Entropy Bounds: New Insights (original) (raw)

Holographic holes and differential entropy

Journal of High Energy Physics, 2014

Recently it has been shown that the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in the bulk of a holographic spacetime has an interpretation as the differential entropy of a particular family of intervals (or strips) in the boundary theory [1, 2]. We first extend this construction to bulk surfaces which vary in time. We then give a general proof of the equality between the gravitational entropy and the differential entropy. This proof applies to a broad class of holographic backgrounds possessing a generalized planar symmetry and to certain classes of higher-curvature theories of gravity. To apply this theorem, one can begin with a bulk surface and determine the appropriate family of boundary intervals by considering extremal surfaces tangent to the given surface in the bulk. Alternatively, one can begin with a family of boundary intervals; as we show, the differential entropy then equals the gravitational entropy of a bulk surface that emerges from the intersection of the neighboring entanglement wedges, in a continuum limit.

Holography, degenerate horizons and entropy

Nuclear Physics B, 2000

We show that a realization of the correspondence AdS 2 /CFT 1 for near extremal Reissner-Nordström black holes in arbitrary dimensional Einstein-Maxwell gravity exactly reproduces, via Cardy's formula, the deviation of the Bekenstein-Hawking entropy from extremality. We also show that this mechanism is valid for Schwarzschild-de Sitter black holes around the degenerate solution dS 2 ×S n. These results reinforce the idea that the Bekenstein-Hawking entropy can be derived from symmetry principles.

Relative entropy and holography

Journal of High Energy Physics, 2013

Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful 2 This inequality can be regarded as a generalized statement of the Bekenstein bound which holds for any region in QFT. This is explained in more detail in the appendix A.4.

Geometric Framework for Entropy in General Relativity

We introduce a covariant and geometrical framework for entropy in relativistic theories. In this framework the entropy of gravitational systems turns out to be a geometric quantity with well-defined cohomological properties arising from the obstruction to foliating spacetime into spacelike hypersurfaces. The framework relies on general covariance within a geometric framework which was recently defined to deal with the variation of conserved quantities generated by Nöther theorem. The definition of gravitational entropy so obtained turns out to be very general: it can be generalized to causal horizons and multiple-horizon spacetimes and it can be applied to define entropy for more exotic singular solutions of Einstein field equations. The same definition is also well-suited for higher dimensions and in the case of alternative gravitational theories (e.g. Chern-Simons theories and Lovelock Gravity).

HOLOGRAPHIC GEOMETRIC ENTROPY AT FINITE TEMPERATURE FROM BLACK HOLES IN GLOBAL ANTI-DE SITTER SPACES

International Journal of Modern Physics A, 2012

Using a holographic proposal for the geometric entropy we study its behavior in the geometry of Schwarzschild black holes in global AdS p for p = 3, 4, 5. Holographically, the entropy is determined by a minimal surface. On the gravity side, due to the presence of a horizon on the background, generically there are two solutions to the surfaces determining the entanglement entropy. In the case of AdS 3 , the calculation reproduces precisely the geometric entropy of an interval of length l in a two dimensional conformal field theory with periodic boundary conditions. We demonstrate that in the cases of AdS 4 and AdS 5 the sign of the difference of the geometric entropies changes, signaling a transition. Euclideanization implies that various embedding of the holographic surface are possible. We study some of them and find that the transitions are ubiquitous. In particular, our analysis renders a very intricate phase space, showing, for some ranges of the temperature, up to three branches. We observe a remarkable universality in the type of results we obtain from AdS 4 and AdS 5 .

Entropy bounds in terms of the w parameter

Journal of High Energy Physics, 2011

In a pair of recent articles [PRL 105 (2010) 041302; JHEP 1103 JHEP (2011 056] two of the current authors have developed an entropy bound for equilibrium uncollapsed matter using only classical general relativity, basic thermodynamics, and the Unruh effect. An odd feature of that bound, S ≤ 1 2 A , was that the proportionality constant, 1 2 , was weaker than that expected from black hole thermodynamics, 1 4 . In the current article we strengthen the previous results by obtaining a bound involving the (suitably averaged) w parameter. Simple causality arguments restrict this averaged w parameter to be ≤ 1. When equality holds, the entropy bound saturates at the value expected based on black hole thermodynamics. We also add some clarifying comments regarding the (net) positivity of the chemical potential. Overall, we find that even in the absence of any black hole region, we can nevertheless get arbitrarily close to the Bekenstein entropy.

Finite dimensions and the covariant entropy bound

2002

We explore the consequences of assuming that the bounded space-time subsets contain a finite number of degrees of freedom. A physically natural hypothesis is that this number is additive for spatially separated subsets. We show that this assumption conflicts with the Lorentz symmetry of Minkowski space since it implies that a conserved current determines the number of degrees of freedom. However, the entanglement across boundaries can lead to a subadditive property for the degrees of freedom of spatially separated sets. We show that this condition and the Poincare symmetry lead to the Bousso covariant entropy bound for Minkowski space.

Minimal length in quantum gravity, equivalence principle and holographic entropy bound

Classical and Quantum Gravity, 2011

A possible discrepancy has been found between the results of a neutron interferometry experiment and Quantum Mechanics. This experiment suggests that the weak equivalence principle is violated at small length scales, which quantum mechanics cannot explain. In this paper, we investigated whether the Generalized Uncertainty Principle (GUP), proposed by some approaches to quantum gravity such as String Theory and Doubly Special Relativity Theories (DSR), can explain the violation of the weak equivalence principle at small length scales. We also investigated the consequences of the GUP on the Liouville theorem in statistical mechanics. We have found a new form of invariant phase space in the presence of GUP. This result should modify the density states and affect the calculation of the entropy bound of local quantum field theory, the cosmological constant, black body radiation, etc. Furthermore, such modification may have observable consequences at length scales much larger than the Planck scale. This modification leads to a √ A-type correction to the bound of the maximal entropy of a bosonic field which would definitely shed some light on the holographic theory.

CFT, holography, and causal entropy bound

Physics Letters B, 2001

The causal entropy bound (CEB) is confronted with recent explicit entropy calculations in weakly and strongly coupled conformal field theories (CFTs) in arbitrary dimension D. For CFT's with a large number of fields, N , the CEB is found to be valid for temperatures not exceeding a value of order M P /N 1 D−2 , in agreement with large N bounds in generic cutoff theories of gravity, and with the generalized second law. It is also shown that for a large class of models including high-temperature weakly coupled CFT's and strongly coupled CFT's with AdS duals, the CEB, despite the fact that it relates extensive quantities, is equivalent to (a generalization of) a purely holographic entropy bound proposed by E. Verlinde.

A gravitational entropy proposal

Classical and Quantum Gravity, 2013

We propose a thermodynamically motivated measure of gravitational entropy based on the Bel-Robinson tensor, which has a natural interpretation as the effective superenergy-momentum tensor of free gravitational fields. The specific form of this measure differs depending on whether the gravitational field is Coulomb-like or wave-like, and reduces to the Bekenstein-Hawking value when integrated over the interior of a Schwarzschild black hole. For scalar perturbations of a Robertson-Walker geometry we find that the entropy goes like the Hubble weighted anisotropy of the gravitational field, and therefore increases as structure formation occurs. This is in keeping with our expectations for the behaviour of gravitational entropy in cosmology, and provides a thermodynamically motivated arrow of time for cosmological solutions of Einstein's field equations. It is also in keeping with Penrose's Weyl curvature hypothesis.