Linear Response, Hamiltonian and Radiative Spinning Two-Body Dynamics

Abstract

Using the spinning, supersymmetric Worldline Quantum Field Theory formalism we compute the momentum impulse and spin kick from a scattering of two spinning black holes or neutron stars up to quadratic order in spin at third post-Minkowskian (PM) order, including radiation-reaction effects and with arbitrarily mis-aligned spin directions. Parts of these observables, both conservative and radiative, are also inferred from lower-PM scattering data by extending Bini and Damour's linear response formula. By solving Hamilton's equations of motion we also use a conservative scattering angle to infer a complete 3PM two-body Hamiltonian including finite-size corrections and misaligned spin-spin interactions. Finally, we describe mappings to the bound two-body dynamics for aligned spin vectors: including a numerical plot of the binding energy for circular orbits compared with numerical relativity, analytic confirmation of the NNLO PN binding energy and the energy loss over successive orbits.

Angular momentum balance in gravitational two-body scattering: Flux, memory, and supertranslation invariance

“…Explicit calculations [5,6,11,14,15] also confirm that the total flux of four-momentum radiated across future null infinity is precisely equal to the total change in the four-momenta of the two bodies. This kind of balance law, in which changes in the mechanical properties of the binary are linked to the flux of outgoing radiation, provides an important consistency check and, in the case of bound orbits, plays a key role in the construction of waveform models [25][26][27][28].…”

“…When it comes to practical calculations, the problem is generally rendered tractable by way of the post-Minkowskian expansion, which-when used alongside a number of powerful techniques adopted from high-energy physics-allows us to solve for each quantity of interest perturbatively in powers of Newton's constant 𝐺 [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. One finds in doing so that gravity manifests as a purely conservative force at first order in the approximation, and that the emission of gravitational waves appears only once we go to higher orders in 𝐺.…”

“…This simple power-counting argument is corroborated by explicit calculations of the four-momentum flux, which were first carried out for the case of two point masses in Refs. [4][5][6][7][8], before being generalized to include tidal interactions [10][11][12] and spin effects [13,14].…”

“…The first is a linear-response relation between the conservative and radiation-reaction parts of the scattering angle [3,36] (see also Ref. [14,37]), which produces the correct result at 𝑂(𝐺 3 ) only if the angular momentum flux starts at 𝑂(𝐺 2 ). The second is a set of explicit solutions to the binary's equations of motion [38,39], which assert that the binary always loses mechanical angular momentum starting at 𝑂(𝐺 2 ).…”

Gravitational Quadratic-in-Spin Hamiltonian at NNNLO in the post-Newtonian framework

“…Therefore, high-precision waveforms incorporating spin contributions are essential to exploit the full potential in GW astronomy. Initial studies extending the classical techniques to include spin were done in [11,12], which was later extended by [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The spin effects in the post-Newtonian formalism were developed using the effective field theory approach in [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”