50 research outputs found
Equivalence problem for the orthogonal webs on the sphere
We solve the equivalence problem for the orthogonally separable webs on the
three-sphere under the action of the isometry group. This continues a classical
project initiated by Olevsky in which he solved the corresponding canonical
forms problem. The solution to the equivalence problem together with the
results by Olevsky forms a complete solution to the problem of orthogonal
separation of variables to the Hamilton-Jacobi equation defined on the
three-sphere via orthogonal separation of variables. It is based on invariant
properties of the characteristic Killing two-tensors in addition to properties
of the corresponding algebraic curvature tensor and the associated Ricci
tensor. The result is illustrated by a non-trivial application to a natural
Hamiltonian defined on the three-sphere.Comment: 32 page
Covariants,joint invariants and the problem of equivalence in the invariant theory of Killing tensors defined in pseudo-Riemannian spaces of constant curvature
The invariant theory of Killing tensors (ITKT) is extended by introducing the
new concepts of covariants and joint invariants of (product) vector spaces of
Killing tensors defined in pseudo-Riemannian spaces of constant curvature. The
covariants are employed to solve the problem of classification of the
orthogonal coordinate webs generated by non-trivial Killing tensors of valence
two defined in the Euclidean and Minkowski planes. Illustrative examples are
provided.Comment: 60 pages. to appear in J. Math. Phy
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
The Dirac Equation Is Separable On The Dyon Black Hole Metric
Using the tetrad formalism, we carry out the separation of variables for the
massive complex Dirac equation in the gravitational and electromagnetic field
of a four-parameter (mass, angular momentum, electric and magnetic charges)
black hole.Comment: 13 page
Comments on higher-spin symmetries
The unconstrained frame-like formulation of an infinite tower of completely
symmetric tensor gauge fields is reviewed and examined in the limit where the
cosmological constant goes to zero. By partially fixing the gauge and solving
the torsion constraints, the form of the gauge transformations in the
unconstrained metric-like formulation are obtained till first order in a weak
field expansion. The algebra of the corresponding gauge symmetries is shown to
be equivalent, at this order and modulo (unphysical) gauge parameter
redefinitions, to the Lie algebra of Hermitian differential operators on R^n,
the restriction of which to the spin-two sector is the Lie algebra of
infinitesimal diffeomorphisms.Comment: 53+1 pages, erratum: minor error in Equ. (106) of Subsect. 5.3.3
correcte
Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
An invariant characterization of the rotationally symmetric R-separable webs
for the Laplace equation in Euclidean space is given in terms of invariants and
covariants of a real binary quartic canonically associated to the
characteristic conformal Killing tensor which defines the webs.Comment: 25 pages, recently submitted to the Journal of Mathematical Physic
Radiative multipole moments of integer-spin fields in curved spacetime
Radiative multipole moments of scalar, electromagnetic, and linearized
gravitational fields in Schwarzschild spacetime are computed to third order in
v in a weak-field, slow-motion approximation, where v is a characteristic
velocity associated with the motion of the source. To zeroth order in v, a
radiative moment of order l is given by the corresponding source moment
differentiated l times with respect to retarded time. At second order in v,
additional terms appear inside the spatial integrals. These are near-zone
corrections which depend on the detailed behavior of the source. At third order
in v, the correction terms occur outside the spatial integrals, so that they do
not depend on the detailed behavior of the source. These are wave-propagation
corrections which are heuristically understood as arising from the scattering
of the radiation by the spacetime curvature surrounding the source. Our
calculations show that the wave-propagation corrections take a universal form
which is independent of multipole order and field type. We also show that in
general relativity, temporal and spatial curvatures contribute equally to the
wave-propagation corrections.Comment: 34 pages, ReVTe
Can Schwarzschildean gravitational fields suppress gravitational waves?
Gravitational waves in the linear approximation propagate in the
Schwarzschild spacetime similarly as electromagnetic waves. A fraction of the
radiation scatters off the curvature of the geometry. The energy of the
backscattered part of an initially outgoing pulse of the quadrupole
gravitational radiation is estimated by compact formulas depending on the
initial energy, the Schwarzschild radius, and the location and width of the
pulse. The backscatter becomes negligible in the short wavelength regime.Comment: 18 pages, Revtex. Added three references; a new comment in Sec. 7;
several misprints corrected. To appear in the Phys. Rev.
Green's function for gravitational waves in FRW spacetimes
A method for calculating the retarded Green's function for the gravitational
wave equation in Friedmann-Roberson-Walker spacetimes, within the formalism of
linearized Einstein gravity is developed. Hadamard's general solution to
Cauchy's problem for second-order, linear partial differential equations is
applied to the FRW gravitational wave equation. The retarded Green's function
may be calculated for any FRW spacetime, with curved or flat spatial sections,
for which the functional form of the Ricci scalar curvature is known. The
retarded Green's function for gravitational waves propagating through a
cosmological fluid composed of both radiation and dust is calculated
analytically for the first time. It is also shown that for all FRW spacetimes
in which the Ricci scalar curvatures does not vanish, , the Green's
function violates Huygens' principle; the Green's function has support inside
the light-cone due to the scatter of gravitational waves off the background
curvature.Comment: 9 pages, FERMILAB-Pub-93/189-
Explicit Kundt type II and N solutions as gravitational waves in various type D and O universes
A particular yet large class of non-diverging solutions which admits a
cosmological constant, electromagnetic field, pure radiation and/or general
non-null matter component is explicitly presented. These spacetimes represent
exact gravitational waves of arbitrary profiles which propagate in background
universes such as Minkowski, conformally flat (anti-)de Sitter, Edgar-Ludwig,
Bertotti-Robinson, and type D (anti-)Nariai or Plebanski-Hacyan spaces, and
their generalizations. All possibilities are discussed and are interpreted
using a unifying simple metric form. Sandwich and impulsive waves propagating
in the above background spaces with different geometries and matter content can
easily be constructed. New solutions are identified, e.g. type D pure radiation
or explicit type II electrovacuum waves in (anti-)Nariai universe. It is also
shown that, in general, there are no conformally flat Einstein-Maxwell fields
with a non-vanishing cosmological constant.Comment: 17 pages, LaTeX 2e. v2: added two references concerning generalized
Kerr-Schild transformations, minor changes in the tex