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

Duality and zero-point length of spacetime

The action for a relativistic free particle of mass $m$ receives a contribution $-mds$ from a path segment of infinitesimal length $ds$. Using this action in a path integral, one can obtain the Feynman propagator for a spinless particle of mass $m$. If one of the effects of quantizing gravity is to introduce a minimum length scale $L_P$ in the spacetime, then one would expect the segments of paths with lengths less than $L_P$ to be suppressed in the path integral. Assuming that the path integral amplitude is invariant under the `duality' transformation $ds\to L_P^2/ds$, one can calculate the modified Feynman propagator. I show that this propagator is the same as the one obtained by assuming that: quantum effects of gravity leads to modification of the spacetime interval $(x-y)^2$ to $(x-y)^2+L_P^2$. This equivalence suggests a deep relationship between introducing a `zero-point-length' to the spacetime and postulating invariance of path integral amplitudes under duality transformations.Comment: Revtex document; 4 page

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.

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

Structure of Lanczos-Lovelock Lagrangians in Critical Dimensions

The Lanczos-Lovelock models of gravity constitute the most general theories of gravity in D dimensions which satisfy (a) the principle of of equivalence, (b) the principle of general co-variance, and (c) have field equations involving derivatives of the metric tensor only up to second order. The mth order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature tensor. The field equations resulting from it become trivial in the critical dimension $D = 2m$ and the action itself can be written as the integral of an exterior derivative of an expression involving the vierbeins, in the differential form language. While these results are well known, there is some controversy in the literature as to whether the Lanczos-Lovelock Lagrangian itself can be expressed as a total divergence of quantities built only from the metric and its derivatives (without using the vierbeins) in $D = 2m$. We settle this issue by showing that this is indeed possible and provide an algorithm for its construction. In particular, we demonstrate that, in two dimensions, $R \sqrt{-g} = \partial_j R^j$ for a doublet of functions $R^j = (R^0,R^1)$ which depends only on the metric and its first derivatives. We explicitly construct families of such R^j -s in two dimensions. We also address related questions regarding the Gauss-Bonnet Lagrangian in $D = 4$. Finally, we demonstrate the relation between the Chern-Simons form and the mth order Lanczos-Lovelock Lagrangian.Comment: 15 pages, no figure

Noether Current, Horizon Virasoro Algebra and Entropy

We provide a simple and straightforward procedure for defining a Virasoro algebra based on the diffeomorphisms near a null surface in a spacetime and obtain the entropy density of the null surface from its central charge. We use the off-shell Noether current corresponding to the diffeomorphism invariance of a gravitational Lagrangian $L(g_{ab},R_{abcd})$ and define the Virasoro algebra from its variation. This allows us to identify the central charge and the zero mode eigenvalue using which we obtain the entropy density of the Killing horizon. Our approach works for all Lanczos-Lovelock models and reproduces the correct Wald entropy. The entire analysis is done off-shell without using the field equations and allows us to define an entropy density for any null surface which acts as a local Rindler horizon for a particular class of observers.Comment: V2: to appear in Phys. Rev.

Combining general relativity and quantum theory: points of conflict and contact

The issues related to bringing together the principles of general relativity and quantum theory are discussed. After briefly summarising the points of conflict between the two formalisms I focus on four specific themes in which some contact has been established in the past between GR and quantum field theory: (i) The role of planck length in the microstructure of spacetime (ii) The role of quantum effects in cosmology and origin of the universe (iii) The thermodynamics of spacetimes with horizons and especially the concept of entropy related to spacetime geometry (iv) The problem of the cosmological constant.Comment: Invited Talk at "The Early Universe and Cosmological Observations: a Critical Review", UCT, Cape Town, 23-25 July,2001; to appear in Class.Quan.Gra