On well-posedness, linear perturbations and mass conservation for axisymmetric Einstein equation

S. DAIN; O. ORTIZ

PHYSICAL REVIEW D

AMER PHYSICAL SOC

Año: 2010 vol. 81 p. 44040 - 44040

1550-7998

For axially symmetric solutions of Einstein equations there exists a gauge  which has the remarkable property that the total mass can be written as a  conserved, positive definite, integral on the spacelike slices. The  mass integral provides a nonlinear control of the variables along the whole  evolution. In this gauge, Einstein equations reduce to a coupled  hyperbolic-elliptic system which is formally singular at the axis.  As a  first step in analyzing this system of equations we study linear  perturbations on flat background. We prove that the linear equations reduce  to a very simple system of equations which provide, thought the mass formula,  useful insight into the structure of the full system. However, the singular  behavior of the coefficients at the axis makes the study of this linear  system difficult from the analytical point of view. In order to understand  the behavior of the solutions, we study the numerical evolution of them. We  provide strong numerical evidence that the system is well-posed and that its  solutions have the expected behavior. Finally, this linear system allows us  to formulate a model problem which is physically interesting in itself, since  it is connected with the linear stability of black hole solutions in axial  symmetry. This model can contribute significantly to solve the nonlinear  problem and at the same time it appears to be tractable.