Summarising my PhD research and thesis

I am excited to share the following extract from the preamble of my thesis (Access my thesis), which endeavours to summarise my PhD research for a non-expert audience (physics/applied mathematics undergraduates and well-informed members of the public).

The last century has shown General Relativity to be outstandingly successful at describing gravity. It has survived every physical measurement that experimentalists and astronomers have devised to test it. Improving on such tests will require probing more extreme regimes of spacetime curvature to assess whether and where General Relativity breaks down.

A new era for testing General Relativity began in 2015 when LIGO made the first detection of gravitational waves [1]. Gravitational waves propagate outward at the speed of light, expanding and contracting space and time as they pass. Einstein predicted their existence due to General Relativity in 1916 [2]. The gravitational waves LIGO first detected were produced billions of years ago by a merging binary black hole system. The detection marked the dawn of a new era for astronomy in two senses:

• the first detection of a whole new class of radiation with which to observe the universe,
• the first direct observation of black holes.

As gravitational wave signals are weak, they are usually hidden beneath detector noise. Extracting the wave signal from detector data requires matched filtering techniques. matched filtering matches signals in the data to waveform templates. This technique relies on using a bank of waveform templates from accurate source models.

Since 2015, there have been dozens of binary black hole detections (and a handful of black hole-neutron star binaries and neutron star binaries) by the LIGO/VIRGO/KAGRA collaboration. These detections have been rich in astrophysical information and produced constraints on alternative theories of gravity. Testing General Relativity further with gravitational wave astronomy will be possible as detectors increase in sensitivity. However, this will require more precise models of the systems that produce gravitational waves. Improving the precision of binary models in General Relativity is the subject of this thesis.

Before I discuss the type of model I have improved, allow me to explain why I chose to pursue a Ph.D. as a relativist: General Relativity is undoubtedly a beautiful theory, birthed out of the simple notion that gravity is fundamentally a manifestation of the curvature of space and time. As the Einstein field equations (EFE) have proven challenging to solve, there is still much to learn and achieve. Lastly, the interaction of General Relativity with observations has just begun, and the next few decades of gravitational wave research have the potential to become a golden age in physics.

Techniques for modelling systems that emit gravitational waves have been of mathematical interest for over half a century. They have recently risen to prominence due to the first detection of gravitational waves. Modelling gravitational wave sources is not a simple task as the equations governing General Relativity, the EFE, are challenging to solve. The Schwarzschild and Kerr solutions are two of the most astrophysically relevant exact solutions to the EFE. These solutions describe stationary black holes; however, being stationary, they do not emit gravitational waves. To model gravitational waves-emitting systems, one must resort to solving the EFE numerically [3] or finding approximate solutions using techniques like perturbation theory [4]. This thesis concentrates on black hole perturbation theory which calculates small perturbations to the Schwarzschild or Kerr solutions to build approximate solutions to the EFE (which emit gravitational waves).

Models alone are insufficient to test physics; we also need observations to test them against. One of the next logical steps in gravitational wave astronomy is building a space-based interferometer. Current plans are for the Laser Interferometer Space Antenna (LISA) to be launched in 2034 [5]; China is also planning two space-based detectors [6,7]. LISA will detect gravitational waves signals in the mHz frequency band (a significantly lower frequency than LIGO), opening up the possibility of detecting new types of gravitational waves sources. The primary application of the work in this thesis will be modelling a key source for LISA, extreme-mass-ratio inspirals (EMRIs). EMRIs occur when a compact stellar-mass object (such as a black hole or neutron star) slowly inspirals into a supermassive black hole.

The evolution of an EMRI from bound orbit to merger is driven by the emission of gravitational waves carrying energy and angular momentum away from the system. The perturbation caused by the presence of the inspiraling object produces these waves. The energy and angular momentum carried away by the waves correspond to the work done (by a force) on the inspiralling object. This force, effectively generated by the presence of the inspiraling object, acts on the object itself. Hence, the effect is known as gravitational self-force. As the self-force is a small effect, it can be calculated with high accuracy using black hole perturbation theory.

