Tail Term Research Articles

Quasilocal contribution to the gravitational self-force

The gravitational self-force on a point particle moving in a vacuum background space-time can be expressed as an integral over the past world line of the particle, the so-called tail term. In this paper, we consider that piece of the self-force obtained by integrating over a portion of the past world line that extends a proper time $\ensuremath{\Delta}\ensuremath{\tau}$ into the past, provided that $\ensuremath{\Delta}\ensuremath{\tau}$ does not extend beyond the normal neighborhood of the particle. We express this quasilocal piece as a power series in the proper time interval $\ensuremath{\Delta}\ensuremath{\tau}$. We argue from symmetries and dimensional considerations that the $O(\ensuremath{\Delta}{\ensuremath{\tau}}^{0})$ and $O(\ensuremath{\Delta}\ensuremath{\tau})$ terms in this power series must vanish, and compute the first two nonvanishing terms which occur at $O(\ensuremath{\Delta}{\ensuremath{\tau}}^{2})$ and $O(\ensuremath{\Delta}{\ensuremath{\tau}}^{3})$. The coefficients in the expansion depend only on the particle's four velocity and on the Weyl tensor and its derivatives at the particle's location. The result may be useful as a foundation for a practical computational method for gravitational self-forces in the Kerr space-time, in which the portion of the tail integral in the distant past is computed numerically from a mode-sum decomposition.

Radiation reaction in Schwarzschild spacetime: Retarded Green’s function via Hadamard-WKB expansion

An analytic method is given for deriving the part of the retarded Green's function ${v(x,x}^{\ensuremath{'}})$ that contributes to the tail term in the radiation reaction force felt by a particle coupled to a massless minimally coupled scalar field. The method gives an expansion of ${v(x,x}^{\ensuremath{'}})$ for small separations of the points ${x,x}^{\ensuremath{'}}$ valid for an arbitrary static spherically symmetric spacetime. It is obtained by using a WKB approximation for the Euclidean Green's function for the massless minimally coupled scalar field and is equivalent to the DeWitt-Schwinger expansion for ${v(x,x}^{\ensuremath{'}}).$ The first few terms in this expansion are displayed here for the case of Schwarzschild spacetime.

Self-force via a Green’s function decomposition

The gravitational field of a particle of small mass \mu moving through curved spacetime is naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through O(\mu). One part is an inhomogeneous field which, near the particle, looks like the \mu/r field distorted by the local Riemann tensor; it does not depend on the behavior of the source in either the infinite past or future. The other part is a homogeneous field and includes the ``tail term''; it completely determines the self force effects of the particle interacting with its own gravitational field, including radiation reaction. Self force effects for scalar, electromagnetic and gravitational fields are all described in this manner.

Robust statistics for deterministic and stochastic gravitational waves in non-Gaussian noise: Frequentist analyses

Gravitational wave detectors will need optimal signal-processing algorithms to extract weak signals from the detector noise. Most algorithms designed to date are based on the unrealistic assumption that the detector noise may be modeled as a stationary Gaussian process. However most experiments exhibit a non-Gaussian ``tail'' in the probability distribution. This ``excess'' of large signals can be a troublesome source of false alarms. This article derives an optimal (in the Neyman-Pearson sense, for weak signals) signal processing strategy when the detector noise is non-Gaussian and exhibits tail terms. This strategy is robust, meaning that it is close to optimal for Gaussian noise but far less sensitive than conventional methods to the excess large events that form the tail of the distribution. The method is analyzed for two different signal analysis problems: (i) a known waveform (e.g., a binary inspiral chirp) and (ii) a stochastic background, which requires a multi-detector signal processing algorithm. The methods should be easy to implement: they amount to truncation or clipping of sample values which lie in the outlier part of the probability distribution.

