Classical gravitational spinning-spinless scattering at $\mathcal{O}(G^{2} S^{\infty})$

Abstract

Making use of the recently-derived, all-spin, opposite-helicity Compton amplitude, we calculate the classical gravitational scattering amplitude for one spinning and one spinless object at O(G 2 ) and all orders in spin. By construction, this amplitude exhibits the spin structure that has been conjectured to describe Kerr black holes. This spin structure alone is not enough to fix all deformations of the Compton amplitude by contact terms, but when combined with considerations of the ultrarelativistic limit we can uniquely assign values to the parameters remaining in the even-in-spin sector. Once these parameters are determined, much of the spin dependence of the amplitude resums into hypergeometric functions. Finally, we derive the eikonal phase for aligned-spin scattering. I. INTRODUCTIONRecent years have seen a large mobilization within the scattering amplitudes community towards describing the gravitational coalescence of compact objects. This stems from the necessity for ever-more precise gravitational wave templates in current and upcoming gravitational wave observatories [1-7], and because scattering amplitudes are eminently suited to calculating classical observables in the post-Minkowskian (PM) expansion [8][9][10][11][12][13][14][15][16]. This huge effort has led to unprecedented precision in the PM description of spinless scattering [17][18][19][20][21][22][23][24][25][26][27][28][29][30], tidal effects [31][32][33][34][35][36][37], and radiation [23][24][25][38][39][40][41]].Yet another pertinent property affecting the motion of the constituents of a binary is their individual rotational angular momenta. The connection between classical rotational angular momentum and quantum spin appearing in scattering amplitudes is by now well understood [11,[42][43][44]. Classical scattering at 1PM is known to all orders in the spin vectors for Kerr black holes [45][46][47][48][49] and general spinning bodies [44]. Dynamics at 2PM have been understood up to quartic order in spin [37,43,46,[50][51][52][53][54][55][56][57]. Until recently, progress past quartic order at 2PM has been restricted owing partly to the absence of a physical opposite-helicity Compton amplitude above this spin order [58]. Several approaches have been taken to remedy these unphysicalities [51,[59][60][61][62]. Results including spin at 3PM have also begun to emerge [63,64]. Recently, refs. [62,65] have pushed the state-of-theart in the scattering of spinning objects at 2PM past the fourth order in spin. In the former work, we applied the

NLO deflections for spinning particles and Kerr black holes

“…In a historical perspective, perhaps one way to deepen our understanding would be to investigate subleading soft theorems for external particles with higher spin. This is certainly an issue that deserves further studies, and indeed different and fascinating works are gaining grounds on this matter [53,54,93,94,[127][128][129]]. An easier task would be to build up on the results of section 5 to incorporate higher spin orders in the light body.…”

Searching for Kerr in the 2PM amplitude

“…We have left this for the follow-up work in ref. [88], in which we compute the aligned-spin eikonal phase based on the more comprehensive (i.e. all-order-in-spin) amplitude presented there.…”

Quantum corrections to frame-dragging in scattering amplitudes

“…While it is known that certain classical spin effects can be captured by quantum spins [16,[35][36][37][38], the description fails to capture terms that are affected by identities dependent on spin representations of the spinning particle, such as trace identities. Therefore, higher-spin (s > 2) particle descriptions are required to capture the full dynamics of classical spinning particles [39][40][41][42][43], which has met with great success in determining the classical two-body dynamics in GR [39,40,[43][44][45][46][47][48][49][50][51]. In this work the semiclassical Compton amplitude [48], which is defined as the Compton amplitude in the classical spin limit (s → ∞ with S = s fixed), is used.…”

“…In this work the semiclassical Compton amplitude [48], which is defined as the Compton amplitude in the classical spin limit (s → ∞ with S = s fixed), is used. Note that amplitude computations at tree level [39-41, 44, 45, 52] and one-loop level [49,51,53,54] with spins in the classical spin limit has been checked against corresponding classical computations, and found to reproduce classical spin dependence to all orders in spin, which can be viewed as an evidence for the validity of the approach.…”