Second- and higher-order perturbations of a spherical spacetime (original) (raw)
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High-order gauge-invariant perturbations of a spherical spacetime
Journal of Physics: Conference Series, 2007
We complete the formulation of a general framework for the analysis of high-order nonspherical perturbations of a four-dimensional spherical spacetime by including a gauge-invariant description of the perturbations. We present a general algorithm to construct these invariants and provide explicit formulas for the case of second-order metric perturbations. We show that the well-known problem of lack of invariance for the first-order perturbations with l = 0, 1 propagates to increasing values of l for perturbations of higher order, owing to mode coupling. We also discuss in which circumstances it is possible to construct the invariants.
On relativistic perturbations of second and higher order
1996
We present the results of a study of the gauge dependence of spacetime perturbations. In particular, we consider gauge invariance in general, we give a generating formula for gauge transformations to an arbitrary order n, and explicit transformation rules at second order.
Physical interpretation of gauge invariant perturbations of spherically symmetric space-times
Physical Review D, 2004
By calculating the Newman-Penrose Weyl tensor components of a perturbed spherically symmetric space-time with respect to invariantly defined classes of null tetrads, we give a physical interpretation, in terms of gravitational radiation, of odd parity gauge invariant metric perturbations. We point out how these gauge invariants may be used in setting boundary and/or initial conditions in perturbation theory.
Nonlinear N-parameter spacetime perturbations: Gauge transformations
Physical Review D, 2004
We introduce N-parameter perturbation theory as a new tool for the study of non-linear relativistic phenomena. The main ingredient in this formulation is the use of the Baker-Campbell-Hausdorff formula. The associated machinery allows us to prove the main results concerning the consistency of the scheme to any perturbative order. Gauge transformations and conditions for gauge invariance at any required order can then be derived from a generating exponential formula via a simple Taylor expansion. We outline the relation between our novel formulation and previous developments.
Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond
Classical and Quantum Gravity, 1997
We consider in detail the problem of gauge dependence that exists in relativistic perturbation theory, going beyond the linear approximation and treating second and higher order perturbations. We first derive some mathematical results concerning the Taylor expansion of tensor fields under the action of one-parameter families (not necessarily groups) of diffeomorphisms. Second, we define gauge invariance to an arbitrary order n. Finally, we give a generating formula for the gauge transformation to an arbitrary order and explicit rules to second and third order. This formalism can be used in any field of applied general relativity, such as cosmological and black hole perturbations, as well as in other spacetime theories. As a specific example, we consider here second order perturbations in cosmology, assuming a flat Robertson-Walker background, giving explicit second order transformations between the synchronous and the Poisson (generalized longitudinal) gauges.
A study of perturbations in scalar–tensor theory using 1 + 3 covariant approach
This work discusses scalar–tensor theories of gravity, with a focus on the Brans–Dicke sub-class, and one that also takes note of the latter's equivalence with f (R) gravitation theories. A 1+3 covariant formalism is used in this case to discuss covariant perturbations on a background Friedmann–Laimaˆıtre–Robertson–Walker (FLRW) spacetime. Linear perturbation equations are developed based on gauge-invariant gradient variables. Both scalar and harmonic decompositions are applied to obtain second-order equations. These equations can then be used for further analysis of the behavior of the perturbation quantities in such a scalar–tensor theory of gravitation. Energy density perturbations are studied for two systems, namely for a scalar fluid-radiation system and for a scalar fluid-dust system, for R n models. For the matter-dominated era, it is shown that the dust energy density perturbations grow exponentially, a result which agrees with those already existing in the literatures. In the radiation-dominated era, it is found that the behavior
A New Approach to Schwarzschild Perturbations
… in Theoretical and …, 1999
Recently 2 we have developed the Quasi-Maxwellian (QM) treatment for dealing with gauge-invariant cosmological perturbations in Friedman-Robertson-Walker model, providing their Hamiltonian description from a complete minimal set of gauge independent variables. The QM framework has been used since the early seventies to analyze perturbations of conformally at geometries uniquely. In this work we show that this approach may go beyond such restricted use. We apply it in order to describe linear perturbations of Schwarzschild vacuum solution, obtaining a simpler formulation than that provided by the usual method which deals with perturbations of the metric tensor (see Chandrasekhar, 3 for instance). For simplicity we will limit ourselves here to the more relevant tensorial case (which should encompass that of gravitational waves). In this vein we construct a convenient frame suggested by the background symmetries, such that the associated congruence has, besides a non null expansion, tensorial kinematic quantities (viz., the shear). The immediate consequence of introducing this property into the realm of the perturbation scheme is that additional constraints arise on the fundamental tensor basis. These new constraints, however, do not concern to eventual perturbations, but are linked with the very existence of a tensor basis referred to the chosen frame, and we exhibit them in order to obtain a gauge independent dynamical system, therefore avoiding the use of the explicit point dependence of such a basis. The resulting perturbations seem to be highly unstable due to cosmological considerations.
Linear perturbations of self-gravitating spherically symmetric configurations
Physical Review D, 2013
We present a new covariant, gauge-invariant formalism describing linear metric perturbation fields on any spherically symmetric background in general relativity. The advantage of this formalism relies in the fact that it does not require a decomposition of the perturbations into spherical tensor harmonics. Furthermore, it does not assume the background to be vacuum, nor does it require its staticity. In the particular case of vacuum perturbations, we derive two master equations describing the propagation of arbitrary linear gravitational waves on a Schwarzschild black hole. When decomposed into spherical harmonics, they reduce to covariant generalizations of the well-known Regge-Wheeler and Zerilli equations. Next, we discuss the general case where the metric perturbations are coupled to matter fields and derive a new constrained wave system describing the propagation of three gauge-invariant scalars from which the complete metric perturbations can be reconstructed. We apply our formalism to the Einstein-Euler system, dividing the fluid perturbations into two parts. The first part, which decouples from the metric perturbations, obeys simple advection equations along the background flow and describes the propagation of the entropy and the vorticity. The second part describes a perturbed potential flow, and together with the metric perturbations it forms a closed wave system.
TURKISH JOURNAL OF PHYSICS
Recently a new approach in constructing the conserved charges in cosmological Einstein's gravity was given. In this new formulation, instead of using the explicit form of the field equations a covariantly conserved rank four tensor was used. In the resulting charge expression, instead of the first derivative of the metric perturbation, the linearized Riemann tensor appears along with the derivative of the background Killing vector fields. Here we give a detailed analysis of the first order and the second order perturbation theory in a gauge-invariant form in cosmological Einstein's gravity. The linearized Einstein tensor is gauge-invariant at the first order but it is not so at the second order, which complicates the discussion. This method depends on the assumption that the first order metric perturbation can be decomposed into gaugevariant and gauge-invariant parts and the gauge-variant parts do not contribute to physical quantities.
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.