The 2PM Hamiltonian for binary Kerr to quartic in spin

Abstract

From the S-matrix of spinning particles, we extract the 2 PM conservative potential for binary spinning black holes up to quartic order in spin operators. An important ingredient is the exponentiated gravitational Compton amplitude in the classical spin-limit for all graviton helicity sectors. The validity of the resulting Hamiltonian is verified by matching to known lower spin order results, as well as direct computation of the 2PM impulse and spin kicks from the eikonal phase and that from the test black hole scattering based on Mathisson-Papapetrou-Dixon equations. Contents 1 Introduction 2 The classical spinning Compton 2.1 The exponentiated representation of matter-graviton coupling 2.2 BCFW construction of Compton amplitude (opposite helicity) 2.3 BCFW construction of Compton amplitude (same helicity) 3 Integral coefficients and the classical potential 3.1 Integral Coefficients 3.1.1 The triangle coefficeints 3.1.2 The box and crossed box coefficients 3.2 Iteration terms 3.2.1 The 3D bubble coefficient 3.2.2 The 3D triangle coefficient: Cancellation of IR divergence 3.3 Conservative spin Hamiltonian 3.4 Matching with other results up to O(S 2 ) 4 Observables 4.1 Observales in the COM frame 4.1.1 Observables from the Hamiltonian EOM 4.1.2 Observables from the eikonal phase 4.2 The background-probe limit 5 Conclusions and outlook A Spin factor composition A.1 Opposite helicity case A.1.1 Helicity configuration 2 + 3 − A.1.2 Helicity configuration 2 − 3 + A.2 Equal helicity case A.2.1 Helicity configuration 2 + 3 + A.2.2 Helicity configuration 2 − 3 − B Classical spinning Compton amplitude for generic Wilson coefficients -i -B.1 The residue integral representation and the spin representation of spinning matter B.2 BCFW construction of Compton amplitude (opposite helicity) B.3 BCFW construction of Compton amplitude (same helicity) B.4 Partial results for integral coefficients in the classical limit B.4.1 Box integral coefficients B.4.2 Triangle integral coefficients C Solving MPD equations iteratively to linear order in test-body spin C.1 Spin vector description of MPD equations C.2 The master perturbation equations C.3 Iterative solution on Kerr-Schild coordinates C.4 Example: equatorial motion with orthogonal initial spin1. All physical channel residues including the t-channel for graviton coupling are correctly captured by this expression to all orders in J.

The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes

“…The impulse is closely related to the classical interaction potential [67-80, 94-96] and the eikonal phase which provide alternative routes linking classical physics and scattering amplitudes which are discussed in more detail in other chapters of this review [91,92]. The important physical effects of spin can easily be incorporated in the impulse and related observables using scattering amplitudes [13] building on a deepening understanding of amplitudes for massive spinning particles [97][98][99][100][101][102][103][104][105][106][107][108][109][110][111][112].…”

“…Type N solutions are purely radiative, and the simplest example is that of pp-waves. All type D or N spacetimes admit the form (110). For type D spacetimes (and also pp-waves), this is guaranteed by the existence of a Killing 2-spinor [139], from which the various objects can be built.…”

“…We will return to this point. More generally, this indicates that we cannot hope for a double copy description of generic dynamical solutions, such as a spacetime representing the scattering of black holes, to be as simple as equation (110).…”

“…Let us discuss first the known instances of an exact double copy. We have already mentioned the Weyl double copy (110). It was introduced in ref.…”

The SAGEX Review on Scattering Amplitudes, Chapter 13: Post-Minkowskian expansion from Scattering Amplitudes

“…The corresponding case of spinning black holes is evidently of great phenomenological importance but the corresponding description in terms of the amplitude approach to general relativity is far more complex compared to the non-spinning case. We shall not be able to cover that fascinating story of spin but refer to some relevant papers here [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. There is also a parallel development to the post-Minkowskian expansion based on the world-line approach [41][42][43][44][45][46][47] that we cannot cover here, but we will make some brief comments on the relation with the velocity cut formalism in section 5.2.…”