# I Is Einstein's elevator different for gravity?

1. Apr 8, 2017

### csullens

It occurs to me that any force which acts on all of your atoms individually (as gravity does) would have the same feature that one would not "feel" accelerated. We only detect acceleration because parts of the system are acted on by different amounts right? When we accelerate in a car, the seat pushes on our back and we "feel" that because the car only accelerates our back at first, creating increased pressure and causing a chain-of-pushing that we feel (if we had a spring with a mass or a pendulum, we could see the acceleration too). But if a car somehow applied a force evenly on every atom in our body, it could accelerate us arbitrarily fast, and we would not be able to detect it. And if we had springs and pendulums with us, they would be uniformly accelerated too and we would not be able to detect the acceleration even with instruments right? So I have a hard time seeing how this thought experiment was at all surprising, and how it differs from any other force which may act on each individual atom/particle through a field.

If you were a charged particle in an electric field, you would not "feel" accelerated either (assuming the charge was uniformly distributed) right? As I ask this question it also occurs to me that I suppose the charge on each atom would have to be proportional to the mass of the atom or else some atoms would be accelerated quicker than others and thus the acceleration would be detected. So is it really just the fact that inertial mass is the same thing as "gravitational charge" ensures this is always true? Is this the insight of this thought experiment? That the same stuff that causes acceleration is the very same stuff that resists it? So all matter is subject to force which is balanced by its own resistance to force, and thus all matter always accelerates by the same amount as all other matter within a uniform gravitational field?

2. Apr 8, 2017

### Staff: Mentor

No. We detect acceleration because it causes effects [Edit: corrected from "stresses" here and below] in objects that would otherwise not be there. Even if you act on every atom in the object the same way, the effects are still there because the equilibrium state of the object is different when it is accelerated than when it is in free fall.

Last edited: Apr 8, 2017
3. Apr 8, 2017

### FactChecker

That's right. A free-fall situation in gravity would feel the same as unaccelerated linear motion. One fundamental consequence of General Relativity is that gravity curves space-time so that a body in free-fall from gravity is following a geodesic space-time path.
What other force is like gravity? Magnetic and electric fields effect positive and negative differently and are easily detected. The thought experiment for gravity goes deeper than that. The fascinating part is how the speed of light can still be constant. The consequences are profound. A massive body causes time to slow down as you get closer. That leads to curved space-time.

4. Apr 8, 2017

### Staff: Mentor

GR explains this by gravity not being a force. Also, "acceleration" in GR means proper acceleration, and any object in free fall has zero proper acceleration. So in GR, objects moving solely under the influence of "gravity" have zero acceleration, which is why it feels the same as unaccelerated linear motion in flat spacetime (i.e., in regions where gravity is negligible).

5. Apr 8, 2017

### csullens

Imagine something like the electric force. If a set of charged particles started accelerating due to a uniform electric field, they would all accelerate uniformly (so long as their weights and charges were the same). As such, the particles looking around at each other wouldn't notice that they were being accelerated at all. How would they be able to detect the force (or the field) if they were inside an elevator of charged particles that was itself being accelerated in the same way. Or imagine any force which acts on particles through a field. If the particles each had the same ratio of mass to "charge" they would be accelerated uniformly and would have no way of knowing that they were being accelerated.

6. Apr 8, 2017

### csullens

If this were true, then it seems like it would violate the Einstein's elevator thought experiment. Also, if you act equally on every atom and subatomic particle, then I really don't think you would create stresses on the object. There would be no tendency for any part of object to move differently than any other part, because each part would be moved by the same amount and by the same acceleration. Since no individual part "wants" to move in relation to any other part, then there would be no added stresses to the system. Imagine just a set of particles not moving in relation to each other, but not bound to each other. If they were each accelerated by the same amount, they would not move in relation to each other.

7. Apr 8, 2017

### Staff: Mentor

Yes, they would. I said in an earlier post that stresses in the object would be present, but that isn't necessarily the case (I've edited the post to correct this); you could apply accelerations to each atom in the object that would maintain their relative distances as they were before the object was accelerated (the technical term for this is "Born rigid acceleration"). But there would be effects that would be present and detectable. The simplest would be that clocks attached to different atoms in the object would run at different rates relative to each other: this could be tested by sending round-trip light signals between the atoms.

8. Apr 8, 2017

### csullens

Can that really be true? Particles that have no relative motion and being accelerated in the same direction and magnitude would have clocks moving at different speeds? How could the clocks decide to move differently when the only difference between the particles is their position in space?

9. Apr 8, 2017

### Staff: Mentor

Yes. It's the basis for Einstein's argument for gravitational time dilation based on the equivalence principle: he showed using SR that there will be time dilation from the bottom to the top of a room being accelerated in flat spacetime, and inferred from the EP that there would then also be time dilation from the bottom to the top of a room sitting at rest on the surface of a planet like the Earth.

10. Apr 8, 2017

### Staff: Mentor

The clocks aren't "moving differently" in the sense of moving relative to each other. The time dilation arises from the way their worldlines "sit" in spacetime, not from any relative motion.

