Is gravitational entropy quantized?

Abstract

In Einstein's gravity, the entropy of horizons is proportional to their area. Several arguments given in the literature suggest that, in this context, both area and entropy should be quantized with an equally spaced spectrum for large quantum numbers. But in more general theories (like, for e.g, in the black hole solutions of Gauss-Bonnet or Lanczos-Lovelock gravity) the horizon entropy is not proportional to area and the question arises as to which of the two (if at all) will have this property. We give a general argument that in all Lanczos-Lovelock theories of gravity, it is the entropy that has equally spaced spectrum. In the case of Gauss-Bonnet gravity, we use the asymptotic form of quasi normal mode frequencies to explicitly demonstrate this result. Hence, the concept of a quantum of area in Einstein Hilbert (EH) gravity needs to be replaced by a concept of quantum of entropy in a more general context. PACS numbers: 04.62.+v, 04.60.-mIt was conjectured by Bekenstein [1] long back that, in a quantum theory, the black hole area would be represented by a quantum operator with a discrete spectrum of eigenvalues. Bekenstein showed that the area of a classical black hole behaves like an adiabatic invariant, and so, according to Ehrenfest's theorem, the corresponding quantum operator must have a discrete spectrum. It was also known that, when a quantum particle is captured by a (non extremal) black hole its area increases by a minimum non-zero value [1,2,3] which is independent of the black hole parameters. This argument also suggests an equidistant spacing of area levels, with a well-defined notion of a quantum of area. The fundamental constants G, c and combine to give a quantity with the dimensions of area A P = (G /c 3 ) = 10 −66 cm 2 , which is quite suggestive [4] and sets the scale in area quantization.In Einstein's gravity, entropy of the horizon is proportional to its area. Hence one could equivalently claim that it is the gravitational entropy which has an equidistant spectrum with a well-defined notion of quantum of entropy. But, when one considers the natural generalization of Einstein gravity by including higher derivative correction terms to the original Einstein-Hilbert action, no such trivial relationship remains valid between horizon area and associated entropy. One such higher derivative theory which has attracted a fair amount of attention is Lanczos-Lovelock (LL) gravity [5], of which the lowest order correction appears as a Gauss-Bonnet (GB) term in D(> 4) dimensions. These lagrangians have the unique feature that the field equations obtained from them are quasi linear, as a result of which, the initial value problem remains well defined. More importantly, several features related to horizon thermodynamics, which were first discovered in the context of Einstein's theory [6], continues * Electronic address: dawood@iucaa.ernet.in † Electronic address: paddy@iucaa.ernet.in ‡ Electronic address: sudipta@iucaa.ernet.in to be valid in LL gravity models [7,8].Black hole solutions in the LL gravity a...

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