Sanved Kolekar

Action principle for the fluid-gravity correspondence and emergent gravity

It has been known for a long time that Einstein’s field equations when projected onto a black hole horizon look very similar to a Navier-Stokes equation in suitable variables. More recently, it was shown that the projection of Einstein’s equation onto any null surface in any spacetime reduces exactly to the Navier-Stokes form when viewed in the freely falling frame. We develop an action principle, the extremization of which leads to the above result, in an arbitrary spacetime. The degrees of freedom varied in the action principle are the null vectors in the spacetime and not the metric tensor. The same action principle was introduced earlier in the context of the emergent gravity paradigm wherein it was shown that the corresponding Lagrangian can be interpreted as the entropy density of spacetime. The current analysis strengthens this interpretation and reinforces the idea that field equations in gravity can be thought of as emergent. We also find that the degrees of freedom on the null surface are equivalent to a fluid with equation of state PA=TS. We demonstrate that the same relation arises in the context of a spherical shell collapsing to form a horizon.

Two Aspects of Black hole entropy in Lanczos-Lovelock models of gravity

We consider two specific approaches to evaluate the black hole entropy which are known to produce correct results in the case of Einstein’s theory and generalize them to Lanczos-Lovelock models. In the first approach (which could be called extrinsic), we use a procedure motivated by earlier work by Pretorius, Vollick, and Israel, and by Oppenheim, and evaluate the entropy of a configuration of densely packed gravitating shells on the verge of forming a black hole in Lanczos-Lovelock theories of gravity. We find that this matter entropy is not equal to (it is less than) Wald entropy, except in the case of Einstein theory, where they are equal. The matter entropy is proportional to the Wald entropy if we consider a specific mth-order Lanczos-Lovelock model, with the proportionality constant depending on the spacetime dimensions D and the order m of the Lanczos-Lovelock theory as (D-2m)/(D-2). Since the proportionality constant depends on m, the proportionality between matter entropy and Wald entropy breaks down when we consider a sum of Lanczos-Lovelock actions involving different m. In the second approach (which could be called intrinsic), we generalize a procedure, previously introduced by Padmanabhan in the context of general relativity, to study off-shell entropy of a class of metrics with horizon using a path integral method. We consider the Euclidean action of Lanczos-Lovelock models for a class of metrics off shell and interpret it as a partition function. We show that in the case of spherically symmetric metrics, one can interpret the Euclidean action as the free energy and read off both the entropy and energy of a black hole spacetime. Surprisingly enough, this leads to exactly the Wald entropy and the energy of the spacetime in Lanczos-Lovelock models obtained by other methods. We comment on possible implications of the result.

Entropy increase during physical processes for black holes in Lanczos-Lovelock gravity

We study quasistationary physical process for black holes within the context of Lanczos-Lovelock gravity. We show that the Wald entropy of the stationary black holes in Lanczos-Lovelock gravity monotonically increases for quasistationary physical processes in which the horizon is perturbed by the accretion of positive energy matter and the black hole ultimately settles down to a stationary state. This result reinforces the physical interpretation of Wald entropy for Lanczos-Lovelock models and takes a step towards proving the analogue of the black hole area increase theorem in a wider class of gravitational theories.

Indistinguishability of thermal and quantum fluctuations

The existence of Davies–Unruh temperature in a uniformly accelerated frame shows that quantum fluctuations of the inertial vacuum state appears as thermal fluctuations in the accelerated frame. Hence thermodynamic experiments cannot distinguish between phenomena occurring in a thermal bath of temperature T in the inertial frame from those in a frame accelerating through inertial vacuum with the acceleration . We show that this indisguishability between quantum fluctuations and thermal fluctuations goes far beyond the fluctuations in the vacuum state. We show by an exact calculation, that the reduced density matrix for a uniformly accelerated observer when the quantum field is in a thermal state of temperature , is symmetric between acceleration temperature and the thermal bath temperature . Thus thermal phenomena cannot distinguish whether (i) one is accelerating with through a bath of temperature or (ii) accelerating with through a bath of temperature T. This shows that thermal and quantum fluctuations in an accelerated frame affect the observer in a symmetric manner. The implications are discussed.

Membrane paradigm and horizon thermodynamics in Lanczos-Lovelock gravity

A bstractWe study the membrane paradigm for horizons in Lanczos-Lovelock models of gravity in arbitrary D dimensions and find compact expressions for the pressure p and viscosity coefficients η and ζ of the membrane fluid. We show that the membrane pressure is intimately connected with Noether charge entropy SWald of the horizon when we consider a specific m-th order Lanczos-Lovelock model, through the relation

Drift, Drag and Brownian motion in the Davies-Unruh bath

{p^{{\left( {\text{m}} \right)}}}A/T = \left[ {\left( {D - {2}m} \right)/\left( {D - {2}} \right)} \right]S_{\text{Wald}}^{{\left( {\text{m}} \right)}}

Gravitational memory for uniformly accelerated observers

, where T is the temperature and A is the area of the horizon. Similarly, the viscosity coefficients are expressible in terms of entropy and quasi-local energy associated with the horizons. The bulk and shear viscosity coefficients are found to obey the relation ζ = −2(D − 3)/(D − 2)η.

Quantum memory for Rindler supertranslations

An interesting feature of the Davies-Unruh effect is that a uniformly accelerated observer sees an isotropic thermal spectrum of particles even though there is a preferred direction in this context, determined by the direction of the acceleration g. We investigate the thermal fluctuations in the Unruh bath by studying the Brownian motion of particles in the bath, especially as regards to isotropy. We find that the thermal fluctuations are anisotropic and induce different frictional drag forces on the Brownian particle depending on whether it has a drift velocity along the direction of acceleration g or in a direction transverse to it. Using the fluctuation-dissipation theorem, we argue that this anisotropy arises due to quantum correlations in the fluctuations at large correlation time scales.

Quantum field theory in the Rindler-Rindler spacetime

Recently, Hawking, Perry and Strominger described a physical process that implants supertranslational hair on a Schwarzschild black hole by an infalling matter shock wave without spherical symmetry. Using the Bondi-Metzner-Sachs-type symmetries of the Rindler horizon, we present an analogous process that implants supertranslational hair on a Rindler horizon by a matter shock wave without planar symmetry, and we investigate the corresponding memory effect on the Rindler family of uniformly linearly accelerated observers. We assume each observer to remain linearly uniformly accelerated through the wave, in the sense of the curved spacetime generalization of the Letaw-Frenet equations. Starting with a family of observers who follow the orbits of a single boost Killing vector before the wave, we find that after the wave has passed, each observer still follows the orbit of a boost Killing vector but this boost differs from trajectory to trajectory, and the trajectory dependence carries a memory of the planar inhomogeneity of the wave. We anticipate this classical memory phenomenon to have a counterpart in Rindler space quantum field theory.

Holography in action

The Rindler horizon in Minkowski spacetime can be implanted with supertranslation hair by a matter shock wave without planar symmetry, and the hair is observable as a supertranslation memory on the Rindler family of uniformly linearly accelerated observers. We show that this classical memory is accompanied by a supertranslation quantum memory that modulates the entanglement between the opposing Rindler wedges in quantum field theory. A corresponding phenomenon across a black hole horizon may play a role in Hawking, Perry, and Strominger’s proposal for supertranslations to provide a solution to the black hole information paradox.