How to Wick rotate generic curved spacetime

Abstract

It is an article of folklore that the collection of ideas identified as Euclidean quantum gravity may be derived from ordinary Lorentzian signature gravity by the procedure of Wick rotation. This note will attempt to shed some light on this relatively ill-understood procedure. I argue that it proves inappropriate and unhelpful to regard Wick rotation in terms of a complex deformation of the time coordinate. Rather, Wick rotation can more usefully be viewed as a complex deformation of the spacetime metric. This simple reformulation of the Wick rotation procedure, while it leaves flat space physics unaffected, has profound implications for quantum gravity.

arXiv commentV1: 11 pages. Old pre-arXiv essay originally written for the 1991 Gravity Research Foundation essay contest. For many years a PDF copy of the essay was available at the GRF website, but many of the older essays have now been removed from that website. Apart from purely cosmetic issues, this arXiv upload is an accurate reflection of that 1991 essay. V2: Extra historical comments; 6 references added

References in corpus (8)

• Quantum Gravity, or The Art of Building Spacetime
• Dynamical Origin of the Lorentzian Signature of Spacetime
• On the emergence of Lorentzian signature and scalar gravity
• Signature change events: A challenge for quantum gravity?
• Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity
• Wick Rotation in the Tangent Space
• Complex lapse, complex action and path integrals
• Black holes: interfacing the classical and the quantum

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• Action and Observer dependence in Euclidean quantum gravity
• Parametrized Path Approach to Vacuum Decay
• Vacuum decay and bubble nucleation in $f(R)$ gravity
• Wick rotations and real GIT
• Scattering in pseudo-Hermitian quantum field theory and causality violation
• Which number system is "best" for describing empirical reality?
• Lorentzian quantum cosmology in novel Gauss-Bonnet gravity from Picard-Lefschetz methods
• Equation of states in the curved spacetime of slowly rotating degenerate stars
• Directed Graph Embeddings in Pseudo-Riemannian Manifolds
• Lorentzian Einstein-Ricci Flows
• Graviton spectrum in simplified Dark Matter models with graviton mediators in the de Sitter space
• A Path Integral for the Chiral-Form Partition Function
• Lorentzian quantum cosmology with $R^2$ correction
• Quantum fields without Wick rotation