Question about Einstein's famous elevator experiment

[Kuruwee]

TL;DR Summary

Something I'm not sure about in Albert's elevator mind experiment

Hi,
In Einstein's famous elevator experiment, someone in the elevator cannot tell if the acceleration they experience is from the gravity of a nearby large mass, or from their own change of velocity under the influence of some external force.

But if there is an external force accelerating the elevator, does not there have to be some other mass that is being accelerated in the opposite direction ( Newtons Laws) ?
If so, the person in the elevator would then also feel the effects of the changing gravity of that other mass as it accelerates away.

The accelerations could still feel the same, but there would need to be a correction to size of the acceleration from the external force to include the gravity from this additional mass.

Does this makes sense or am I missing something ??

 

[Sagittarius A-Star]

The accelerations could still feel the same, but there would need to be a correction to size of the acceleration from the external force to include the gravity from this additional mass.

No. In the elevator thought experiment in case of the accelerated elevator, usually flat spacetime is assumed. That's an idealization.

 

[Ibix]

You're missing something.

There are many ways to tell if you're in a gravitational field or accelerating in deep space, but only if the region of spacetime you consider is large. The thought experiment is supposed to be in a small region. You are considering a small space and a long time, long enough to detect changes in propellant mass through gravity, so are ignoring the "small region" restriction.

Formally, the equivalence principle only applies to pointlike objects and infinitely short times, because the mathematical statement is that coordinates can be chosen to make the first derivatives of the metric vanish at a point but not the second derivatives. But given that there is always a finite precision to measurements, in practice it also applies in small regions, such as a short time in a small elevator.

 

[Kuruwee]

You're missing something.

There are many ways to tell that you're in a gravitational field or accelerating in deep space, but only if the region of spacetime you consider is large. The thought experiment is supposed to be in a small region. You are considering a small space and a long time, long enough to detect changes in propellant mass through gravity, so are ignoring the "small region" restriction.

Formally, the equivalence principle only applies to pointlike objects and infinitely short times, because the mathematical statement is that coordinates can be chosen to make the first derivatives of the metric vanish at a point but not the second derivatives. But given that there is always a finite precision to measurements, in practice it also applies in small regions, such as a short time in a small elevator.

Hi,
Let me re-phrase: Assume a point mass and a single instant ( or infinitely short time period). Velocity is zero, but there is acceleration. I agree it is impossible to tell if the acceleration comes from the gravity of a local mass or from a local force. The point mass can't tell.

What I am saying is that if the acceleration is due to a instantaneous local force, then there must be an additional local mass also accelerating in the opposite direction ( velocity zero in that instant). If that is true, and not assumed zero, the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass.

As I see it, all this is still very local and at infinitely short time intervals.

Really appreciate your thoughts on this...thanks

 

[PeroK]

Hi,
Let me re-phrase: Assume a point mass and a single instant ( or infinitely short time period). Velocity is zero, but there is acceleration. I agree it is impossible to tell if the acceleration comes from the gravity of a local mass or from a local force. The point mass can't tell.

What I am saying is that if the acceleration is due to a instantaneous local force, then there must be an additional local mass also accelerating in the opposite direction ( velocity zero in that instant). If that is true, and not assumed zero, the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass.

As I see it, all this is still very local and at infinitely short time intervals.

Really appreciate your thoughts on this...thanks

The elevator thought experiment is an idealised scenario. It doesn't include an analysis of how you could accelerate an elevator through empty space; nor, how you could create a uniform gravitational field. Both of these things are impossible idealisations. What it says is that there is no fundamental difference between a) a room on the surface of the Earth; and b) a room being accelerated in flat spacetime. And, if you carry out local experiments, then they will give the same results.

In any case, there is the following contrast between Newtonian physics and relativity regarding the real scenario where you are standing on the surface on the Earth:

In Newtonian physics, there are two forces acting on you: the gravitational force from the Earth acting downwards and the normal force from the Earth's surface acting upwards. These forces are balanced and you have zero acceleration.

