Einstein's Elevator: Gravity vs Acceleration

[Jim Hammond]

In Einstein's thought experiment (if I understand it correctly) a person in a sealed elevator in space accelerating at 9.807 m/s2 would not be able to tell the difference between that and the effect of gravity on Earth.
Is there any sort of experiment the person in the elevator could do that would tell him he is feeling acceleration rather than gravity from Earth? For instance, I think that if you hung two weights from long strings some distance apart on Earth they would not be parallel. But in the elevator they would be.

Sorry if this has been asked before, but I couldn't find the answer on this forum.

 

[Orodruin]

Is there any sort of experiment the person in the elevator could do that would tell him he is feeling acceleration rather than gravity from Earth?

The statement presupposes that the elevator is small enough that any effects of spacetime curvature are negligible. Therefore

For instance, I think that if you hung two weights from long strings some distance apart on Earth they would not be parallel. But in the elevator they would be.

is not applicable as you have required an elevator large enough to see the effects of the curvature. You are correct in that these long strings would behave differently, but the point is that a point mass cannot tell the difference.

 

[Bandersnatch]

I think the thought experiment Einstein used for his equivalence principle talks specifically about uniform gravitational fields and uniform acceleration.
But you're right, that given sufficiently precise instruments, a person in a non-uniform gravitational field - like that of Earth - could measure tidal forces (like you described, or measure how the strength of gravity changes with height) and by this measurement distinguish their state from being uniformly accelerated.

 

[PeterDonis]

I think the thought experiment Einstein used for his equivalence principle talks specifically about uniform gravitational fields and uniform acceleration.
But you're right, that given sufficiently precise instruments, a person in a non-uniform gravitational field - like that of Earth - could measure tidal forces (like you described, or measure how the strength of gravity changes with height) and by this measurement distinguish their state from being uniformly accelerated.

Actually, the term "uniform" here is a bit misleading. The weights hanging in an accelerating spaceship in flat spacetime would be parallel, yes; but the strength of "gravity" inside the ship would not be constant with height.

 

[FactChecker]

You are correct. Keep in mind that Einstein's thought experiments are just meant to be motivation for his formal work. They are not perfect and should not be taken too literally. His formal work involved mathematics that was then tested experimentally.

 

[m4r35n357]

For instance, I think that if you hung two weights from long strings some distance apart on Earth they would not be parallel. But in the elevator they would be.

The whole point of the scenario is that the lift is small enough in spacetime extent for these effects to be unmeasurable!

 

[Jim Hammond]

Thanks for the replies. But guys and gals, explain to me like I'm a 6 year old please! My grasp of physics is tentative at best. I'm trying to get my head around some of these concepts by picking up my studies where I left off in college more years ago than I care to admit.
I'm not as interested in the thought experiment itself as in the effect gravity on Earth would have. Is the non-parallel distance between the strings on Earth measurable in some way? Can you think of an experiment where this can be tested?

 

[Orodruin]

Is the non-parallel distance between the strings on Earth measurable in some way?

Yes, but it violates the premise on which the elevator equivalence is built. The equivalence is local (small scale) and you need a big system (large scale) to measure the difference.*

* What is "small" and "large" will depend on the possible accuracy of your measurements.

 

[Ibix]

The formal mathematical statement that the equivalence principle hints at makes it clear that the equivalence principle is only strictly true at a point. But it also makes clear that spacetime is "near enough" flat in any small region that you can't tell the difference between "at rest on a planet's surface" and "accelerating in a lift in deep space". Your experiment is simply using a large enough region that you can see the errors.

It's closely analogous to the fact that you can treat your kitchen floor as flat and tile it with square tiles. But if you keep on adding tiles and extending the area covered you'll eventually find that they won't quite fit. If you look really, really closely you could in principle spot the curvature of your floor. But it's near enough that you can ignore it. You're just deciding that the flat approximation isn't precise enough for your purposes.

 

[Bandersnatch]

Can you think of an experiment where this can be tested?

