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

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.

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

Gravity: A New Holographic Perspective

A general paradigm for describing classical (and semiclassical) gravity is presented. This approach brings to the centre-stage a holographic relationship between the bulk and surface terms in a general class of action functionals and provides a deeper insight into several aspects of classical gravity which have no explanation in the conventional approach. After highlighting a series of unresolved issues in the conventional approach to gravity, I show that (i) principle of equivalence, (ii) general covariance and (iii)a reasonable condition on the variation of the action functional, suggest a generic Lagrangian for semiclassical gravity of the form $L=Q_a^{bcd}R^a_{bcd}$ with $\nabla_b Q_a^{bcd}=0$. The expansion of $Q_a^{bcd}$ in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein-Hilbert action and the first order correction is given by the Gauss-Bonnet action. Any such Lagrangian can be decomposed into a surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector. Hence the field equations are invariant under the transformation $T_{ab} \to T_{ab} + \lambda g_{ab}$ and gravity does not respond to the changes in the bulk vacuum energy density. The cosmological constant arises as an integration constant in this approach. The implications are discussed.Comment: Plenary talk at the International Conference on Einstein's Legacy in the New Millennium, December 15 - 22, 2005, Puri, India; to appear in the Proceedings to be published in IJMPD; 16 pages; no figure

A new perspective on Gravity and the dynamics of Spacetime

The Einstein-Hilbert action has a bulk term and a surface term (which arises from integrating a four divergence). I show that one can obtain Einstein's equations from the surface term alone. This leads to: (i) a novel, completely self contained, perspective on gravity and (ii) a concrete mathematical framework in which the description of spacetime dynamics by Einstein's equations is similar to the description of a continuum solid in the thermodynamic limit.Comment: Based on the Essay selected for Honorable Mention in the Gravity Research Foundation Essay Contest, 2005; to appear in the special issue of IJMP

Hawking radiation in different coordinate settings: Complex paths approach

We apply the technique of complex paths to obtain Hawking radiation in different coordinate representations of the Schwarzschild space-time. The coordinate representations we consider do not possess a singularity at the horizon unlike the standard Schwarzschild coordinate. However, the event horizon manifests itself as a singularity in the expression for the semiclassical action. This singularity is regularized by using the method of complex paths and we find that Hawking radiation is recovered in these coordinates indicating the covariance of Hawking radiation as far as these coordinates are concerned.Comment: 18 pages, 2 figures, Uses IOP style file; final version; accepted in Class. Quant. Gra

Vacuum Fluctuations of Energy Density can lead to the observed Cosmological Constant

The energy density associated with Planck length is $\rho_{uv}\propto L_P^{-4}$ while the energy density associated with the Hubble length is $\rho_{ir}\propto L_H^{-4}$ where $L_H=1/H$. The observed value of the dark energy density is quite different from {\it either} of these and is close to the geometric mean of the two: $\rho_{vac}\simeq \sqrt{\rho_{uv} \rho_{ir}}$. It is argued that classical gravity is actually a probe of the vacuum {\it fluctuations} of energy density, rather than the energy density itself. While the globally defined ground state, being an eigenstate of Hamiltonian, will not have any fluctuations, the ground state energy in the finite region of space bounded by the cosmic horizon will exhibit fluctuations $\Delta\rho_{\rm vac}(L_P, L_H)$. When used as a source of gravity, this $\Delta \rho$ should lead to a spacetime with a horizon size $L_H$. This bootstrapping condition leads naturally to an effective dark energy density $\Delta\rho\propto (L_{uv}L_H)^{-2}\propto H^2/G$ which is precisely the observed value. The model requires, either (i) a stochastic fluctuations of vacuum energy which is correlated over about a Hubble time or (ii) a semi- anthropic interpretation. The implications are discussed.Comment: r pages; revtex; comments welcom

Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity

I show that the principle of equipartition, applied to area elements of a surface which are in equilibrium at the local Davies-Unruh temperature, allows one to determine the surface number density of the microscopic spacetime degrees of freedom in any diffeomorphism invariant theory of gravity. The entropy associated with these degrees of freedom matches with the Wald entropy for the theory. This result also allows one to attribute an entropy density to the spacetime in a natural manner. The field equations of the theory can then be obtained by extremising this entropy. Moreover, when the microscopic degrees of freedom are in local thermal equilibrium, the spacetime entropy of a bulk region resides on its boundary.Comment: v1: 20 pages; no figures. v2: Sec 4 added; 23 page

Why do we observe a small but non zero cosmological constant ?

The current observations seem to suggest that the universe has a positive cosmological constant of the order of $H_0^2$ while the most natural value for the cosmological constant will be $L_P^{-2}$ where $L_P = (G\hbar/c^3)^{1/2}$ is the Planck length. This reduction of the cosmological constant from $L_P^{-2}$ to $L_P^{-2}(L_PH_0)^2$ may be interpreted as due to the ability of quantum micro structure of spacetime to readjust itself and absorb bulk vacuum energy densities. Being a quantum mechanical process, such a cancellation cannot be exact and the residual quantum fluctuations appear as the ``small'' cosmological constant. I describe the features of a toy model for the spacetime micro structure which could allow for the bulk vacuum energy densities to be canceled leaving behind a small residual value of the the correct magnitude. Some other models (like the ones based on canonical ensemble for the four volume or quantum fluctuations of the horizon size) lead to an insignificantly small value of $H_0^2(L_PH_0)^n$ with $n=0.5-1$ showing that obtaining the correct order of magnitude for the residual fluctuations in the cosmological constant is a nontrivial task, becaue of the existence of the small dimensionless number $H_0L_P$ .Comment: couple of references added; matches with published versio

Gravity as elasticity of spacetime: a paradigm to understand horizon thermodynamics and cosmological constant

It is very likely that the quantum description of spacetime is quite different from what we perceive at large scales, $l\gg (G\hbar/c^3)^{1/2}$. The long wave length description of spacetime, based on Einstein's equations, is similar to the description of a continuum solid made of a large number of microscopic degrees of freedom. This paradigm provides a novel interpretation of coordinate transformations as deformations of "spacetime solid" and allows one to obtain Einstein's equations as a consistency condition in the long wavelength limit. The entropy contributed by the microscopic degrees of freedom reduces to a pure surface contribution when Einstein's equations are satisfied. The horizons arises as "defects" in the "spacetime solid" (in the sense of well defined singular points) and contributes an entropy which is one quarter of the horizon area. Finally, the response of the microstructure to vacuum energy leads to a near cancellation of the cosmological constant, leaving behind a tiny fluctuation which matches with the observed value.Comment: This essay received an ``honorable mention'' in the 2004 Essay Competition of the Gravity Research Foundation; accepted for publication in IJMP