Abstract
We derive the static Schwarzschild-Tangherlini metric by extracting the classical contributions from the multiloop vertex functions of a graviton emitted from a massive scalar field. At each loop order the classical contribution is proportional to a unique master integral given by the massless sunset integral. By computing the scattering amplitudes up to three-loop order in general dimension, we explicitly derive the expansion of the metric up to the fourth post-Minkowskian order $O({G}_{N}^{4})$ in four, five and six dimensions. There are ultraviolet divergences that are cancelled with the introduction of higher-derivative nonminimal couplings. The standard Schwarzschild-Tangherlini is recovered by absorbing their effects by an appropriate coordinate transformation induced from the de Donder gauge condition.
Highlights
General relativity is a theory for the action of gravity in space and time
We show that in five dimensions one needs to consider higher dimensional of nonminimal couplings δð2ÞSct at the third post-Minkowskian order and δð3ÞSct at the fourth postMinkowskian
General relativity is coordinate system invariant, our analysis shows that there is a preferred coordinate system when extracting the classical geometry from scattering amplitudes in the de Donder gauge
Summary
General relativity is a theory for the action of gravity in space and time. The dynamics of the gravitational field is constrained by the Einstein’s classical field equations. The relation between the quantum theory of gravity and the classical Einstein’s theory of general relativity has received a new interpretation with the understanding [8,9,10,11,12,13] that an appropriate (and subtle) ħ → 0 limit of quantum multiloop scattering gravitational amplitudes lead to higher GN-order classical gravity contributions. Considering the importance of such approach for the evaluation of the post-Minkowskian expansion for the gravitational two-body scattering [14,15,16,17,18,19,20], we use the procedure given in [12] for extracting the classical contributions from the multiloop vertex function of a graviton emission from a massive scalar field to recover the Schwarzschild-Tangherlini metric in various dimensions. The Appendix A contains formulas for the Fourier transforms used in the text, and Appendix B the vertices for the scattering amplitude computations
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