovsky, Maxim Eingorn, Alexander Zhuk
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     In this paper we consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, e.g., a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, e.g., astrophysical objects such as our Sun. We suppose that these masses are pressureless in the external space but they have relativistic pressure in the internal space. We show that metric perturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over the internal space. This means, first, that KK modes corresponding to the metric fluctuations are absent and, second, particles should be only in the ground quantum state with respect to the internal space. In our opinion, these results look very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, i.e. in KK modes. We also get formulae for the metric correction terms which enable to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes and the perihelion advance.

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     • in the absence of matter sources, the metric is the direct sum of the internal and external metric
     • metric perturbations caused by several gravitating bodies (like our sun, etc.) are independent of internal coordinates, which implies that the masses are uniformly smeared over the internal space
     • therefore, KK modes corresponding to the metric fluctuations are absent
     • and furthermore, the particles associated with the perturbing masses should be in internal ground state (unnatural, since T ≠ 0)
     • p. 7: a high symmetry of internal space means a suppression of the extra degrees of freedom
   • [ pdf | 1311.0220v2 [hep-th] | Adv. in HEP server ]
8. • Wave propagation in electromagnetic systems with a linear response
   • Dec, 2013 | Y. Itin
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     n.a.

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     • mathematically precise tensor decomposition into irreducible parts using Young tableaux
   • [ pdf ]
9. • On the Localization of Energy-Momentum and Spin in Classical Field Theory
   • Oct 9, 2013 | F.W. Hehl
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     n.a.

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     • ...
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10. • The angular momentum controversy: What's it all about and does it matter?
    • Sep 18, 2013 | Elliot Leader, Cédric Lorcé
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      The general question, crucial to an understanding of the internal structure of the nucleon, of how to split the total angular momentum of a photon or gluon into spin and orbital contributions is one of the most important and interesting challenges faced by gauge theories like Quantum Electrodynamics and Quantum Chromodynamics. This is particularly challenging since all QED textbooks state that such an splitting cannot be done for a photon (and a fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the world are engaged in measuring what they believe is the gluon spin! This question has been a subject of intense debate and controversy, ever since, in 2008, it was claimed that such a gauge-invariant split was, in fact, possible. We explain in what sense this claim is true and how it turns out that one of the main problems is that such a decomposition is not unique and therefore raises the question of what is the most natural or physical choice. The essential requirement of measurability does not solve the ambiguities and leads us to the conclusion that the choice of a particular decomposition is essentially a matter of taste and convenience. In this review, we provide a pedagogical introduction to the question of angular momentum decomposition in a gauge theory, present the main relevant decompositions and discuss in detail several aspects of the controversies regarding the question of gauge invariance, frame dependence, uniqueness and measurability. We stress the physical implications of the recent developments and collect into a separate section all the sum rules and relations which we think experimentally relevant. We hope that such a review will make the matter amenable to a broader community and will help to clarify the present situation.

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    • [ pdf | 1309.4235 [hep-th] ]
11. • Allgemeine Relativitätstheorie - Vorlesungsskript
    • Mar 4, 2013 | Haye Hinrichsen
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      n.a.

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      • ...
    • [ pdf ]
12. • A Possible Intuitive Derivation of the Kerr Metric in Orthogonal Form Based On Ellipsoidal Metric Ansatz
    • Oct 22, 2012 | Branislav D. Nikolic, Milan R. Pantic
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      In this paper we show that it is possible to derive the Kerr solution in an alternative, intuitive way, based on physical reasoning and starting from an orthogonal metric ansatz having manifest ellipsoidal space-time symmetry (ellipsoidal symmetry). This is possible because both flat metric in oblate spheroidal (ellipsoidal) coordinates and Kerr metric in Boyer-Lindquist coordinates can be rewritten in such a form that the difference between the two is only in the time-time and radial-radial metric tensor components, just as is the case with Schwarzschild metric and flat metric in spherical coordinates.

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    • [ pdf | 1210.5922 [gr-qc] ]
13. • On the Origin of Gravity and the Laws of Newton
    • Jan 6, 2010 | Erik Verlinde
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      Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.

