The work described on this page was carried out at Washington University, St. Louis by members of WUGRAV.

According to general relativity, binary stellar systems emit gravitational radiation, and thereby lose energy and angular momentum. This dissipation causes the orbit to circularize and decay, with ever decreasing orbital period and orbital separation. Eventually, systems containing compact objects such as neutron stars or black holes, will decay to such tight, high-velocity orbits that a final burst of gravitational radiation will result, accompanied by a rapid inspiral toward coalescence. In 2015, gravitational waves from the final ``death-spiral'' of two black holes were detected by the LIGO interferometric gravitational observatories. Three other black hole binary insipiral were detected in 2015 and 2017, one of them detected joinly by LIGO and the Virgo observatory in Italy. And in August 2017 a binary neutron star inspiral was detected both by the LIGO-Virgo instruments and by the Fermi gamma-ray telescope. The space interferometer LISA ,scheduled for launch around 2034, will open the low-frequency gravitational-wave window. Detection and study of the gravitational waves from such systems has initiated a new era of gravitational-wave astronomy and provided important new tests of general relativity, especially in its highly dynamical, strong-field regimes.

Extracting the characteristic inspiralling-binary signal from the noisy data in an array of detectors, and using the signal to obtain useful information about the sources, such as their directions, distances, masses, spins, and so on involves a process of ``optimal matched filtering'' of a set of theoretical ``template waveforms'' against the output of the detectors. This method requires that the theoretical templates be a sufficiently accurate representation of the waves from the hypothesized source that it match the observed signal to within a fraction of a phase over the tens to thousands of gravitational-wave cycles observed in the sensitive bandwidth.

For most of the binary inspiral phase, (v/c)^2 ~ (Gm/rc^2) is small, and so post-Newtonian (PN) approximations to general relativity can be used to describe the orbit and gravitational waveform. However, in order to calculate a template of sufficient accuracy, higher-order post-Newtonian corrections must be taken into account. Corrections through at least third PN (3PN) order have been calculated.

During our time at Washington University (until our move to the University of Florida in 2012), our calculations of higher-order post-Newtonian effects were carried out using a method called DIRE, Direct Integration of the Relaxed Einstein equations. It is based on the 1975 Epstein-Wagoner (EW) approach, in which Einstein's equations are cast in their so-called ``relaxed'' form, as a flat spacetime wave equation with a non-compact source consisting of matter and the stress-energy of the gravitational fields themselves. The solutions are expressed as integral equations, which can be iterated order-by-order. In the near zone, we evaluate the integrals by a slow-motion, multipole expansion. In the far zone, we evaluate the integrals over the past-light cone using a technique developed by our group (Will and Wiseman 1996, Pati and Will 2000). In this new method, there are NO divergent integrals, at least through 3.5 PN order. Furthermore, the effects of gravitational-wave ``tails'' and of propagation of the radiation along the true cones of the spacetime are obtained. For a pedagogical introduction to DIRE, post-Minkowskian theory and post-Newtonian theory, see the book Gravity: Newtonian, post-Newtonian, Relativistic by Poisson and Will

Significant progress has been made in calculating and studying the implications of higher-order post-Newtonian effects in the behavior of and gravitational waves from inspiralling binaries (see, for example, the Living Review by Luc Blanchet). Among the results achieved by members of WUGRAV at Washington University were:

Gravitational Waveforms to 2PN Order

We calculated the 2PN contributions to the waveform and the energy flux (Blanchet, Damour, Iyer, Will and Wiseman 1995 ; Will and Wiseman 1996 ; Blanchet, Iyer, Will and Wiseman 1996 ). The BIWW 2PN waveform was used in early phases of LIGO's data analysis software. The more modern versions incorporate higher order PN terms, sophisticated resummations techniques (such as the ''effective one-body'' approach of Buonanno and Damour), and phenomenological waveforms that connect smoothly to results from numerical relativity for the merger and ringdown phases.

Equations of Motion: 3.5PN Radiation Reaction

We developed the formalism of DIRE through 3.5 PN order (Pati and Will 2000), and calculated the equations of motion for binary systems through 2.5PN order, and including 3.5PN order terms (Pati and Will 2002). 

Spin Effects and Radiation Reaction

We calculated the effects of spin on gravitational radiation reaction, and the effects of radiation reaction on the precession of spin (Will 2005, Wang and Will 2007). This requires use of the 2.5 PN and 3.5 PN terms in the equations of motion for rotating fluid bodies. We found that radiation reaction had no effect on the magnitude of the body's spins, and at most could cause a spin-spin induced precession.  We also showed that the equations of motion including spin-induced radiation reaction could be derived from an assumption of energy and angular momentum balance, modulo a set of arbitrary coefficients that reflect the effects of gauge freedom in the equations of motion (Zeng and Will 2007)

Gravitational Radiation Recoil

We calculated the momentum radiated by an inspiraling black-hole binary system and the resulting recoil velocity, to 2PN order (Blanchet, Qusailah and Will 2005).  The results (eg. a maximum of 250 km/s for a mass ratio of  0.38) were in amazing agreement with the recoil velocity calculated using numerical relativity up to the point of merger, over a range of mass ratios. By combining 2PN techniques with a ``Close-Limit Approximation'' for treating the perturbed, newly formed black hole, we (Le Tiec, Blanchet and Will 2010) showed that black hole ringdown radiation following the merger reduces the kick somewhat. Our final results for the total kick (2PN plus ringdown) agreed completely with results from full numerical relativity.

The Strong Equivalence Principle to 2PN Order

We showed that the motion of non-spinning bodies of finite size in a binary system is independent of their internal structure, specifically of binding energy effects that scale as (1/size of the bodies), through 2PN order (Mitchell and Will 2007)

Interface between post-Newtonian Theory and Numerical Relativity

We developed a post-Newtonian diagnostic tool that can help elucidate the physical content of the numerically generated initial states for binary inspiral (Mora and Will, 2002, 2003; Berti, Iyer and Will 2006, 2007).


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