Wheeler Gauge Research Articles

Gravitational waves on charged black hole backgrounds in modified gravity

The stability of Reissner–Nördstrom black holes with an extremal mass–charge relation was determined by calculating the propagation speed of gravitational waves on this background in an effective field theory (EFT) of gravity. New results for metric components are shown, along with the corresponding new extremal relation, part of which differs by a global factor of 2 from the past published work. This new relation further develops the existing constraints on EFT parameters. The radial propagation speed for gravitational waves in the Regge–Wheeler gauge was calculated linearly for all perturbations, yielding exact luminality for all dimension-4 operators. The dimension-6 radial speed modifications introduce no constraints on the sign of the modified theory parameters from causality arguments, while the deviation from classical theories vanishes at both horizons. The angular speed was found to be altered for the dimension-4 operators, with possible new constraints on the modified theory being suggested from causality arguments. Results are consistent with existing literature on Schwarzschild black hole backgrounds, with some EFT terms becoming active only in non-vacuum spacetimes such as Reissner–Nördstrom black holes.

Odd-parity gravity waves from the Regge–Wheeler gauge perspective in the f(R, T) theory

We mainly intend to discuss the odd-parity (or axial) modes of gravity (or gravitational) waves (GWs) in the Regge–Wheeler formalism. Since, there are plenty of astrophysical data that lead us to a space-time structure entered in a speedy expansion epoch, we perform our analysis in the f(R, T) theory of gravity, where the gravitational Lagrangian is written as an arbitrary function of the Ricci scalar R and the trace of the stress-energy tensor T, from the rainbow formalism of gravity perspective. In our study, we start with perturbing the background metric, background material distribution and the four-velocity components via the Regge–Wheeler gauge. Subsequently, formulating the corresponding field equations for both unperturbed and perturbed forms of the spatially flat conformal Friedmann–Lemaitre–Robertson–Walker metric, we solve them simultaneously to find out exact expressions of the unknown perturbation functions. Also, we focus our attention on Huygens’ principle to discuss additional features of the odd-parity modes of GWs.

The Radiation-Dominated Universe in Nonstandard Cosmology

We continue the recent study of our model theory of low-density cosmology without dark matter. We assume a purely radiative spherically symmetric background and treat matter as anisotropic perturbations. Einstein’s equations for the background are solved numerically. We find two irregular singular points, one is the Big Bang and the other a Big Crunch. The radiation temperature continues to decrease for another 0.21 Hubble times and then starts to increase towards infinity. Then we derive the four evolution equations for the anisotropic perturbations. In the Regge- Wheeler gauge there are three metric perturbations and a radial velocity perturbation. The solution of these equations allow a detailed discussion of the cosmic evolution of the model universe under study.

Axial gravitational waves in Bianchi I universe

In this paper, we have studied the propagation of axial gravitational waves in Bianchi I universe using the Regge–Wheeler gauge. In this gauge, there are only two nonzero components of [Formula: see text] in the case of axial waves: [Formula: see text] and [Formula: see text]. The field equations in absence of matter have been derived both for the unperturbed as well as axially perturbed metric. These field equations are solved simultaneously by assuming the expansion scalar [Formula: see text] to be proportional to the shear scalar [Formula: see text] (so that [Formula: see text], where [Formula: see text], [Formula: see text] are the metric coefficients and [Formula: see text] is an arbitrary constant), and the wave equation for the perturbation parameter [Formula: see text] has been derived. We used the method of separation of variables to solve for this parameter, and have subsequently determined [Formula: see text]. We then discuss a few special cases to interpret the results. We find that the anisotropy of the background spacetime is responsible for the damping of the gravitational waves as they propagate through this spacetime. The perturbations depend on the values of the angular momentum [Formula: see text]. The field equations in the presence of matter reveal that the axially perturbed spacetime leads to perturbations only in the azimuthal velocity of the fluid leaving the matter field undisturbed.

Indirect (source-free) integration method. II. Self-force consistent radial fall

We apply our method of indirect integration, described in Part I, at fourth order, to the radial fall affected by the self-force (SF). The Mode-Sum regularization is performed in the Regge–Wheeler gauge using the equivalence with the harmonic gauge for this orbit. We consider also the motion subjected to a self-consistent and iterative correction determined by the SF through osculating stretches of geodesics. The convergence of the results confirms the validity of the integration method. This work complements and justifies the analysis and the results appeared in [Int. J. Geom. Meth. Mod. Phys. 11 (2014) 1450090].

