105 research outputs found
Axisymmetric stationary solutions with arbitrary multipole moments
In this paper, the problem of finding an axisymmetric stationary spacetime
from a specified set of multipole moments, is studied. The condition on the
multipole moments, for existence of a solution, is formulated as a convergence
condition on a power series formed from the multipole moments. The methods in
this paper can also be used to give approximate solutions to any order as well
as estimates on each term of the resulting power series.Comment: 12 page
Calculation of, and bounds for, the multipole moments of stationary spacetimes
In this paper the multipole moments of stationary asymptotically flat
spacetimes are considered. We show how the tensorial recursion of Geroch and
Hansen can be replaced by a scalar recursion on R^2. We also give a bound on
the multipole moments. This gives a proof of the "necessary part" of a long
standing conjecture due to Geroch.Comment: 11 page
Static spacetimes with prescribed multipole moments; a proof of a conjecture by Geroch
In this paper we give sufficient conditions on a sequence of multipole
moments for a static spacetime to exist with precisely these moments. The proof
is constructive in the sense that a metric having prescribed multipole moments
up to a given order can be calculated. Since these sufficient conditions agree
with already known necessary conditions, this completes the proof of a long
standing conjecture due to Geroch.Comment: 29 page
Approximate twistors and positive mass
In this paper the problem of comparing initial data to a reference solution
for the vacuum Einstein field equations is considered. This is not done in a
coordinate sense, but through quantification of the deviation from a specific
symmetry. In a recent paper [T. B\"ackdahl, J.A. Valiente Kroon, Phys. Rev.
Lett. 104, 231102 (2010)] this problem was studied with the Kerr solution as a
reference solution. This analysis was based on valence 2 Killing spinors. In
order to better understand this construction, in the present article we analyse
the analogous construction for valence 1 spinors solving the twistor equation.
This yields an invariant that measures how much the initial data deviates from
Minkowski data. Furthermore, we prove that this invariant vanishes if and only
of the mass vanishes. Hence, we get a proof of the positivity of mass.Comment: 18 pages, corrected typos, updated reference
Report on workshop A1: Exact solutions and their interpretation
I report on the communications and posters presented on exact solutions and
their interpretation at the GRG18 Conference, Sydney.Comment: 9 pages, no figures. Many typos corrected. Report submitted to the
Proceedings of GR18. To appear in CQ
Static axisymmetric space-times with prescribed multipole moments
In this article we develop a method of finding the static axisymmetric
space-time corresponding to any given set of multipole moments. In addition to
an implicit algebraic form for the general solution, we also give a power
series expression for all finite sets of multipole moments. As conjectured by
Geroch we prove in the special case of axisymmetry, that there is a static
space-time for any given set of multipole moments subject to a (specified)
convergence criterion. We also use this method to confirm a conjecture of
Hernandez-Pastora and Martin concerning the monopole-quadropole solution.Comment: 14 page
Spinors: a Mathematica package for doing spinor calculus in General Relativity
The "Spinors" software is a "Mathematica" package which implements
2-component spinor calculus as devised by Penrose for General Relativity in
dimension 3+1. The "Spinors" software is part of the "xAct" system, which is a
collection of "Mathematica" packages to do tensor analysis by computer. In this
paper we give a thorough description of "Spinors" and present practical
examples of use.Comment: 12 pages, one figure. Typos corrected and references added. To appear
in Computer Physics Communication
Decay of solutions to the Maxwell equation on the Schwarzschild background
A new Morawetz or integrated local energy decay estimate for Maxwell test
fields on the exterior of a Schwarzschild black hole spacetime is proved. The
proof makes use of a new superenergy tensor defined in terms of the
Maxwell field and its first derivatives. The superenergy tensor, although not
conserved, yields a conserved higher order energy current . The tensor vanishes for the static Coulomb field, and
the Morawetz estimate proved here therefore yields integrated decay for the
Maxwell field to the Coulomb solution on the Schwarzschild exterior.Comment: 15 pages, updated reference
On the construction of a geometric invariant measuring the deviation from Kerr data
This article contains a detailed and rigorous proof of the construction of a
geometric invariant for initial data sets for the Einstein vacuum field
equations. This geometric invariant vanishes if and only if the initial data
set corresponds to data for the Kerr spacetime, and thus, it characterises this
type of data. The construction presented is valid for boosted and non-boosted
initial data sets which are, in a sense, asymptotically Schwarzschildean. As a
preliminary step to the construction of the geometric invariant, an analysis of
a characterisation of the Kerr spacetime in terms of Killing spinors is carried
out. A space spinor split of the (spacetime) Killing spinor equation is
performed, to obtain a set of three conditions ensuring the existence of a
Killing spinor of the development of the initial data set. In order to
construct the geometric invariant, we introduce the notion of approximate
Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the
initial hypersurface and satisfy a certain second order elliptic equation
---the approximate Killing spinor equation. This equation arises as the
Euler-Lagrange equation of a non-negative integral functional. This functional
constitutes part of our geometric invariant ---however, the whole functional
does not come from a variational principle. The asymptotic behaviour of
solutions to the approximate Killing spinor equation is studied and an
existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte
- …