Confusion on Einstein's Continuously Accelerating Rocket

[Schmonad]

I’ve been thinking about Einstein’s gedankenexperiment (because I’m meta like that) regarding the sealed room on Earth, and the sealed room on a steadily accelerating rocket. And the result of this thought process is… I’m confused, probably because I do not correctly understand the parameters of the experiment. My understanding was, there is supposed to be no test one can perform within the sealed room, to determine if you are “stationary” on Earth, or on a rocket constantly accelerating at 9.8 m/s. I am confused because I can think of two(ish) such tests. So I’d be much obliged if some kind person could explain why these tests would not work / or how I am misunderstanding the assertion.

1. This requires some exceptionally precise instruments, or perhaps a room with exceptionally high ceilings. In a gravity well, there are a number differences that would appear when measuring something on the floor versus the ceiling. Put a clock on both ceiling and floor, and then see if one starts to go faster than the other, weigh things at different heights, measure red/blue shifting rates at different heights. Would a rocket not have uniform g-forces within any point within the room?
2. Drop something really massive. Let’s say our subject is Wile E. Coyote, and he has an enormous anvil suspended from the roof of the room. For the sake of simplicity, we’ll say the anvil has mass equal to the rest of all other parts of the rocket combined. When the anvil is dropped, while it is falling, it is no different than if the anvil had been jettisoned, thus temporarily the mass of the rocket is effectively halved. As the force propelling the rocket would remain unchanged, acceleration would momentarily double, creating a measurable jerk (a second jerk if you’re counting Wile E. as the first). Then an opposite jerk as it hit the floor. If you were on Earth, no such doubling of weight would be felt.

 

[phinds]

Really good, valid, thinking on your part.

Your first one is correct. The experiment is only valid "locally", where "locally" prohibits thinking about tall ceilings

Your second one is cheating a bit. The experiment is only WITHIN the closed room not taking the room itself as part of the experiment (you can't go breaking off parts of the room).

 

[PeterDonis]

Put a clock on both ceiling and floor, and then see if one starts to go faster than the other, weigh things at different heights, measure red/blue shifting rates at different heights. Would a rocket not have uniform g-forces within any point within the room?

Uniform g-forces (meaning uniform felt acceleration) does not imply no time dilation between ceiling and floor, nor does it imply no redshift/blueshift with change in height. (It does imply no change in weight with height, but that will be true on Earth as well, provided the height of the room is small enough. See further comments below.) There will indeed be time dilation and redshift/blueshift between the floor and ceiling of an accelerating rocket, just as in a room that is stationary on the Earth's surface. Einstein showed this using some simple thought experiments.

Regarding the change in g-force with height, this will actually happen in an accelerating rocket too, as long as we stipulate that the floor and ceiling of the rocket remain at rest relative to each other (which seems like an obvious stipulation to make, but it turns out to require that the ceiling has less g-force than the floor when relativistic effects are taken into account). For a rocket whose floor is accelerating at 1 g, the distance between ceiling and floor would have to be an appreciable fraction of a light-year for the difference in acceleration between floor and ceiling to be noticeable, so in practical terms the change is negligible. In any case, this is certainly not a local measurement, so it falls outside the conditions of Einstein's thought experiment. But it's worth noting that the change in g-force with height is *different* for a very tall accelerating rocket than it is for, say, a tall building on the Earth's surface; so if we don't restrict ourselves to local measurements, measuring the variation of g-force with height is one way to distinguish the accelerating rocket from the room or building on the Earth's surface.

When the anvil is dropped, while it is falling, it is no different than if the anvil had been jettisoned, thus temporarily the mass of the rocket is effectively halved. As the force propelling the rocket would remain unchanged, acceleration would momentarily double, creating a measurable jerk (a second jerk if you’re counting Wile E. as the first). Then an opposite jerk as it hit the floor. If you were on Earth, no such doubling of weight would be felt.

