We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations (for spins $s=0,1,2$). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.
Odd-type spin 2 perturbations of Einstein's equation can be reduced to the scalar Regge-Wheeler equation. We show that the weighted norms of solutions are in L^2 of time and space. This result uses commutator methods and applies uniformly to all relevant spherical harmonics.
Reconsideration of the Regge-Wheeler equation is processed by using the Painlevé coordinate and "good" timelier to define the initial time. We find that: the Regge-Wheeler equation could has positive imaginary frequency. Because the Regge-Wheeler equation is the odd (angular) perturbation to the Schwarzschild black hole, the conclusion is that the Schwarzschild black hole is unstable with respect to the rotating perturbation.
We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and …
We derive the equations governing the linear stability of Kerr-Newman spacetime to coupled electromagnetic-gravitational perturbations. The equations generalize the celebrated Teukolsky equation for curvature perturbations of Kerr, and the Regge-Wheeler equation for metric perturbations of Reissner-Nordström. Because of the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields", as put by Chandrasekhar, the stability of Kerr-Newman spacetime can not be obtained through standard decomposition in modes. Due to the impossibility to decouple the modes of the gravitational …
So-called "dirty" black holes are those surrounded by non-zero stress-energy, rather than vacuum. The presence of the non-zero stress-energy modifies key features of the black hole, such as the surface gravity, Regge-Wheeler equation, linear stability, and greybody factors in a rather nontrivial way. Working within the inverse-Cowling approximation, (effectively the test-field limit), we shall present general forms for the Regge-Wheeler equation for linearized spin 0, spin 1, and axial spin 2 perturbations on an arbitrary static spherically symmetric background spacetime. Using very general fe…
We solve exactly the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered.
We study linear perturbations of the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological model in the Regge-Wheeler formalism which is a standard framework to study perturbations of spherically-symmetric black holes. In particular, we show that the general solution of linear perturbation equations can be given in terms of two copies of a master scalar satisfying scalar wave equation on the FLRW background (with a Regge-Wheeler/Zerilli type potential) thus representing two gravitational degrees of freedom, and one scalar satisfying a transport type equation representing (conformal) matter p…
The Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in linear approximation. We present its exact solutions in terms of the confluent Heun's functions, the basic properties of the general solution, novel analytical approach and numerical techniques for study of different boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects. We depict in more detail the exact solutions of Regge-Wheeler equation in the Schwarzschild black hole interior and on Kruscal-Szekeres manifold.
An efficient approach to tensor perturbation calculations by proper use of computer algebra methods is described, reaching the sufficient generality required for a comprehensive analysis of the Schwarzschild and Reissner-Nordstroem metric stability.
The Regge-Wheeler equation describes axial perturbations of Schwarzschild metric in linear approximation. Teukolsky Master Equation describes perturbations of Kerr metric in the same approximation. We present here unified description of all classes of exact solutions to these equations in terms of the confluent Heun's functions. Special attention is paid to the polynomial solutions, which yield novel applications of Teukolsky Master Equation for description of relativistic jets and astrophysical explosions.
We consider the vector wave equation on the Schwarzschild spacetime, which can be considered as coming from the harmonic (or Lorenz) gauge fixed Maxwell equations. After a separation of variables, the radial mode equations form a complicated system of coupled linear ODEs. We outline a precise abstract strategy to decouple this system into triangular form, where the diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations, with $s=0$ or $1$. This strategy is then implemented to give an explicit transformation of the radial mode equations (with nonzero frequency and angular momentum) i…
Perturbation theory of rotating black holes is described in terms of the Weyl scalars $ψ_4$ and $ψ_0$; each satisfying the Teukolsky's complex master wave equation with spin $s=\mp2$, 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 nonrotating limit. We propose a generalization of the Regge-Wheeler gauge for Kerr background in the Newman-Penrose language, and discuss the approach for building up the perturbed spacetime of a rotating black hole. We also provide both-way rel…
We study the propagation of axial gravitational waves in Friedman 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.