Performance analysis and comparison of blind to non-blind least-squares equalization with respect to effective channel overmodeling

The object of this work is the study of a direct blind equalization algorithm which appeared recently in the literature. It is a least-squares (LS) equalization method in the blind context, assuming a linear FIR communication channel and a linear equalizer. If channel order is known, blind LS equalizers can be constructed that entirely suppress intersymbol interference in noiseless signal transmission. In practice, though, channels may be comprised of a few “big” consecutive taps, which we call “significant part”, surrounded by a lot of smaller leading and/or trailing “tail” terms. In such an environment, channel order is harder to define while the value used by the algorithm is critical to its performance. We carry out both theoretical analysis, making use of perturbation theory arguments, and simulations for the cases where channel order determination procedure has yielded an estimate greater than (“effective overmodeling”) or equal to the order of the significant part. Our purpose is to compare the performance of blind LS algorithm with that of its non-blind counterpart. We conclude that (a) when channel does not possess leading tail terms, blind LS is robust to effective overmodeling, meaning that it behaves very much like non-blind LS, and (b) when leading tail terms are present, blind LS will generally not work satisfactorily in the effective overmodeling scenario. In either case, when the order of the significant part is identified correctly and the actual significant parts of subchannels are sufficiently diverse, the algorithm behaves well.

Second post-Newtonian gravitational wave polarizations for compact binaries in elliptical orbits

The second post-Newtonian (2PN) contribution to the `plus' and `cross' gravitational wave polarizations associated with gravitational radiation from non-spinning, compact binaries moving in elliptic orbits is computed. The computation starts from our earlier results on 2PN generation, crucially employs the 2PN accurate generalized quasi-Keplerian parametrization of elliptic orbits by Damour, Sch\"afer and Wex and provides 2PN accurate expressions modulo the tail terms for gravitational wave polarizations incorporating effects of eccentricity and periastron precession.

New approach to electromagnetic wave tails on a curved spacetime

We present an alternative method for constructing the exact and approximate solutions of electromagnetic wave equations whose source terms are arbitrary order multipoles on a curved spacetime. The developed method is based on the higher-order Green's functions for wave equations which are defined as distributions that satisfy wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. The constructed solution is applied to the study of various geometric effects on the generation and propagation of electromagnetic wave tails to first order in the Riemann tensor. Generally the received radiation tail occurs after a time delay which represents geometrical backscattering by the central gravitational source. It is shown that the truly nonlocal wave-propagation correction (the tail term) takes a universal form which is independent of multipole order. In a particular case, if the radiation pulse is generated by the source during a finite time interval, the tail term after the primary pulse is entirely determined by the energy-momentum vector of the gravitational field source: the form of the tail term is independent of the multipole structure of the gravitational source. We apply the results to a compact binary system and conclude that under certain conditions the tail energy can be a noticeable fraction of the primary pulse energy. We argue that the wave tails should be carefully considered in energy calculations of such systems.

Electromagnetic wave tails on curved spacetime and their observational implications

We calculate the tail term of the electromagnetic potential of a pulsed source in arbitrary bounded motion in a weak gravitational field using a higher-order Green's function approach and demonstrate that generally the received radiation tail arrives after a time delay which represents geometrical backscattering by the central gravitational source. We apply the results to a compact binary system and conclude that under certain conditions the tail energy can be a noticeable fraction of the direct pulse energy.

Second post-Newtonian radiative evolution of the relative orientations of angular momenta in spinning compact binaries

The radiative evolution of the relative orientations of the spin and orbital angular momentum vectors ${\bf S}_{{\bf 1}}, {\bf S}_{{\bf 2}}$ and ${\bf L}$, characterizing a binary system on eccentric orbit is studied up to the second post-Newtonian order. As an intermediate result, all Burke-Thorne type instantaneous radiative changes in the spins are shown to average out over a radial period. It is proved that spin-orbit and spin-spin terms contribute to the radiative angular evolution equations, while Newtonian, first and second post-Newtonian terms together with the leading order tail terms do not. In complement to the spin-orbit contribution, given earlier, the spin-spin contribution is computed and split into two-body and self-interaction parts. The latter provide the second post-Newtonian order corrections to the 3/2 order Lense-Thirring description.

Complete algorithms for calculating the higher-order fundamental solutions of wave equations

Complete recurrent algorithms for calculating the higher-order fundamental solutions of covariant linear wave equations for scalar and tensor wave fields on an arbitrary curved spacetime are derived. The higher-order fundamental solutions are the distributions that satisfy the wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. Like the classical Green's function for a scalar wave equation, the higher-order fundamental solutions contain the terms which have support on, and only on, the lightcone as well as tail terms which have support inside the lightcone. With the help of the higher-order fundamental solutions found it is possible to compute the exact multipole solutions of wave equations in a form convenient for practical computations. As applications we consider the exact field of a dipole source of variable strength travelling in an arbitrary curved spacetime and the tail term of scalar multipole waves in the Friedman dust-dominated universe for the case of minimal coupling.