You have misunderstood what "constantly accelerating rocket" means. It does *not* mean a constant force; it means a constant felt acceleration. The force exerted by the rocket engine is whatever force is required to maintain a constant acceleration, even if that force has to vary in time. In other words, it implies precise control over the rocket engine so that its thrust varies from instant to instant in exactly the right way to keep the acceleration felt by the crew constant, even while things like anvils dropping are happening inside. (Obviously this is an idealization; no real rocket engine can be controlled that precisely.)

The reason we define things this way is that, if the felt acceleration in the rocket is not constant, the situation can't be compared to an observer stationary on the Earth's surface, because there, the felt acceleration *is* constant. In other words, the point of the thought experiment is to compare what happens in an accelerating rocket that, by hypothesis, can't be distinguished from standing on the surface of the Earth by felt acceleration alone; the question is whether there are any *other* ways to distinguish the two cases by purely local measurements, given that the felt acceleration is the same.

 

[PeterDonis]

Your first one is correct.

I'm not sure it is. I think the OP was suggesting that there would be no time dilation or redshift/blueshift between the floor and ceiling of the rocket, whereas there would be in a room on Earth. That's not correct, as I noted in my response. (And you don't need a very tall rocket to make the time dilation or redshift/blueshift measurable; the height change used in the Pound-Rebka experiment was less than the length of the Space Shuttle orbiter, for example, so you could do a redshift/blueshift experiment inside the Space Shuttle with its engines firing at 1 g and get a non-null result.)

 

[phinds]

I'm not sure it is. I think the OP was suggesting that there would be no time dilation or redshift/blueshift between the floor and ceiling of the rocket, whereas there would be in a room on Earth. That's not correct, as I noted in my response. (And you don't need a very tall rocket to make the time dilation or redshift/blueshift measurable; the height change used in the Pound-Rebka experiment was less than the length of the Space Shuttle orbiter, for example, so you could do a redshift/blueshift experiment inside the Space Shuttle with its engines firing at 1 g and get a non-null result.)

Yeah, you're right. I was focused only on the "weighing things" which is what I've seen before as the test for the difference in a tall room between earth/rocket.

 

[Schmonad]

I'm not sure it is. I think the OP was suggesting that there would be no time dilation or redshift/blueshift between the floor and ceiling of the rocket, whereas there would be in a room on Earth. That's not correct, as I noted in my response. (And you don't need a very tall rocket to make the time dilation or redshift/blueshift measurable; the height change used in the Pound-Rebka experiment was less than the length of the Space Shuttle orbiter, for example, so you could do a redshift/blueshift experiment inside the Space Shuttle with its engines firing at 1 g and get a non-null result.)

I was actually trying to imply that the rate of shift would be different when measured from floor to half way point, than when measure from half way point to ceiling. At least on Earth. Given given my (apparently mistaken?) understanding that g-forces would be uniform throughout the rocket, I assumed this effect would not occur on the rocket. But, I was cramming a bit much in a short paragraph and not being terribly clear.

But, thank you both for the explanations! I believe I have a slightly firmer grasp. Though, I'm still a bit trouble about the uniformity of the acceleration in the rocket. A gravity well is inherently non-uniform. But the rocket is also equally non-uniform? Because if they have differences in the rates they change at, and if you had sufficiently precise enough instruments, I feel like you'd have to start defining "local" as Planck length in order for the results to be uniform. But I think this is just me missing the point I guess. The point being that any effects that can be demonstrated in one, could be demonstrated in the other, just at possibly slightly different rates.

 

[stevendaryl]

I’ve been thinking about Einstein’s gedankenexperiment (because I’m meta like that) regarding the sealed room on Earth, and the sealed room on a steadily accelerating rocket. And the result of this thought process is… I’m confused, probably because I do not correctly understand the parameters of the experiment. My understanding was, there is supposed to be no test one can perform within the sealed room, to determine if you are “stationary” on Earth, or on a rocket constantly accelerating at 9.8 m/s. I am confused because I can think of two(ish) such tests. So I’d be much obliged if some kind person could explain why these tests would not work / or how I am misunderstanding the assertion.