We present numerical results for the gravitational self-force and redshift invariant calculated in the Regge-Wheeler and Easy gauges for circular orbits in a Schwarzschild background, utilizing the regularization framework introduced by Pound, Merlin, and Barack. The numerical calculation is performed in the frequency domain and requires the integration of a single second-order ODE, greatly improving computation times over more traditional Lorenz gauge numerical methods. A sufficiently high-order, analytic expansion of the Detweiler-Whiting singular field is gauge-transformed to both the Regge…
The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the $r^p$ hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerilli variables decay as $t^{-\frac{3}{2}}$ in fixed regions of $r$.
Analytic solutions of the Regge-Wheeler equation are presented in the form of series of hypergeometric functions and Coulomb wave functions which have different regions of convergence. Relations between these solutions are established. The series solutions are given as the Post-Minkowskian expansion with respect to a parameter $ε\equiv 2Mω$, $M$ being the mass of black hole. This expansion corresponds to the post-Newtonian expansion when they are applied to the gravitational radiation from a particle in circular orbit around a black hole. These solutions can also be useful for numerical comput…
The Regge-Wheeler equation for black-hole gravitational waves is analyzed for large negative imaginary frequencies, leading to a calculation of the cut strength for waves outgoing to infinity. In the--limited--region of overlap, the results agree well with numerical findings [Class. Quantum Grav._20_, L217 (2003)]. Requiring these waves to be outgoing into the horizon as well subsequently yields an analytic formula for the highly damped Schwarzschild quasinormal modes,_including_ the leading correction. Just as in the WKB quantization of, e.g., the harmonic oscillator, solutions in different r…
We present a method of post-Newtonian expansion to solve the homogeneous Regge-Wheeler equation which describes gravitational waves on the Schwarzschild spacetime. The advantage of our method is that it allows a systematic iterative analysis of the solution. Then we obtain the Regge-Wheeler function which is purely ingoing at the horizon in closed analytic form, with accuracy required to determine the gravitational wave luminosity to (post)$^{4}$-Newtonian order (i.e., order $v^8$ beyond Newtonian) from a particle orbiting around a Schwarzschild black hole. Our result, valid in the small-mass …
In ``Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds'' (math-ph/0002030), ``Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates'' (gr-qc/0310091), and ``The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation'' (gr-qc/0310066), local decay estimates were proven for the (decoupled) Schrodinger, wave, and Regge-Wheeler equations on the Schwarzschild manifold, using commutator methods. Here, we correct a step in the commutator argument. The corrected argument works either for…
We present a characteristic initial value approach to calculating the Green function of the Regge-Wheeler and Zerilli equations. We combine well-known numerical methods with newly derived initial data to obtain a scheme which can in principle be generalised to any desired order of convergence. We demonstrate the approach with implementations up to sixth-order in the grid spacing. By combining the results of our numerical code with late-time tail expansions and methods of subtracting the direct part of the Green function, we show that the scalar self-force in Schwarzschild spacetime can be comp…
We study master variables in the Regge-Wheeler-Zerilli formalism. We show that a specific choice of new variables is suitable for studying perturbation theory from the viewpoint of radiation reaction calculations. With explicit definition of the improved master variables in terms of components of metric perturbations, we present the master equations, with source terms, and metric reconstruction formulas. In the scheme using these new variables, we do not need any time and radial integrations except for solving the master equation. We also show that the master variable for even parity modes whi…
The well-known Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in the linear approximation. From a mathematical point of view it presents a particular case of the confluent Heun equation and can be solved exactly, due to recent mathematical developments. We present the basic properties of its general solution. A novel analytical approach and numerical techniques for study the boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects are developed.
