GR to Naive Physics: Is it Possible?

[Simon Bridge]

I get asked a lot of questions like "what is the gravitational field due to xyz curved spacetime" ... and I don't think the concepts match well the way the querant is thinking.

However, I was wondering if there is a sort-of way to get to something like that.

I would normally think of a gravitational field as the potential energy function in some coordinate system ... that would be, the amount of work has to be done to get a test mass from at rest at some (reference) place to at rest in another place.

There's a way to work that out from GR and I forget ... it's the sort of thing that used to be given to students to show why it is problematic and not all that useful. Still, if going to use GR to do something in real life, you want to know how much energy will be needed to do something like change orbits. So it's doable.

I thought rather than figure it out myself, ask to see if there is a shortcut for common situations other people know about, or if there is a resource set up for this style of thinking.

Am I making sense?

 

[PeterDonis]

I would normally think of a gravitational field as the potential energy function in some coordinate system ... that would be, the amount of work has to be done to get a test mass from at rest at some (reference) place to at rest in another place.

This works, but only in a restricted class of spacetimes, the stationary spacetimes. A stationary spacetime has a timelike Killing vector field, and you can treat each of the integral curves of that timelike KVF as labeling a "point in space", and the norm of the timelike KVF along each integral curve is the "potential energy" at that point in space. Then everything works just as you describe.

However, there are many spacetimes of interest which are not stationary; the most commonly used ones are the FRW spacetimes of cosmology. In a non-stationary spacetime, there is no well-defined "potential energy", so what you describe does not work.