1. This requires some exceptionally precise instruments, or perhaps a room with exceptionally high ceilings. In a gravity well, there are a number differences that would appear when measuring something on the floor versus the ceiling. Put a clock on both ceiling and floor, and then see if one starts to go faster than the other, weigh things at different heights, measure red/blue shifting rates at different heights. Would a rocket not have uniform g-forces within any point within the room?
2. Drop something really massive. Let’s say our subject is Wile E. Coyote, and he has an enormous anvil suspended from the roof of the room. For the sake of simplicity, we’ll say the anvil has mass equal to the rest of all other parts of the rocket combined. When the anvil is dropped, while it is falling, it is no different than if the anvil had been jettisoned, thus temporarily the mass of the rocket is effectively halved. As the force propelling the rocket would remain unchanged, acceleration would momentarily double, creating a measurable jerk (a second jerk if you’re counting Wile E. as the first). Then an opposite jerk as it hit the floor. If you were on Earth, no such doubling of weight would be felt.

The equivalence principle is really about a limiting case. If the size of a room is so small that there is a negligible change in gravity (or g-forces) between the ceiling and the floor, and if the room is accelerating continuously during the experiment, then no experiment can tell the difference between a room on the Earth and a room accelerating in space.

To make the comparison closer, instead of imagining a room that is sitting on the Earth, consider a room that is using a rocket to hover at a constant height above the ground. Then your anvil experiment would affect the Earth case exactly like the outer space case.

You're certainly right, that sufficiently precise measurements of gravity could detect a difference in gravitational force between the ceiling and the floor, and this difference could tell you that the gravity is most likely due to a planet, rather than acceleration. The variation of gravity with location is called "tidal forces". The equivalence principle is usually stated as "In the absence of measurable tidal forces..."

 

[Nugatory]

But the rocket is also equally non-uniform? Because if they have differences in the rates they change at, and if you had sufficiently precise enough instruments, I feel like you'd have to start defining "local" as Planck length in order for the results to be uniform.

Yes, the rocket is equally non-uniform, but no, the Planck length has nothing to do with anything here. As far as special and general relativity (which quantize neither time nor space) are concerned "local" means that the closer two points are the smaller the non-local effects between them get, so that we can make these effects arbitrarily small by considering a small enough volume of space.

 

[WannabeNewton]

A gravity well is inherently non-uniform.

No. Any gravitational field is uniform on small enough length scales. The equivalence principle is only valid in these length scales. In actual GR calculations the equivalence principle is only valid at a single event on the worldline of any observer in a given space-time. Away from this event, or equivalently away from the observer worldline, it is only valid approximately to certain orders under a Taylor expansion in the distance from the worldline. The exact order to which the expansion provides a good approximation of the equivalence principle depends on various factors such as the acceleration/rotation of the observer and length scales of curvature variations of the space-time.

 

[PeterDonis]

I was actually trying to imply that the rate of shift would be different when measured from floor to half way point, than when measure from half way point to ceiling. At least on Earth.

Ah, ok. In principle this is true (and it's true in the rocket as well as on Earth), but whether it's detectable depends on how high the room or the rocket is. For a small enough height, the shift is the same from floor to halfway point as from halfway point to ceiling; that's because the amount of shift with height depends on the g-force, and for a small enough height the g-force is uniform (more precisely, changes in the g-force are not detectable)--again, in both the rocket and the room on Earth.

Given given my (apparently mistaken?) understanding that g-forces would be uniform throughout the rocket

G-forces are uniform throughout the rocket if the height is small enough; but they're also uniform throughout the room on Earth if the height is small enough--where "small enough" just means that over the height of the rocket or the room, the change in g-force is too small to be detectable. But the really key point is that time dilation and redshift/blueshift from floor to ceiling do *not* require a non-uniform g-force; they are present even if the g-force is uniform.