Recently, the Effective Field Theory (EFT) of perturbations on an arbitrary background metric with a timelike scalar profile was formulated in the context of scalar-tensor theories. Here, we generalize the dictionary between the coefficients in the EFT action and those in covariant theories to accommodate shift- and reflection-symmetric quadratic higher-order scalar-tensor theories, including DHOST as well as U-DHOST. We then use the EFT action to study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike scalar profile. Finally, …
Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime
Institut für Theoretische Physik, Universität Leipzig（莱比锡大学理论物理研究所）；School of Mathematics and Statistics, University College Dublin（都柏林大学学院数学与统计学院）；Centro Brasileiro de Pesquisas Físicas (CBPF)（巴西物理学研究中心）
An analytic solution of the Regge-Wheeler (RW) equation has been found via the Frobenius method at the regular singularity of the horizon 2M, in the form of a time and radial coordinate dependent series. The RW partial differential equation, derived from the Einstein field equations, represents the first order perturbations of the Schwarzschild metric. The known solutions are numerical in time domain or approximate and asymptotic for low or high frequencies in Fourier domain. The former is of scarce relevance for comprehension of the geodesic equations for a body in the black hole field, while…
We discuss the gravitational self-force on a particle in a black hole space-time. For a point particle, the full (bare) self-force diverges. It is known that the metric perturbation induced by a particle can be divided into two parts, the direct part (or the S part) and the tail part (or the R part), in the harmonic gauge, and the regularized self-force is derived from the R part which is regular and satisfies the source-free perturbed Einstein equations. In this paper, we consider a gauge transformation from the harmonic gauge to the Regge-Wheeler gauge in which the full metric perturbation c…
Waveform stability for the piecewise step approximation of Regge-Wheeler potential
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（基础物理与数学科学学院，杭州高等研究院，UCAS）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Interdisciplinary Center for Theoretical Study and Department of Modern Physics, University of Science and Technology of China（中国科学技术大学理论交叉研究中心和现代物理系）；Institute of Fundamental Physics and Quantum Technology, Ningbo University（宁波大学基础物理与量子技术研究所）
Constructing an exact correspondence between a black hole model, formed from the most simple solution of Einstein's equations, and a particular moving mirror trajectory, we investigate a new model that preserves unitarity. The Bogoliubov coefficients in 1+1 dimensions are computed analytically. The key modification limits the origin of coordinates (moving mirror) to sub-light asymptotic speed. Effective continuity across the metric ensures that there is no information loss. The black hole emits thermal radiation and the total evaporation energy emitted is finite without backreaction, consisten…
We use the, covariant and gauge-invariant, 1+1+2 formalism developed by Clarkson and Barrett, and develop new techniques, to decouple electromagnetic (EM) perturbations on arbitrary locally rotationally symmetric (LRS) space-times. Ultimately, we derive 3 decoupled complex equations governing 3 complex scalars. One of these is a new Regge-Wheeler (RW) equation generalized for LRS space-times, whereas the remaining two are new generalizations of the Bardeen-Press (BP) equations. This is achieved by first using linear algebra techniques to rewrite the first-order Maxwell equations in a new compl…
We obtain a fourth order accurate numerical algorithm to integrate the Zerilli and Regge-Wheeler wave equations, describing perturbations of nonrotating black holes, with source terms due to an orbiting particle. Those source terms contain the 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).
