How solid is the time-independence assumption in Schwarzschild?

[DonaldBR]

I was working through the derivation of the Schwarzschild solution and hit a wall trying to properly justify the assumption of a static, spherically symmetric vacuum. My textbook just states it as given, but how do we know the time-independence isn't itself an approximation that breaks the solution's physical validity in some regimes?

[PaulQW]

I wrestled with that assumption too. I started with a metric that could depend on r and t, then imposed spherical symmetry and a vacuum. The equations pressured me to set time dependence to zero; if you keep it, you keep extra terms that don’t cancel unless you add matter or radiation. It felt like a symmetry argument more than a proof I could point to quickly, and I kept circling back to the idea that the exterior kind of “wants” to be static.

[Gary_M]

In the exterior vacuum, Birkhoff's theorem is the move you can lean on. spherical symmetry and no matter means no time dependence; the solution is static and unique up to the mass parameter. So the time-independence isn’t just an approximation there—it's a theorem for the region outside the source.

[Scott.M]

That said, real systems aren't perfectly symmetric or vacuum. If the interior changes shape or emits gravitational waves, the exterior can feel it in principle, though far away it settles back to Schwarzschild-like behavior. I tried a toy pulsating shell and saw only tiny, transient effects at large radius.

[Richard92]

Is the real issue about whether the exterior vacuum is truly time independent, or about how you set the boundary between interior and exterior?