Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations
- Facultad de Matematica, Astronomia y Fisica, FaMAF, Universidad Nacional de Cordoba, Instituto de Fisica Enrique Gaviola, IFEG, CONICET, Ciudad Universitaria (5000) Cordoba (Argentina) and Max Planck Institute for Gravitational Physics (Albert-Einstein-Institut) Am Muehlenberg 1 D-14476 Potsdam (Germany)
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 a flat background. We prove that the linear equations reduce to a very simple system of equations which provide, though 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.
- OSTI ID:
- 21409326
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 81, Issue 4; Other Information: DOI: 10.1103/PhysRevD.81.044040; (c) 2010 The American Physical Society; ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
Similar Records
Well-posedness of the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's field equations
3D simulations of Einstein's equations: Symmetric hyperbolicity, live gauges, and dynamic control of the constraints