Abstract
The Effective One-Body formalism of the gravitational two-body problem in general relativity is reconsidered in the light of recent scattering amplitude calculations. Based on the kinematic relationship between momenta and the effective potential, we consider an energy-dependent effective metric describing the scattering in terms of an Effective One-Body problem for the reduced mass. The identification of the effective metric simplifies considerably in isotropic coordinates when combined with a redefined angular momentum map. While the effective energy-dependent metric as expected is not unique, solutions can be chosen perturbatively in the Post-Minkowskian expansion without the need to introduce non-metric corrections. By a canonical transformation, our condition maps to the one based on the standard angular momentum map. Expanding our metric around the Schwarzschild solution we recover the solution based on additional non-metric contributions.
Highlights
Since the aim of this paper is to explore some of the consequences of calculating classical general relativity observables with modern scattering-amplitude methods, we begin this section with an elementary introduction to the effective one-body (EOB) formalism, phrased in a manner that may be more accessible to particle physicists
If our objective is to identify a class of metrics that reproduce the scattering angle of the actual two-body problem using an EOB formalism, there is nothing to prevent us from pursuing this approach
Ri≥1 which shows how to express the qi-coefficients in terms of the hi-coefficients of this paper. To summarize this part: We have shown the equivalence between our remodeled EOB formalism in isotropic coordinates and the conventionally used formalism that separates out all nonquadratic energy-momentum terms in a function Q which is added to the mass-shell condition
Summary
Recent advances in the scattering amplitude-based approach to the post-Minkowskian expansion of classical general relativity [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] have demonstrated that this new approach holds the promise of significantly changing the efficiency of computations in general relativity. Full third-order post-Minkowskian amplitude calculations of the scattering of two massive objects are available [7,14,15,16,20,29], and the first results for fourth post-Minkowskian order have already appeared [17] This amplitude-based approach generically computes one observable; the scattering angle in what we can call the hyperbolic regime of the two-body problem in gravity. Choosing an angular momentum map that differs from the one conventionally used [30,31] connects most straightforwardly to the scattering amplitude-based approach to general relativity, and we end up describing the reduced problem in terms of a massive object in an effective metric that only reduces to the Schwarzschild metric in the probe. Expanding our metric around the Schwarzschild metric can rephrase the solution in terms of the combination of a Schwarzschild metric plus additional nonmetric terms, finding complete agreement with the solution given in that form by Damour in Ref. [3]
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