Einstein's equations as a thermodynamic identity: The cases of stationary axisymmetric horizons and evolving spherically symmetric horizons

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 TdS = dE + PdV. 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.Comment: revtex; 6 pages; no figure

Response of Unruh-DeWitt detector with time-dependent acceleration

It is well known that a detector, coupled linearly to a quantum field and accelerating through the inertial vacuum with a constant acceleration $g$, will behave as though it is immersed in a radiation field with temperature $T=(g/2\pi)$. We study a generalization of this result for detectors moving with a time-dependent acceleration $g(\tau)$ along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, $g(\tau)$, to linear order in the parameter $\eta = \dot g / g^2$. In this case, we have three length scales in the problem: $g^{-1}, (\dot g/g)^{-1}$ and $\omega^{-1}$ where $\omega$ is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When $\omega^{-1} \ll g^{-1} \ll (\dot g/g)^{-1}$, the rate of transition of the detector corresponds to a slowly varying temperature $T(\tau) = g(\tau)/2 \pi$, as one would have expected. (b) However, when $g^{-1}\ll \omega^{-1} \ll (\dot g/g)^{-1}$, we find that the spectrum is modified \textit{even at the order $\mathcal{O}(\eta)$}. This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since $(\dot g/g) \ll \omega$ and we certainly do not have deviation from the thermal spectrum when $\dot g =0$. This result shows that there is a subtle discontinuity in the behaviour of detectors with $\dot g = 0$ and $\dot g/g^2$ being arbitrarily small. We corroborate this result by evaluating the detector response for a particular trajectory which admits an analytic expression for the poles of the Wightman function.Comment: v1, 7 pages, no figures; v2, an Acknowledgment and some clarifying comments added, matches version accepted for publication in Physics Letters

Thermodynamics of horizons from a dual quantum system

It was shown recently that, in the case of Schwarschild black hole, one can obtain the correct thermodynamic relations by studying a model quantum system and using a particular duality transformation. We study this approach further for the case a general spherically symmetric horizon. We show that the idea works for a general case only if we define the entropy S as a congruence ("observer") dependent quantity and the energy E as the integral over the source of the gravitational acceleration for the congruence. In fact, in this case, one recovers the relation S=E/2T between entropy, energy and temperature previously proposed by one of us in gr-qc/0308070. This approach also enables us to calculate the quantum corrections of the Bekenstein-Hawking entropy formula for all spherically symmetric horizons.Comment: 5 pages; no figure

Zero-point length from string fluctuations

One of the leading candidates for quantum gravity, viz. string theory, has the following features incorporated in it. (i) The full spacetime is higher dimensional, with (possibly) compact extra-dimensions; (ii) There is a natural minimal length below which the concept of continuum spacetime needs to be modified by some deeper concept. On the other hand, the existence of a minimal length (or zero-point length) in four-dimensional spacetime, with obvious implications as UV regulator, has been often conjectured as a natural aftermath of any correct quantum theory of gravity. We show that one can incorporate the apparently unrelated pieces of information - zero-point length, extra-dimensions, string T-duality - in a consistent framework. This is done in terms of a modified Kaluza-Klein theory that interpolates between (high-energy) string theory and (low-energy) quantum field theory. In this model, the zero-point length in four dimensions is a ``virtual memory'' of the length scale of compact extra-dimensions. Such a scale turns out to be determined by T-duality inherited from the underlying fundamental string theory. From a low energy perspective short distance infinities are cut off by a minimal length which is proportional to the square root of the string slope, i.e. \sqrt{\alpha^\prime}. Thus, we bridge the gap between the string theory domain and the low energy arena of point-particle quantum field theory.Comment: 7 pages, Latex, no figures, one reference adde

Emergent perspective of Gravity and Dark Energy

There is sufficient amount of internal evidence in the nature of gravitational theories to indicate that gravity is an emergent phenomenon like, e.g, elasticity. Such an emergent nature is most apparent in the structure of gravitational dynamics. It is, however, possible to go beyond the field equations and study the space itself as emergent in a well-defined manner in (and possibly only in) the context of cosmology. In the first part of this review, I describe various pieces of evidence which show that gravitational field equations are emergent. In the second part, I describe a novel way of studying cosmology in which I interpret the expansion of the universe as equivalent to the emergence of space itself. In such an approach, the dynamics evolves towards a state of holographic equipartition, characterized by the equality of number of bulk and surface degrees of freedom in a region bounded by the Hubble radius. This principle correctly reproduces the standard evolution of a Friedmann universe. Further, (a) it demands the existence of an early inflationary phase as well as late time acceleration for its successful implementation and (b) allows us to link the value of late time cosmological constant to the e-folding factor during inflation.Comment: 38 pages; 5 figure

