Abstract
We present a general framework with which the Schwarzschild-Tangherlini metric of a point particle in arbitrary dimensions can be derived from a scattering amplitude to all orders in the gravitational constant, ${G}_{N}$, in covariant gauge (i.e. ${R}_{\ensuremath{\xi}}$ gauge) with a generalized de Donder--type gauge function, ${G}_{\ensuremath{\sigma}}$. The metric is independent of the covariant gauge parameter $\ensuremath{\xi}$ and obeys the classical gauge condition ${G}_{\ensuremath{\sigma}}=0$. We compute the metric with the generalized gauge choice explicitly to second order in ${G}_{N}$ where gravitational self-interactions become important and these results verify the general framework to one-loop order. Interestingly, after generalizing to arbitrary dimension, a logarithmic dependence on the radial coordinate appears in space-time dimension $D=5$.
Highlights
The classical limit of effective quantum gravity is a successful description of general relativity
Quantum field theoretic methods are used to derive results in classical general relativity [1,2,3,4,5,6,7,8,9]. In this approach gravitational interactions are mediated by spin-2 gravitons and general relativity is recast in the language of quantum field theory [10]
In space-time dimension D 1⁄4 5 we find the curious appearance of a logarithmic dependence on the radial variable at second order in GN
Summary
The classical limit of effective quantum gravity is a successful description of general relativity. We analyze the quantum field theoretic expansion of the Schwarzschild-Tangherlini metric from a series of Feynman diagrams with an ever-increasing number of loops. Such an all-order expansion was suggested in [2] where it was shown how the loop integrals can be reduced in the classical limit. We explain how the arbitrary scale introduced corresponds to a coordinate transformation which is allowed because of redundant gauge freedom From this explanation it is expected that the appearance of logarithmic dependence is limited to D 1⁄4 5 at second and higher orders in GN and D 1⁄4 4 at third and higher orders in GN. In Appendix C we go through the alternative derivation of the expansion of the Schwarzschild-Tangherlini metric
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