Gravitational perturbations of the Schwarzschild spacetime: A practical covariant and gauge-invariant formalism

The method of geodesic deviations has been applied to derive accurate analytic approximations to geodesics in Schwarzschild space-time. The results are used to construct analytic expressions for the source terms in the Regge-Wheeler and Zerilli-Moncrief equations, which describe the propagation of gravitational waves emitted by a compact massive object moving in the Schwarzschild background space-time. The wave equations are solved numerically to provide the asymptotic form o...

Starting from Newton's gravitational theory, we give a general introduction into the spherically symmetric solution of Einstein's vacuum field equation, the Schwarzschild(-Droste) solution, and into one specific stationary axially symmetric solution, the Kerr solution. The Schwarzschild solution is unique and its metric can be interpreted as the exterior gravitational field of a spherically symmetric mass. The Kerr solution is only unique if the multipole moments of its mass ...

A relativistic model for the emission of gravitational waves from an initially unperturbed Schwarzschild black hole, or spherical collapsing configuration, is completely integrated. The model consists basically of gravitational perturbations of the Robinson-Trautman type on the Schwarzschild spacetime. In our scheme of perturbation, gravitational waves may extract mass from the collapsing configuration. Robinson-Trautmann perturbations also include another mode of emission of...

A one-parameter family of time-symmetric initial data for the radial infall of a particle into a Schwarzschild black hole is constructed within the framework of black-hole perturbation theory. The parameter measures the amount of gravitational radiation present on the initial spacelike surface. These initial data sets are then evolved by integrating the Zerilli-Moncrief wave equation in the presence of the particle. Numerical results for the gravitational waveforms and their ...

The aim of this paper is to present a governing equation for first order axial metric perturbations of general, not necessarily static, spherically symmetric spacetimes. Under the non-restrictive assumption of axisymmetric perturbations, the governing equation is shown to be a two-dimensional wave equation where the wave function serves as a twist potential for the axisymmetry generating Killing vector. This wave equation can be written in a form which is formally a very simp...

A gauge-invariant treatment of the monopole- ($l=0$) and dipole ($l=1$) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the $l=1$ odd modes and the additional mass parameter perturbations of the Schwarzschild mass in ...

We apply a formulation of Einstein's general relativity with only cubic interactions for deriving the metric of a Schwarzschild black hole to all orders in perturbation theory. This cubic interactions formulation coupled to effective worldline action of a massive point particle allows to derive a recursion relation for the form factors of the off-shell graviton emission current. The unique solution to the recursion relation leads to the Schwarzschild black-hole solution in fo...

We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first order perturbations. This provides a formalism to address the validity of many first order calculations of interest in astrophysics.

In this article we analyze the behavior of a non-relativistic spinless particle near the event horizon of a Schwarzschild-like black hole. In this way, the Schr\"odinger covariant equation that describes the particle is obtained from the Galilean covariance technique. The Schr\"odinger equation in a Schwarzschild-like spacetime is solved analytically and its solutions are given in terms of the confluent Heun function. As a relevant result, we discovered that the energy levels...

In this short note we shall demonstrate that given a smooth solution $\gamma$ to the linearised Einstein equations on Schwarzschild which is supported on the $l\geq 2$ spherical harmonics and expressed relative to a transverse and traceless gauge then one can construct from it a smooth solution to the sourced Maxwell equations expressed relative to a generalised Lorentz gauge. Here the Maxwell current is constructed from those gauge-invariant combinations of the components of...