Two-body problem in curved spacetime: exploring gravitational wave transient cases
Authors: Yershov V N , Raikov A A , Popova E A 
Published: June 2023

Abstract
Gravitational-Wave Transient Catalogues (GWTC) from the LIGO-Virgo-KAGRA collaborations (LVC and LVK) contain almost a hundred gravitational wave (GW) detection cases. We explore them from the perspective of the two-body problem in curved spacetime, starting with the first case, GW150914, which marks the GW discovery [1]. In this paper, the LVC authors estimated the characteristic (chirp) mass of the binary blackhole system emitted this signal. Their calculation was based on Numerical-Relativity (NR) templates and presumably accounted fully for the non-linearity of GR. The same team later presented an alternative analysis of GW150914 [2], using the quadrupole post-Newtonian (PN) approximation of GR. Both analyses gave similar results, despite being based on quite different assumptions about the linearity or non-linearity of the coordinate reference frame near the GW source. Here we revisit the PN-analysis of GW150914 for which we use less noisy input GW frequencies, as we have filtered them by reading them from the time-frequency map of GW150914. As in paper [2], our result also agrees with the NR-based chirp mass value published in [1]. Additionally, we apply the PN-approximation formalism to the rest of the GWTC cases, finding that practically all of their PN-approximated chirp masses coincide with the published NR-based values from GWTC. In our view, this implies that the NR-based theory, which is currently in use for processing GW signals, does not fully account for the difference between the source and detector reference frames because the PN-approximation, which is used for the comparison, does not account for this difference by design, given the flat-spacetime initial assumptions of this approximation. We find that the basis of this issue lies in the source-to-detector coordinate transformation. For example, when obtaining the equation of motion of a coalescing binary system by integrating its energy-momentum tensor and varying the corresponding reduced action functional, the lapse and shift functions are not involved within the Arnowitt-Deser-Misner (ADM) parametrisation scheme, which is typically used for the NR-based calculation of GW waveforms A similar non-involvement of the lapse and shift functions is known to occur in the description of motion of an orbiter around a Schwarzschild blackhole. Here the GR expression for the orbital angular frequency, as seen by a remote observer, coincides with the Keplerian non-relativistic formula until the very last orbits before the plunge phase (although being fully GR-compliant). This non-involvement of the time lapse function renders the source-to-detector coordinate transformation suitable for building GW waveforms corresponding to the detector frame. However, the inverse (detector-to-source) transformation requires the derivatives of GW frequencies to be known in the source reference frame. The lack of this knowledge leads to a systematic error in the estimated chirp masses of GW sources. The corresponding luminosity distances of these sources also turn out to be overestimated.