The effects of anti-correlation on gravitational clustering

We use non-linear scaling relations (NSRs) to investigate the effects arising from the existence of negative correlations on the evolution of gravitational clustering in an expanding universe. It turns out that such anti-correlated regions have important dynamical effects on {\it all} scales. In particular, the mere existence of negative values for the linear two-point correlation function \xib_L over some range of scales starting from $l = L_o$, implies that the non-linear correlation function is bounded from above at {\it all} scales $x < L_o$. This also results in the relation \xib \propto x^{-3}, at these scales, at late times, independent of the original form of the correlation function. Current observations do not rule out the existence of negative \xib for $200 h^{-1}$ Mpc \la \xib \la 1000 h^{-1} Mpc; the present work may thus have relevance for the real Universe. The only assumption made in the analysis is the {\it existence} of the NSR; the results are independent of the form of the NSR as well as of the stable clustering hypothesis.Comment: 11 pages, 6 figures. Accepted for publication in MNRA

Cosmic Information, the Cosmological Constant and the Amplitude of primordial perturbations

A unique feature of gravity is its ability to control the information accessible to any specific observer. We quantify the notion of cosmic information ('CosmIn') for an eternal observer in the universe. Demanding the finiteness of CosmIn requires the universe to have a late-time accelerated expansion. Combining the introduction of CosmIn with generic features of the quantum structure of spacetime (e.g., the holographic principle), we present a holistic model for cosmology. We show that (i) the numerical value of the cosmological constant, as well as (ii) the amplitude of the primordial, scale invariant, perturbation spectrum can be determined in terms of a single free parameter, which specifies the energy scale at which the universe makes a transition from a pre-geometric phase to the classical phase. For a specific value of the parameter, we obtain the correct results for both (i) and (ii). This formalism also shows that the quantum gravitational information content of spacetime can be tested using precision cosmology.Comment: 9 pages; 1 figur

CosMIn: The Solution to the Cosmological Constant Problem

The current acceleration of the universe can be modeled in terms of a cosmological constant. We show that the extremely small value of \Lambda L_P^2 ~ 3.4 x 10^{-122}, the holy grail of theoretical physics, can be understood in terms of a new, dimensionless, conserved number CosMIn (N), which counts the number of modes crossing the Hubble radius during the three phases of evolution of the universe. Theoretical considerations suggest that N ~ 4\pi. This single postulate leads us to the correct, observed numerical value of the cosmological constant! This approach also provides a unified picture of cosmic evolution relating the early inflationary phase to the late-time accelerating phase.Comment: ver 2 (6 pages, 2 figures) received Honorable Mention in the Gravity Research Foundation Essay Contest, 2013; to appear in Int.Jour.Mod.Phys.

Entropy of Horizons, Complex Paths and Quantum Tunneling

In any spacetime, it is possible to have a family of observers following a congruence of timelike curves such that they do not have access to part of the spacetime. This lack of information suggests associating a (congruence dependent) notion of entropy with the horizon that blocks the information from these observers. While the blockage of information is absolute in classical physics, quantum mechanics will allow tunneling across the horizon. This process can be analysed in a simple, yet general, manner and we show that the probability for a system with energy $E$ to tunnel across the horizon is $P(E)\propto\exp[-(2\pi/\kappa)E)$ where $\kappa$ is the surface gravity of the horizon. If the surface gravity changes due to the leakage of energy through the horizon, then one can associate an entropy $S(M)$ with the horizon where $dS = [ 2\pi / \kappa (M) ] dM$ and $M$ is the active gravitational mass of the system. Using this result, we discuss the conditions under which, a small patch of area $\Delta A$ of the horizon contributes an entropy $(\Delta A/4L_P^2)$, where $L_P^2$ is the Planck area.Comment: published versio