Would Birkhoff's Theorem also apply for a field with negative energy density?

Question

Birkhoff's Theorem says that any spherically symmetric solution to the field equations must be the Schwarzschild metric. I'm wondering whether that could be generalized to a field with negative energy density?

Answer 1

No, it wouldn't. Birkhoff's Theorem states not that any spherically symmetric solution to the field equations must be the Schwarzschild solution, but rather that any spherically symmetric solution to the vacuum field equations must be the Schwarzschild solution. If there is any non-negative energy density on spacetime and the Einstein Field Equations are assumed to hold, then that spacetime can't be Schwarzschild spacetime.

Notice that the Schwarzschild metric has $R_{\mu\nu} = 0$, and hence $R = 0$. As a consequence, the Einstein Equations lead to $$\begin{align} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} &= 8 \pi T_{\mu\nu}, \\ 0 &= 8 \pi T_{\mu\nu}, \\ T_{\mu\nu} &= 0. \end{align}$$

If the stress-energy tensor is anything other than identically zero (including if one considers a non-vanishing cosmological constant), then the Schwarzschild metric is not a solution to the Einstein equations on that region where $T_{\mu\nu} \neq 0$.