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      • p. 3: gravity might be explained as an entropic force caused by a change in the amount of information associated with a given matter/energy distribution
      • p. 3: gravity allows action at a distance even though there is no mediating force field
    • [ pdf | 1001.0785 [gr-qc] | JHEP server ]
14. • The Shape and Topology of the Universe
    • Feb 15, 2008 | Jean-Pierre Luminet
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      What is the shape of the Universe? Is it curved or flat, finite or infinite ? Is space "wrapped around" to create ghost images of faraway cosmic sources? We review how tessellations allow to build multiply-connected 3D Riemannian spaces useful for cosmology. We discuss more particularly the proposal of a finite, positively curved, dodecahedral space for explaining some puzzling features of the cosmic microwave background radiation, as revealed by the 2003-2006 WMAP data releases.

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    • [ pdf | 0802.2236 [astro-ph] ]
15. • Can Gravitons Be Detected?
    • Dec 2, 2006 | Tony Rothman, Stephen Boughn
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      Freeman Dyson has questioned whether any conceivable experiment in the real universe can detect a single graviton. If not, is it meaningful to talk about gravitons as physical entities? We attempt to answer Dyson's question and find it is possible concoct an idealized thought experiment capable of detecting one graviton; however, when anything remotely resembling realistic physics is taken into account, detection becomes impossible, indicating that Dyson's conjecture is very likely true. We also point out several mistakes in the literature dealing with graviton detection and production.

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      • gravitational radiation cannot be in thermodynamical equilibrium with its surroundings
      • primordial black holes could make up most of the universe's dark matter
      • σ(ν) = 10²⁰ σ(γ)
    • [ pdf | gr-qc/9605010 | Found. of Phys. server ]
16. • Notes on Differential Geometry
    • May 12, 2003 | Domenico Giulini
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      These notes present various concepts in differential geometry from the elegant and unifying point of view of principal bundles and their associated vector bundles.

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    • [ pdf ]
17. • Cosmic Topology
    • Jan 9, 2003 | Marc Lachièze-Rey, Jean-Pierre Luminet
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      General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi- rather than simply-connected. We review the main mathematical properties of multi-connected spaces, and the different tools to classify them and to analyse their properties. Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models. We briefly discuss some implications of multi-connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe. We consider in details the properties of the cosmological models where space is multi-connected, with emphasis towards observable effects. We then review the analyses of observational results obtained in this context, to search for a possible signature of multi-connectedness, or to constrain the models. They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,., or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them.

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    • [ pdf | gr-qc/9605010 | Phys. Rep. server ]
18. • Rotating, charged, and uniformly accelerating mass in general relativity
    • May 1, 1976 | J. F. Plebanski, M. Demianski
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      A new general class of solutions of the Einstein-Maxwell equations is presented. It depends on seven arbitrary parameters that group in a natural way into three complex parameters m + in, a + ib, e + ig, and the cosmological constant ?. They correspond to mass, NUT parameter, angular momentum per unit mass, acceleration, and electric and magnetic charge. The metric is in general stationary and axially symmetric. These solutions are of type D and contain as special cases all known solutions of type D belonging to this class. The known solutions are recovered by performing limiting transitions. An appropriate limit of our solutions describes an electromagnetic field in flat spacetime. We investigate the properties of that field. Its singular region corresponds in general to two circles moving with uniform acceleration in the positive and negative directions along the axis of symmetry. One can easily extend our solutions to the complex domain. Then it turns out that the metric can be written in a double Kerr-Schild form.

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    • [ Ann. Phys. server ]
19. • Black Hole Equilibrium States
    • 1972 | B. Carter
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      ...

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    • [ pdf (part 1) | pdf (part 2) ]
20. • Empty-Space Generalization of the Schwarzschild Metric
    • Nov 5, 1962 | E. Newman, L. Tamburino and T. Unti
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      A new class of empty-space metrics is obtained, one member of this class being a natural generalization of the Schwarzschild metric. This latter metric contains one arbitrary parameter in addition to the mass. The entire class is the set of metrics which are algebraically specialized (contain multiple-principle null vectors) such that the propagation vector is not proportional to a gradient. These metrics belong to the Petrov class type I degenerate.

