The 24th Capra meeting on Radiation Reaction in General Relativity

We revisit the scattering of massless waves of helicity $|h|=0,\frac{1}{2},1,2$ in Schwarzschild and Kerr backgrounds, in the long-wavelength regime. The Newman-Penrose scattering amplitudes arising from the Black Hole Perturbation Theory (BHPT) framework are found in agreement with the classical limit of QFT amplitudes at finite values of the scattering angle and arbitrary spin orientation. The latter amplitudes are obtained from on-shell methods and describe the $2\to 2$ scattering of a massless particle of helicity $|h|$ with a massive particle of arbitrary spin $S$, where $S=0$ corresponds to the Schwarzschild case. The effect of the black hole spin in the polarization of the waves is found in agreement with previous analysis. Finally, unitarity constraints based on partial amplitudes and positive time delay are also discussed.

Future gravitational wave detectors will map out and characterize every binary merger in the history of the universe. The possibilities for new and unexpected scientific discoveries from this wealth of data is staggering, but hinges crucially on complementary advances in our theoretical understanding of the nature of gravitational wave sources. However, the path from Einstein’s equation to precision binary dynamics is notoriously difficult, and conventional methods do not scale to the demands of future detectors. I will describe our recent efforts in solving the relativistic two body problem using tools from quantum field theory.

We revisit the old problem of the self-force on a particle moving in a weak-field spacetime in the context of renewed interest in gravitational two-body scattering. We calculate the scalar, electromagnetic, and gravitational self-force on a particle moving on a straight-line trajectory in the spacetime of a Newtonian star and use these results to find the associated correction to the scattering angle in each case. In the gravitational case we must also take into account the motion of the star via a ``matter-mediated'' force on the particle, which acts at the same perturbative order as the gravitational self-force.

Hyperbolic-type scattering orbits are excellent probes of the strong-field regime around black holes, and their analysis can inform the construction of an accurate two-body Hamiltonian. In particular, it has been shown that knowledge of the scattering angle through linear order in the mass ratio completely determines the 4PM Hamiltonian. With this motivation in mind, we describe a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. The method is based on a time-domain metric reconstruction from a Hertz potential in a radiation gauge. The crucial ingredient in this formulation are certain jump conditions that (each multipole mode of) the Hertz potential must satisfy along the orbit, in a 1+1-dimensional multipole reduction of the problem. We show a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime, and present a full numerical implementation for a scatter orbit.

Hyperbolic scattering orbits, able to penetrate deep into the sub-ISCO region even at relatively low energies, provide an excellent probe of the strong-field regime outside black holes. Self-force calculations of the scatter angle can greatly advance the development of post-Minkowskian theory and of the EOB model of binary dynamics. We develop a frequency-domain method for calculating the 1st order scalar self-force acting on a charge moving along a hyperbolic Schwarzschild geodesic, outlining the formulation of the problem, the challenges faced and our attempted solutions. Particular attention will be paid to issues faced by the usual method of extended homogeneous solutions (EHS) used to circumvent the Gibbs phenomenon.

We consider the motion of a spinning neutron star with astrophysically relevant speed in the gravity field of a massive black hole. The orbital dynamic is described by the MPD equations which include up to quadrupole interaction. We compare the orbits of the neutron star under geodesic motion and under MPD equations and show that the difference in the orbital motion can translate into a variation of pulse-arrival-time. Such a difference is within the observational limit of the radio telescopes like SKA and FAST.

"We calculate the first order metric perturbation to a Schwarzschild background spacetime induced by a spinning secondary body in the Regge-Wheeler and Zerilli gauges. In particular we specialise to a secondary with spin (anti-)aligned to the total orbital angular momentum in a quasi-circular orbit. From the metric perturbation we can calculate gauge invariant self-force quantities such as Detweiler’s redshift invariant and compare with known PN results. In doing so we present the first strong field calculation of a conservative self-force quantity with a spinning secondary and emphasise: 1) The treatment of the additional spin term to the singular field by deriving additional tensor harmonic regularisation parameters. 2) Parametrising at fixed frequency and practically extracting the linear in spin contribution to the metric perturbation."

I will revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles around a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, I obtain three non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, thereby completing the two quasi-constants of motion found by Rüdiger with one new independent quasi-constant of motion. Finally, I will discuss the implications for the motion of spinning particles in the Kerr geometry.

We use the frequency and time domain Teukolsky formalism to calculate gravitational wave fluxes from a spinning body on a bound eccentric equatorial orbit around a Kerr black hole. The spinning body is represented as a point particle following the pole-dipole approximation of the Mathisson-Papapetrou-Dixon equations. Reformulating these equations we are not only able to find the trajectory of a spinning particle in terms of its constants of motion, but also to provide a method to calculate the azimuthal and the radial frequency of this trajectory. Using these orbital quantities, we introduce the machinery to calculate through the frequency domain Teukolsky formalism the energy and the angular momentum fluxes at infinity, and at the horizon, along with the gravitational strain at infinity. We crosscheck the results obtained from the frequency domain approach with the results obtained from a time domain Teukolsky equation solver called Teukode.

The Mathisson-Papapetrou-Dixon (MPD) equations describe the motion of an extended test body in general relativity. This system of equations, though, is underdetermined and has to be accompanied by constraining supplementary conditions, even in its simplest version, which is the pole-dipole approximation corresponding to a spinning test body. In particular, imposing a spin supplementary condition (SSC) fixes the center of the mass of the spinning body, i.e. the centroid of the body. In the present study, we examine whether characteristic features of the centroid of a spinning test body, moving in a circular equatorial orbit around a massive black hole, are preserved under the transition to another centroid of the same physical body, governed by a different SSC. For this purpose, we establish an analytical algorithm for deriving the orbital frequency of a spinning body, moving in the background of an arbitrary, stationary, axisymmetric spacetime with reflection symmetry, for the Tulczyjew-Dixon, the Mathisson-Pirani and the Ohashi-Kyrian-Semerak SSCs. Then, we focus on the Schwarzschild black hole background and a power series expansion method is developed, in order to investigate the discrepancies in the orbital frequencies expanded in power series of the spin among the different SSCs. Lastly, by employing the fact that the position of the centroid and the measure of the spin alters under the centroid's transition, we impose proper corrections to the power expansion of the orbital frequencies, which allows to improve the convergence between the SSCs. Our concluding argument is that when we shift from one circular equatorial orbit to another in the Schwarzschild background, under the change of a SSC, the convergence between the SSCs holds only up to certain powers in the spin expansion, and it cannot be achieved for the whole power series.