A fourth-order indirect integration method for black hole perturbations: even modes

On the basis of a recently proposed strategy of finite element integration in time domain for partial differential equations with a singular source term, we present a fourth-order algorithm for non-rotating black hole perturbations in the Regge–Wheeler gauge. Herein, we address even perturbations induced by a particle plunging in. The forward time value at the upper node of the (r*, t) grid cell is obtained by an algebraic sum of (i) the preceding node values of the same cell, (ii) analytic expressions, related to the jump conditions on the wavefunction and its derivatives and (iii) the values of the wavefunction at adjacent cells. In this approach, the numerical integration does not deal with the source and potential terms directly, for cells crossed by the particle world line. This scheme has also been applied to circular and eccentric orbits and it will be the object of a forthcoming publication.

Two approaches for the gravitational self-force in black hole spacetime: Comparison of numerical results

Recently, two independent calculations have been presented of finite-mass ("self-force") effects on the orbit of a point mass around a Schwarzschild black hole. While both computations are based on the standard mode-sum method, they differ in several technical aspects, which makes comparison between their results difficult--but also interesting. Barack and Sago [Phys. Rev. D {\bf 75}, 064021 (2007)] invoke the notion of a self-accelerated motion in a background spacetime, and perform a direct calculation of the local self force in the Lorenz gauge (using numerical evolution of the perturbation equations in the time domain); Detweiler [Phys. Rev. D {\bf 77}, 124026 (2008)] describes the motion in terms a geodesic orbit of a (smooth) perturbed spacetime, and calculates the metric perturbation in the Regge--Wheeler gauge (using frequency-domain numerical analysis). Here we establish a formal correspondence between the two analyses, and demonstrate the consistency of their numerical results. Specifically, we compare the value of the conservative $O(\mu)$ shift in $u^t$ (where $\mu$ is the particle's mass and $u^t$ is the Schwarzschild $t$ component of the particle's four-velocity), suitably mapped between the two orbital descriptions and adjusted for gauge. We find that the two analyses yield the same value for this shift within mere fractional differences of $\sim 10^{-5}$--$10^{-7}$ (depending on the orbital radius)--comparable with the estimated numerical error.

Quantum amplitudes in black-hole evaporation: Spins 1 and 2

Quantum amplitudes for s = 1 Maxwell fields and for s = 2 linearised gravitational-wave perturbations of a spherically symmetric Einstein/massless scalar background, describing gravitational collapse to a black hole, are treated by analogy with the previous treatment of s = 0 scalar-field perturbations of gravitational collapse at late times. Both the spin-1 and the spin-2 perturbations split into parts with odd and even parity. Their detailed angular behaviour is analysed, as well as their behaviour under infinitesimal coordinate transformations and their linearised field equations. In general, we work in the Regge–Wheeler gauge, except that, at a certain point, it becomes necessary to make a gauge transformation to an asymptotically flat gauge, such that the metric perturbations have the expected fall-off behaviour at large radii. In both the s = 1 and s = 2 cases, we isolate suitable ‘coordinate’ variables which can be taken as boundary data on a final space-like hypersurface Σ F . (For simplicity of exposition, we take the data on the initial surface Σ I to be exactly spherically symmetric.) The (large) Lorentzian proper-time interval between Σ I and Σ F , measured at spatial infinity, is denoted by T. We then consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action S class ( 2 ) , on the assumption that the time interval T has been rotated into the complex: T → | T| exp (−i θ), for 0 < θ ⩽ π/2. This complexified classical boundary-value problem is expected to be well-posed, in contrast to the boundary-value problem in the Lorentzian-signature case ( θ = 0), which is badly posed, since it refers to hyperbolic or wave-like field equations. Following Feynman, we recover the Lorentzian quantum amplitude by taking the limit as θ → 0 + of the semi-classical amplitude exp ( i S class ( 2 ) ) . The boundary data for s = 1 involve the (Maxwell) magnetic field, while the data for s = 2 involve the magnetic part of the Weyl curvature tensor. These relations are also investigated, using 2-component spinor language, in terms of the Maxwell field strength ϕ AB = ϕ ( AB) and the Weyl spinor Ψ ABCD = Ψ ( ABCD) . The magnetic boundary conditions are related to each other and to the natural s = 1 2 boundary conditions by supersymmetry.

