Falling Through a Hole in the Earth: GR Effects on Motion

[Adrian F]

Hi, folks.

There's a somewhat popular hypothetical situation that involves a person falling into a hole dugg all the way through the Earth or any planet, passing straight to its center. Now, I understand quite well that the resulting motion would be periodical and the reasoning behind it. My question (or questions rather) is regarding to what a person would "feel" during this trip.

According to General Relativity (please correct me if I'm wrong), free falling inside a gravitational field is equivalent to uniform motion for a point particle (removing tidal effects). There is no way to distinguish one from the other without any external references. Is this true for any gravitational field or only a uniform field?

Now, my primary question is this: Assuming there's no air, a perfectly symetrical and uniform body (though I think the latter is not strictly necessary) and that the person is isolated from the outside (e.g. inside a box), would a person falling through this hole feel anything different from floating freely inside the cabin? I think that with sufficiently precise instruments the person could measure tidal effects, but would the person feel anything different in different points of the trajectory?

I hope I explained my self sufficiently well, my English is not the greatest.

Thanks in advance.

 

[rootone]

Are you asking if the falling person will experience things differently if they are enclosed inside a capsule?
I don't see why that would make any difference to the basic physics.
They will miss the view of the planet center while passing through, (or any view of anything unless the capsule is transparent).
They also might lightly bounce off the capsule walls at the end of each oscillation period, (not sure about that, it probably depends on the density of the capsule material).

 

[Adrian F]

No, I'm asking if the person would feel anything different from floating freely inside cabin, as if the cabin were in free space. The cabin is just a mean to isolate the person from external visual references.

I don't think the person (I'll asume is a he) would bounce at the ends, as both he and the cabin are accelerating always at the same rate, therefore there should be no relative motion between them in any point of the trajectory. My guess is that he wouldn't know he's oscilating and he'd feel practically the same as if the cabin were in free space.

My question boils down to this: are tidal effects big enough for a person to feel them? Or rather is the person big enough to feel the tidal effects? I believe the answer is no, but I'd like to hear the experts opinion.

 

[rootone]

OK, well there are people here who are much more expert than I am, so let's see what they say.
I am inclined to agree with you that their experience would likely be much the same if the capsule was floating in open space.
As for tidal effects. I doubt these would be noticable if falling through an Earth size body, but they could be if falling through a very massive body.

 

[jbriggs444]

A good first step would be to calculate the magnitude of the tidal forces that the person would be subject to.

Start with the tidal force that you are subject to right now. Assume that the acceleration of gravity at the Earth's surface is 10 meters per second², that the radius of the Earth is 6000 km and that you are 2 meters tall. Approximately what is the difference between the acceleration of gravity at your head and at your feet?

 

[PeterDonis]

I'm asking if the person would feel anything different from floating freely inside cabin, as if the cabin were in free space.

If we ignore tidal effects, no, they wouldn't. The question is, can we ignore tidal effects? The way to figure that out is, as jbriggs444 said, to calculate how big the tidal effects are.

 

[Adrian F]

A good first step would be to calculate the magnitude of the tidal forces that the person would be subject to.

Start with the tidal force that you are subject to right now. Assume that the acceleration of gravity at the Earth's surface is 10 meters per second², that the radius of the Earth is 6000 km and that you are 2 meters tall. Approximately what is the difference between the acceleration of gravity at your head and at your feet?

Alright. If I got it right, the acceleration of gravity at the top of my head would be 9.9999994 m/s² using those values. That's a 0.000006 m/s² difference. I guess that if you don't feel that difference in the surface, it's even less likely that you would feel it near the center or at any other point. I should have been able to think about it that way. Thank you, guys.

Basically a person falling through the hole would have no way (without instruments or external visual references) of knowing if he's falling and oscilating inside the hole or floating about in free space. And if he know he's falling and oscilating, he'd have no way of knowing (again without instruments and references) where he is in the trajectory.

 

[Janus]

Alright. If I got it right, the acceleration of gravity at the top of my head would be 9.9999994 m/s² using those values. That's a 0.000006 m/s² difference. I guess that if you don't feel that difference in the surface, it's even less likely that you would feel it near the center or at any other point. I should have been able to think about it that way. Thank you, guys.

Basically a person falling through the hole would have no way (without instruments or external visual references) of knowing if he's falling and oscilating inside the hole or floating about in free space. And if he know he's falling and oscilating, he'd have no way of knowing (again without instruments and references) where he is in the trajectory.

But consider the following: At the moment his feet pass the center, head is 2 meters from the center. Since we are assuming a uniform body, we also know that acceleration due to gravity increased linearly with distance from the center. Thus his head would experience an acceleration of 10/3000 = ~0.00333 m/s and his feet 0 m/s. That a differential that is 555 times more than standing on the surface.

In addition, outside the Earth the differential works to stretch him head to toe, but inside the hole, it works to compress him. So theoretically, he can tell the difference, it is just a matter if he can do so practically.

 

[DrGreg]

Thus his head would experience an acceleration of 10/3000 = ~0.00333 m/s and his feet 0 m/s. That a differential that is 555 times more than standing on the surface.

I think you forgot to convert km to m.

 

[Adrian F]

But consider the following: At the moment his feet pass the center, head is 2 meters from the center. Since we are assuming a uniform body, we also know that acceleration due to gravity increased linearly with distance from the center. Thus his head would experience an acceleration of 10/3000 = ~0.00333 m/s and his feet 0 m/s. That a differential that is 555 times more than standing on the surface.

In addition, outside the Earth the differential works to stretch him head to toe, but inside the hole, it works to compress him. So theoretically, he can tell the difference, it is just a matter if he can do so practically.

I think you forgot to convert km to m.

Yes. That'd be 10/3000000 or 3,33 x 10^-6 m/s². I highly doubt anyone would be able to feel a compression (or tension) effect due to tidal forces of that magnitud.

By the way, one of my questions remains unanswered, is the equivalence of free fall and inertial motion true for any gravitational field? Say a caotic field, with big changes in small distances. I'm talking about a point particle. Obviously a body with dimensions would experience tidal forces.

 

[Janus]

I think you forgot to convert km to m.

Oops, your right, I did.

 

[Nugatory]

free fall and inertial motion true for any gravitational field?

Always true if you can neglect tidal effects, never true when the tidal effects are not negligible.

There is no gravitational field that is completely free of tidal effects; a completely uniform gravitational field is not a solution of Einstein field equations (and thus the equivalence between accelerated motion and gravity applies only across a small enough region that tidal effects are negligible). However, there are many gravitational fields for which the tidal effects are negligible; Adrian's and Janus's calculations above are a good example (and the fact that we can misplace three decimal orders of magnitude without noticing just reinforces how negligible the effect is).