41,700 research outputs found
Connecting the exterior gravitational field with the energy momentum tensor of axially symmetric compact objects
A method to construct interior axially symmetric metrics that appropriately
match with any vacuum solution of the Weyl family is developed in
Hernandez-Pastora etal. (Class Quantum Gravity 33:235005, 2016). It was
shown,for the case of some vacuum solutions, that the simplestsolution for the
interior metric leads to sources with well behaved energy conditions. Now, we
integrate the field equa-tions to obtain the interior metric functions in terms
of theanisotropies and pressures of the source. As well, the compatible
equations of state for these global models are calculated. The interior metric
and the suitable energy momentum tensor describing the source are constructed
in terms of the exterior metric functions. At the boundary of the compact
object,the behaviour of a pressure Tm, defined from the energy momentum tensor,
is shown to be related with the exterior gravitational field. This fact allows
us to explore the differences arising at the matter distribution when the
sphericalsymmetry of the global metric is dropped. Finally, an equation derived
from the matching conditions is obtained whichallows us to calculate the Weyl
coefficients of the exteriormetric as source integrals. Hence the Relativistic
MultipoleMoments of the global model can be expressed in terms ofthe matter
distribution of the source
Linearized multipole solutions and their representation
The monopole solution of the Einstein vacuum field equations (Schwarzschild`s
solution) in Weyl coordinates involves a metric function that can be
interpreted as the gravitational potential of a bar of length with
constant linear density. The question addressed in this work is whether similar
representations can be constructed for Weyl solutions other than the
spherically symmetric one.
A new family of static solutions of the axisymmetric vacuum field equations
generalizing the M-Q solution is developed. These represent slight
deviations from spherical symmetry in terms of the relativistic multipole
moments (RMM) we wish the solution to contain. A Newtonian object referred to
as a dumbbell can be used to describe these solutions in a simple form by means
of the density of this object, since the physical properties of the
relativistic solution are characterized by its behaviour. The density profile
of the dumbbell, which is given in terms of the RMM of the solution, allows us
to distinguish general multipole Weyl solutions from the constant-density
Schwarzschild solution. The range of values of the multipole moments that
generate positive-definite density profiles are also calculated. The bounds on
the multipole moments that arise from this density condition are identical to
those required for a well-behaved infinite-redshift surface .Comment: 32 pages, 2 figure
On the Solutions of infinite systems of linear equations
New theorems about the existence of solution for a system of infinite linear
equations with a Vandermonde type matrix of coefficients are proved. Some
examples and applications of these results are shown. In particular, a kind of
these systems is solved and applied in the field of the General Relativity
Theory of Gravitation. The solution of the system is used to construct a
relevant physical representation of certain static and axisymmetric solution of
the Einstein vacuum equations. In addition, a newtonian representation of these
relativistic solutions is recovered. It is shown as well that there exists a
relation between this application and the classical Haussdorff moment problem.Comment: Accepted for publication in General Relativity and Gravitatio
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