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The general field equation for GR is

R

where I am setting G = 1 and c = 1.

Also, the vacuum solution is

R

But it seems to me that this "vacuum" solution must hold even when there is matter present. Pick a point within a planet. Then excavate an infinitesimal vacuum chamber about the point. That can't affect the solution there because the local contribution to the solution is infinitesimal. Therefore, whether there is a vacuum at a point or not makes no difference, the vacuum solution still holds. R

So why can't the field equation then be simplified to this by replacing R

- 1/2 g

That makes it look a lot like Gauss's law of gravity, which does not add in a zero term representing a vacuum solution.

∇

This makes more sense to me. The solution is determined by the sources only, not by a zero valued vacuum solution.

Did Einstein glue together two equations into one for brevity? If so, why doesn't Gauss's law need another equation? And why would a vacuum solution be needed in addition to the source term in GR?

R

_{ab}- 1/2 g_{ab}R = 8πT_{ab}where I am setting G = 1 and c = 1.

Also, the vacuum solution is

R

_{ab}= 0But it seems to me that this "vacuum" solution must hold even when there is matter present. Pick a point within a planet. Then excavate an infinitesimal vacuum chamber about the point. That can't affect the solution there because the local contribution to the solution is infinitesimal. Therefore, whether there is a vacuum at a point or not makes no difference, the vacuum solution still holds. R

_{ab}= 0 is a constraint on curvature that is everywhere satisfied.So why can't the field equation then be simplified to this by replacing R

_{ab}with 0?- 1/2 g

_{ab}R = 8πT_{ab}That makes it look a lot like Gauss's law of gravity, which does not add in a zero term representing a vacuum solution.

∇

^{2}[itex]\varphi[/itex] = 4πρThis makes more sense to me. The solution is determined by the sources only, not by a zero valued vacuum solution.

Did Einstein glue together two equations into one for brevity? If so, why doesn't Gauss's law need another equation? And why would a vacuum solution be needed in addition to the source term in GR?

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