Axiomatic approach to electromagnetic and gravitational radiation reaction of particles in curved spacetime

The problem of determining the electromagnetic and gravitational ``self-force'' on a particle in a curved spacetime is investigated using an axiomatic approach. In the electromagnetic case, our key postulate is a ``comparison axiom,'' which states that whenever two particles of the same charge $e$ have the same magnitude of acceleration, the difference in their self-force is given by the ordinary Lorentz force of the difference in their (suitably compared) electromagnetic fields. We thereby derive an expression for the electromagnetic self-force which agrees with that of DeWitt and Brehme as corrected by Hobbs. Despite several important differences, our analysis of the gravitational self-force proceeds in close parallel with the electromagnetic case. In the gravitational case, our final expression for the (reduced order) equations of motion shows that the deviation from geodesic motion arises entirely from a ``tail term,'' in agreement with recent results of Mino et al. Throughout the paper, we take the view that ``point particles'' do not make sense as fundamental objects, but that ``point particle equations of motion'' do make sense as means of encoding information about the motion of an extended body in the limit where not only the size but also the charge and mass of the body go to zero at a suitable rate. Plausibility arguments for the validity of our comparison axiom are given by considering the limiting behavior of the self-force on extended bodies.

On Huygens’ Principle for the Hodge-de Rham Equations with Lorentzian Gauge

In an arbitrary curved space-time the Hodge-de Rham equations with Lorentzian gauge are studied. Using the spinor calculus and propositions on the curvature tensors, espe-cially on Hall’s canonical forms of Ricci tensors, some properties of the tail terms with respect to second order differential operators are proved. Finally, all Huygens’ operators are explicitly determined.

Absolute space-time measurements and gravitation

A method of direct («absolute») space-time measurements by means of free trajectories is proposed, and its application to general relativity is discussed. Only the existence and order of the intersections of trajectories are used. Top speed is defined independently at all space-time points and for all pairs of bodies. The ratios of (infinite) oscillation numbers of photons (top-speed particles) provide all the means for geometry: topology, differentiable manifold structure etc. Metric is replaced with these ratios in that a correspondence of affine parameters on different null geodesics can be obtained. So, one is able to define an absolute integration and to construct the Green’s functional demanding the field-determining path integral to be finite and expressed solely via oscillation numbers. In particular, the simple geometric structure and immediate physical meaning of the so-called «tail term» become evident in connection with Huygens’ principle.

Post-Newtonian Expansion of the Ingoing-Wave Regge-Wheeler Function

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 limit of one body, gives an important guideline for the study of coalescing compact binaries. In particular, it provides basic formulas to analytically calculate detailed waveforms and luminosity, including the tail terms to (post)$~3$-Newtonian order, which should be reproduced in any other post-Newtonian calculations.

Post-Newtonian Expansion of the Ingoing-Wave Regge-Wheeler Function

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 limit of one body, gives an important guideline for the study of coalescing compact binaries. In particular, it provides basic formulas to analytically calculate detailed waveforms and luminosity, including the tail terms to (post)$^3$-Newtonian order, which should be reproduced in any other post-Newtonian calculations.

Gravitational radiation from a particle in circular orbit around a black hole. I. Analytical results for the nonrotating case.

