Exact Solutions of Einstein's Field Equations: Notation
01 Jan 2003-
About: The article was published on 2003-01-01. It has received 236 citations till now. The article focuses on the topics: Einstein notation & Notation for differentiation.
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TL;DR: The role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description, and various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories are reviewed.
Abstract: Over the past decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f(T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f(T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations, or alternatively, the Big Bang singularity can be avoided due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f(T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f(T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f(R) gravity, trying to enlighten the subject of which formulation might be more suitable for quantization ventures and cosmological applications.
697 citations
TL;DR: In this article, the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry was studied. And the same result was obtained from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches.
Abstract: We study the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry. It models a unitary evolution in which the field theory starts in a pure state, its vacuum, and undergoes a perturbation that brings it far from equilibrium. The entanglement entropy in this set up provides a measurement of the quantum entanglement in the system. Using holographic techniques we recover the same result obtained before from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches. Namely, entanglement in 2-dimensional conformal field theories propagates with velocity v2 = 1 [1]. Both in quantum quenches and in the Vaidya model equilibration is only achieved at the local level. Remarkably, the holographic derivation of this last fact requires information from behind the apparent horizon generated in the process of gravitational collapse described by the Vaidya geometry. In the early stages of the evolution the apparent horizon seems however to play no relevant role with regard to the entanglement entropy. We speculate on the possibility of deriving a thermalization time for occupation numbers from our analysis.
402 citations
Cites background from "Exact Solutions of Einstein's Field..."
...There is a family of exact solutions to the Einstein equations which describes the collapse of null dust to form a black hole, known as Vaidya metrics [33]....
[...]
TL;DR: In this article, the authors extend their previous analysis of Riemannian four-manifolds and show that they admit rigid supersymmetry to theories that do not possess a U(1)姫 R TAMADRA symmetry.
Abstract: We extend our previous analysis of Riemannian four-manifolds $ \mathcal{M} $
admitting rigid supersymmetry to $ \mathcal{N} $
= 1 theories that do not possess a U(1)
R
symmetry. With one exception, we find that $ \mathcal{M} $
must be a Hermitian manifold. However, the presence of supersymmetry imposes additional restrictions. For instance, a supercharge that squares to zero exists, if the canonical bundle of the Hermitian manifold $ \mathcal{M} $
admits a nowhere vanishing, holomorphic section. This requirement can be slightly relaxed if $ \mathcal{M} $
is a torus bundle over a Riemann surface, in which case we obtain a supercharge that squares to a complex Killing vector. We also analyze the conditions for the presence of more than one supercharge. The exceptional case occurs when $ \mathcal{M} $
is a warped product S
3
×
$ \mathbb{R} $
, where the radius of the round S
3 is allowed to vary along $ \mathbb{R} $
. Such manifolds admit two supercharges that generate the superalgebra OSp(1|2). If the S
3 smoothly shrinks to zero at two points, we obtain a squashed four-sphere, which is not a Hermitian manifold.
311 citations
Cites background from "Exact Solutions of Einstein's Field..."
...The fact that the three Killing vectors are orthogonal implies that hab(τ) = r(τ) (2)δab, see for instance [46]....
[...]
TL;DR: In this article, the relationship between classical solutions of non-Abelian gauge theory and gravity was examined, and a general class of gauge theory solutions that double copy to gravity was proposed, namely those involving stationary Kerr-Schild metrics.
Abstract: Recently, a perturbative duality between gauge and gravity theories (the double copy) has been discovered, that is believed to hold to all loop orders. In this paper, we examine the relationship between classical solutions of non-Abelian gauge theory and gravity. We propose a general class of gauge theory solutions that double copy to gravity, namely those involving stationary Kerr-Schild metrics. The Schwarzschild and Kerr black holes (plus their higher-dimensional equivalents) emerge as special cases. We also discuss plane wave solutions. Furthermore, a recently examined double copy between the self-dual sectors of Yang-Mills theory and gravity can be reinterpreted using a momentum-space generalisation of the Kerr-Schild framework.
250 citations
TL;DR: In this paper, two new classes of dyonic anti-de Sitter black hole solutions of four-dimensional maximal SO(8)-gauged supergravity were presented, and their thermodynamics were studied.
