# Entropy and Quantum Gravity

## Abstract

**:**

**Figure 1.**Schematic diagrams contrasting our approach to open systems (

**a**) with that on the traditional “environment-induced decoherence” paradigm (

**b**).

**Our first set of results**concerns the impressive quantitative agreement between the results of Strominger and Vafa [36] and subsequent authors for the entropy of extremal and near-extremal black holes and the original Hawking entropy formulae and also between the semi-qualitative results of Susskind [14] and of Horowitz and Polchinski [15,16] for the entropy of, say, Schwarzschild black holes and the original Hawking entropy formula for those. These results clearly indicate that string theory is capable of providing an understanding of black hole entropy. But there are unsatisfactory puzzling issues too: Strominger and Vafa obtain the entropy as the logarithm of the degeneracy of an energy-level. Yet (to quote our paper [17]) the degeneracy of the nth energy level of the textbook Hydrogen atom Hamiltonian is ${n}^{2}$ but we would not conclude that the Hydrogen atom has an entropy of $klog{n}^{2}$! There is a related unsatisfactory puzzling issue in the work of Susskind and of Horowitz and Polchinski. They derive the entropy of a Schwarzschild black hole up to a small unknown constant with an argument which we now sketch. (In what follows, we take ℏ and c to equal 1. Following [15,16] we assume we can work with (1+3)-dimensional strings; ℓ stands for the string length scale, g for the string coupling constant and G for Newton’s constant, related to g and ℓ by $G={g}^{2}{\ell}^{2}$.) Horowitz and Polchinski assume that, as one scales ℓ up and g down from their physical values, keeping $G={g}^{2}{\ell}^{2}$ fixed, a Schwarzschild black hole of mass M will go over to a long string with roughly the same energy, $\u03f5=M$. The density of states of such a long string, for small enough g, is known, very roughly (i.e., omitting an inverse-power prefactor) to take the exponential form, ${\sigma}_{\mathrm{ls}}\left(\u03f5\right)={C}_{\mathrm{ls}}{e}^{\ell \u03f5}$ (${C}_{\mathrm{ls}}$ a constant with the dimensions of inverse energy of the same order of magnitude as ℓ). Horowitz and Polchinski then say that the “logarithm” of this is approximately given by $\ell \u03f5$ and propose that k times this should be equated with the entropy, S of a (Schwarzschild) black hole provided that one does the equating when, to within an order of magnitude or so, $\ell =GM$. Combining these latter two equations (and replacing ϵ by M) they arrive at the conclusion that the entropy of the black hole will be a moderately sized constant times $kG{M}^{2}$ which agrees, up to an undetermined value for the constant, with the Hawking value, $4\pi kG{M}^{2}$ for the entropy of a black hole.

**Our second set of results**concerns the AdS/CFT correspondence [39] which is usually thought to be a full equivalence between a quantum gravity theory in the bulk of Anti de Sitter space (AdS) and a conformal field theory (CFT) on the AdS conformal boundary. By considering states of quantum gravity which contain black holes and which are modelled classically by the Schwarzschild-Anti-de Sitter (Schwarwzschild-AdS) spacetime, and by arguing that it is correct to describe these states as in our above discussed description of black hole equilibrium states in terms of a pure total state, we have argued in [33] and [34] that the AdS/CFT correspondence is, instead, a bijection between the boundary CFT and just the matter degrees of freedom of the bulk AdS quantum gravity theory. As explained in those papers, this seems to offer a resolution to a puzzle raised [1] by Arnsdorf and Smolin: The puzzle arises, if one adopts the usual view of full equivalence, from the fact that Rehren has shown in [40,41] that any CFT on the conformal boundary of AdS is also equivalent, under a natural form of fixed-background holography which he introduced in these papers, to a quantum field theory on the AdS bulk (satisfying vanishing boundary conditions at the conformal boundary and) obeying an appropriate version of commutativity at spacelike separation. Such a commutativity condition would seem to be appropriate for a bulk theory involving matter, but not for one involving gravity.

**Figure 3.**(= “Figure 3” of Reference [34]. Reproduced here with kind permission from Springer Science + Business Media.) Schematic diagram of the four wedges of the region of 1 + 1 Minkowski space between the two components of a hyperbolic boundary (i.e., the curve $uv=-1$, in the indicated double-null coordinates, u and v) which may be thought of as a pair of accelerated mirrors. Shown are lines of constant phase of (the restriction to Region IV of) an initially right-moving plane wave. The wave reflects off the mirror in Region I and so do its lines of constant phase, piling up towards the horizon, ${\mathcal{H}}_{B}$ ($v=0$). We argue that a similar pile-up occurs in the Schwarzschild-AdS spacetime leading to the instability of the ${\mathcal{H}}_{A}$ and ${\mathcal{H}}_{B}$ horizons there.

## Conflicts of Interest

## References and Notes

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Kay, B.S. Entropy and Quantum Gravity. *Entropy* **2015**, *17*, 8174-8186.
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Kay BS. Entropy and Quantum Gravity. *Entropy*. 2015; 17(12):8174-8186.
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Kay, Bernard S. 2015. "Entropy and Quantum Gravity" *Entropy* 17, no. 12: 8174-8186.
https://doi.org/10.3390/e17127873