A New Approach to Schwarzschild Perturbations

2001

A new gauge-invariant approach for describing cosmological perturbations is developed. It is based on a physically motivated splitting of the stress-energy tensor of the perturbation into two parts -the bare perturbation and the complementary perturbation associated with stresses in the background gravitational field induced by the introduction of the bare perturbation. The complementary perturbation of the stress-energy tensor is explicitly singled out and taken to the left side of the perturbed Einstein equations so that the bare stress-energy tensor is the sole source for the perturbation of the metric tensor and both sides of these equations are gauge invariant with respect to infinitesimal coordinate transformations. For simplicity we analyze the perturbations of the spatially-flat Friedmann-Lemaître-Robertson-Walker (FLRW) dust model. A cosmological gauge can be chosen such that the equations for the perturbations of the metric tensor are completely decoupled for the h 00 , h 0i , and h ij metric components and explicitly solvable in terms of retarded integrals.

Classical and Quantum Gravity, 2006

The theory of gauge-invariant non-spherical metric perturbations of Schwarzschild black hole spacetimes is now well established. Yet, as different notations and conventions have been used throughout the years, the literature on the subject is often confusing and sometimes confused. The purpose of this paper is to review and collect the relevant expressions related to the Regge-Wheeler and Zerilli equations for the odd and even-parity perturbations of a Schwarzschild spacetime. Special attention is paid to the form they assume in the presence of matter-sources and, for the two most popular conventions in the literature, to the asymptotic expressions and gravitational-wave amplitudes. Besides pointing out some inconsistencies in the literature, the expressions collected here could serve as a quick reference for the calculation of the perturbations of Schwarzschild black hole spacetimes driven by generic sources and for those approaches in which gravitational waves are extracted from numerically generated spacetimes.

Physical Review D, 2006

Gauge invariant treatments of the second order cosmological perturbation in a four dimensional homogeneous isotropic universe filled with the perfect fluid are completely formulated without any gauge fixing. We derive all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. These equations imply that the tensor and the vector mode of the second order metric perturbations may be generated by the scalar-scalar mode coupling of the linear order perturbations as the result of the non-linear effects of the Einstein equations.

Classical and Quantum Gravity, 2012

Scalar perturbations of Friedmann-Lemaitre cosmologies can be analyzed in a variety of ways using Einstein's field equations, the Ricci and Bianchi identities, or the conservation equations for the stress-energy tensor, and possibly introducing a timelike reference congruence. The common ground is the use of gauge invariants derived from the metric tensor, the stress-energy tensor, or from vectors associated with a reference congruence, as basic variables. Although there is a complication in that there is no unique choice of gauge invariants, we will show that this can be used to advantage.

Physical Review D, 2009

Using recently developed efficient symbolic manipulations tools, we present a general formalism to study arbitrary second-order perturbations of a Schwarzschild black hole. The formalism is both covariant (independent of the background coordinates) and gauge invariant. In particular, we construct the second order Zerilli and Regge-Wheeler equations under the presence of any two first-order modes, reconstruct the perturbed metric in terms of the master scalars, and compute the radiated energy at null infinity. The results of this paper enable systematic studies of generic second order perturbations of the Schwarzschild spacetime. In particular, studies of mode-mode coupling and non-linear effects in gravitational radiation, the non-linear stability of the Schwarzschild spacetime, or the geometry of the black hole horizon.

Physical Review D, 2006

Journal of Cosmology and Astroparticle Physics, 2013

We compute the third order gauge invariant action for scalar-graviton interactions in the Jordan frame. We demonstrate that the gauge invariant action for scalar and tensor perturbations on one physical hypersurface only differs from that on another physical hypersurface via terms proportional to the equation of motion and boundary terms, such that the evolution of non-Gaussianity may be called unique. Moreover, we demonstrate that the gauge invariant curvature perturbation and graviton on uniform field hypersurfaces in the Jordan frame are equal to their counterparts in the Einstein frame. These frame independent perturbations are therefore particularly useful in relating results in different frames at the perturbative level. On the other hand, the field perturbation and graviton on uniform curvature hypersurfaces in the Jordan and Einstein frame are non-linearly related, as are their corresponding actions and n-point functions.

General Relativity and Gravitation, 2010

Classical and Quantum …, 1997

The propagation of gravitational waves or tensor perturbations in a perturbed Friedmann -Robertson -Walker universe filled with a perfect fluid is re -examined. It is shown that while the shear and magnetic part of the Weyl tensor satisfy linear, homogeneous second order wave equations, for perfect fluids with a γ -law equation of state satisfying 2 3 < γ < 2, the electric part of the Weyl tensor satisfies a linear homogeneous third order equation. Solutions to these equations are obtained for a flat Friedmann -Robertson -Walker background and we discuss implications of this result.

Journal of Cosmology and Astroparticle Physics

In a previous work the authors have solved the Einstein equations of General Relativity for a class of metrics with constant spatial curvature, where it was found a non vanishing Weyl tensor in the presence of a primordial magnetic field with an anisotropic pressure component. Here, we perform the perturbative analysis of this model in order to study the gravitational stability under linear scalar perturbations. For this purpose, we take the Quasi-Maxwellian formalism of General Relativity as our framework, which offers a naturally covariant and gauge-invariant approach to deal with perturbations that are directly linked to observational quantities. We then compare this scenario with the perturbed dust-dominated Friedmann model emphasizing how the growth of density perturbations are enhanced in our case.