44 research outputs found
Brownian motion meets Riemann curvature
The general covariance of the diffusion equation is exploited in order to
explore the curvature effects appearing on brownian motion over a d-dimensional
curved manifold. We use the local frame defined by the so called Riemann normal
coordinates to derive a general formula for the mean-square geodesic distance
(MSD) at the short-time regime. This formula is written in terms of
invariants that depend on the Riemann curvature tensor. We study the
n-dimensional sphere case to validate these results. We also show that the
diffusion for positive constant curvature is slower than the diffusion in a
plane space, while the diffusion for negative constant curvature turns out to
be faster. Finally the two-dimensional case is emphasized, as it is relevant
for the single particle diffusion on biomembranes.Comment: 16 pages and 3 figure
Epicyclic frequencies derived from the effective potential: simple and practical formulae
We present and discuss a short and simple derivation of orbital epicyclic
frequencies for circular geodesic orbits in stationary and axially symmetric
spacetimes. Such spacetimes include as special cases analytically known black
hole Kerr and Schwarzschild spacetimes, as well as the analytic Hartle-Thorne
spacetime and all numerically constructed spacetimes relevant for rotating
neutron stars. Our derivation follows directly from energy and angular momentum
conservation and it uses the concept of the effective potential. It has never
been published, except for a few special cases, but it has already become a
part of the common knowledge in the field.Comment: Invited lecture at the conference "From X-ray Binaries to Quasars:
Black Hole Accretion on All Mass Scales", 13-15 July, 2004, Amsterda
The heat kernel for deformed spheres
We derive the asymptotic expansion of the heat kernel for a Laplace operator
acting on deformed spheres. We calculate the coefficients of the heat kernel
expansion on two- and three-dimensional deformed spheres as functions of
deformation parameters. We find that under some deformation the conformal
anomaly for free scalar fields on and is canceled.Comment: 10 pages, latex, no figure
Exotic spacetimes, superconducting strings with linear momentum, and (not quite) all that
We derive the general exact vacuum metrics associated with a stationary (non
static), non rotating, cylindrically symmetric source. An analysis of the
geometry described by these vacuum metrics shows that they contain a subfamily
of metrics that, although admitting a consistent time orientation, display
"exotic" properties, such as "trapping" of geodesics and closed causal curves
through every point. The possibility that such spacetimes could be generated by
a superconducting string, endowed with a neutral current and momentum, has
recently been considered by Thatcher and Morgan. Our results, however, differ
from those found by Thatcher and Morgan, and the discrepancy is explained. We
also analyze the general possibility of constructing physical sources for the
exotic metrics, and find that, under certain restrictions, they must always
violate the dominant energy condition (DEC). We illustrate our results by
explicitly analyzing the case of concentric shells, where we find that in all
cases the external vacuum metric is non exotic if the matter in the shells
satisfies the DEC.Comment: 13 pages with no figures. Accepted in PR
Spectral Action for Robertson-Walker metrics
We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our
previous method of computation of the spectral action based on the Poisson
summation formula. We show how to compute directly the spectral action for the
general case of Robertson-Walker metrics. We check the terms of the expansion
up to a_6 against the known universal formulas of Gilkey and compute the
expansion up to a_{10} using our direct method
Bianchi I Quantum cosmology in the Bergmann-Wagoner theory
The Wheeler-DeWitt equation is considered in the context of generalized
scalar-tensor theories of gravitation for Bianchi type I cosmology. Exact
solutions are found for two selfinteracting potentials and arbitary coupling
function. The WKB wavefunctions are obtained and a family of solutions
satisfying the Hawking-Page regularity conditions of wormholes are found.Comment: 12 pages, Latex fil
Multiple reflection expansion and heat kernel coefficients
We propose the multiple reflection expansion as a tool for the calculation of
heat kernel coefficients. As an example, we give the coefficients for a sphere
as a finite sum over reflections, obtaining as a byproduct a relation between
the coefficients for Dirichlet and Neumann boundary conditions. Further, we
calculate the heat kernel coefficients for the most general matching conditions
on the surface of a sphere, including those cases corresponding to the presence
of delta and delta prime background potentials. In the latter case, the
multiple reflection expansion is shown to be non-convergent.Comment: 21 pages, corrected for some misprint
Non stationary Einstein-Maxwell fields interacting with a superconducting cosmic string
Non stationary cylindrically symmetric exact solutions of the
Einstein-Maxwell equations are derived as single soliton perturbations of a
Levi-Civita metric, by an application of Alekseev inverse scattering method. We
show that the metric derived by L. Witten, interpreted as describing the
electrogravitational field of a straight, stationary, conducting wire may be
recovered in the limit of a `wide' soliton. This leads to the possibility of
interpreting the solitonic solutions as representing a non stationary
electrogravitational field exterior to, and interacting with, a thin, straight,
superconducting cosmic string. We give a detailed discussion of the
restrictions that arise when appropiate energy and regularity conditions are
imposed on the matter and fields comprising the string, considered as `source',
the most important being that this `source' must necessarily have a non-
vanishing minimum radius. We show that as a consequence, it is not possible,
except in the stationary case, to assign uniquely a current to the source from
a knowledge of the electrogravitational fields outside the source. A discussion
of the asymptotic properties of the metrics, the physical meaning of their
curvature singularities, as well as that of some of the metric parameters, is
also included.Comment: 14 pages, no figures (RevTex
Index-free Heat Kernel Coefficients
Using index-free notation, we present the diagonal values of the first five
heat kernel coefficients associated with a general Laplace-type operator on a
compact Riemannian space without boundary. The fifth coefficient appears here
for the first time. For a flat space with a gauge connection, the sixth
coefficient is given too. Also provided are the leading terms for any
coefficient, both in ascending and descending powers of the Yang-Mills and
Riemann curvatures, to the same order as required for the fourth coefficient.
These results are obtained by directly solving the relevant recursion
relations, working in Fock-Schwinger gauge and Riemann normal coordinates. Our
procedure is thus noncovariant, but we show that for any coefficient the
`gauged' respectively `curved' version is found from the corresponding
`non-gauged' respectively `flat' coefficient by making some simple covariant
substitutions. These substitutions being understood, the coefficients retain
their `flat' form and size. In this sense the fifth and sixth coefficient have
only 26 and 75 terms respectively, allowing us to write them down. Using
index-free notation also clarifies the general structure of the heat kernel
coefficients. In particular, in flat space we find that from the fifth
coefficient onward, certain scalars are absent. This may be relevant for the
anomalies of quantum field theories in ten or more dimensions.Comment: 38 pages, LaTe
Closed Strings with Low Harmonics and Kinks
Low-harmonic formulas for closed relativistic strings are given. General
parametrizations are presented for the addition of second- and third-harmonic
waves to the fundamental wave. The method of determination of the
parametrizations is based upon a product representation found for the finite
Fourier series of string motion in which the constraints are automatically
satisfied. The construction of strings with kinks is discussed, including
examples. A procedure is laid out for the representation of kinks that arise
from self-intersection, and subsequent intercommutation, for harmonically
parametrized cosmic strings.Comment: 39, CWRUTH-93-