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Gravitational Self-Force and Post-Newtonian Methods for Gravitational Wave Detection

Periodic Reporting for period 2 - GravityWaveWindow (Gravitational Self-Force and Post-Newtonian Methods for Gravitational Wave Detection)

Reporting period: 2018-04-01 to 2019-03-31

Gravitational waves; ripples in spacetime as predicted by Einstein’s theory of relativity. Einstein saw masses as sources of energy that curved spacetime around them pulling everything inwards like the incline of a hill. When extremely compact objects like black holes or neutron stars get too close, a violent interaction occurs between their fields as they merge - this clash of curvature results in smaller waves rippling out across the universe. The first detection of gravitational waves occurred in September 2015 and marked the birth of gravitational wave astronomy - not only giving us a new window through which to look at the universe but it marks an united effort by researchers across the globe in accomplishing a common goal - insight into the world in which we all live.

To enable detection, one must calculating gravitational waveforms. The current Earth detectors, aLIGO and aVIRGO are sensitive to binaries (black holes and/or neutron stars) with similar masses. Waveforms in this regime are well known, however, a key aim of gravitational wave detectors is to test Einstein’s theory of general relativity. To do so conclusively, one needs waveforms for alternative theories; these currently exist only to first post-Newtonian accuracy (1PN), an approximations that assumes a slow-moving system. However, for gravitational wave detection, it is well established that one requires second order post-Newtonian (2PN) calculations – this carries over to waveforms for alternative theories of gravity. We are currently calculating the 2PN waveform for scalar-tensor gravity, a theory that differs from General Relativity by only a varying Gravitational `constant’. This, combined with the detection data of the binary neutron star can be used to test general relativity in the strong field regime.

All gravitational wave detectors are currently on Earth and hence limited to higher-frequency gravitational waves by the tectonic plate movements – for low frequency, one most have space-based detectors, like LISA, the future ESA detector. LISA will detect EMRI’s, Extreme Mass Ratio Inspirals - these are thought to form when a `small' stellar mass black hole (the same mass as our sun) is bumped into the grasp of a supermassive black hole (like that living at the centre of our galaxy). Unfortunately, we do not have EMRI waveforms. The self-force, which expands in the mass ratio, is perfect for tackling this problem. At zero order, the smaller body behaves massless and follows the curved spacetime of the larger black hole. At first order, it deviates due to interaction with its own field. This field has a singularity at the particle; however, one can model a singular field, that captures this singularity without affecting motion, and remove it. A more accurate singular field allows a more efficient (increased accuracy and speed) self-force calculation – in a field of producing banks of waveforms with costly computations, this is a necessity for self-force calculations.

To successfully produce waveforms, the self-force community requires highly accurate first order calculations (enhanced by accurate singular fields), second order calculations and a method to evolve the orbit. On calculating the self-force, one knows how much the body’s own field pushes it off its `natural’ path, depicted by the supermassive black hole. One then includes this force on the particle to calculate the self-force at the next step – hence evolving the orbit. This project deals with production of the singular field for first order self-force and orbit evolution codes - this allows more efficient calculations. This project is also involves the second order self-force - a calculation that has not yet been accomplished. In this manner, this project is assisting, on every front, the effort of self-force calculations towards EMRI waveforms.
We have calculated the 3PN scalar field terms and need only calculate the multipole moments to obtain the 2PN scalar field and energy flux - the last missing pieces for 2PN waveforms. In doing so, we have set up a more general program, which could be applied to similar alternate theories of gravity. Progress reports on this work have been given in a number of conferences, most recently the April American Physical Society Meeting. We have also identified several follow-on theories for future work.

In self-force, we have developed a covariant code for the singular field. The results of this code in general are analytical expressions for the singular field and its corresponding regularisation parameters. These allow for the safe removal of a divergence which slows down numerical calculations. For every additional order of accuracy that one calculates the singular field, the faster the convergence of the numerical codes designed by collaborators. Therefore, any calculation that wants to avail of faster runtime for self-force calculations in the Lorenz gauge, will be use these results. Our article outlining accelerated self-force calculations in the scalar case has been published by Classical and Quantum Gravity. We have made this article available on and the parameters available on We have modified the code for scalar generic Kerr orbits to complement work being carried out by the Chapel Hill group - this is currently complete and in preparation for publication. We now only require covariant expressions to calculate similar parameters for Schwarzschild non-geodesic gravity and Kerr gravity while all ingredients are complete for Kerr non-geodesic scalar parameters and gauge invariant quantities - these are very close to completion.

In second order self-force, we are following the standard format of in black hole perturbation theory and extending to second order. Similar work has been done in the Regge Wheeler gauge; we are building on this by keeping the calculations gauge free. This will give a clear indication of a) which gauge to proceed in and b) what is required when working with modes of finite l to ensure the problem remains well posed. We are part way through this calculation, progress of this work has been reported at the CAPRA meeting for radiation reaction.
We are calculating the first 2PN waveform for an alternative theory of gravity and hence enabling the first gravitational wave comparison tests using data from the ground based detectors. We are also producing previously unknown parameters that will speed up self-force calculations drastically as well as carrying out the first second order calculations outside the Lorenz gauge. All of these calculations impact the world of gravitational wave astronomy which in turn is continuing to inform and shape our knowledge of the universe.

Currently there are 3 detectors, two in the U.S. and one in Europe, however Japan is nearing on completion of their detector and India has also begun building their detector, while the European Space Agency has committed to launching one in space - this is truly a global community. And one where experimentalists, data analysts and theoretical physicists are all working together for a common goal. The detection of gravitational waves has captured the public's imaginations, not only did it win the 2017 Nobel Prize in Physics but it was referenced in newspapers, comics and even real estate advertising campaigns. The continuing information and discoveries that await us, will no doubt build on this already magnificent base.