Gravitational field of a Schwarzschild black hole and a rotating mass ring

Abstract

The linear perturbation of the Kerr black hole has been discussed by using the Newman--Penrose and the perturbed Weyl scalars, ${\psi }_0$ and ${\psi }_4$ can be obtained from the Teukolsky equation. In order to obtain the other Weyl scalars and the perturbed metric, a formalism was proposed by Chrzanowski and by Cohen and Kegeles (CCK) to construct these quantities in a radiation gauge via the Hertz potential. As a simple example of the construction of the perturbed gravitational field with this formalism, we consider the gravitational field produced by a rotating circular ring around a Schwarzschild black hole. In the CCK method, the metric is constructed in a radiation gauge via the Hertz potential, which is obtained from the solution of the Teukolsky equation. Since the solutions ${\psi }_0$ and ${\psi }_4$ of the Teukolsky equations are spin-2 quantities, the Hertz potential is determined up to its monopole and dipole modes. Without these lower modes, the constructed metric and Newman--Penrose Weyl scalars have unphysical jumps on the spherical surface at the radius of the ring. We find that the jumps of the imaginary parts of the Weyl scalars are cancelled when we add the angular momentum perturbation to the Hertz potential. Finally, by adding the mass perturbation and choosing the parameters which are related to the gauge freedom, we obtain the perturbed gravitational field which is smooth except on the equatorial plane outside the ring. We discuss the implication of these results to the problem of the computation of the gravitational self-force to the point particles in a radiation gauge.

arXiv comment15pages, 9 figures, Accepted by Physical Review D

References in corpus (11)

• Possibility of Direct Measurement of the Acceleration of the Universe Using 0.1 Hz Band Laser Interferometer Gravitational Wave Antenna in Space
• Beyond LISA: Exploring Future Gravitational Wave Missions
• Gravitational self force and gauge transformations
• Gravitational Self-force in a Radiation Gauge
• Reconstruction of inhomogeneous metric perturbations and electromagnetic four-potential in Kerr spacetime
• EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole
• Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a Radiation Gauge
• Gravitational self-force from radiation-gauge metric perturbations
• Reconstruction of Black Hole Metric Perturbations from Weyl Curvature
• On finding fields and self-force in a gauge appropriate to separable wave equations
• Metric of a tidally perturbed spinning black hole

Cited by in corpus (4)

• The Kerr Metric
• Completion of metric reconstruction for a particle orbiting a Kerr black hole
• Perturbation of a Schwarzschild black hole due to a rotating thin disc
• Point particles and Appell's solutions on the axis of Kerr black hole for arbitrary spin in terms of the Debye potentials