Validity of Birkhoff's theorem in flat Minkowski spacetime

[Volterra]

TL;DR Summary

I would like to know/discuss, if Birkhoffs theorem is valid in flat Minkowski spacetime

I have a theoretical question regarding the validity of Birkhoff's theorem in flat Minkowski spacetime (noting that this is a hypothetical scenario, as mass curves spacetime).

Common derivations of Birkhoff's theorem are based on the assumption that the Ricci tensor
R_ij = R_kij^k (Einstein summation convention)
must vanish to satisfy Einstein's vacuum solution.

In flat spacetime, all components of the Riemann curvature tensor trivially vanish. Consequently, all components of the Ricci tensor in flat spacetime also vanish, regardless of any potential time dependence.
From this, one might conclude that Birkhoff's theorem is not applicable or meaningful in flat spacetime.
Is this conclusion correct, or am I overlooking an important aspect? How is this viewed in the forum?

 

[Ibix]

Birkhoff's theorem just says that the only spherically symmetric static vacuum spacetime is Schwarzschild. Flat spacetime satisfies Birkhoff's theorem, which is fine because Minkowski spacetime is Schwarzschild spacetime with $M=0$.

 

[PeterDonis]

Common derivations of Birkhoff's theorem are based on the assumption that the Ricci tensor
R_ij = R_kij^k (Einstein summation convention)
must vanish to satisfy Einstein's vacuum solution.

It's not an assumption, it's a definition. "Vacuum solution" is defined as the stress-energy tensor being zero, hence the Einstein tensor is zero by the Einstein Field Equation, hence the Ricci tensor is zero.

In flat spacetime, all components of the Riemann curvature tensor trivially vanish. Consequently, all components of the Ricci tensor in flat spacetime also vanish

Yes, which means flat spacetime is a vacuum solution.

regardless of any potential time dependence.

What does time dependence have to do with anything? No time dependence is assumed in the derivation of Birkhoff's Theorem.

The fact that there is a fourth Killing vector field (in addition to the three that come with spherical symmetry) in a spherically symmetric vacuum spacetime is a conclusion of the theorem, not an assumption. Flat spacetime satisfies that conclusion of the theorem. (Note that the theorem does not say that the fourth KVF must be timelike everywhere, so even its presence does not necessarily say anything about "time dependence".)

From this, one might conclude that Birkhoff's theorem is not applicable or meaningful in flat spacetime.

I don't see why. See above.