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    • [ pdf J. Math. Phys. server ]

Useful Books

Here is a list of books that I find useful for my studies. They are listed separately, and not as part of the literature list above which is organized into topics. This is due to them covering broad ranges of topics, but also due to them not being freely available online. Therefore they are merely listed as a reference and for sake of completeness.

1. • Gauge Theories of Gravitation: A Reader with Commentaries
   • 2012 | Milutin Blagojevic, Friedrich W. Hehl
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     During the last five decades, gravity, as one of the fundamental forces of nature, has been formulated as a gauge field theory of the Weyl-Cartan-Yang-Mills type. The resulting theory, the Poincar\'e gauge theory of gravity, encompasses Einstein's gravitational theory as well as the teleparallel theory of gravity as subcases. In general, the spacetime structure is enriched by Cartan's torsion and the new theory can accommodate fermionic matter and its spin in a perfectly natural way.
     The present reprint volume contains articles from the most prominent proponents of the theory and is supplemented by detailed commentaries of the editors. This guided tour starts from special relativity and leads, in its first part, to general relativity and its gauge type extensions a la Weyl and Cartan. Subsequent stopping points are the theories of Yang-Mills and Utiyama and, as a particular vantage point, the theory of Sciama and Kibble. Later, the Poincar\'e gauge theory and its generalizations are explored and specific topics, such as its Hamiltonian form and exact solutions, are studied.
     This guide to the literature on gauge theories of gravity is intended to be a stimulating introduction to the field of classical gauge theories of gravity.

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     Chapter 18

     • p. 588: "Just as an electric charge carries a Coulomb field, E_a, with itself, a dislocation line is surrounded by a stress field, σ_ab."
     • p. 588: "An electric charge as a scalar is the source of a vector field, E_a, a dislocation line with a vector 'charge' δb_a, is the source of a tensor field, σ_ab, of 2nd rank."
   • [ pdf | 1210.3775 [gr-qc] ]
2. • Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity
   • Aug 20, 2009 | Leonard Parker, David Toms
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     n.a.

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     • p. 27ff: nice summary of non-Abelian SU(n) gauge theories
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3. • Theoretische Physik 3: Klassische Feldtheorie
   • 2009 | Florian Scheck
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     n.a.

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     • ...
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4. • Problems & Solutions in Group Theory for Physicists
   • 2004 | Zhong-Qi Ma, Xiao-Yan Gu
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     n.a.

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     • ...
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5. • Vector Bundles and K-Theory
   • 2003 | Allen Hatcher
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     The plan is for this to be a fairly short book focusing on topological K-theory and containing also the necessary background material on vector bundles and characteristic classes. Here is a provisional Table of Contents. At present only about half of the book is in good enough shape to be posted online, approximately 110 pages. This is available as a pdf file here. (I have reformatted this with narrower margins for a better reading experience on devices like an iPad, but for a paper copy with more standard size margins try printing at 85-90 per cent of full size.)

     What is included in this installment is:

     • Chapter 1, containing basics about vector bundles.
     • Part of Chapter 2, introducing K-theory, then proving Bott periodicity in the complex case and Adams' theorem on the Hopf invariant, with its famous applications to division algebras and parallelizability of spheres. Not yet written is the proof of Bott Periodicity in the real case, with its application to vector fields on spheres.
     • Most of Chapter 3, constructing Stiefel-Whitney, Chern, Euler, and Pontryagin classes and establishing their basic properties.
     • Part of Chapter 4 on the stable J-homomorphism. What is written so far is just the application of complex K-theory, using the Chern character, to give a lower bound on the order of the image of the stable J-homomorphism.

     Much of this material is already well covered in other sources, notably the classic books of Atiyah (K-theory) and Milnor & Stasheff (Characteristic Classes). These books are still in print, although they have become somewhat expensive.

