How does GR deal with pointlike objects with mass?

[JC_Silver]

TL;DR

In GR, you cannot tell the difference between being in a gravitational field or in a accelerated frame of reference, can the same be said if you are in a field caused by a pointlike particle?

GR is not my field so I apologise if this is something basic, but I've been reading Wald's book on GR and while I could finish the book before asking questions about the subject, I feel I might forget the questions I have by the time I finish reading.

My question as stated on the title is about a common example used to explain how relativity can be viewed as both an observer being in a gravitational field or in an accelerated frame of reference. Usually the example is an astronaut in a ship or elevator but for the purposes of stating my question I'm going to use a planet.
Below I have a sketch I made showing someone falling in a gravitational field vs someone moving in a straightline with the floor accelerating on them:

Both are fine reference frames, but planets are round, so if we add a second observer on the other side of the planet, what happens? Well, nothing really because the same way we argue that spacetime is contracting, we can say the planet is stretching, and all observers agree they can't tell weather they are in a gravitational field or if they are in an accelerated frame of reference. Drawing below:

Now, what happens if I change the planet they are falling into for a particle of any mass, that can generate gravity even if small? The particle can't "grow" like the planet can to justify a change in reference frame. With one observer, he can argue that the particle is accelerating towards him, but with two observers, both would disagree on that. By noticing that they are both falling to the same point, they should be able to tell they are in a gravitational field instead of an accelerated reference frame, right?
Did I get something wrong? Should I finish Wald's book before asking more questions?

 

[Ibix]

My question as stated on the title is about a common example used to explain how relativity can be viewed as both an observer being in a gravitational field or in an accelerated frame of reference

The statement is that you can't tell the difference between being stationary on a planet's surface and being under acceleration in flat spacetime. It is only true strictly at a point, but it's also approximately true for small regions where tidal gravity is negligible. Once you start looking on the scale of whole planets (or on length scales comparable to the size of your source in general), you can't neglect tidal gravity and attempting to reconcile the global picture with the equivalence principle won't work

Thus we don't need to invoke "expanding planets" or anything of the sort.

 

[PeroK]

The equivalence principle in this case is for local experiments over a short period of time. The equivalence between local gravity and an accelerating reference frame, in particular, is an idealised approximation when compared with the gravity around a planet.

 

[pervect]

I would say that your characterization of GR as saying that "the planet is stretching" is flawed. But lets move on to your question, perhaps the answer to that will help clear up your confusion and lead you twoards a less-flawed understanding of what GR is saying.

The closest approximation that GR has to a "pointlike particle" would be a black hole. Google, and the formula 2GM/c^2, says that the Schwarzschild radius of a black hole with an Earth mass would be slightly under 9 millimeters.

Then GR is saying you could imagine an Earth sized planet with a circumference of 40km as being a sphere of low mass at a Scharzschild r coordiante of about 6.4km surrounding the black hole, whose event horizon has an R coordiante of about 4.5 milimeters. The idea of radial distance to the center of the black hole isn't well defined inside the event horizon. The circumference and the Schwarzschild R coordinates are well defined, however.

It is then a true statement that some force needs to be exerted on a pointlike object to keep it's Scharzschild R coordiante at a constat value of 6.4km, assuming the particle is stationary and not orbiting - i.e. that the angular Schwarzschild coordiantes are also constant. This force could be thrust from a rocket, or the outward force resulting from a very strong hollow sphere in compression.

I'm not sure how much this will help you, as the geometry of a black hole is not especially intuitive, but a black hole is the closest thing GR has to a "point particle".

 

[PeterDonis]

a field caused by a pointlike particle?

There is no such solution in GR. As @pervect says, the closest you can get is a black hole, but a black hole is nothing like what you would imagine a "point particle" to be based on your Newtonian intuitions. GR is not Newtonian gravity.

 

[JC_Silver]

Thanks everyone! Hopefully as I study further I'll come back with more questions in the future!