A gravity well is inherently non-uniform.

Over large enough distances, yes. But if the distance is small enough, the non-uniformities are not detectable, yet physical effects like time dilation and redshift/blueshift *are* detectable. That's the key point: as I said above, those effects do not require non-uniformity in the g-force; they just require a g-force to be present (even if it's uniform).

But the rocket is also equally non-uniform?

Over large enough distances, the rocket is non-uniform, but the non-uniformity is *different* than it is in the case of a gravity well.

Because if they have differences in the rates they change at, and if you had sufficiently precise enough instruments, I feel like you'd have to start defining "local" as Planck length in order for the results to be uniform.

In principle, yes, if you had sufficiently precise instruments, you could detect non-uniformities over extremely small distances, and you could use that to distinguish the rocket from the room on Earth. But that "sufficiently precise" is the key. See further comments below.

The point being that any effects that can be demonstrated in one, could be demonstrated in the other, just at possibly slightly different rates.

That's not quite the point. The point is that there are effects, like time dilation and redshift/blueshift from the floor to the ceiling of the room, that only require the *presence* of a g-force; they don't depend on whether the g-force is uniform or non-uniform, or what kind of non-uniformities it has. So you can't use these effects to distinguish one type of g-force (a gravity well) from another type of g-force (an accelerating rocket). This in turn allows us to use special relativity, as applied to an accelerating rocket, to "bootstrap" ourselves to a theory that includes gravity, by exploiting the fact that, with regard to these certain effects, g-force due to gravity "looks the same" as g-force due to acceleration.

In principle, as I said above, you could distinguish the two cases (gravity vs. acceleration) with sufficiently precise instruments; but how precise the instruments have to be depends on how the specific non-uniformities in the two cases differ. In practice, we find that we can treat the two cases as being the same (i.e., the difference in non-uniformities is too small to matter) over regions of significant extent in space and time, quite enough to apply the theory of general relativity to cases where we need to apply it.

 

[PeterDonis]

Yes, the rocket is equally non-uniform

No, it isn't. The rocket is accelerating in flat spacetime; the gravity well is in curved spacetime. This means the non-uniformities in the two cases are different. How large a range of distances or times you need to sample to see the difference is another question; but, for example, in the case of the Earth, the g-force would be perceptibly smaller a thousand miles above the Earth's surface, say, whereas a thousand-mile-long rocket with 1-g acceleration at its floor would have 1-g acceleration at its ceiling to a very, very good approximation (since a thousand miles is still much, much, much less than a light-year, which is the distance scale on which the g-force varies significantly for a 1-g accelerating rocket).

 

[Schmonad]

No. Any gravitational field is uniform on small enough length scales. The equivalence principle is only valid in these length scales. In actual GR calculations the equivalence principle is only valid at a single event on the worldline of any observer in a given space-time. Away from this event, or equivalently away from the observer worldline, it is only valid approximately to certain orders under a Taylor expansion in the distance from the worldline. The exact order to which the expansion provides a good approximation of the equivalence principle depends on various factors such as the acceleration/rotation of the observer and length scales of curvature variations of the space-time.

Do you mean effectively uniform? As if there was actually zero change, would not all increases in length scale be multiplying by zero? Sounds like I shall have to do some reading on worldlines to comprehend the rest of your point though.

 

[WannabeNewton]

Do you mean effectively uniform?

Yes it's an approximation in some neighborhood of a point in the gravitational field that becomes exact in the limit as the neighborhood shrinks to the point. We do the same thing when studying conductors in electrostatics for example.

 

[Nugatory]

A gravity well is inherently non-uniform.

No. Any gravitational field is uniform on small enough length scales...

I think the two of you may be talking at cross purposes here. I'm understanding Schmonad to be saying that if you examine the gravitational field at a sufficiently small scale, its non-uniformity will appear. If that's what he's trying to say, then he's come to understand the situation as you describe it and you should be saying "Yes" above.