     Eventually I intend for the book to include things that aren't readily accessible elsewhere, such as the full story on the stable J homomorphism. What is posted now is Version 2.1, dated May 2009. This is a very minor revision of Version 2.0 from January 2003. Unfortunately I haven't been able to find time to work on the book much since then. When I do get back to serious work on it, the first thing I plan to do is extend the proof of Bott periodicity from the complex case to the real case following the argument in Atiyah's book, using Clifford algebras.

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   • [ pdf | personal website ]
6. • Gravitation and Gauge Symmetries
   • 2002 | Milutin Blagojevic
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     n.a.

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     • ...
   • [ pdf ]
7. • Quantum Gravity in 2+1 Dimensions
   • 1998 | Steven Carlip
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     n.a.

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     • ...
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8. • Geometrodynamics of Gauge Fields
   • 1987 | Eckehard W. Mielke
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     n.a.

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     • ...
   • [ pdf ]
9. • Tensor Analysis for Physicists
   • 1953 | J. A. Schouten
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     n.a.

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     • The coordinate expression for a general connection with torsion and non-metricity is derived. Note that if the connection is considered an independent variable, a decomposition of the connection is "unwarranted" (cf. Hehl, Gauge Theory of Gravity and Spacetime)
   • [ pdf ]

Gravitoelectromagnetism (GEM)

1. • Gravitomagnetism and the significance of the curvature scalar invariants
   • Mar 11, 2016 | L. Filipe O. Costa, Lode Wylleman, José Natário
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     The curvature invariants have been subject of recent interest due to the debate concerning the notions of intrinsic/extrinsic frame-dragging, the use of the electromagnetic analogy in such classification, and the question of whether there is a fundamental difference between the gravitomagnetic field arising from the translational motion of the sources, detected with Lunar Laser Raging and in the observations of binary pulsars, and the gravitomagnetic field produced by the rotation of the Earth, detected in the LAGEOS Satellites data and by the Gravity Probe-B mission. In this work we clarify both the algebraic and physical meaning of the curvature invariants and their electromagnetic counterparts. The structure of the invariants of the astrophysical setups of interest is studied in detail, and its relationship with the gravitomagnetic effects is dissected. Finally, a new classification for intrinsic/extrinsic gravitomagnetism is put forth.

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   • [ pdf | 1603.03143 [gr-qc] | inspire ]

Torsion and Cosmology

1. • Self-accelerating Universe in modified gravity with dynamical torsion
   • Jun 8, 2016 | V. Nikiforova, S. Randjbar-Daemi, V. Rubakov
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     We consider a model belonging to the class of Poincare gauge gravities. The model is free of ghosts and gradient instabilities about Minkowski and torsionless Einstein backgrounds. We find that at zero cosmological constant, the model admits a self-accelerating solution with non-Riemannian connection. Small value of the effective cosmological constant is obtained at the expense of the hierarchy between the dimensionless couplings.

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   • [ pdf | 1606.02565 [hep-th] ]
2. • Infrared Modified Gravity with Dynamical Torsion
   • May 22, 2009 | V. Nikiforova, S. Randjbar-Daemi, V. Rubakov
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     We continue the recent study of the possibility of constructing a consistent infrared modification of gravity by treating the vierbein and connection as independent dynamical fields. We present the generalized Fierz--Pauli equation that governs the propagation of a massive spin-2 mode in a model of this sort in the backgrounds of arbitrary torsionless Einstein manifolds. We show explicitly that the number of propagating degrees of freedom in these backgrounds remains the same as in flat space-time. This generalizes the recent result that the Boulware--Deser phenomenon does not occur in de Sitter and anti-de Sitter backgrounds. We find that, at least for weakly curved backgrounds, there are no ghosts in the model. We also briefly discuss the interaction of sources in flat background.

