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One of the things I've always wondered is how to prove that the Einstein field equations preserve the sign of the determinant of the Minkowski metric tensor:

$$\text{sgn}( \det (g)) = -1$$

Does anyone know how to do it?

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2 Answers 2

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The proof would go like this:

  1. First, break the EFEs into a 3+1 formulation, so that they are an initial value problem. Then the EFEs turn into two "constraint" and two "evolution" equations. See http://arxiv.org/abs/1304.1960. Theorem 1 in the above reference: a smooth spatial slice that satisfies the constraint equations uniquely determines an everywhere-smooth spacetime (other theorems - see Baumgarte and Shapiro's book on numerical relativity for example - show that this spacetime is the solution to the evolution equations given the initial constraint-satisfying slice).
  2. The metric tensor is a quadratic form. Therefore, by Sylvester's law of inertia for quadratic forms https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia, its signature is preserved under diffeomorphisms, notably including changes of basis or dragging along curves, provided the manifold itself is smooth. But from (1) the manifold implied by the EFEs to develop from smooth initial data is smooth.

If the spacetime is not smooth, these conditions could fail (but then we wouldn't be dealing with 'normal' GR). Some quantum gravity or cosmology ideas appeal to this.

Perhaps your question was more simply "how do we know solutions to the EFEs have continuous signature?". For spacetimes without discontinuities this is established by (2) immediately. (1) establishes that physically meaningful spacetimes should not have discontinuities provided the EFEs furnish the governing physics.

Edit: Note that while the overall signature is preserved, the specific coordinate which gets the minus sign can very well change. This occurs for example when traversing a SC black hole: having crossed the Killing horizon, the "time" and "distance from singularity" coordinates exchange signs.

Edit 2: Note Bruce Greetham's answer is essentially a more succinct phrasing of the above; I'm not sure why it is getting downvoted.

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I believe this question has been answered correctly at EFE and Local Minkowski.

The essential point of the argument is that if you start in a local coordinate system which is locally Minkowski (i.e. a local free fall frame as demanded by Einstein's principle of Equivalence), then evolution under the Einstein field equations will perform a continuous change of the metric.

This argument is a general argument which does not depend on the specific form of the Einstein Field Equation.

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  • $\begingroup$ What does "singular metric" mean? In the answer to the link you provided, he just said "it is a theorem" without actually giving it, providing a reference, or even providing a name for that theorem. $\endgroup$
    – QuantumDot
    Jul 31, 2016 at 18:42

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