Einstein Spacetimes with Constant Weyl Eigenvalues

TL;DR

This paper classifies Einstein vacuum spacetimes with constant Weyl eigenvalues, showing they are either homogeneous or Kundt spacetimes, and provides new insights into their geometric properties and specific solutions.

Contribution

It proves that Einstein spacetimes with constant Weyl eigenvalues are limited to homogeneous or Kundt spacetimes, extending previous results and analyzing Petrov types I, II, and D.

Findings

Type II & D spacetimes have a fixed non-zero eigenvalue of 2Λ/3.

The only Petrov type I solution is a known homogeneous vacuum.

New algebraic relations between Weyl tensor components and optical scalars are derived.

Abstract

Einstein spacetimes (that is vacuum spacetimes possibly with a non-zero cosmological constant {\Lambda}) with constant non-zero Weyl eigenvalus are considered. For type Petrov II & D this assumption allows one to prove that the non-repeated eigenvalue necessarily has the value 2{\Lambda}/3 and it turns out that the only possible spacetimes are some Kundt-waves considered by Lewandowski which are type II and a Robinson-Bertotti solution of type D. For Petrov type I the only solution turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are Kundt space- times. This result is similar to that of Coley et al. who proved their result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant. Some preliminary results are also presented for Petrov Type I vacua in which either only one of the Weyl eigenvalues is constant or in which the ratios of the Weyl eigenvalues are constants. In particular in each case there is a simple algebraic relation between the Newman-Penrose Weyl tensor components $\Psi_2$ & $\Psi_0 (=\Psi_4)$ and the 'cross-ratio' of the optical scalars {\kappa}{\nu}-{\sigma}{\lambda} of the associated principal null tetrad of the Weyl tensor.