On the Linear Regime of the Characteristic formulation of General relativity in the Minkowski and Schwarzschild's Backgrounds

Abstract

We present here the linear regime of the Einstein's field equations in the characteristic formulation. Through a simple decomposition of the metric variables in spin-weighted spherical harmonics, the field equations are expressed as a system of coupled ordinary differential equations. The process for decoupling them leads to a simple equation for $J$ - one of the Bondi-Sachs metric variables - known in the literature as the master equation. Then, this last equation is solved in terms of Bessel's functions of the first kind for the Minkowski's background, and in terms of the Heun's function in the Schwarzschild's case. In addition, when a matter source is considered, the boundary conditions across the time-like world tubes bounding the source are taken into account. These boundary conditions are computed for all multipole modes. Some examples as the point particle binaries in circular and eccentric orbits, in the Minkowski's background are shown as particular cases of this formalism.

132 pages, 17 figures

References in corpus (9)

• Numerical Relativity and Astrophysics
• Gravitational Wave Extraction in Simulations of Rotating Stellar Core Collapse
• Strategies for the Characteristic Extraction of Gravitational Waveforms
• Numerical relativity with characteristic evolution, using six angular patches
• On characteristic initial data for a star orbiting a black hole
• Numerical evolution of radiative Robinson-Trautman spacetimes
• Extended two-dimensional characteristic framework to study nonrotating black holes
• Global energy conservation in nonlinear spherical characteristic evolutions
• Gravitational Waves, Sources, and Detectors