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

Perturbation theory of vacuum spherically-symmetric spacetimes is a crucial tool to understand the dynamics of black hole perturbations. Spherical symmetry allows for an expansion of the perturbations in scalar, vector, and tensor harmonics. The resulting perturbative equations are decoupled for modes with different parity and different harmonic numbers. Moreover, for each harmonic and parity, the equations for the perturbations can be decoupled in terms of (gauge-invariant) ...

In order to derive the precise gravitational waveforms for extreme mass ratio inspirals (EMRI), we develop a formulation for the second order metric perturbations produced by a point particle moving in the Schwarzschild spacetime. The second order waveforms satisfy a wave equation with an effective source build up from products of the first order perturbations and its derivatives. We have explicitly regularized this source at the horizon and at spatial infinity. We show that ...

We study linear gravitational perturbations of Schwarzschild spacetime by solving numerically Regge-Wheeler-Zerilli equations in time domain using hyperboloidal surfaces and a compactifying radial coordinate. We stress the importance of including the asymptotic region in the computational domain in studies of gravitational radiation. The hyperboloidal approach should be helpful in a wide range of applications employing black hole perturbation theory.

The explicit form of perturbation equation for the $\Psi_4$ Weyl scalar, containing the matter source terms, is derived for general type D spacetimes. It is described in detail the particular case of the Schwarzschild spacetime using in-going penetrating coordinates. As a practical application, we focused on the emission of gravitational waves when a black hole is perturbed by a surrounding dust-like fluid matter. The symmetries of the spacetime and the simplicity of the matt...

This is the Part III paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory and the proposal on the gauge-invariant treatments for $l=0,1$ mode perturbations on the Schwarzschild background spacetime in [K.~Nakamura, arXiv:2110.13508 [gr-qc]], we examine the problem whether the $l=0,1$ even-mode solutions derived in the Part II paper [K.~...

The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction o...

Modelling the free fall (and radiative phenomenology) of a massive particle, charged or not, in a static and spherically symmetric black hole is a classic, good relativistic dare that produced a remarkable series of papers, mainly in the seventies of the past century. Some formal topics about the mathematical machinery required to perform the task are unfortunately still not very clear; however, with the help of modern computer algebra techniques, some results can at least be...

In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, remain uniformly bounded and decay to a linearized Kerr metric on the exterior region. We employ Hodge decomposition to split the solution into closed and co-clo...

We revisit the classic problem of gravitational wave emission by a test particle plunging into a Schwarzschild black hole both in the frequency-domain Regge-Wheeler-Zerilli formalism and in the semirelativistic approximation. We use, and generalize, a transformation due to Nakamura, Sasaki, and Shibata to improve the falloff of the source term of the Zerilli function. The faster decay improves the numerical convergence of quantities of interest, such as the energy radiated at...

We compute the propagation and scattering of linear gravitational waves off a Schwarzschild black hole using a numerical code which solves a generalization of the Zerilli equation to a three dimensional cartesian coordinate system. Since the solution to this problem is well understood it represents a very good testbed for evaluating our ability to perform three dimensional computations of gravitational waves in spacetimes in which a black hole event horizon is present.