You hang a weighed string down at one of the poles. You hang another at the equator. They'll be at 90 degrees to one another, because they'll always point towards the centre of the planet. If you hang a string at X latitude, and another at Y latitude, while keeping the same longitude, the angle between the strings will be (the modulus of) the difference between X and Y.
You can even express the size of a typical elevator in terms of the difference in latitude of its edges. It will be a very small number, obviously, and that's also what the angle between the strings will be. Now you just need an instrument of sufficient precision to observe this angle.

 

[Ibix]

Can you think of an experiment where this can be tested?

Hang two strings. If they are a distance d apart then the angle between them will be $d/R$ if they are on the surface of a planet of radius R, and zero in an elevator. Multiply by $180/\pi$ if you want the angle in degrees.

This kind of thing is very important - the tides happen the way they do because the equivalence principle isn't globally applicable. Should you have the misfortune to fall into a black hole, a failure of the equivalence principle will spaghettify you (assuming you somehow survive the extreme radiation environment). But the errors on a human scale from neglecting tidal forces are minuscule.

 

[FactChecker]

Is the non-parallel distance between the strings on Earth measurable in some way? Can you think of an experiment where this can be tested?

You are absolutely correct. One should be able to measure the shorter distance at the lower end of the strings. I don't know how difficult that particular measurement would really be, but there is a lot of evidence that gravity pulls toward the center of planets and stars.

PS. There are tiny adjustments that need to be made due to motion of the planets and stars, but that is another issue in General Relativity.

 

[PeterDonis]

Hang two strings. If they are a distance $d$ apart then the angle between them will be $d/R$ if they are on the surface of a planet of radius $R$, and zero in an elevator. Multiply by $180/\pi$ if you want the angle in degrees.

It's worth noting that the angle equaling $d/R$ is strictly true only on a non-rotating, spherical planet. On a rotating planet like the Earth, a hanging string does not always point directly at the center of the Earth because of centrifugal force. I haven't done the math in detail, but at the level of accuracy needed to measure the convergence of strings that are a few meters apart, I wouldn't be surprised if the centrifugal effects are also significant, and they would change the expected angle somewhat.

 

[David Byrden]

Well, I don't know a way to test the acceleration itself, but to test whether you're on Earth (and away from the equator) you can use Foucault's Pendulum.

David

 

[Ibix]

Well, I don't know a way to test the acceleration itself, but to test whether you're on Earth (and away from the equator) you can use Foucault's Pendulum.

There are two points to be made here. The first is that this won't work on a non-rotating planet - it's a feature of the rotation, not of gravity-versus-acceleration. Second, you could make a Foucault pendulum work in a rocket accelerating along a helical (I think, without doing any maths) path. Again, this is because the pendulum precesses due to the rotation.

So I don't think this is actually diagnostic of anything other than that you are undergoing circular motion.

 

[Orodruin]

There are two points to be made here. The first is that this won't work on a non-rotating planet - it's a feature of the rotation, not of gravity-versus-acceleration. Second, you could make a Foucault pendulum work in a rocket accelerating along a helical (I think, without doing any maths) path. Again, this is because the pendulum precesses due to the rotation.

So I don't think this is actually diagnostic of anything other than that you are undergoing circular motion.

More importantly, it is again a non-local feature.

 

[Ibix]

More importantly, it is again a non-local feature.

I wrote that in my first draft but deleted it. I agree it's an extended measure, but it's not probing tidal gravity at all. So, nitpickingly, is it really important whether or not it's a local measure?

 

[David Byrden]

More importantly, it is again a non-local feature.

Now it's getting interesting. I was under the impression that Foucault's would work the same if made arbitrarily small. Was I wrong?

 

[Ibix]

Now it's getting interesting. I was under the impression that Foucault's would work the same if made arbitrarily small. Was I wrong?

Spatially small, yes, but you need to wait quite a long time to be able to detect any rotation of the swing direction. And this is spacetime we're talking about. The extent in time matters too.