The Huygens principle and cosmological gravitational waves in the Regge–Wheeler gauge

We study the propagation of axial gravitational waves in Friedmann universes. The evolution equation is obtained in the Regge–Wheeler gauge. The gravitational waves obey the Huygens principle in the radiation-dominated era, but in the matter-dominated universe their propagation depends on their wavelengths, with the scale fixed essentially by the Hubble radius. Short waves practically satisfy the Huygens principle while long waves can backscatter off the curvature of a spacetime.

A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit

We obtain a fourth-order accurate numerical algorithm to integrate the Zerilli and Regge–Wheeler wave equations, describing perturbations of non-rotating black holes, with source terms due to an orbiting particle. Those source terms contain Dirac's delta and its first derivative. We also re-derive the source of the Zerilli and Regge–Wheeler equations for more convenient definitions of the waveforms that allow direct metric reconstruction (in the Regge–Wheeler gauge).

Reconstruction of black hole metric perturbations from Weyl curvature: II. The Regge–Wheeler gauge

Perturbation theory of rotating black holes is described in terms of the Weyl scalars ψ4 and ψ0, each satisfying Teukolsky's complex master wave equation with spin s = ∓2, and respectively representing outgoing and ingoing radiation. We explicitly construct the metric perturbations out of these Weyl scalars in the Regge–Wheeler gauge in the non-rotating limit. We propose a generalization of the Regge–Wheeler gauge for a Kerr background in Newman–Penrose language and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide two-way relationships between waveforms defined in the metric and curvature approaches in the time domain, also known as (inverse) Chandrasekhar transformations, generalized to include matter.

EVOLUTION EQUATIONS FOR SLOWLY ROTATING STARS

We present a hyperbolic formulation of the evolution equations describing nonradial perturbations of slowly rotating relativistic stars in the Regge–Wheeler gauge. We demonstrate the stability properties of the new evolution set of equations and compute the polar w-modes for slowly rotating stars.

Back Reaction of Gravitational Radiation on the Schwarzschild Black Hole

We address some of the issues that appear in the study of back reaction in Schwarzschild backgrounds. Our main object is the effective energy-momentum tensor (EEMT) of gravitational perturbations. It is commonly held that only asymptotically flat or radiation gauges can be employed for these purposes. We show that the traditional Regge–Wheeler gauge for perturbations of the Schwarzschild metric can also be used for calculating physical quantities both at the horizon and at infinity, even if the metric components themselves diverges there. In particular, components of the EEMT obey the same asymptotic behaviour as the stress-energy tensor of a scalar field in the Schwarzschild background. We obtain a well-defined inner product for gravitational waves, and show how it leads to a finite normalization prescription. We also use the G rt equation to compute the monopole contribution to the mass-energy carried by the gravitational waves.

A spherical mass shell as a `total reflector' for gravitational waves

For normal bulk material the reflection coefficient for gravitational waves is usually much smaller than 1. Therefore, a spherical mass shell (mass M, radius R) can serve as a `container' for gravitational radiation at most in the extreme case R = 2M. If, however, the shell material is especially adapted to the properties of the wave, total reflection can occur due to interference effects, i.e. suitable mass shells can totally reflect weak gravitational waves of arbitrary frequency and angular momentum l coming from the outside; for waves inside the shell total reflection is only possible for a series of discrete -values, dependent on R and l. This is proven by first-order perturbation theory on the basis of the Schwarzschild mass shell. Perturbation of the exterior Schwarzschild metric is treated in Regge - Wheeler gauge. Connection of this exterior metric to a (suitably transformed) flat interior metric at r = R according to Israel's method leads to a unique relation between the amplitudes of the ingoing and outgoing wave (total reflection!). For , the shell material fulfils the dominant energy condition. The logarithmic phase of the asymptotic gravitational wave (in radiation gauge) compares nicely with the phase in relativistic Coulomb scattering: different sign of the logarithmic term due to gravitational attraction, and a factor 2 due to the spin of the `graviton'.