Negative potential energy and negative mass

[PeterDonis]

That sounds like we could get an answer this way but it would be limited to a Cauchy surface that includes your initial conditions.

No, it isn't. Data specified on a Cauchy surface is sufficient to determine the entire spacetime geometry.

That's what I mean with "static".

As PAllen said, that is not the correct definition of "static". The class of spacetimes that have a Cauchy surface includes many spacetimes which are not static, or even stationary.

Hawking & Ellis lays all of this out in detail. It is advanced, but definitely worth reading if you want to understand the most general theorems we have on global properties of spacetimes.

 

[Mister T]

What makes you sure that the external field can be sitched off gradually without letting the objects moving too fast and emitting gravitational waves?

It doesn't matter what happened in the past to create the condition described. There could have been waves created.

I was addressing your objection that the scenario required instantaneous switching by describing one way in which it didn't.