5,110 research outputs found
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
Thermodynamics and/of Horizons: A Comparison of Schwarzschild,RINDLER and desitter Spacetimes
The notions of temperature, entropy and `evaporation', usually associated
with spacetimes with horizons, are analyzed using general approach and the
following results, applicable to different spacetimes, are obtained at one go.
(i) The concept of temperature associated with the horizon is derived in a
unified manner and is shown to arise from purely kinematic considerations. (ii)
QFT near any horizon is mapped to a conformal field theory without introducing
concepts from string theory. (iii) For spherically symmetric spacetimes (in
D=1+3) with a horizon at r=l, the partition function has the generic form
, where and . This
analysis reproduces the conventional result for the blackhole spacetimes and
provides a simple and consistent interpretation of entropy and energy for
deSitter spacetime. (iv) For the Rindler spacetime the entropy per unit
transverse area turns out to be (1/4) while the energy is zero. (v) In the case
of a Schwarzschild black hole there exist quantum states (like Unruh vacuum)
which are not invariant under time reversal and can describe blackhole
evaporation. There also exist quantum states (like Hartle-Hawking vacuum) in
which temperature is well-defined but there is no flow of radiation to
infinity. In the case of deSitter universe or Rindler patch in flat spacetime,
one usually uses quantum states analogous to Hartle-Hawking vacuum and obtains
a temperature without the corresponding notion of evaporation. It is, however,
possible to construct the analogues of Unruh vacuum state in the other cases as
well. Associating an entropy or a radiating vacuum state with a general horizon
raises conceptual issues which are briefly discussed.Comment: Invited talk at the Workshop "Interface of gravitational and quantum
realms"; to appear in Mod.Phys.Letts. A; 18 pages; one eps figure embedde
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
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
Nonlinear evolution of density perturbations using approximate constancy of gravitational potential
During the evolution of density inhomogeneties in an , matter
dominated universe, the typical density contrast changes from to . However, during the same time, the typical
value of the gravitational potential generated by the perturbations changes
only by a factor of order unity. This significant fact can be exploited to
provide a new, powerful, approximation scheme for studying the formation of
nonlinear structures in the universe. This scheme, discussed in this paper,
evolves the initial perturbation using a Newtonian gravitational potential
frozen in time. We carry out this procedure for different intial spectra and
compare the results with the Zeldovich approximation and the frozen flow
approximation (proposed by Mattarrese et al. recently). Our results are in far
better agreement with the N-body simulations than the Zeldovich approximation.
It also provides a dynamical explanation for the fact that pancakes remain thin
during the evolution. While there is some superficial similarity between the
frozen flow results and ours, they differ considerably in the velocity
information. Actual shell crossing does occur in our approximation; also there
is motion of particles along the pancakes leading to further clumping. These
features are quite different from those in frozen flow model. We also discuss
the evolution of the two-point correlation function in various approximations.Comment: 10 pages, TeX, 6 figures available on request, IUCAA -14/93(
Corrections for mailing error
A New Statistical Indicator to Study Nonlinear Gravitational Clustering and Structure Formation
In an expanding universe, velocity field and gravitational force field are
proportional to each other in the linear regime. Neither of these quantities
evolve in time and these can be scaled suitably so that the constant of
proportionality is unity and velocity and force field are equal. The Zeldovich
approximation extends this feature beyond the linear regime, until formation of
pancakes. Nonlinear clustering which takes place {\it after} the breakdown of
Zeldovich approximation, breaks this relation and the mismatch between these
two vectors increases as the evolution proceeds. We suggest that the difference
of these two vectors could form the basis for a powerful, new, statistical
indicator of nonlinear clustering. We define an indicator called velocity
contrast, study its behaviour using N-Body simulations and show that it can be
used effectively to delineate the regions where nonlinear clustering has taken
place. We discuss several features of this statistical indicator and provide
simple analytic models to understand its behaviour. Particles with velocity
contrast higher than a threshold have a correlation function which is biased
with respect to the original sample. This bias factor is scale dependent and
tends to unity at large scales.Comment: 12 pages, 8 figures, LaTeX with uuencoded figures, uses MN.sty and
epsf.sty; Discussion has been enlarged to clarify a few points. Introduction
has been added. Some figures have change
Critical Index and Fixed Point in the Transfer of Power in Nonlinear Gravitational Clustering
We investigate the transfer of power between different scales and coupling of
modes during non-linear evolution of gravitational clustering in an expanding
universe. We start with a power spectrum of density fluctuations that is
exponentially damped outside a narrow range of scales and use numerical
simulations to study evolution of this power spectrum. Non-Linear effects
generate power at other scales with most power flowing from larger to smaller
scales. The ``cascade'' of power leads to equipartition of energy at smaller
scales, implying a power spectrum with index . We find that such a
spectrum is produced in the range for density contrast
. This result continues to hold even when small scale power is added to
the initial power spectrum. Semi-analytic models for gravitational clustering
suggest a tendency for the effective index to move towards a critical index
in this range. For n<n_c, power in this range grows faster than
linear rate, while if n>n_c, it grows at a slower rate - thereby changing the
index closer to n_c. At scales larger than the narrow range of scales with
initial power, a k^4 tail is produced. We demonstrate that non-linear small
scales do not effect the growth of perturbations at larger scales.Comment: Title changed. Added two figures and some discussion. Postscript file
containing all the figures is available at
http://www.ast.cam.ac.uk/~jasjeet/papers/powspec.ps.gz Accepted for
publication in the MNRA
Vacuum Fluctuations of Energy Density can lead to the observed Cosmological Constant
The energy density associated with Planck length is while the energy density associated with the Hubble length is
where . 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: .
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 . When used as a source of gravity, this should
lead to a spacetime with a horizon size . This bootstrapping condition
leads naturally to an effective dark energy density 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
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