Spacetimes with horizons show a resemblance to thermodynamic systems and one can associate the notions of temperature and entropy with them. In the case of Einstein-Hilbert gravity, it is possible to interpret Einstein's equations as the thermodynamic identity T dS = dE + P dV for a spherically symmetric spacetime and thus provide a thermodynamic route to understand the dynamics of gravity. We study this approach further and show that the field equations for LanczosLovelock action in a spherically symmetric spacetime can also be expressed as T dS = dE +P dV with S and E being given by expressions previously derived in the literature by other approaches. The Lanczos-Lovelock Lagrangians are of the form L = Q bcd a R a bcd with ∇ b Q bcd a = 0. In such models, the expansion of Q bcd a in terms of the derivatives of the metric tensor determines the structure of the theory and higher order terms can be interpreted quantum corrections to Einstein gravity. Our result indicates a deep connection between the thermodynamics of horizons and the allowed quantum corrections to standard Einstein gravity, and shows that the relation T dS = dE + P dV has a greater domain of validity that Einstein's field equations.

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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...

There is an intriguing analogy between the gravitational dynamics of the horizons and thermodynamics. In case of general relativity, as well as for a wider class of Lanczos-Lovelock theories of gravity, it is possible to interpret the field equations near any spherically symmetric horizon as a thermodynamic identity T dS = dE + P dV . We study this approach further and generalize the results to two more generic cases within the context of general relativity: (i) stationary axis-symmetric horizons and (ii) time dependent evolving horizons . In both the cases, the near horizon structure of Einstein equations can be expressed as a thermodynamic identity under the virtual displacement of the horizon. This result demonstrates the fact that the thermodynamic interpretation of gravitational dynamics is not restricted to spherically symmetric or static horizons but is quite generic in nature and indicates a deeper connection between gravity and thermodynamics.

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