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   • [ pdf | 0905.3732 [hep-th] ]

Black hole physics

1. • Can Superconducting Cosmic Strings Piercing Seed Black Holes Generate Supermassive Black Holes in the Early Universe?
   • May 7, 2015 | Matthew J. Lake, Tiberiu Harko
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     The discovery of a large number of supermassive black holes (SMBH) at redshifts z>6, when the Universe was only 900 million years old, raises the question of how such massive compact objects could form in a cosmologically short time interval. Each of the standard scenarios proposed, involving rapid accretion of seed black holes or black hole mergers, faces severe theoretical difficulties in explaining the short-time formation of supermassive objects. In this work we propose an alternative scenario for the formation of SMBH in the early Universe, in which energy transfer from superconducting cosmic strings piercing small seed black holes is the main physical process leading to rapid mass increase. As a toy model, the accretion rate of a seed black hole pierced by two antipodal strings carrying constant current is considered. Using an effective action approach, which phenomenologically incorporates a large class of superconducting string models, we estimate the minimum current required to form SMBH with masses of order M=2×109M? by z=7.085. This corresponds to the mass of the central black hole powering the quasar ULAS J112001.48+064124.3 and is taken as a test case scenario for early-epoch SMBH formation. For GUT scale strings, the required fractional increase in the string energy density, due to the presence of the current, is of order 10?7, so that their existence remains consistent with current observational bounds on the string tension. In addition, we consider an "exotic" scenario, in which an SMBH is generated when a small seed black hole is pierced by a higher-dimensional F?string, predicted by string theory. We find that both topological defect strings and fundamental strings are able to carry currents large enough to generate early-epoch SMBH via our proposed mechanism.

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     • antipodal strings carry oppositely charged (null) currents, with equal magnitude and opposite direction
     • the emission of electromagnetic radiation from superconducting GUT strings plays a negligibly small role in the determining string loop lifetimes
     • in their estimate of the mass transfer they only consider the rest mass of the charges, even though they are describing null currents
     • they consider a string-BH system to form simultaneously, there is no notion of the string merging with the BH
   • [ pdf | 1505.01584 [hep-th] | inspire ]
2. • Near-horizon Kerr Magnetosphere
   • Feb 4, 2016 | S. E. Gralla, A. Lupsasca, A. Strominger
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     We exploit the near-horizon conformal symmetry of rapidly spinning black holes to determine universal properties of their magnetospheres. Analytic expressions are derived for the limiting form of the magnetosphere in the near-horizon region. The symmetry is shown to imply that the black hole Meissner effect holds for free Maxwell fields but is generically violated for force-free fields. We further show that in the extremal limit, near-horizon plasma particles are infinitely boosted relative to accretion flow. Active galactic nuclei powered by rapidly spinning black holes are therefore natural sites for high-energy particle collisions.

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   • [ pdf | 1602.01833 [hep-th] | inspire ]
3. • Fourth-order split monopole perturbation solutions to the Blandford-Znajek mechanism
   • Mar 17, 2015 | Zhen Pan, Cong Yu
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     The Blandford-Znajek (BZ) mechanism describes a physical process for the energy extraction from a spinning black hole (BH), which is believed to power a great variety of astrophysical sources, such as active galactic nuclei (AGNs) and Gamma ray bursts (GRBs). The only known analytic solution to the BZ mechanism is a split monopole perturbation solution up to O(a2), where a is the spin parameter of a Kerr black hole. In this paper, we extend the monopole solution to higher order ?O(a4). We carefully investigate the structure of the BH magnetosphere, including the angular velocity of magnetic field lines ?, the toroidal magnetic field B? as well as the poloidal electric current I. In addition, the relevant energy extraction rate E? and the stability of this high-order monopole perturbation solution are also examined.

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   • [ pdf | 1503.05248 [hep-th] | inspire ]
4. • The Black Hole Meissner Effect and Blandford-Znajek Jets
   • Mar 4, 2014 | Robert F. Penna
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     Spinning black holes tend to expel magnetic fields. In this way they are similar to superconductors. It has been a persistent concern that this black hole "Meissner effect" could quench jet power at high spins. This would make it impossible for the rapidly rotating black holes in Cyg X-1 and GRS 1915+105 to drive Blandford-Znajek jets. We give a simple geometrical argument why fields which become entirely radial near the horizon are not expelled by the Meissner effect and may continue to power jets up to the extremal limit. A simple and natural example is a split-monopole field. We stress that ordinary Blandford-Znajek jets are impossible if the Meissner effect operates and expels the field. Finally, we note that in our general relativistic magnetohydrodynamic simulations of black hole jets, there is no evidence that jets are quenched by the Meissner effect. The simulated jets develop a large split monopole component spontaneously which supports our proposal for how the Meissner effect is evaded and jets from rapidly rotating black holes are powered in nature.

