8 research outputs found
Asymptotic simplicity and static data
The present article considers time symmetric initial data sets for the vacuum
Einstein field equations which in a neighbourhood of infinity have the same
massless part as that of some static initial data set. It is shown that the
solutions to the regular finite initial value problem at spatial infinity for
this class of initial data sets extend smoothly through the critical sets where
null infinity touches spatial infinity if and only if the initial data sets
coincide with static data in a neighbourhood of infinity. This result
highlights the special role played by static data among the class of initial
data sets for the Einstein field equations whose development gives rise to a
spacetime with a smooth conformal compactification at null infinity.Comment: 25 page
A rigidity property of asymptotically simple spacetimes arising from conformally flat data
Given a time symmetric initial data set for the vacuum Einstein field
equations which is conformally flat near infinity, it is shown that the
solutions to the regular finite initial value problem at spatial infinity
extend smoothly through the critical sets where null infinity touches spatial
infinity if and only if the initial data coincides with Schwarzschild data near
infinity.Comment: 37 page
Conformal structures of static vacuum data
In the Cauchy problem for asymptotically flat vacuum data the solution-jets
along the cylinder at space-like infinity develop in general logarithmic
singularities at the critical sets at which the cylinder touches future/past
null infinity. The tendency of these singularities to spread along the null
generators of null infinity obstructs the development of a smooth conformal
structure at null infinity. For the solution-jets arising from time reflection
symmetric data to extend smoothly to the critical sets it is necessary that the
Cotton tensor of the initial three-metric h satisfies a certain conformally
invariant condition (*) at space-like infinity, it is sufficient that h be
asymptotically static at space-like infinity. The purpose of this article is to
characterize the gap between these conditions. We show that with the class of
metrics which satisfy condition (*) on the Cotton tensor and a certain
non-degeneracy requirement is associated a one-form with conformally
invariant differential . We provide two criteria: If is real
analytic, is closed, and one of it integrals satisfies a certain
equation then h is conformal to static data near space-like infinity. If h is
smooth, is asymptotically closed, and one of it integrals satisfies a
certain equation asymptotically then h is asymptotically conformal to static
data at space-like infinity.Comment: 68 pages, typos corrected, references and details adde
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
Bumpy black holes from spontaneous Lorentz violation
We consider black holes in Lorentz violating theories of massive gravity. We
argue that in these theories black hole solutions are no longer universal and
exhibit a large number of hairs. If they exist, these hairs probe the
singularity inside the black hole providing a window into quantum gravity. The
existence of these hairs can be tested by future gravitational wave
observatories. We generically expect that the effects we discuss will be larger
for the more massive black holes. In the simplest models the strength of the
hairs is controlled by the same parameter that sets the mass of the graviton
(tensor modes). Then the upper limit on this mass coming from the inferred
gravitational radiation emitted by binary pulsars implies that hairs are likely
to be suppressed for almost the entire mass range of the super-massive black
holes in the centers of galaxies.Comment: 40 pages, 4 figure
Conformal-thin-sandwich initial data for a single boosted or spinning black hole puncture
Sequences of initial-data sets representing binary black holes in
quasi-circular orbits have been used to calculate what may be interpreted as
the innermost stable circular orbit. These sequences have been computed with
two approaches. One method is based on the traditional
conformal-transverse-traceless decomposition and locates quasi-circular orbits
from the turning points in an effective potential. The second method uses a
conformal-thin-sandwich decomposition and determines quasi-circular orbits by
requiring the existence of an approximate helical Killing vector. Although the
parameters defining the innermost stable circular orbit obtained from these two
methods differ significantly, both approaches yield approximately the same
initial data, as the separation of the binary system increases. To help
understanding this agreement between data sets, we consider the case of initial
data representing a single boosted or spinning black hole puncture of the
Bowen-York type and show that the conformal-transverse-traceless and
conformal-thin-sandwich methods yield identical data, both satisfying the
conditions for the existence of an approximate Killing vector.Comment: 13 pages, 2 figure
On conserved quantities, symmetries and radioactive properties of peeling and non-peeling (polyhomogeneous) asymptotically flat spacetimes
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