11. Apr 8, 2017

### csullens

Wouldn't such time dilation be due to the different curvature of space time in the different locations though? If you were in a uniformly curved space-time, such that the curvature at the top of the elevator was the same at the bottom (at least by approximation) then this effect would disappear right? It's known that the thought experiment of the elevator falls apart if you don't allow for the elevator to be in a uniform gravitational field. If the bottom of the elevator is accelerated more strongly than the top because you are close enough to a gravitational source then you can detect the gravity inside the elevator and the thought experiment fails.

12. Apr 8, 2017

### Staff: Mentor

No. Spacetime is flat in this scenario. In the presence of gravity spacetime as a whole is curved, but the equivalence principle is applied to a small enough patch that the curvature is not observable, so it looks like flat spacetime.

13. Apr 8, 2017

### csullens

Well, I'm talking about the case of free-fall, not sitting on the surface resisting the pull of gravity. By Einstein's own thought experiment, the clocks in various places inside the elevator should all agree, otherwise you could use the clocks to conclude that your elevator is actually in a gravitational field rather than floating in space. I feel like this is even a bit circular because you are using Einsteins conclusions to help explain this experiment. He started by saying that there is no measurable difference between free fall in a gravitational field and floating freely in deep space right? Then he investigated carefully the implications of such a claim. From that he deduced things about clocks changing speed and massive bodies bending space-time, he didn't start there. What I'm saying is that it seems that the only thing that makes gravity have this property that Einstein so carefully explored, is that gravity accelerates all masses equally. If it were true that some particles accelerated faster than others, then you would be able to detect the difference between free-fall in a gravitation field versus floating in space by simply watching the behavior of these different particles. But the only requirement for all masses to accelerate by the same amount is that the force applied is proportional to the mass. If there were another "charge" which was proportional to the mass of the particle, it would react to a force field in exactly the same way. There would be no way to tell the difference between those particles freely accelerating due to their "charge" versus freely floating in space (please don't use the conclusions of GM to contradict this, because Einstein didn't know the conclusions when he created the thought experiment). Again, it seems to me it comes back to the fact that the gravitational "charge" is mass. And the resistance to a change in inertia also mass. It seems to me that this is the equivalence which must be at the heart of GM. If gravitational mass were not the same as inertial mass, then F = minertiala = mgravitationalg would not reduce to a = g, but rather a = mgravitationalg/minertial and the equivalence principle would be out the window.

14. Apr 8, 2017

### Staff: Mentor

I thought you were talking about a hypothetical force that is not gravity, but acts equally on all atoms in an object. If you're talking about gravity, then we already know experimentally that objects moving solely under gravity have zero acceleration--accelerometers attached to them read zero.

If the elevator is in free fall, yes. But if the elevator is in free fall, no instruments can detect acceleration because there is no acceleration--accelerometers read zero.

You appear to be confusing two different concepts that the term "acceleration" can refer to. In GR, these concepts are called "proper acceleration" and "coordinate acceleration". Proper acceleration is the one I've been talking about, because it's the one that's directly measurable by accelerometers and is the same regardless of which coordinates you adopt. Coordinate acceleration depends on the coordinates you adopt, and in any relativistic theory no physical quantity can depend on the coordinates you adopt.

As an example of the two concepts, consider the classic apple that Newton saw falling from a tree. In coordinates in which the Earth and the tree are fixed, the apple has nonzero coordinate acceleration; but in coordinates in which the apple is fixed and the Earth and the tree are moving, the apple has zero coordinate acceleration. But in either set of coordinates, the apple has zero proper acceleration, and the Earth and the tree have nonzero proper acceleration.

He did, but he also said something else: that there is no measurable difference between being at rest in a gravitational field and accelerating uniformly in flat spacetime. And it's the latter statement that let him figure out things about gravity. See below.

From the second of the two statements I gave above, yes. Clocks inside a rocket accelerating in flat spacetime go at different rates--one higher up goes faster than one lower down. Einstein deduced from that that clocks inside a room at rest on the surface of the Earth would also go at different rates--one higher up goes faster than one lower down.

No; the above is not enough to get to spacetime curvature. Spacetime curvature is tidal gravity; it requires globally looking at how the "gravitational field" varies from place to place. A measure of how much harder this is than the basic insight given above is that Einstein had the basic insight given above in 1907, two years after he published SR, but it took him eight more years, until 1915, to find the correct equation--the Einstein Field Equation--that tells us how massive bodies bend spacetime.

This is using "accelerates" in the sense of coordinate acceleration, which is the wrong sense. A better way to say it is the way I said it before: an object moving solely under gravity is in free fall, with zero proper acceleration; and this is true regardless of the object's mass. In GR, this tells us that the trajectory of such a body is determined by the geometry of spacetime, not by any property of the body.

In GR, gravity is not a force. A "force" in GR is defined as something that causes nonzero proper acceleration. The reason GR does this is that, as I said above, proper acceleration is independent of your choice of coordinates, and we want "force" in GR to also be independent of coordinates, like any physical quantity must be in a relativistic theory. (Note, btw, that this definition of "force" also applies in special relativity.)