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   • [ pdf | 1403.0938 [hep-th] | inspire ]

Cosmic strings

1. • Dynamics and Properties of Chiral Cosmic Strings
   • Jun 5, 2001 | M. Pickles, A.C. Davis
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     Chiral cosmic strings naturally arise in many particle physics models, in particular in supersymmetric theories with a D-term. These strings have a single fermion zero mode in the core. We derive the general equation of motion for such strings. In Minkowski space we give the self-intersections for an arbitary varying current on the loop, showing that the self-intersection probability is dominated by the fraction of loop with maximal charge. We show how to relate the charge to the fermion condensation temperature, arguing that strings which become current carrying at formation will automatically have a maximal charge. Any daughter loops produced are likely to have the same charge as the parent loop. Possible models for chiral cosmic strings are also discussed and consequences for D-term inflation mentioned.

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   • [ pdf | hep-ph/0106079 | Phys. Lett. B server | inspire ]
2. • Dynamics and properties of chiral cosmic strings in Minkowski space
   • May 25, 2000 | A.C. Davis, T.W.B. Kibble, M. Pickles, D.A. Steer
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     Chiral cosmic strings are produced naturally at the end of inflation in supersymmetric models where the symmetry is broken via a D-term. Consequently in such theories, where both inflation and cosmic strings contribute to the density and CMBR (microwave background) perturbations, it is necessary to understand the evolution of chiral cosmic string networks. We study the dynamics of chiral cosmic strings in Minkowski space and comment on a number of differences with those of Nambu-Goto strings. To do this we follow the work of Carter and Peter who showed that the equations of motion for chiral cosmic strings reduce to a wave equation and two constraints, only one of which is different from the familiar Nambu-Goto constraints. We study chiral string loop solutions consisting of many harmonics and determine their self-intersection probabilities, and comment on the possible cosmological significance of these results.

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   • [ pdf | astro-ph/0005514 | Phys. Rev. D server | inspire ]
3. • Dynamics of superconducting strings with chiral currents
   • April 28, 2000 | J.J. Blanco-Pillado, Ken D. Olum, Alexander Vilenkin
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     We rederive, using an elementary formalism, the general solution to the equations of motion for a superconducting string with a chiral (null) neutral current, earlier obtained by Carter and Peter. We apply this solution to show that the motion of such string loops is strictly periodic and analyze cusp-like behavior and vorton solutions of arbitrary shape. We argue that this solution can be used to approximately describe the dynamics of superconducting strings with small non-chiral currents. We use this description to estimate the electromagnetic radiation power from such strings.

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   • [ pdf | astro-ph/0004410 | Phys. Rev. D server | inspire ]
4. • Dynamics and integrability property of the chiral string model
   • May 5, 1999 | Brandon Carter, Patrick Peter
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     The effect of fermionic string conductivity by purely right (or purely left) moving ``zero modes'' is shown to be governed by a simple Lagrangian characterising a certain ``chiral'' (null current carrying) string model whose dynamical equations of motion turn out to be explicitly integrable in a flat spacetime background.

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   • [ pdf | hep-th/9905025 | Phys. Lett. B server | inspire ]
5. • Abelian Higgs Hair for Black Holes
   • May 22, 1995 | Ana Achucarro, Ruth Gregory, Konrad Kuijken
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     We find evidence for the existence of solutions of the Einstein and Abelian Higgs field equations describing a black hole pierced by a Nielsen-Olesen vortex. This situation falls outside the scope of the usual no-hair arguments due to the non-trivial topology of the vortex configuration and the special properties of its energy-momentum tensor. By a combination of numerical and perturbative techniques we conclude that the black hole horizon has no difficulty in supporting the long range fields of the Nielsen Olesen string. Moreover, the effect of the vortex can in principle be measured from infinity, thus justifying its characterization as black hole ``hair".

