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EINSTEIN’S EQUATIONS IN THE PRESENCE OF SIGNATURE CHANGE

Abstract

INTRODUCTION Classical cosmological models containing an initial region of Euclidean signature joined to a final region with the usual Lorentzian signature were introduced by Ellis et al.1,2 A basic feature of this work is the use of the Darmois junction conditions at the surface where the signature changes. This assumption has been questioned by Hayward,3 who prefers to assume the stronger conditions appropriate for quantum cosmology. We argue here in favor of the Darmois approach by deriving these junction conditions from the Einstein-Hilbert action. What are Einstein’s equations in the presence of signature change? Formal computation quickly goes astray: A signature-changing metric is necessarily degenerate at the hypersurface of signature change. The Geroch-Traschen conditions4 for the existence of a distributional curvature tensor thus fail to be satisfied, and it is not clear whether a preferred connection exists. Supposing that a suitable distributional connection is available, the distributional curvature tensor could be readily constructed, but it would still be unclear at best how to reverse its trace with the degenerate metric to obtain a distributional Einstein tensor. We adopt instead a variational approach, and begin with the natural generalization of the Einstein-Hilbert action to signature change, subtracting the standard surface term used in the nondegenerate case in the presence of boundaries. We choose to work with a discontinuous metric, as this permits the introduction of a frame which is orthonormal almost everywhere. Having made these choices, we find that the variations proceed unchanged from the degenerate case, and we recover the identical result: The Darmois conditions ~continuity of the extrinsic curvature! ensure the absence of a surface layer, and the Lanczos equation relates the discontinuity of the extrinsic curvature to the surface stress tensor. The former result agrees with one of Embacher’s variational principles;5 the latter result is new.

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PBM_hg3Du

Reference

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