G W Gibbons 1999 Class. Quantum Grav. 16 L71 doi:10.1088/0264-9381/16/10/103
G W Gibbons
Show affiliationsIn 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.
83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries
Issue 10 (October 1999)
Received 26 May 1999
G W Gibbons 1999 Class. Quantum Grav. 16 L71
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