In relativity, there is only one force acting on you: the normal force from the Earth's surface is acceletrating you upwards.

You can argue all day about the details of the elevator experiment, but it doesn't affect this fundamental difference between Newtonian gravity and general relativity.

 

[Ibix]

the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass.

I'm not sure what you mean by "the equivalence math". I think you just mean that, to calculate the acceleration of a rocket I should take not take thrust over mass, but also subtract off the gravitational attraction between rocket and propellant. I guess so, in Newtonian terms anyway, but so what? That isn't really important to the equivalence principle.

 

[Dale]

What I am saying is that if the acceleration is due to a instantaneous local force, then there must be an additional local mass also accelerating in the opposite direction ( velocity zero in that instant). If that is true, and not assumed zero, the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass

What you are describing here is usually neglected for simplification. But it wouldn’t actually change the equivalence math, it would just make the analysis more complicated.

Suppose that we have two rockets. Neither one is firing its engines. One is not in a gravitational field and the other is in a uniform gravitational field. The one in the gravitational field is accelerating in free fall. The one not in a gravitational field is not accelerating. No local measurement allows them to distinguish.

Now suppose that we have two rockets. Each is firing its engines. One is not in a gravitational field and the other is in a uniform gravitational field. The one in the gravitational field is motionless and not accelerating. The one not in a gravitational field is accelerating. No local measurement allows them to distinguish, including measurements of the gravitational effects of the propellant on the ship. This is what is called the strong equivalence principle.

 

[Mister T]

TL;DR Summary: Something I'm not sure about in Albert's elevator mind experiment

But if there is an external force accelerating the elevator, does not there have to be some other mass that is being accelerated in the opposite direction ( Newtons Laws) ?
If so, the person in the elevator would then also feel the effects of the changing gravity of that other mass as it accelerates away.

You could imagine a rocket pulling the elevator using a long tether. The exhaust gasses and the rocket itself exert a gravitational force, but you can make those effects negligible. Just keep repeating the experiment with ever longer tethers until it's effects are too small to register on your measuring instruments.

These kinds of considerations are present in most if not all thought experiments, and real experiments, too.

 

[PeterDonis]

In Einstein's famous elevator experiment, someone in the elevator cannot tell if the acceleration they experience is from the gravity of a nearby large mass, or from their own change of velocity under the influence of some external force.

No. The person in the elevator knows that the proper acceleration they experience (i.e., they feel weight) comes from an "external force"--a non-gravitational force that is pushing on them. In the usual version, the "elevator" is a rocket whose engine is firing.

What the person in the elevator doesn't know, assuming they only make observations confined to the elevator and that the elevator is small enough, is whether the elevator is accelerating freely in flat spacetime, far from any gravitating bodies, or whether it is hovering at rest in the gravitational field of a nearby large mass.

 

[PeterDonis]

if the acceleration is due to a instantaneous local force, then there must be an additional local mass also accelerating in the opposite direction

In the case I described, where the local force is a rocket whose engine is firing, there is mass accelerating in the opposite direction: the rocket exhaust.

If that is true, and not assumed zero, the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass.

In the usual formulation, it is assumed that the rocket and everything involved with it have negligible mass, in terms of producing a gravitational field--or, in GR terms, curving spacetime. It is assumed that within the small local patch of spacetime occupied by the rocket during the experiment, spacetime is flat. This version of the equivalence principle is usually called the "Einstein equivalence principle", and applies to any non-gravitational experiment that you can run inside the elevator.

There is, however, as @Dale said, what is called the "strong equivalence principle", also satisfied by GR, in which you can even do a gravitational experiment inside the rocket--for example, you could do a Cavendish experiment to measure the gravitational constant--and the results will be the same whether the rocket is accelerating freely far from all gravitating masses, or hovering in the gravitational field of a nearby large mass. In this formulation, you could also include in your analysis, if you wanted to, the mass of the rocket itself, of you, the observer, and of the rocket exhaust (and anything else inside the rocket), and the strong equivalence principle would tell you that the net result of all those factors, taken together, would still look the same whether the rocket was accelerating freely far from all gravitating masses, or hovering in the gravitational field of a nearby large mass.

