Failure of Standard Conservation Laws at a Classical Change of Signature

TL;DR

This paper demonstrates that standard conservation laws break down at a classical change of spacetime signature, requiring modifications to the divergence theorem and resulting in non-conservation of matter and current across the signature change surface.

Contribution

It derives modified identities for conservation laws at a signature change and shows that standard conservation cannot be maintained without unrealistic restrictions.

Findings

Divergence theorem fails at signature change without modification

Conservation of matter and current is violated at the change surface

Standard conservation laws are only recoverable under unrealistic conditions

Abstract

The Divergence Theorem as usually stated cannot be applied across a change of signature unless it is re-expressed to allow for a finite source term on the signature change surface. Consequently all conservation laws must also be `modified', and therefore insistence on conservation of matter across such a surface cannot be physically justified. The Darmois junction conditions normally ensure conservation of matter via Israel's identities for the jump in the energy-momentum density, but not when the signature changes. Modified identities are derived for this jump when a signature change occurs, and the resulting surface effects in the conservation laws are calculated. In general, physical vector fields experience a jump in at least one component, and a source term may therefore appear in the corresponding conservation law. Thus current is also not conserved. These surface effects are a consequence of the change in the character of physical law. The only way to recover standard conservation laws is to impose restrictions that no realistic cosmological model can satisfy.