If the Earth was made of gruyere cheese...

[geordief]

We have an object approaching the surface of the Earth (no atmosphere) and instead  of crashing into the surface it enters a hole/tunnel in the "gruyere" that is by luck or design the exact spiral shape that allows the object to spiral into the centre of the Earth without physical contact.

The next step is to "melt all the cheese " so that all the mass is together in a very small region of the  centre of the "Earth"

Another object (same size,same mass,same shape etc etc) approaches this second "Earth" at the same angle and speed of approach.

 

Do the two objects follow more or less identical trajectories  in GR?

I think ,but could be wrong that they may well do so and that perhaps they would not under Newtonian gravity.

 

But I don't know,myself .

[Genady]

I think the trajectories would be different in both, the GR and the Newtonian gravity. Why do you think they would be identical in GR?

[geordief]

5 minutes ago, Genady said:

I think the trajectories would be different in both, the GR and the Newtonian gravity. Why do you think they would be identical in GR?

Not sure why I thought that.It was probably just a combination of a guess and  a vague recollection of things I had heard along those lines in the past.

So you are fairly sure that the two trajectories would be different both in GR and Newtonian gravity ?(it is only really the GR case I was interested in)

 

 

[Genady]

27 minutes ago, geordief said:

Not sure why I thought that.It was probably just a combination of a guess and  a vague recollection of things I had heard along those lines in the past.

So you are fairly sure that the two trajectories would be different both in GR and Newtonian gravity ?(it is only really the GR case I was interested in)

 

 

Yes, I'm quite sure. The spacetime metric outside a non-rotating spherically symmetric body is the Schwarzschild one. Not inside, though. The first object would have a bigger part of its trajectory inside the Earth than the second one. So, their trajectories would go through different spacetime metrics. 

[Markus Hanke]

6 hours ago, geordief said:

it is only really the GR case I was interested in

As Genady quite correctly said - the geometries in exterior vacuum and in the interior of bodies are not the same (in fact, they are quite different), so the trajectories will not in general be the same.

What will be the same though is the situation at some distance from the central body - for some exterior and sufficiently distant observer who measures the gravitational effect of the body, there will be no difference between the body being extended and “Gruyère”, or melted down and small and compact (provided the entire situation is spherically symmetric).