From Euclidean to Lorentzian General Relativity: The Real Way
J. Fernando Barbero G.
TL;DR
The paper addresses how to relate Euclidean and Lorentzian solutions of the Einstein equations by introducing a real Wick-rotation-like mechanism via a two-parameter action with $\alpha$ and $\beta$. The approach produces a family of vacuum GR solutions whose signature can be Euclidean or Lorentzian, with the physical spacetime metric recovered from an auxiliary Euclidean metric through a specific reconstruction formula, and the Schwarzschild solution used to demonstrate the method. A key result is that, for generic $(\alpha,\beta)$, the field equations yield GR solutions up to a signature-determining mapping to $g^{\text{Eins}}_{ab}$ or its equivalent, while certain parameter choices lead to degenerate or purely Euclidean theories. The framework enables exploration of signature change, the problem of time, and potential quantum-gravity applications, though it leaves open questions about analyticity in $(\alpha,\beta)$ and the full scope of admissible spacetimes; the Schwarzschild example illustrates a concrete bridge between Euclidean and Lorentzian pictures with a mass rescaling factor $\bar M=2^{-1/2}|\alpha(\alpha+2\beta)|^{1/4}M$.
Abstract
We study in this paper a new approach to the problem of relating solutions to the Einstein field equations with Riemannian and Lorentzian signatures. The procedure can be thought of as a "real Wick rotation". We give a modified action for general relativity, depending on two real parameters, that can be used to control the signature of the solutions to the field equations. We show how this procedure works for the Schwarzschild metric and discuss some possible applications of the formalism in the context of signature change, the problem of time, black hole thermodynamics...