We calculate the odd-parity, radiative ($\ell \ge 2$) parts of the metric perturbation in Lorenz gauge caused by a small compact object in eccentric orbit about a Schwarzschild black hole. The Lorenz gauge solution is found via gauge transformation from a corresponding one in Regge-Wheeler gauge. Like the Regge-Wheeler gauge solution itself, the gauge generator is computed in the frequency domain and transferred to the time domain. The wave equation for the gauge generator has a source with a compact, moving delta-function term and a discontinuous non-compact term. The former term allows the m…
We derive the solutions of gravitational waves in the future (F) expanding and the past (P) shrinking Kanser spacetimes as well as in the left (L) and right (R) Rindler wedges in the Regge-Wheeler gauge. The solutions for all metric components are obtained in an analytic form in each region. We identify the master variables, which are equivalent to massless scalar fields, to describe the gravitational degrees of freedom for the odd parity and even parity modes under the transformation in the two-dimensional plane symmetric space. Then, the master variables are quantized, and we develop the qua…
It is shown that the dynamical evolution of perturbations on a static spacetime is governed by a standard pulsation equation for the extrinsic curvature tensor. The centerpiece of the pulsation equation is a wave operator whose spatial part is manifestly self-adjoint. In contrast to metric formulations, the curvature-based approach to gravitational perturbation theory generalizes in a natural way to self-gravitating matter fields. For a certain relevant subspace of perturbations the pulsation operator is symmetric with respect to a positive inner product and therefore allows spectral theory to…
We prove sharp pointwise $t^{-3}$ decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates $t^{-4}$, and $t^{-6}$, respectively. We proceed by decomposition into angular momentum $\ell$ and summation of the decay estimates on the Regge-Wheeler equation for fixed $\ell$. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the consta…
On the instability of the fundamental mode of the Regge-Wheeler effective potential
Zhejiang Guangsha Vocational and Technical University of Construction（浙江广厦建设职业技术大学）；Hefei Institutes of Physical Science, Chinese Academy of Sciences（中国科学院合肥物质科学研究院）
In this paper, a candidate for pregeometry, Ponzano-Regge spin networks, will be examined in the context of the pregeometric philosophy of Wheeler. Ponzano and Regge were able to construct a theory for 3-dimensional quantum gravity based on 3nj-symbols, obtaining the path integral over the metric in the semiclassical limit. However, extension of this model to 4-dimensions has proven to be difficult. It will be shown that the building blocks for 4-dimensional spacetime are already present in the Ponzano-Regge formalism using a reinterpretation of the theory based on the pregeometric hypotheses …
The asymptotics of some spin foam amplitudes for a quantum 4-simplex is known to display rapid oscillations whose frequency is the Regge action. In this note, we reformulate this result through a difference equation, asymptotically satisfied by these models, and whose semi-classical solutions are precisely the sine and the cosine of the Regge action. This equation is then interpreted as coming from the canonical quantization of a simple constraint in Regge calculus. This suggests to lift and generalize this constraint to the phase space of loop quantum gravity parametrized by twisted geometrie…
A Regge--Wheeler analysis is performed for a novel black hole mimicker `the regular black hole with asymptotically Minkowski core', followed by an approximation of the permitted quasi-normal modes for propagating waveforms. A first-order WKB approximation is computed for spin zero and spin one perturbations of the candidate spacetime. Subsequently, numerical results analysing the respective fundamental modes are compiled for various values of the $a$ parameter (which quantifies the distortion from Schwarzschild spacetime), and for various multipole numbers $\ell$. Both electromagnetic spin one…
We show that the radial Teukolsky equation (in the frequency domain) with sources that extend to infinity has well-behaved solutions. To prove that, we follow Poisson approach to regularize the non-rotating hole, and extend it to the rotating case. To do so we use the Chandrasekhar transformation among the Teukolsky and Regge-Wheeler-like equations, and express the integrals over the source in terms of solutions to the homogeneous Regge-Wheeler-like equation, to finally regularize the resulting integral. We then discuss the applicability of these results.
