Two loop and all loop finite 4-metrics

TL;DR

This paper investigates the quantum corrections of certain Lorentzian and Riemannian 4-metrics, showing that some classes of Lorentzian metrics have no quantum corrections at all loop orders, unlike Riemannian metrics.

Contribution

It demonstrates that specific Petrov type III or N Lorentzian 4-metrics and certain Kundt class spacetimes are free of quantum corrections at all loop orders, contrasting with Riemannian metrics.

Findings

Ricci flat Lorentzian 4-metrics of Petrov types III or N have no two-loop counter terms.

All invariants vanish for certain Kundt class spacetimes, leading to no quantum corrections.

Complete non-singular Riemannian metrics only have vanishing two-loop counter terms if flat.

Abstract

In pure Einstein theory, Ricci flat Lorentzian 4-metrics of Petrov types III or N have vanishing counter terms up to and including two loops. Moreover for pp-waves and type-N spacetimes of Kundt's class which admit a non-twisting, non expanding, null congruence all possible invariants formed from the Weyl tensor and its covariant derivatives vanish. Thus these Lorentzian metrics suffer no quantum corrections to all loop orders. By contrast for complete non-singular Riemannian metrics the two loop counter term vanishes only if the metric is flat.