 

[Nugatory]

What I am saying is that if the acceleration is due to a instantaneous local force, then there must be an additional local mass also accelerating in the opposite direction ( velocity zero in that instant). If that is true, and not assumed zero, the equivalence math would need to include a very small adjustment for the local gravitational force on our point mass from the propellent mass.

It would be good exercise to calculate about what this adjustment might be. Assume for definiteness that whatever we’re using to accelerate the elevator is subjecting it one 1G for ten seconds, make plausible assumptions about how much additional mass will be moving in the opposite direction, estimate to just one significant digit what sort of correction will needed. For this one-sig-digit calculation Newton’s law of gravity is sufficient.

 

[Grinkle]

OP - you are conflating engineering and physics. Don't concern yourself with the implementation of the accelerated elevator - that is not relevant to the physics of the thought experiment.

 

[Kuruwee]

In the case I described, where the local force is a rocket whose engine is firing, there is mass accelerating in the opposite direction: the rocket exhaust.


In the usual formulation, it is assumed that the rocket and everything involved with it have negligible mass, in terms of producing a gravitational field--or, in GR terms, curving spacetime. It is assumed that within the small local patch of spacetime occupied by the rocket during the experiment, spacetime is flat. This version of the equivalence principle is usually called the "Einstein equivalence principle", and applies to any non-gravitational experiment that you can run inside the elevator.

There is, however, as @Dale said, what is called the "strong equivalence principle", also satisfied by GR, in which you can even do a gravitational experiment inside the rocket--for example, you could do a Cavendish experiment to measure the gravitational constant--and the results will be the same whether the rocket is accelerating freely far from all gravitating masses, or hovering in the gravitational field of a nearby large mass. In this formulation, you could also include in your analysis, if you wanted to, the mass of the rocket itself, of you, the observer, and of the rocket exhaust (and anything else inside the rocket), and the strong equivalence principle would tell you that the net result of all those factors, taken together, would still look the same whether the rocket was accelerating freely far from all gravitating masses, or hovering in the gravitational field of a nearby large mass.

I thank you all for your thoughtful and helpful comments.

I agree that the "rocket exhaust" does not impact the Equivalence of the two scenarios. In fact I'm counting on it.

Imagine the rocket (or elevator) on the surface of a large mass. Inside the rocket you feel acceleration, let's just say at 9.8 ms².

Then in open space, far from any gravity, I accelerate my rocket to perfectly match what I felt on the surface of the large mass, so I feel the 9.8ms² acceleration.

Inside my rocket, I now feel exactly the same as I did on the mass. I cannot tell the difference. Equivalence. But to an external observer my rocket isn't accelerating at exactly 9.8ms². An external observer would see the rocket in open space is accelerating at 9.8ms² less the miniscule acceleration caused by the exhaust mass needed to accelerate my rocket. ( I cannot assume my rocket has zero mass, as it reacts to gravity. Because it has mass, it requires some exhaust/propellant mass for the rocket to accelerate. )

So to the rocket, the acceleration due to gravity and due to rocket are both 9.8 ms².
To the observer the acceleration that was needed to match gravity is very slightly less than 9.8 ms².

If this is true, in the real world, the effect would be most visible at areas of very low gravitational acceleration acting on very large masses. (I think?)

Look forward to your thoughts...

 

[Ibix]

But to an external observer my rocket isn't accelerating at exactly 9.8ms²

Yeah it is - the external observer is affected by the gravity of the exhaust too.

The concept you are groping towards is the distinction between proper acceleration (the quantity you measure with an accelerometer attached to yourself) and coordinate acceleration, which is the rate if change of speed relative to some arbitrary standard of rest. Consider jumping out of a spaceship hovering above the moon (or some other airless body). You are in freefall, and accelerometers (e.g. the ones in your phone, or just a weighing scale) will read zero. However, you could use a radar set carried with you to measure your increasing speed towards the surface of the moon. The former measure is proper acceleration, and can be measured in a closed box. The latter is coordinate acceleration which you can only measure with respect to some arbitrary stationary object. You'll get a different rate of coordinate acceleration if you turn your radar set around and bounce pulses off the Earth, for example, because you chose a different definition of what "not accelerating" looks like.