In this paper we consider the field equations for linearized gravity and other integer spin fields on the Kerr spacetime, and more generally on spacetimes of Petrov type D. We give a derivation, using the GHP formalism, of decoupled field equations for the linearized Weyl scalars for all spin weights and identify the gauge source functions occuring in these. For the spin weight 0 Weyl scalar, imposing a generalized harmonic coordinate gauge yields a generalization of the Regge-Wheeler equation. Specializing to the Schwarzschild case, we derive the gauge invariant Regge-Wheeler and Zerilli equa…
We calculate the eccentricity dependence of the high-order post-Newtonian (PN) expansion of the spin-precession invariant $ψ$ for eccentric-orbit extreme-mass-ratio inspirals with a Schwarzschild primary. The series is calculated in first-order black hole perturbation theory through direct analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a code written in \textsc{Mathematica}. Modes with small values of $l$ are found via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by…
Beginning with the pioneering work of Regge and Wheeler (Phys. Rev. 108, 1957), there have been many studies of perturbations away from the Schwarzschild spacetime background. In particular several authors (e.g. Moncrief, Ann. Phys 88, 1974) have investigated gauge invariant quantities of the Regge-Wheeler (RW) gauge. Steven Detweiler also investigated perturbations of Schwarzschild in his own gauge, which he denoted the "easy (EZ) gauge", and which he was in the process of adapting for use in the second-order self-force problem. We present here a compilation of some of his working results, ar…
We study odd parity perturbations of spherically symmetric black holes with time-dependent scalar hair in shift-symmetric higher-order scalar-tensor theories. The analysis is performed in a general way without assuming the degeneracy conditions. Nevertheless, we end up with second-order equations for a single master variable, similarly to cosmological tensor modes. We thus identify the general form of the quadratic Lagrangian for the odd parity perturbations, leading to a generalization of the Regge-Wheeler equation. We also investigate the structure of the effective metric for the master vari…
We study the metric perturbations induced by a classical spinning particle moving along a circular orbit on a Schwarzschild background, limiting the analysis to effects which are first order in spin. The particle is assumed to move on the equatorial plane and has its spin aligned with the $z$-axis. The metric perturbations are obtained by using two different approaches, i.e., by working in two different gauges: the Regge-Wheeler gauge (using the Regge-Wheeler-Zerilli formalism) and a radiation gauge (using the Teukolsky formalism). We then compute the linear-in-spin contribution to the first-o…
We derive a geometrical version of the Regge-Wheeler and Zerilli equations, which allows us to study gravitational perturbations on an arbitrary spherically symmetric slicing of a Schwarzschild black hole. We explain how to obtain the gauge-invariant part of the metric perturbations from the amplitudes obeying our generalized Regge-Wheeler and Zerilli equations and vice-versa. We also give a general expression for the radiated energy at infinity, and establish the relation between our geometrical equations and the Teukolsky formalism. The results presented in this paper are expected to be usef…
Gravitational perturbations due to a point particle moving on a static black hole background are naturally described in Regge-Wheeler gauge. The first-order field equations reduce to a single master wave equation for each radiative mode. The master function satisfying this wave equation is a linear combination of the metric perturbation amplitudes with a source term arising from the stress-energy tensor of the point particle. The original master functions were found by Regge and Wheeler (odd parity) and Zerilli (even parity). Subsequent work by Moncrief and then Cunningham, Price and Moncrief …
We consider general perturbations of a Schwarzschild black holes in the context of f(R) gravity. A reduced set of frame independent master variables are determined, which obey two closed wave equations - one for the transverse, trace-free (tensor) perturbations and the other for the additional scalar degree of freedom which characterise fourth-order theories of gravity. We show that for the tensor modes, the underlying dynamics in f(R) gravity is governed by a modified Regge-Wheeler tensor which obeys the same Regge-Wheeler equation as in General Relativity. We find that the possible sources o…
We present a discrete form of the Wheeler-DeWitt equation for quantum gravitation, based on the lattice formulation due to Regge. In this setup the infinite-dimensional manifold of 3-geometries is replaced by a space of three-dimensional piecewise linear spaces, with the solutions to the lattice equations providing a suitable approximation to the continuum wave functional. The equations incorporate a set of constraints on the quantum wavefunctional, arising from the triangle inequalities and their higher dimensional analogs. The character of the solutions is discussed in the strong coupling (l…
