4,367 research outputs found

### 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

### 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

### 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.

### 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

### 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

### 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

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