435 research outputs found
On some geometric features of the Kramer interior solution for a rotating perfect fluid
Geometric features (including convexity properties) of an exact interior
gravitational field due to a self-gravitating axisymmetric body of perfect
fluid in stationary, rigid rotation are studied. In spite of the seemingly
non-Newtonian features of the bounding surface for some rotation rates, we
show, by means of a detailed analysis of the three-dimensional spatial
geodesics, that the standard Newtonian convexity properties do hold. A central
role is played by a family of geodesics that are introduced here, and provide a
generalization of the Newtonian straight lines parallel to the axis of
rotation.Comment: LaTeX, 15 pages with 4 Poscript figures. To be published in Classical
and Quantum Gravit
MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems
CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric
systems of linear equations. When these methods are applied to an incompatible
system (that is, a singular symmetric least-squares problem), CG could break
down and SYMMLQ's solution could explode, while MINRES would give a
least-squares solution but not necessarily the minimum-length (pseudoinverse)
solution. This understanding motivates us to design a MINRES-like algorithm to
compute minimum-length solutions to singular symmetric systems.
MINRES uses QR factors of the tridiagonal matrix from the Lanczos process
(where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where
rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned
systems (singular or not), MINRES-QLP can give more accurate solutions than
MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better
estimates of the solution and residual norms, the matrix norm, and the
condition number.Comment: 26 pages, 6 figure
Estimates on Green functions of second order differential operators with singular coefficients
We investigate the Green functions G(x,x^{\prime}) of some second order
differential operators on R^{d+1} with singular coefficients depending only on
one coordinate x_{0}. We express the Green functions by means of the Brownian
motion. Applying probabilistic methods we prove that when x=(0,{\bf x}) and
x^{\prime}=(0,{\bf x}^{\prime}) (here x_{0}=0) lie on the singular hyperplanes
then G(0,{\bf x};0,{\bf x}^{\prime}) is more regular than the Green function of
operators with regular coefficients.Comment: 16 page
Hadamard regularization of the third post-Newtonian gravitational wave generation of two point masses
Continuing previous work on the 3PN-accurate gravitational wave generation
from point particle binaries, we obtain the binary's 3PN mass-type quadrupole
and dipole moments for general (not necessarily circular) orbits in harmonic
coordinates. The final expressions are given in terms of their ``core'' parts,
resulting from the application of the pure Hadamard-Schwartz (pHS) self-field
regularization scheme, and augmented by an ``ambiguous'' part. In the case of
the 3PN quadrupole we find three ambiguity parameters, xi, kappa and zeta, but
only one for the 3PN dipole, in the form of the particular combination
xi+kappa. Requiring that the dipole moment agree with the center-of-mass
position deduced from the 3PN equations of motion in harmonic coordinates
yields the relation xi+kappa=-9871/9240. Our results will form the basis of the
complete calculation of the 3PN radiation field of compact binaries by means of
dimensional regularization.Comment: 33 pages, to appear in Phys. Rev.
Dimensional regularization of the third post-Newtonian gravitational wave generation from two point masses
Dimensional regularization is applied to the computation of the gravitational
wave field generated by compact binaries at the third post-Newtonian (3PN)
approximation. We generalize the wave generation formalism from isolated
post-Newtonian matter systems to d spatial dimensions, and apply it to point
masses (without spins), modelled by delta-function singularities. We find that
the quadrupole moment of point-particle binaries in harmonic coordinates
contains a pole when epsilon = d-3 -> 0 at the 3PN order. It is proved that the
pole can be renormalized away by means of the same shifts of the particle
world-lines as in our recent derivation of the 3PN equations of motion. The
resulting renormalized (finite when epsilon -> 0) quadrupole moment leads to
unique values for the ambiguity parameters xi, kappa and zeta, which were
introduced in previous computations using Hadamard's regularization. Several
checks of these values are presented. These results complete the derivation of
the gravitational waves emitted by inspiralling compact binaries up to the
3.5PN level of accuracy which is needed for detection and analysis of the
signals in the gravitational-wave antennas LIGO/VIRGO and LISA.Comment: 60 pages, LaTeX 2e, REVTeX 4, 10 PostScript files (1 figure and 9
Young tableaux used in the text
Gravitational Larmor formula in higher dimensions
The Larmor formula for scalar and gravitational radiation from a pointlike
particle is derived in any even higher-dimensional flat spacetime. General
expressions for the field in the wave zone and the energy flux are obtained in
closed form. The explicit results in four and six dimensions are used to
illustrate the effect of extra dimensions on linear and uniform circular
motion. Prospects for detection of bulk gravitational radiation are briefly
discussed.Comment: 5 pages, no figure
Matched Asymptotic Expansion for Caged Black Holes - Regularization of the Post-Newtonian Order
The "dialogue of multipoles" matched asymptotic expansion for small black
holes in the presence of compact dimensions is extended to the Post-Newtonian
order for arbitrary dimensions. Divergences are identified and are regularized
through the matching constants, a method valid to all orders and known as
Hadamard's partie finie. It is closely related to "subtraction of
self-interaction" and shows similarities with the regularization of quantum
field theories. The black hole's mass and tension (and the "black hole
Archimedes effect") are obtained explicitly at this order, and a Newtonian
derivation for the leading term in the tension is demonstrated. Implications
for the phase diagram are analyzed, finding agreement with numerical results
and extrapolation shows hints for Sorkin's critical dimension - a dimension
where the transition turns second order.Comment: 28 pages, 5 figures. v2:published versio
Standard and geometric approaches to quantum Liouville theory on the pseudosphere
We compare the standard and geometric approaches to quantum Liouville theory
on the pseudosphere by performing perturbative calculations of the one and two
point functions up to the third order in the coupling constant. The choice of
the Hadamard regularization within the geometric approach leads to a
discrepancy with the standard approach. On the other hand, we find complete
agreement between the results of the standard approach and the bootstrap
conjectures for the one point function and the auxiliary two point function.Comment: 31 pages, LaTeX, 8 figure
Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems
An effective characterization of chaotic conservative Hamiltonian systems in
terms of the curvature associated with a Riemannian metric tensor derived from
the structure of the Hamiltonian has been extended to a wide class of potential
models of standard form through definition of a conformal metric. The geodesic
equations reproduce the Hamilton equations of the original potential model
through an inverse map in the tangent space. The second covariant derivative of
the geodesic deviation in this space generates a dynamical curvature, resulting
in (energy dependent) criteria for unstable behavior different from the usual
Lyapunov criteria. We show here that this criterion can be constructively used
to modify locally the potential of a chaotic Hamiltonian model in such a way
that stable motion is achieved. Since our criterion for instability is local in
coordinate space, these results provide a new and minimal method for achieving
control of a chaotic system
Gravitational radiation in d>4 from effective field theory
Some years ago, a new powerful technique, known as the Classical Effective
Field Theory, was proposed to describe classical phenomena in gravitational
systems. Here we show how this approach can be useful to investigate
theoretically important issues, such as gravitational radiation in any
spacetime dimension. In particular, we derive for the first time the
Einstein-Infeld-Hoffman Lagrangian and we compute Einstein's quadrupole formula
for any number of flat spacetime dimensions.Comment: 32 pages, 10 figures. v2: Factor in eq. (3.11) fixed. References
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