Complete classification of second-order symmetric spacetimes

TL;DR

This paper provides a complete classification of four-dimensional second-order symmetric spacetimes in Lorentzian geometry, focusing on Brinkmann metrics and including known plane wave solutions.

Contribution

It offers the first comprehensive classification of second-order symmetric Lorentzian spacetimes, extending the understanding beyond locally symmetric cases.

Findings

Classification of all 4D second-order symmetric spacetimes.

Identification of Brinkmann metrics as the relevant class.

Inclusion of known plane wave solutions.

Abstract

As a difference with the positive-definite Riemannian case, in the Lorentzian case there exists proper second-order symmetric spacetimes, i.e., those with vanishing second covariant derivative of the Riemannian tensor ($R_{\lambda\mu\nu\rho;\alpha;\beta}=0$) which are not locally symmetric ($R_{\lambda\mu\nu\rho;\alpha}\neq 0$). In fact, they lie in the class of Brinkmann metrics, and include some known plane waves. Our purpose is to classify these spacetimes in the 4 dimensional case.