A Wick-rotatable metric is purely electric

TL;DR

This paper proves that any metric permitting a standard Wick-rotation to a Riemannian metric must have a purely electric Riemann and Weyl tensor, linking Wick-rotation properties to tensor classifications.

Contribution

It establishes a necessary condition for Wick-rotatable metrics, showing they must have purely electric curvature tensors, a novel connection between Wick-rotation and tensor properties.

Findings

Wick-rotatable metrics have purely electric Riemann tensors.

Wick-rotatable metrics have purely electric Weyl tensors.

The result applies to metrics of arbitrary dimension and signature.

Abstract

We show that a metric of arbitrary dimension and signature which allows for a standard Wick-rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor.