3,952 research outputs found
Duality and zero-point length of spacetime
The action for a relativistic free particle of mass receives a
contribution from a path segment of infinitesimal length . Using
this action in a path integral, one can obtain the Feynman propagator for a
spinless particle of mass . If one of the effects of quantizing gravity is
to introduce a minimum length scale in the spacetime, then one would
expect the segments of paths with lengths less than to be suppressed in
the path integral. Assuming that the path integral amplitude is invariant under
the `duality' transformation , 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 to . 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
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 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 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 . 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, for a doublet of functions 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 . Finally, we demonstrate the
relation between the Chern-Simons form and the mth order Lanczos-Lovelock
Lagrangian.Comment: 15 pages, no figure
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
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
Ideal Gas in a strong Gravitational field: Area dependence of Entropy
We study the thermodynamic parameters like entropy, energy etc. of a box of
gas made up of indistinguishable particles when the box is kept in various
static background spacetimes having a horizon. We compute the thermodynamic
variables using both statistical mechanics as well as by solving the
hydrodynamical equations for the system. When the box is far away from the
horizon, the entropy of the gas depends on the volume of the box except for
small corrections due to background geometry. As the box is moved closer to the
horizon with one (leading) edge of the box at about Planck length (L_p) away
from the horizon, the entropy shows an area dependence rather than a volume
dependence. More precisely, it depends on a small volume A*L_p/2 of the box,
upto an order O(L_p/K)^2 where A is the transverse area of the box and K is the
(proper) longitudinal size of the box related to the distance between leading
and trailing edge in the vertical direction (i.e in the direction of the
gravitational field). Thus the contribution to the entropy comes from only a
fraction O(L_p/K) of the matter degrees of freedom and the rest are suppressed
when the box approaches the horizon. Near the horizon all the thermodynamical
quantities behave as though the box of gas has a volume A*L_p/2 and is kept in
a Minkowski spacetime. These effects are: (i) purely kinematic in their origin
and are independent of the spacetime curvature (in the sense that Rindler
approximation of the metric near the horizon can reproduce the results) and
(ii) observer dependent. When the equilibrium temperature of the gas is taken
to be equal to the the horizon temperature, we get the familiar A/L_p^2
dependence in the expression for entropy. All these results hold in a D+1
dimensional spherically symmetric spacetime.Comment: 19 pages, added some discussion, matches published versio
Cosmological production of H_2 before the formation of the first galaxies
Previous calculations of the pregalactic chemistry have found that a small
amount of H_2, x[H_2]=n[H_2]/n[H] = 2.6e-6, is produced catalytically through
the H^-, H_2^+, and HeH^+ mechanisms. We revisit this standard calculation
taking into account the effects of the nonthermal radiation background produced
by cosmic hydrogen recombination, which is particularly effective at destroying
H^- via photodetachment. We also take into consideration the non-equilibrium
level populations of H_2^+, which occur since transitions among the
rotational-vibrational levels are slow compared to photodissociation. The new
calculation predicts a final H_2 abundance of x[H_2] = 6e-7 for the standard
cosmology. This production is due almost entirely to the H^- mechanism, with ~1
per cent coming from HeH^+ and ~0.004 per cent from H_2^+. We evaluate the
heating of the diffuse pregalactic gas from the chemical reactions that produce
H_2 and from rotational transitions in H_2, and find them to be negligible.Comment: 13 pages, 5 figures, MNRAS submitte
Hypothesis of path integral duality: Applications to QED
We use the modified propagator for quantum field based on a ``principle of
path integral duality" proposed earlier in a paper by Padmanabhan to
investigate several results in QED. This procedure modifies the Feynman
propagator by the introduction of a fundamental length scale. We use this
modified propagator for the Dirac particles to evaluate the first order
radiative corrections in QED. We find that the extra factor of the modified
propagator acts like a regulator at the Planck scales thereby removing the
divergences that otherwise appear in the conventional radiative correction
calculations of QED. We find that:(i) all the three renormalisation factors
, , and pick up finite corrections and (ii) the modified
propagator breaks the gauge invariance at a very small level of
. The implications of this result to generation of the
primordial seed magnetic fields are discussed.Comment: 15 pages, LaTeX2e (uses ijmpd.sty); To appear in IJMP-D; References
adde
Limit to General Relativity in f(R) theories of gravity
We discuss two aspects of f(R) theories of gravity in metric formalism. We
first study the reasons to introduce a scalar-tensor representation for these
theories and the behavior of this representation in the limit to General
Relativity, f(R)--> R. We find that the scalar-tensor representation is well
behaved even in this limit. Then we work out the exact equations for
spherically symmetric sources using the original f(R) representation, solve the
linearized equations, and compare our results with recent calculations of the
literature. We observe that the linearized solutions are strongly affected by
the cosmic evolution, which makes very unlikely that the cosmic speedup be due
to f(R) models with correcting terms relevant at low curvatures.Comment: 8 pages; small changes to match published version (some comments,
references added, title corrected); to appear in Phys.Rev.
The hypothesis of path integral duality II: corrections to quantum field theoretic results
In the path integral expression for a Feynman propagator of a spinless
particle of mass , the path integral amplitude for a path of proper length
connecting events and in a spacetime
described by the metric tensor is . In a recent paper, assuming the path integral amplitude to be
invariant under the duality transformation ,
Padmanabhan has evaluated the modified Feynman propagator in an arbitrary
curved spacetime. He finds that the essential feature of this `principle of
path integral duality' is that the Euclidean proper distance
between two infinitesimally separated spacetime events is replaced by . In other words, under the duality principle the spacetime
behaves as though it has a `zero-point length' , a feature that is
expected to arise in a quantum theory of gravity. In the Schwinger's proper
time description of the Feynman propagator, the weightage factor for a path
with a proper time is . Invoking Padmanabhan's `principle of
path integral duality' corresponds to modifying the weightage factor
to . In this paper, we use this modified
weightage factor in Schwinger's proper time formalism to evaluate the quantum
gravitational corrections to some of the standard quantum field theoretic
results in flat and curved spacetimes. We find that the extra factor
acts as a regulator at the Planck scale thereby `removing' the
divergences that otherwise appear in the theory. Finally, we discuss the wider
implications of our analysis.Comment: 26 pages, Revte
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