Among the most promising and interesting sources of gravitational waves for interferometric detectors, such as the ground-based Laser Interferometer Gravitational-wave Observatory (LIGO)/VIRGO system and the proposed space-based Laser Gravitational-Wave Observatory in Space (LAGOS), is the last several minutes of inspiral of a compact binary (one made of neutron stars and/or black holes). This paper is the first in a series that will carry out detailed calculations relevant to such binaries, in the case where one body is a small-mass black hole or neutron star and the other is a much more massive black hole, and the orbit is circular (aside from its gradual inspiral). These papers will focus primarily on the emitted waveforms and especially their phasing---which is crucial for extraction of information from the detectors' measurements. This first paper is restricted to the case where the massive black hole is nonrotating. The paper begins by bringing the already well-developed formalisms for computing the waveforms (the "Regge-Wheeler" and "Teukolsky" formalisms) into a combined form that is particularly well suited both for high accuracy numerical calculations (to be carried out in paper II), and for analytic calculations (this paper). Then analytic solutions to the formalism's equations are found in the limiting case of orbits with large radii ${r}_{0}$, and correspondingly small values of $v={(\frac{M}{{r}_{0}})}^{\frac{1}{2}}={(M\ensuremath{\Omega})}^{\frac{1}{3}}$. Here $M$ is the mass of the large black hole; $v$ and $\ensuremath{\Omega}$ are, respectively, the orbital linear and angular velocities as measured far from the hole. In particular, (i) the leading-order (in $v$) contribution of each spherical harmonic to the waveforms and to the energy loss is computed analytically, and (ii) the full wave-forms and full energy loss are computed analytically up through fractional corrections of order ${v}^{3}$ beyond Newtonian, i.e. up through ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian order. It is shown that propagation of the waves through the intermediate zone (which connects the near zone to the wave zone) distorts the waveforms and changes their power (and hence phasing), at ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian order, in ways that have not previously been computed---except abstractly and nonconcretely as formal "tail terms" in the waves. It is demonstrated that these ${\mathrm{post}}^{\frac{3}{2}}$-Newtonian corrections will be of considerable importancee for the extraction of information from the waveforms that LIGO/VIRGO expects to measure.

Representation of pure fluid coexistence curves by series expansion

We give a systematic recipe for constructing the coexistence curve associated with a pure fluid described by an analytical equation of state. The novelty of our result lies not in its theoretical basis, which is well known, but in the high degree of accuracy that it will routinely yield over a broad range of density and temperature for a relatively modest expenditure of computing effort. Our results are quite general: they may be applied to any analytical equation of state. The effectiveness of our method is illustrated by applying our general solution to four widely known model equations of state: van der Waals, Dieterici, Soave-Redlich-Kwong, and Carnahan-Starling with van der Waals tail term. These models show a considerable diversity of coexistence curve shapes, and our method yields accurate representations for each model. An investigation is also initiated of the representation of experimental coexistence curves that exhibit pronounced “non-classical” behavior over an appreciable range of densities.

The motion of a charged particle in a spacetime with a conformal metric

It is shown to the lowest-order expansion in the geodetic interval that the DeWitt-Brehme tail term (1960) is given by an integral containing a covariant derivative of the Weyl tensor. Thus to lowest order the DeWitt-Brehme tail term inherits some of the conformal properties of the Weyl tensor, for example it vanishes in conformally flat spacetimes.

Explizite Darstellungsformeln GREENscher Funktionen von kovarianten Wellengleichungen im schwachen Gravitationsfeld II. Linearisierte EINSTEINsche Gravitationsgleichungen

AbstractThe propagation equations for small perturbations of a background gravitational field satisfying the EINSTEIN equations are considered. For the perturbation potential the covariantly generalized EINSTEIN‐HILBERT gauge is chosen. With the aid of the method used in [10], bitensor GREEN's functions for the propagation equations in a weak vacuum field are given explicitly. The tail term is obtained to be an integral of the first‐order RIEMANN curvature tensor. As an application of the formulae, GREEN's functions for perturbations of the SCHWARZSCHILD metric are calculated to first order in the mass parameter.

Explizite Darstellungsformeln GREENscher Funktionen von kovarianten Wellengleichungen im schwachen Gravitationsfeld. II. Linearisierte EINSTEINsche Gravitationsgleichungen

The propagation equations for small perturbations of a background gravitational field satisfying the EINSTEIN equations are considered. For the perturbation potential the covariantly generalized EINSTEIN-HILBERT gauge is chosen. With the aid of the method used in [10], bitensor GREEN's functions for the propagation equations in a weak vacuum field are given explicitly. The tail term is obtained to be an integral of the first-order RIEMANN curvature tensor. As an application of the formulae, GREEN's functions for perturbations of the SCHWARZSCHILD metric are calculated to first order in the mass parameter.