A two-dimensional representation of four-dimensional gravitational waves

TL;DR

This paper presents a 2D representation of 4D gravitational waves using Einstein equations with symmetries, providing new insights into Birkhoff's theorem and the nature of gravitational waves in reduced dimensions.

Contribution

It introduces a novel 2D metric-dilaton gravity framework for 4D gravitational waves with symmetries, offering new proofs and explanations for their properties.

Findings

New 2D representation of 4D gravitational waves

Proof of Birkhoff theorem for plane-symmetric spacetimes

Scalar fields encode gravitational wave information

Abstract

The Einstein equation in D dimensions, if restricted to the class of space-times possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in 2 dimensions with n scalar fields. For n = 2, this results reduces to the known reduction of certain 4-dimensional metrics which include gravitational waves. Here, we give such a representation which leads to a new proof of the Birkhoff theorem for plane-symmetric space--times, and which leads to an explanation, in which sense two (spin zero-) scalar fields in 2 dimensions may incorporate the (spin two-) gravitational waves in 4 dimensions. (This result should not be mixed up with well--known analogous statements where, however, the 4-dimensional space-time is supposed to be spherically symmetric, and then, of course, the equivalent 2-dimensional picture cannot mimic any gravitational waves.) Finally, remarks on hidden symmetries in 2 dimensions are made.