Your external observer can measure less than $9.8\mathrm{ms^{-2}}$ if they make some measure like coordinate acceleration relative to a body that remains unaccelerated and was initially at rest with respect to the rocket. But that's an arbitrary choice, and not the kind of acceleration the equivalence principle cares about.

 

[PeroK]

So to the rocket, the acceleration due to gravity and due to rocket are both 9.8 ms².
To the observer the acceleration that was needed to match gravity is very slightly less than 9.8 ms².

To an outside observer, the rocket's acceleration would be $\frac a {\gamma^3}$ where $\gamma$ is the gamma factor associated with the rocket's instantaneous speed. And $a$ is the proper acceleration of the rocket.

This is why a rocket could theoretically accelerate indefinitely in its own reference frame, while never exceeding the speed of light in any external reference frame.

This is why it's important to learn physics in a coherent way and study the basics.

 

[Dale]

I thank you all for your thoughtful and helpful comments.

I agree that the "rocket exhaust" does not impact the Equivalence of the two scenarios. In fact I'm counting on it.

Imagine the rocket (or elevator) on the surface of a large mass. Inside the rocket you feel acceleration, let's just say at 9.8 ms².

Then in open space, far from any gravity, I accelerate my rocket to perfectly match what I felt on the surface of the large mass, so I feel the 9.8ms² acceleration.

Inside my rocket, I now feel exactly the same as I did on the mass. I cannot tell the difference. Equivalence. But to an external observer my rocket isn't accelerating at exactly 9.8ms². An external observer would see the rocket in open space is accelerating at 9.8ms² less the miniscule acceleration caused by the exhaust mass needed to accelerate my rocket. ( I cannot assume my rocket has zero mass, as it reacts to gravity. Because it has mass, it requires some exhaust/propellant mass for the rocket to accelerate. )

So to the rocket, the acceleration due to gravity and due to rocket are both 9.8 ms².
To the observer the acceleration that was needed to match gravity is very slightly less than 9.8 ms².

If this is true, in the real world, the effect would be most visible at areas of very low gravitational acceleration acting on very large masses. (I think?)

Look forward to your thoughts...

This would be equivalent under the Einstein equivalence principle but not the strong equivalence principle. GR respects the strong equivalence principle so it predicts the difference that you are describing here. Meaning, this does not conflict with GR. It simply shows that there is a difference between the different equivalence principles.

 

[cianfa72]

This would be equivalent under the Einstein equivalence principle but not the strong equivalence principle.

Could you be more precise about the difference between weak and strong equivalence principle ? Thanks.

 

[Dale]

Could you be more precise about the difference between weak and strong equivalence principle ? Thanks.

The weak equivalence principle says that all non-gravitating test particles in free fall are equivalent.

The Einstein equivalence principle says that all local non-gravitational experiments are equivalent.

The strong equivalence principle says that all local experiments are equivalent.

The difference between the strong and the weak equivalence principle is that the strong equivalence principle is not limited to objects that are small enough to not have their own gravitational field and the experiments are not limited to measuring free-fall. When most people speak of "the equivalence principle" they are usually referring to the Einstein equivalence principle. The OP has described an experiment that shows the difference between the Einstein equivalence principle and the strong equivalence principle, which is also satisfied by GR.

Inside my rocket, I now feel exactly the same as I did on the mass. I cannot tell the difference. Equivalence. But to an external observer my rocket isn't accelerating at exactly 9.8ms². An external observer would see the rocket in open space is accelerating at 9.8ms² less the miniscule acceleration caused by the exhaust mass needed to accelerate my rocket.

