A two-dimensional representation of four-dimensional gravitational waves

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.

12 pages, LaTeX, no figures, Int. J. Mod. Phys. D in print

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• Probing Quantum Gravity Through Exactly Soluble Midi-Superspaces I
• On the Completeness of the Black Hole Singularity in 2d Dilaton Theories
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• Graded Poisson-Sigma Models and Dilaton-Deformed 2D Supergravity Algebra
• The classical solutions of two-dimensional gravity
• The Complete Solution of 2D Superfield Supergravity from graded Poisson-Sigma Models and the Super Pointparticle
• Quantization of 2D dilaton supergravity with matter
• Congruence Convergence in pp-wave Spacetime