# New invariants for the gravitational two-body problem

### Abstract

Our perception of the Universe, at present, is based on "sight". Ever since Galileo spied the moons of Jupiter in 1610, we have been driven to improve the sensitivity and resolution of telescopes. These are our eyes on the Universe, working across the electromagnetic spectrum. Yet, with only "eyes", our perception is limited; much of the Universe remains dark, or shrouded from view behind clouds of dust and gas.

What if we could "hear" the Universe? This is not as fanciful as it may seem. Einstein's theory of General Relativity predicts the existence of "gravitational waves": ripples in the fabric of spacetime which propagate at the speed of light. Einstein's theory, now a century old, has passed every experimental test so far devised. Indeed, the existence of gravitational waves was confirmed by careful decade-long observations of a pulsar's orbit (brilliant work which won the Nobel prize in 1993).

It is widely anticipated that we stand on the cusp of developing "hearing". Enhanced detectors such as LIGO and VIRGO, coming online in the next few years, should enable us to "hear" gravitational waves for the first time. Gravitational waves are generated by the most powerful processes in the Universe, such as the orbits of supermassive black holes. Unlike our "eyes", our "ears" will be only weakly directional, and thus will hear a wide variety of sources and ambient noise. The key challenge will be to separate the interesting sources from the background. The challenge is akin to trying to listen to a friend over the hubbub at a noisy party: ears are not sufficient, a highly-evolved brain is essential too! To succeed, it is crucial to know precisely what we are listening for. Specifically, we need highly accurate models of gravitational wave signals emanating from compact binaries.

In this project, I aim to improve and upgrade our modelling of such signals. To achieve this aim, I must address a foundational problem in relativity: predicting the motion of two compact bodies, such as black holes, moving under mutual gravitational attraction. Why hasn't such a simple problem been "solved" already? Because it's not so simple! Einstein's theory describes, simultaneously, how the stage affects the actors and how the actors affect the stage. In the words of John Wheeler, "Matter tells space how to curve. Space tells matter how to move". In other words, finding "exact" mathematical solutions in dynamical scenarios is hard or impossible! Instead, we must develop and apply a range of numerical and approximation tools.

My key claim is that there are certain "invariants", related to physically-observable quantities (such as redshift, precession angle; tidal stress, etc.), that are yet to be computed for the gravitational two-body problem over its 100-year history. Invariants are crucial, as they allow us to compare, calibrate and enhance the various mathematical methods currently in use. Fancifully, I imagine these invariants to be part of a Rosetta stone for translation between mathematical "languages". I propose to explore the idea by, first, calculating the invariants (itself a difficult task) and, then, investigating their role in three other approaches to the same problem. I hope to work with leading teams in Canada, France and the US, and to help establish the UK as a leader in this exciting area.

What if we could "hear" the Universe? This is not as fanciful as it may seem. Einstein's theory of General Relativity predicts the existence of "gravitational waves": ripples in the fabric of spacetime which propagate at the speed of light. Einstein's theory, now a century old, has passed every experimental test so far devised. Indeed, the existence of gravitational waves was confirmed by careful decade-long observations of a pulsar's orbit (brilliant work which won the Nobel prize in 1993).

It is widely anticipated that we stand on the cusp of developing "hearing". Enhanced detectors such as LIGO and VIRGO, coming online in the next few years, should enable us to "hear" gravitational waves for the first time. Gravitational waves are generated by the most powerful processes in the Universe, such as the orbits of supermassive black holes. Unlike our "eyes", our "ears" will be only weakly directional, and thus will hear a wide variety of sources and ambient noise. The key challenge will be to separate the interesting sources from the background. The challenge is akin to trying to listen to a friend over the hubbub at a noisy party: ears are not sufficient, a highly-evolved brain is essential too! To succeed, it is crucial to know precisely what we are listening for. Specifically, we need highly accurate models of gravitational wave signals emanating from compact binaries.

In this project, I aim to improve and upgrade our modelling of such signals. To achieve this aim, I must address a foundational problem in relativity: predicting the motion of two compact bodies, such as black holes, moving under mutual gravitational attraction. Why hasn't such a simple problem been "solved" already? Because it's not so simple! Einstein's theory describes, simultaneously, how the stage affects the actors and how the actors affect the stage. In the words of John Wheeler, "Matter tells space how to curve. Space tells matter how to move". In other words, finding "exact" mathematical solutions in dynamical scenarios is hard or impossible! Instead, we must develop and apply a range of numerical and approximation tools.

