We show that the large-N limits of certainconformal field theories in various dimensions includein their Hilbert space a sector describing supergravityon the product of anti-de Sitter spacetimes, spheres, and other compact manifolds. This is shown bytaking some branes in the full M/string theory and thentaking a low-energy limit where the field theory on thebrane decouples from the bulk. We observe that, in this limit, we can still trust thenear-horizon geometry for large N. The enhancedsupersymmetries of the near-horizon geometry correspondto the extra supersymmetry generators present in thesuperconformal group (as opposed to just the super-Poincaregroup). The 't Hooft limit of 3 + 1 N = 4 super-Yang–Mills at the conformal pointis shown to contain strings: they are IIB strings. Weconjecture that compactifications of M/string theory on various anti-de Sitterspacetimes is dual to various conformal field theories.This leads to a new proposal for a definition ofM-theory which could be extended to include fivenoncompact dimensions.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1998/0002/0002/ATMP-1998-0002-0002-a001.pdf

The Large N limit of superconformal field theories and supergravity

http://inspirehep.net/record/499969/files/arXiv:hep-th_9905111.pdf

Large N Field Theories, String Theory and Gravity

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

With an aim towards understanding the time-dependence of entanglement entropy in generic quantum field theories, we propose a covariant generalization of the holographic entanglement entropy proposal of hep-th/0603001. Apart from providing several examples of possible covariant generalizations, we study a particular construction based on light-sheets, motivated in similar spirit to the covariant entropy bound underlying the holographic principle. In particular, we argue that the entanglement entropy associated with a specified region on the boundary in the context of the AdS/CFT correspondence is given by the area of a co-dimension two bulk surface with vanishing expansions of null geodesics. We demonstrate our construction with several examples to illustrate its reduction to the holographic entanglement entropy proposal in static spacetimes. We further show how this proposal may be used to understand the time evolution of entanglement entropy in a time varying QFT state dual to a collapsing black hole background. Finally, we use our proposal to argue that the Euclidean wormhole geometries with multiple boundaries should be regarded as states in a non-interacting but entangled set of QFTs, one associated to each boundary.

/pdf/a-covariant-holographic-entanglement-entropy-proposal-7emwnougsz.pdf

A covariant holographic entanglement entropy proposal

We consider $f(R,T)$ modified theories of gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar $R$ and of the trace of the stress-energy tensor $T$. We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the stress-energy tensor. Generally, the gravitational field equations depend on the nature of the matter source. The field equations of several particular models, corresponding to some explicit forms of the function $f(R,T)$, are also presented. An important case, which is analyzed in detail, is represented by scalar field models. We write down the action and briefly consider the cosmological implications of the $f(R,{T}^{\ensuremath{\phi}})$ models, where ${T}^{\ensuremath{\phi}}$ is the trace of the stress-energy tensor of a self-interacting scalar field. The equations of motion of the test particles are also obtained from a variational principle. The motion of massive test particles is nongeodesic, and takes place in the presence of an extra-force orthogonal to the four velocity. The Newtonian limit of the equation of motion is further analyzed. Finally, we provide a constraint on the magnitude of the extra acceleration by analyzing the perihelion precession of the planet Mercury in the framework of the present model.

$f(R,T)$ gravity

1. Why (2+1)-dimensional gravity? 2. Classical general relativity in 2+1 dimensions 3. A field guide to the (2+1)-dimensional spacetimes 4. Geometric structures and Chern-Simons theory 5. Canonical quantization in reduced phase space 6. The connection representation 7. Operator algebras and loops 8. The Wheeler-DeWitt equation 9. Lorentzian path integrals 10. Euclidian path integrals and quantum cosmology 11. Lattice methods 12. The (2+1)-dimensional black hole 13. Next steps Appendix A. The topology of manifolds Appendix B. Lorentzian metrics and causal structure Appendix C. Differential geometry and fiber bundles References Index.

Quantum Gravity in 2+1 Dimensions

Many recent attempts to calculate black hole entropy from first principles rely on conformal field theory techniques. By examining the logarithmic corrections to the Cardy formula, I compute the first-order quantum correction to the Bekenstein-Hawking entropy in several models, including those based on asymptotic symmetries, horizon symmetries and certain string theories. Despite very different physical assumptions, these models all give a correction proportional to the logarithm of the horizon size, and agree qualitatively with recent results from `quantum geometry' in 3 + 1 dimensions. There are some indications that even the coefficient of the correction may be universal, up to differences that depend on the treatment of angular momentum and conserved charges.

https://iopscience.iop.org/article/10.1088/0264-9381/17/20/302/pdf

Logarithmic corrections to black hole entropy, from the Cardy formula

Restricted to a black hole horizon, the ``gauge'' algebra of surface deformations in general relativity contains a Virasoro subalgebra with a calculable central charge. The fields in any quantum theory of gravity must transform accordingly, i.e., they must admit a conformal field theory description. Applying Cardy's formula for the asymptotic density of states, I use this result to derive the Bekenstein-Hawking entropy. This method is universal\char22{}it holds for any black hole, and requires no details of quantum gravity\char22{}but it is also explicitly statistical mechanical, based on counting microscopic states.

Black Hole Entropy from Conformal Field Theory in Any Dimension

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 non-vanishing 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.

https://iopscience.iop.org/article/10.1088/0264-9381/12/12/005/pdf

The (2 + 1)-dimensional black hole

The problem of reconciling general relativity and quantum theory has fascinated and bedeviled physicists for more than 70?years. Despite recent progress in string theory and loop quantum gravity, a complete solution remains out of reach. I review the status of the continuing effort to quantize gravity, emphasizing the underlying conceptual issues and the various attempts to come to grips with them.

https://cds.cern.ch/record/517743/files/0108040.pdf

Steven Carlip

Papers

Quantum Gravity in 2+1 Dimensions

Logarithmic corrections to black hole entropy, from the Cardy formula

Black Hole Entropy from Conformal Field Theory in Any Dimension

The (2 + 1)-dimensional black hole

Quantum Gravity: a Progress Report