Specifically, the inside observer can perform non-gravitational experiments inside the rocket to test the Einstein equivalence principle. The outside observer, looking specifically at the gravitational attraction between the exhaust and the rocket, is using the strong equivalence principle.

 

[cianfa72]

The Einstein equivalence principle says that all local non-gravitational experiments are equivalent.

Releasing a free body inside the (proper) accelerating rocket is a local non-gravitational experiment as viewed locally in the rest rocket frame ?

 

[Dale]

Releasing a free body inside the (proper) accelerating rocket is a local non-gravitational experiment as viewed locally in the rest rocket frame ?

Yes. Although one of the reasons for the strong equivalence principle is that the distinction between gravitational and non-gravitational experiments is not always clear.

 

[cianfa72]

Which could be another type of local non-gravitational experiment to perform ?

 

[Dale]

Which could be another type of local non-gravitational experiment to perform ?

Anything with EM, testing Maxwell's equations, etc.

 

[Sagittarius A-Star]

Which could be another type of local non-gravitational experiment to perform ?

measuring the "gravitational" red/blueshift

 

[cianfa72]

Now suppose that we have two rockets. Each is firing its engines. One is not in a gravitational field and the other is in a uniform gravitational field. The one in the gravitational field is motionless and not accelerating. The one not in a gravitational field is accelerating.

The acceleration of rockets in both cases should be their proper acceleration as measured by an accelerometer attached to each of them. Both rockets share the same proper acceleration (in each of the two cases). In some sense it is an absolute/invariant physical quantity.

 

[Dale]

The acceleration of rockets in both cases should be their proper acceleration as measured by an accelerometer attached to each of them. Both rockets share the same proper acceleration (in each of the two cases). In some sense it is an absolute/invariant physical quantity.

Their proper acceleration is the same. Their coordinate acceleration is different. No measurable outcomes depend on the coordinate acceleration.

 

[cianfa72]

Their proper acceleration is the same. Their coordinate acceleration is different. No measurable outcomes depend on the coordinate acceleration.

As coordinate acceleration (i.e. the derivative of the spacelike coordinates of the rocket w.r.t. the timelike coordinate taken as rocket's path parameter) you mean the acceleration of the rockets in any coordinate chart that cover the spacetime region involved.

 

[cianfa72]

From Sean Carroll lectures on GR I found the following claim:

In fact, it is hard to imagine theories which respect the WEP but violate the EEP.
Consider a hydrogen atom, a bound state of a proton and an electron. Its mass is actually
less than the sum of the masses of the proton and electron considered individually, because
there is a negative binding energy — you have to put energy into the atom to separate the
proton and electron. According to the WEP, the gravitational mass of the hydrogen atom is
therefore less than the sum of the masses of its constituents; the gravitational field couples
to electromagnetism (which holds the atom together) in exactly the right way to make the
gravitational mass come out right. This means that not only must gravity couple to rest
mass universally, but to all forms of energy and momentum — which is practically the claim
of the EEP.​

In this sentence I think the first occurrences of the term mass (in Italic underline) should be actually understood as "inertial mass". Then the claim in bold (since according to WEP the gravitational mass of the hydrogen atom has to be equal to its inertial mass) from a logical point of view should actually be:

"According to the WEP, the gravitational mass of the hydrogen atom is therefore less than the sum of the gravitational masses of its constituents"

Does it makes sense ?

 

[Dale]

I think that the unqualified term mass in that statement is indeed intended to be inertial mass. But I think that they are consistent in that usage and that the final use of the unqualified term mass is also intended to be inertial mass and not switching to gravitational mass.

But since those are equivalent even in the weak equivalence principle, does it make a difference either way?

 

[cianfa72]

But I think that they are consistent in that usage and that the final use of the unqualified term mass is also intended to be inertial mass and not switching to gravitational mass.

Yes, right. It also makes sense from a logical point of view.

Anyway, as you said, there is no difference since they are equivalent thanks to WEP.

 

[PeterDonis]

The OP question has been addressed. Thread is now closed. Thanks to all who participated.