EMRIs offer excellent testing grounds for General Relativity. The compact object spends a long time (~2 years) near the supermassive black hole. In this region, the spacetime curvature is stronger (known as strong-field) than the regimes in which we usually test General Relativity (e.g., the solar system). Additionally, the compact object achieves relativistic velocities (~ 0.3 c) during an inspiral. Moreover, EMRIs are detectable for ~10^5 space-filling orbits before the inspiraling object plunges into the supermassive black hole. Hence, LISA observations of EMRIs will be excellent maps of the spacetime around a supermassive black hole. For a review of the possible tests of General Relativity and astrophysics that can be performed with EMRIs, see Refs. [8,9,10]. To summarize, EMRIs are expected to be sensitive to violations of the no-hair theorem, extra degrees of freedom beyond General Relativity, dark matter, and astrophysical effects.

Precise extraction of an EMRI’s parameters from LISA measurements will require highly accurate waveform models. These models must be calculated to what is known as post-adiabatic order [11] accuracy. This high accuracy requirement can be understood by considering that an EMRI will typically be in the LISA frequency band for hundreds of thousands of orbits. Hence, even a small error in modelling a single orbit will cause the accumulation of a significant error throughout the whole inspiral. These highly accurate models require, amongst other things, contributions from the second-perturbative-order self-force [11].

This thesis presents methods to help calculate the second-order self-force. Recently, the first second-order self-force results have been published [12,13,14]. However, these results specialize to a Schwarzschild black hole (which has no spin). Astrophysical supermassive black holes are expected to have significant spin. Hence, astrophysically accurate self-force models need to be for a Kerr black hole. Here, I develop new second-order methods, applicable for self-force calculations in Kerr. I derive an equation for calculating a second-order quantity in Kerr, from which the self-force can likely be extracted. I also help solve this equation in Schwarzschild and will soon calculate the second-order self-force (and compare the results with Ref. [14]).

[1] Abbott, Benjamin P., et al. “Observation of gravitational waves from a binary black hole merger.” Physical review letters 116.6 (2016): 061102.

[2] Einstein, Albert. “Näherungsweise integration der feldgleichungen der gravitation.” Albert Einstein: Akademie‐Vorträge: Sitzungsberichte der Preußischen Akademie der Wissenschaften 1914–1932 (2005): 99-108.

[3] Arnowitt, Richard, Stanley Deser, and Charles W. Misner. “Republication of: The dynamics of general relativity.” General Relativity and Gravitation 40.9 (2008): 1997-2027.

[4] Poisson, Eric, and Clifford M. Will. Gravity: Newtonian, post-newtonian, relativistic. Cambridge University Press, 2014.

[5] Danzmann, K. The {ESA} website, The Laser Interferometer Space Antenna. https://www.elisascience.org/articles/lisa-mission, 2019

[6] Luo, Jun, et al. “TianQin: a space-borne gravitational wave detector.” Classical and Quantum Gravity 33.3 (2016): 035010.

[7] Ruan, Wen-Hong, et al. “Taiji program: Gravitational-wave sources.” International Journal of Modern Physics A 35.17 (2020): 2050075.

[8] Barausse, Enrico, et al. “Prospects for fundamental physics with LISA.” General Relativity and Gravitation 52.8 (2020): 1-33.

[9] Arun, K. G., et al. “New horizons for fundamental physics with LISA.” Living Reviews in Relativity 25.1 (2022): 1-148.

[10] Amaro-Seoane, Pau, et al. “Astrophysics with the Laser Interferometer Space Antenna.” arXiv preprint arXiv:2203.06016 (2022).

[11] Hinderer, Tanja, and Eanna E. Flanagan. “Two-timescale analysis of extreme mass ratio inspirals in Kerr spacetime: Orbital motion.” Physical Review D 78.6 (2008): 064028.

[12] Pound, Adam, et al. “Second-order self-force calculation of gravitational binding energy in compact binaries.” Physical review letters 124.2 (2020): 021101.

[13] Warburton, Niels, et al. “Gravitational-wave energy flux for compact binaries through second order in the mass ratio.” Physical Review Letters 127.15 (2021): 151102.

[14] Wardell, Barry, et al. “Gravitational waveforms for compact binaries from second-order self-force theory.” arXiv preprint arXiv:2112.12265 (2021).