My key claim is that there are certain "invariants", related to physically-observable quantities (such as redshift, precession angle; tidal stress, etc.), that are yet to be computed for the gravitational two-body problem over its 100-year history. Invariants are crucial, as they allow us to compare, calibrate and enhance the various mathematical methods currently in use. Fancifully, I imagine these invariants to be part of a Rosetta stone for translation between mathematical "languages". I propose to explore the idea by, first, calculating the invariants (itself a difficult task) and, then, investigating their role in three other approaches to the same problem. I hope to work with leading teams in Canada, France and the US, and to help establish the UK as a leader in this exciting area.

### Planned Impact

The proposed research is focused on the development and application of mathematical methods to tackle the historic problem of motion in Einstein's theory of General Relativity (GR). The motivation is to enhance the search for gravitational waves generated by inspiralling pairs of black holes and neutron stars. The primary impact is academic, and the primary pathway is described in the Case for Support.

Wider economical and societal impacts will be indirect, arising primarily from diffusion of improved methods in applied mathematics and scientific computing; enhanced teaching and public engagement; and inspiration of the next generation of UK researchers. Over the long term, fundamental research, which underpins industrial progress, has unanticipated and wide-ranging impacts. For instance, the accuracy of Global Positioning Systems (GPS) relies, in part, on accurate calculations in General Relativity.

In Pathways to Impact I describe concrete steps to achieve impact under three themes: (I1) ``Towards gravitational wave detection''; (I2) ``Towards multimessenger astronomy''; and (I3) ``Relativity and imagination''. Theme I1, the core of this programme, will benefit mathematical physics, primarily through dissemination of high-quality new research with an enduring influence. Theme I2 will impact wider astronomy communities, primarily through inter-disciplinary opportunities and the shaping of future capabilities. Theme I3 will seek to engage and involve anyone who is interested in the Universe and our place within it.

Wider economical and societal impacts will be indirect, arising primarily from diffusion of improved methods in applied mathematics and scientific computing; enhanced teaching and public engagement; and inspiration of the next generation of UK researchers. Over the long term, fundamental research, which underpins industrial progress, has unanticipated and wide-ranging impacts. For instance, the accuracy of Global Positioning Systems (GPS) relies, in part, on accurate calculations in General Relativity.

In Pathways to Impact I describe concrete steps to achieve impact under three themes: (I1) ``Towards gravitational wave detection''; (I2) ``Towards multimessenger astronomy''; and (I3) ``Relativity and imagination''. Theme I1, the core of this programme, will benefit mathematical physics, primarily through dissemination of high-quality new research with an enduring influence. Theme I2 will impact wider astronomy communities, primarily through inter-disciplinary opportunities and the shaping of future capabilities. Theme I3 will seek to engage and involve anyone who is interested in the Universe and our place within it.

## People |
## ORCID iD |

Sam Richard Dolan (Principal Investigator) |

Shipley J
(2016)

*Binary black hole shadows, chaotic scattering and the Cantor set*in Classical and Quantum Gravity
Dolan S
(2015)

*Bound states of the Dirac equation on Kerr spacetime*in Classical and Quantum Gravity
Akcay S
(2017)

*Spin-orbit precession for eccentric black hole binaries at first order in the mass ratio*in Classical and Quantum Gravity
Dempsey D
(2016)

*Waves and null congruences in a draining bathtub*in International Journal of Modern Physics D
Leite L
(2016)

*Absorption of massless scalar field by rotating black holes*in International Journal of Modern Physics D
Crispino L
(2015)

*Scattering from charged black holes and supergravity*in Physical Review D
Ponglertsakul S
(2016)

*Stability of gravitating charged-scalar solitons in a cavity*in Physical Review D
Dolan S
(2017)

*Rainbow scattering in the gravitational field of a compact object*in Physical Review D
Nolan P
(2015)

*Octupolar invariants for compact binaries on quasicircular orbits*in Physical Review D
Dolan S
(2015)

*Stability of black holes in Einstein-charged scalar field theory in a cavity*in Physical Review D
Benone C
(2017)

*Addendum to "Absorption of a massive scalar field by a charged black hole"*in Physical Review D
Dolan S
(2016)

*Stable photon orbits in stationary axisymmetric electrovacuum spacetimes*in Physical Review D
Dolan S
(2015)

*Tidal invariants for compact binaries on quasicircular orbits*in Physical Review D