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   • [ pdf | gr-qc/9505039 | Phys. Rev. D server | inspire ]
6. • Euclidean Black Hole Vortices
   • Dec 20, 1991 | Fay Dowker, Ruth Gregory, Jennie Traschen
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     We argue the existence of solutions of the Euclidean Einstein equations that correspond to a vortex sitting at the horizon of a black hole. We find the asymptotic behaviours, at the horizon and at infinity, of vortex solutions for the gauge and scalar fields in an abelian Higgs model on a Euclidean Schwarzschild background and interpolate between them by integrating the equations numerically. Calculating the backreaction shows that the effect of the vortex is to cut a slice out of the Euclidean Schwarzschild geometry. Consequences of these solutions for black hole thermodynamics are discussed.

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7. • Dynamics and Properties of Chiral Cosmic Strings
   • Jan 5, 1987 | David N. Spergel, Tsvi Piran, Jeremy Goodman
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     This paper introduces a covariant classical action for a superconducting string and derives the equations of motion for the string and its charge carriers. A formula for the electromagnetic power emitted by the moving string is calculated and is applied in the near-cusp region. The string emits beamed electromagnetic radiation, as well as relativistic particles, from the near-cusp region. A net current along the string can arise from either primordial flux threading the string loop or from a net asymmetry in the quantum number of the charge carriers. The string's motion in an external magnetic field can produce an oscillating current whose amplitude grows secularly. The implications of these results are discussed in the context of current astrophysical motins.

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8. • Electromagnetic Radiation from Superconducting Cosmic Stringss
   • Nov 24, 1986 | Mukunda Aryal, L.H. Ford, Alexander Vilenkin
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     The power of electromagnetic radiation from a current-carrying loop of superconducting string is much greater than that given by a naive estimate. Most of the radiation originates near the cusps where the string velocity reaches the speed of light, and is emitted in short and sharply directed pulses.

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9. • Cosmic Strings and Black Holes
   • Jul 29, 1986 | Mukunda Aryal, L.H. Ford, Alexander Vilenkin
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     The metric for a Schwarzschild black hole with a cosmic string passing through it is discussed. The thermodynamics of such an object is considered, and it is shown that S=(1/4)A, where S is the entropy and A is the horizon area. It is noted that the Schwarzschild mass parameter M, which is the gravitational mass of the system, is no longer identical to its energy. A solution representing a pair of black holes held apart by strings is discussed. It is nearly identical to a static, axially symmetric solution given long ago by Bach and Weyl. It is shown how these solutions, which were formerly a mathematical curiosity, may be given a more physical interpretation in terms of cosmic strings.

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10. • Superconducting Strings
    • Aug 14, 1984 | Edward Witten
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      It is known that certain spontaneously broken gauge theories give rise to stable strings or vortex lines. In this paper it is shown that under certain conditions such strings behave like superconducting wires whose passage through astrophysical magnetic fields would generate a variety of striking and perhaps observable effects. The superconducting charge carriers may be either bosons (if a charged Higgs field has an expectation value in the core of the string) or fermions (if charged fermions are trapped in zero modes along the string, as is known to occur in certain circumstances). They might be observable as synchrotron sources or as sources of high-energy cosmic rays. If the charge carriers are ordinary quarks and leptons, the strings have important baryon number violating interactions with magnetic fields; such a string, traversing a galactic magnetic field of 10 ?6 G, creates baryons (or antibaryons) at a rate of order 10 12 particles/cm of string per second.

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11. • Topology of Cosmic Domains and Strings
    • Mar 11, 1976 | Tom W. B. Kibble
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      The possible domain structures which can arise in the universe in a spontaneously broken gauge theory are studied. It is shown that the formation of domain wall, strings or monopoles depends on the homotopy groups of the manifold of degenerate vacua. The subsequent evolution of these structures is investigated. It is argued that while theories generating domain walls can probably be eliminated (because of their unacceptable gravitational effects), a cosmic network of strings may well have been formed and may have had important cosmological effects.

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