Abstract: We present two new classes of dyonic anti--de Sitter black hole solutions of four-dimensional maximal $\mathcal{N}=8$, SO(8) gauged supergravity. They are (1) static black holes of $\mathcal{N}=2$, $\mathrm{U}(1{)}^{4}$ gauged supergravity with four electric and four magnetic charges, with spherical, planar or hyperbolic horizons; and (2) rotating black holes of $\mathcal{N}=2$, $\mathrm{U}(1{)}^{2}$ gauged supergravity with two electric and two magnetic charges. We study their thermodynamics, and point out that the formulation of a consistent thermodynamics for dyonic anti--de Sitter black holes is dependent on the existence of boundary conditions for the gauge fields. We identify several distinct classes of boundary conditions for gauge fields in $\mathrm{U}(1{)}^{4}$ supergravity. We study a general family of metrics containing the rotating solutions, and find Killing-Yano tensors with torsion in two conformal frames, which underlie separability.
143 citations
Cites background or methods from "Exact Solutions of Einstein's Field..."
...[50], but its physics was studied later [51–55]....
[...]
...[50], and much used in the AdS/CFT correspondence....
[...]
References
More filters
TL;DR: In this paper, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description.
Abstract: Over recent decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T) gravity, where f (T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations-or, alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to illuminate the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications.
969 citations
TL;DR: In this article, the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry was studied. And the same result was obtained from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches.
Abstract: We study the evolution of entanglement entropy in a 2-dimensional equilibration process that has a holographic description in terms of a Vaidya geometry. It models a unitary evolution in which the field theory starts in a pure state, its vacuum, and undergoes a perturbation that brings it far from equilibrium. The entanglement entropy in this set up provides a measurement of the quantum entanglement in the system. Using holographic techniques we recover the same result obtained before from the study of processes triggered by a sudden change in a parameter of the hamiltonian, known as quantum quenches. Namely, entanglement in 2-dimensional conformal field theories propagates with velocity v2 = 1 [1]. Both in quantum quenches and in the Vaidya model equilibration is only achieved at the local level. Remarkably, the holographic derivation of this last fact requires information from behind the apparent horizon generated in the process of gravitational collapse described by the Vaidya geometry. In the early stages of the evolution the apparent horizon seems however to play no relevant role with regard to the entanglement entropy. We speculate on the possibility of deriving a thermalization time for occupation numbers from our analysis.
402 citations
TL;DR: In this article, the authors extend their previous analysis of Riemannian four-manifolds and show that they admit rigid supersymmetry to theories that do not possess a U(1)姫 R TAMADRA symmetry.
Abstract: We extend our previous analysis of Riemannian four-manifolds $ \mathcal{M} $
admitting rigid supersymmetry to $ \mathcal{N} $
= 1 theories that do not possess a U(1)
R
symmetry. With one exception, we find that $ \mathcal{M} $
must be a Hermitian manifold. However, the presence of supersymmetry imposes additional restrictions. For instance, a supercharge that squares to zero exists, if the canonical bundle of the Hermitian manifold $ \mathcal{M} $
admits a nowhere vanishing, holomorphic section. This requirement can be slightly relaxed if $ \mathcal{M} $
is a torus bundle over a Riemann surface, in which case we obtain a supercharge that squares to a complex Killing vector. We also analyze the conditions for the presence of more than one supercharge. The exceptional case occurs when $ \mathcal{M} $
is a warped product S
3
×
$ \mathbb{R} $
, where the radius of the round S
3 is allowed to vary along $ \mathbb{R} $
. Such manifolds admit two supercharges that generate the superalgebra OSp(1|2). If the S
3 smoothly shrinks to zero at two points, we obtain a squashed four-sphere, which is not a Hermitian manifold.
311 citations
TL;DR: In this article, the relationship between classical solutions of non-Abelian gauge theory and gravity was examined, and a general class of gauge theory solutions that double copy to gravity was proposed, namely those involving stationary Kerr-Schild metrics.
Abstract: Recently, a perturbative duality between gauge and gravity theories (the double copy) has been discovered, that is believed to hold to all loop orders. In this paper, we examine the relationship between classical solutions of non-Abelian gauge theory and gravity. We propose a general class of gauge theory solutions that double copy to gravity, namely those involving stationary Kerr-Schild metrics. The Schwarzschild and Kerr black holes (plus their higher-dimensional equivalents) emerge as special cases. We also discuss plane wave solutions. Furthermore, a recently examined double copy between the self-dual sectors of Yang-Mills theory and gravity can be reinterpreted using a momentum-space generalisation of the Kerr-Schild framework.
272 citations
TL;DR: In this article, the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions was studied.
Abstract: We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information.
198 citations