51 research outputs found
Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime
The conformal structure of the Schwarzschild-de Sitter spacetime is analysed
using the extended conformal Einstein field equations. To this end, initial
data for an asymptotic initial value problem for the Schwarzschild-de Sitter
spacetime is obtained. This initial data allows to understand the singular
behaviour of the conformal structure at the asymptotic points where the
horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary.
Using the insights gained from the analysis of the Schwarzschild-de Sitter
spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations
close to the Schwarzschild- de Sitter spacetime in the asymptotic region. We
show that small enough perturbations of asymptotic initial data for the
Schwarzschild de-Sitter spacetime give rise to a solution to the Einstein field
equations which exists to the future and has an asymptotic structure similar to
that of the Schwarzschild-de Sitter spacetime.Comment: Accepted version in Ann. Henri Poincar\'e. Title change: "Conformal
properties of the Schwarzschild-de Sitter spacetime" to "Perturbations of the
asymptotic region of the Schwarzschild-de Sitter spacetime". Sections
reorganised. 64 pages, 10 figure
On the nonexistence of conformally flat slices in the Kerr and other stationary spacetimes
It is proved that a stationary solutions to the vacuum Einstein field
equations with non-vanishing angular momentum have no Cauchy slice that is
maximal, conformally flat, and non-boosted. The proof is based on results
coming from a certain type of asymptotic expansions near null and spatial
infinity --which also show that the developments of Bowen-York type of data
cannot have a development admitting a smooth null infinity--, and from the fact
that stationary solutions do admit a smooth null infinity
Asymptotic properties of the development of conformally flat data near spatial infinity
Certain aspects of the behaviour of the gravitational field near null and
spatial infinity for the developments of asymptotically Euclidean, conformally
flat initial data sets are analysed. Ideas and results from two different
approaches are combined: on the one hand the null infinity formalism related to
the asymptotic characteristic initial value problem and on the other the
regular Cauchy initial value problem at spatial infinity which uses Friedrich's
representation of spatial infinity as a cylinder. The decay of the Weyl tensor
for the developments of the class of initial data under consideration is
analysed under some existence and regularity assumptions for the asymptotic
expansions obtained using the cylinder at spatial infinity. Conditions on the
initial data to obtain developments satisfying the Peeling Behaviour are
identified. Further, the decay of the asymptotic shear on null infinity is also
examined as one approaches spatial infinity. This decay is related to the
possibility of selecting the Poincar\'e group out of the BMS group in a
canonical fashion. It is found that for the class of initial data under
consideration, if the development peels, then the asymptotic shear goes to zero
at spatial infinity. Expansions of the Bondi mass are also examined. Finally,
the Newman-Penrose constants of the spacetime are written in terms of initial
data quantities and it is shown that the constants defined at future null
infinity are equal to those at past null infinity.Comment: 24 pages, 1 figur
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
On the existence and convergence of polyhomogeneous expansions of zero-rest-mass fields
The convergence of polyhomogeneous expansions of zero-rest-mass fields in
asymptotically flat spacetimes is discussed. An existence proof for the
asymptotic characteristic initial value problem for a zero-rest-mass field with
polyhomogeneous initial data is given. It is shown how this non-regular problem
can be properly recast as a set of regular initial value problems for some
auxiliary fields. The standard techniques of symmetric hyperbolic systems can
be applied to these new auxiliary problems, thus yielding a positive answer to
the question of existence in the original problem.Comment: 10 pages, 1 eps figur
Can one detect a non-smooth null infinity?
It is shown that the precession of a gyroscope can be used to elucidate the
nature of the smoothness of the null infinity of an asymptotically flat
spacetime (describing an isolated body). A model for which the effects of
precession in the non-smooth null infinity case are of order is
proposed. By contrast, in the smooth version the effects are of order .
This difference should provide an effective criterion to decide on the nature
of the smoothness of null infinity.Comment: 6 pages, to appear in Class. Quantum Gra
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
Does asymptotic simplicity allow for radiation near spatial infinity?
A representation of spatial infinity based in the properties of conformal
geodesics is used to obtain asymptotic expansions of the gravitational field
near the region where null infinity touches spatial infinity. These expansions
show that generic time symmetric initial data with an analytic conformal metric
at spatial infinity will give rise to developments with a certain type of
logarithmic singularities at the points where null infinity and spatial
infinity meet. These logarithmic singularities produce a non-smooth null
infinity. The sources of the logarithmic singularities are traced back down to
the initial data. It is shown that is the parts of the initial data responsible
for the non-regular behaviour of the solutions are not present, then the
initial data is static to a certain order. On the basis of these results it is
conjectured that the only time symmetric data sets with developments having a
smooth null infinity are those which are static in a neighbourhood of infinity.
This conjecture generalises a previous conjecture regarding time symmetric,
conformally flat data. The relation of these conjectures to Penrose's proposal
for the description of the asymptotic gravitational field of isolated bodies is
discussed.Comment: 22 pages, 4 figures. Typos and grammatical mistakes corrected.
Version to appear in Comm. Math. Phy
Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants
A discussion of polyhomogeneity (asymptotic expansions in terms of and
) for zero-rest-mass fields and gravity and its relation with the
Newman-Penrose (NP) constants is given. It is shown that for spin-
zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms
in the asymptotic expansion appear naturally if the field does not obey the
``Peeling theorem''. The terms that give rise to the slower fall-off admit a
natural interpretation in terms of advanced field. The connection between such
fields and the NP constants is also discussed. The case when the background
spacetime is curved and polyhomogeneous (in general) is considered. The free
fields have to be polyhomogeneous, but the logarithmic terms due to the
connection appear at higher powers of . In the case of gravity, it is
shown that it is possible to define a new auxiliary field, regular at null
infinity, and containing some relevant information on the asymptotic behaviour
of the spacetime. This auxiliary zero-rest-mass field ``evaluated at future
infinity ()'' yields the logarithmic NP constants.Comment: 19 page
Time asymmetric spacetimes near null and spatial infinity. I. Expansions of developments of conformally flat data
The Conformal Einstein equations and the representation of spatial infinity
as a cylinder introduced by Friedrich are used to analyse the behaviour of the
gravitational field near null and spatial infinity for the development of data
which are asymptotically Euclidean, conformally flat and time asymmetric. Our
analysis allows for initial data whose second fundamental form is more general
than the one given by the standard Bowen-York Ansatz. The Conformal Einstein
equations imply upon evaluation on the cylinder at spatial infinity a hierarchy
of transport equations which can be used to calculate in a recursive way
asymptotic expansions for the gravitational field. It is found that the the
solutions to these transport equations develop logarithmic divergences at
certain critical sets where null infinity meets spatial infinity. Associated to
these, there is a series of quantities expressible in terms of the initial data
(obstructions), which if zero, preclude the appearance of some of the
logarithmic divergences. The obstructions are, in general, time asymmetric.
That is, the obstructions at the intersection of future null infinity with
spatial infinity are different, and do not generically imply those obtained at
the intersection of past null infinity with spatial infinity. The latter allows
for the possibility of having spacetimes where future and past null infinity
have different degrees of smoothness. Finally, it is shown that if both sets of
obstructions vanish up to a certain order, then the initial data has to be
asymptotically Schwarzschildean to some degree.Comment: 32 pages. First part of a series of 2 papers. Typos correcte
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