[Paper Review] $S$-matrix approach to general gravity and beyond

Pierre Vanhove|arXiv (Cornell University)|Apr 20, 2021

Pulsars and Gravitational Waves Research57 references4 citations

TL;DR

This paper develops an S-matrix approach to extract classical gravity from quantum scattering amplitudes, demonstrating that the Schwarzschild-Tangherlini black hole metric in arbitrary dimensions can be derived from one-loop graviton exchange amplitudes. By taking the classical limit (ħ → 0) of these amplitudes, the method reproduces the full metric, including logarithmic corrections from UV-finite quantum effects, validating the link between quantum scattering and classical general relativity in higher dimensions.

ABSTRACT

In this text we outline the motivation for developping a quantum $S$-matrix approach for the classical gravitational two-body scattering. As an application we briefly present the derivation of black-hole metrics in various dimensions.

Motivation & Objective

• To establish a quantum S-matrix framework for extracting classical gravitational observables from scattering amplitudes.
• To test the validity of the S-matrix approach in deriving classical black hole metrics beyond four dimensions.
• To understand how quantum corrections, such as logarithmic terms, emerge in the classical metric through UV-finite amplitude regularization.
• To extend the method to include spin and charge for deriving Kerr-Newman and Reissner-Nordström metrics.
• To provide a systematic, dimensionally general method for computing classical gravity observables from quantum field theory amplitudes.

Proposed method

• The S-matrix approach computes quantum scattering amplitudes for two massive bodies exchanging gravitons in general spacetime dimensions.
• The classical limit (ħ → 0) with q²/ħ fixed extracts the post-Minkowskian potential, encoding the conservative dynamics of binary systems.
• Amplitudes are regulated using higher-derivative non-minimal couplings to cancel ultraviolet divergences, generating finite-size logarithmic corrections.
• The resulting metric components are matched to the Schwarzschild-Tangherlini form via coordinate redefinitions, absorbing logarithmic terms into the radial function.
• The method is applied to d = 4, 5, 6 dimensions, reproducing known black hole metrics up to O(Gₙ⁴) order.
• The approach generalizes to include spin and charge by computing vertex functions for spinning and charged particles.

Experimental results

Research questions

• RQ1 Can the classical Schwarzschild-Tangherlini metric in arbitrary dimensions be derived from quantum scattering amplitudes?
• RQ2 How do quantum corrections, such as logarithmic terms, manifest in the classical metric despite being finite-size effects?
• RQ3 What is the role of UV regularization via higher-derivative couplings in generating finite-size corrections to the metric?
• RQ4 Can the S-matrix approach be extended to derive rotating and charged black hole metrics like Kerr-Newman and Reissner-Nordström?
• RQ5 How does the S-matrix method connect post-Minkowskian dynamics with classical gravity in higher dimensions?

Key findings

• The S-matrix approach successfully reproduces the Schwarzschild-Tangherlini metric in d = 4, 5, 6 dimensions from one-loop graviton exchange amplitudes.
• Logarithmic corrections in the metric components, such as log(rC₃ / Gₙm), arise from UV-finite cancellations in regulated amplitudes and are reabsorbed via coordinate redefinitions.
• The finite-size effects do not alter the physical form of the static metric, confirming the robustness of the classical solution.
• The method confirms the classical limit of quantum amplitudes yields the correct conservative potential and scattering angle in post-Minkowskian theory.
• The approach is generalizable to higher-loop amplitudes and can be extended to include spin and charge for deriving Kerr-Newman and Reissner-Nordström metrics.
• The derivation validates the use of scattering amplitudes as a tool to extract classical gravity in arbitrary dimensions, including quantum corrections.