Linear Perturbations for the Vacuum Axisymmetric Einstein Equations |
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Authors: | Sergio Dain Mart��n Reiris |
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Institution: | 1. Facultad de Matem??tica, Astronom??a y F??sica, FaMAF, Universidad Nacional de C??rdoba, Instituto de F??sica Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria, 5000, C??rdoba, Argentina 2. Max Planck Institute for Gravitational Physics, (Albert Einstein Institute), Am M??hlenberg 1, 14476, Potsdam, Germany
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Abstract: | In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved,
positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution.
In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to
the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods.
To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation
around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral
transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the
non-linear case. |
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