This is true in Newtonian gravity, yes. But in GR, there is no such thing as "gravitational charge" in the sense you are using the term here, because, as above, gravity is not a force. And because it is not a force, "inertial mass" has nothing to do with determining the trajectory of a body moving solely under gravity either. As I said above, it's the geometry of spacetime that determines the trajectory. In other words, in GR, the "equivalence between inertial and gravitational mass" is not even a question, because there are not two separate concepts, there's only one, and it only comes into play when we subject an object to a force in the GR sense, i.e., something other than gravity.

I think you mean "GR" (General Relativity), not "GM". The equivalence which is at the heart of GR is the equivalence between a small enough patch of curved spacetime (in which gravity is present) and a small patch of flat spacetime (in which there is no gravity). This allows us to apply all the laws of special relativity in a small enough patch of spacetime even if the spacetime is curved, i.e., even if gravity is present. That means that we can learn a lot about how things work when gravity is present by applying our SR knowledge of how things work when gravity is not present. The only thing we need to add is figuring out the relationship between different small patches in a curved spacetime, i.e., in finding a global spacetime geometry that describes how all the small patches, in each of which the laws of SR apply, fit together.

15. Apr 8, 2017

### csullens

Before I say anything else, I want be sure to thank you for your very thoughtful responses and you attempt to help me understand.

It seems to me that you are having difficulty pretending that you don't know the conclusions of GR to get into the mind of Einstein as he initially pondered the elevator thought experiment (before even he knew the conclusions of GR). What I'm trying to get at is, what was it about the elevator thought experiment that prompted Einstein to develop GR. At the time he had essentially the old notions of Newtonian gravity still, and this is the context in which he pondered the falling elevator. Gravity at the time was perfectly analogous to the electric force mathematically (I'm sure you are familiar with the equations), except that the electric force repels like charges, whereas with gravity it attracts. But the mechanics were the same. The force proportional to the charge and inversely proportional to the square of the distance for each. Given that perspective, and without knowing ahead of time the conclusions of GR, Einstein was able to realize that there's no way to tell the difference between falling and floating in space, (and as you pointed out, accelerating in space or sitting still in a gravitational field) was able to figure out that there was something worth exploring there, and was eventually able to create GR.

All you need to know to conclude that you can't tell the difference between falling due to a force and floating in space, is that all objects will be accelerated uniformly, including every single atom inside the object. Not knowing anything about GR, one can conclude that a force which accelerates every single atom simultaneously, by the same amount and in the same direction (with a newtonian conception of forces, gravity, and acceleration), would be undetectable by the particles being accelerated if they were locked in a box and couldn't see around them. Similarly, if the particles were not allowed to "fall" (because the box was being held in place), they would all collide with the side of the box, and this would be indistinguishable from being in a rocket ship. Now if you had a bunch of negative charges sitting in a uniform electric field, you WOULD be able to tell the difference between floating in space and being accelerated by the field, but ONLY because some charges may be heavier than others and the heavy ones would accelerate more slowly and you would see relative movement between the particles. But if it were the case that charge was, for some reason, proportional to mass, you would have to conclude the exact same thing about charges accelerating in an electric field. There would be no way to tell the difference, and it seems that whatever Einstein concluded from that fact, he would have had to have similar conclusions about electric fields. But Einstein, it seems, noticed that there was something different about gravity than other forces (despite the obvious mathematical similarity at the time); that all massive particles are affected in proportion to their mass, and for this force alone, the acceleration is independent of "charge" and mass because they cancel out leaving only acceleration. I think he must have had an insight about that key feature. That there was a deep connection explaining why the resistance to acceleration (inertia) would happen to also be the "charge" creating gravitational attraction, a feature not shared by any other known force, and which is absolutely essential for the equivalence principle to hold.

16. Apr 8, 2017

### FactChecker

Positively charged particles would go one way and negatively charges ones would go the other way. That makes it very detectable and completely different from gravity. As @PeterDonis pointed out in post #4, GR says that gravity causes a curve in space-time and that the motion of objects in free fall is considered unaccelerated. So gravity is not a force like others. It works through curving space-time.

17. Apr 8, 2017

### csullens

Yes I realize that. I was hoping you might be able to imagine a force, like the electric force, but imagine that there was just one charge. It's not the fact that there are two different electric charges that kept Einstein from reaching conclusions about electric forces that would otherwise have been analogous to what he did with the gravitational force right? Again, I'm trying to think about the insight that Einstein had when he first came up with the thought experiment. Invoking the curving of space-time misses the point since Einstein didn't start out thinking about curved space-time and conclude the equivalence principle. It was the other way around right?

18. Apr 8, 2017

### Staff: Mentor

It's not just that fact - there are many other differences between the behavior of electromagnetism and gravity that preclude that conclusion. However, there are forces other than electromagnetism that behave locally exactly like gravity, so do encourage analogy with gravity. These are the "fictitious" forces such as centrifugal and coriolis forces, and the key insight from the elevator is that Newtonian gravitational forces can be treated as another of them.

The equivalence principle leads to the conclusion that the Newtonian gravitational force is locally indistinguishable from a "fictitious" force such as centrifugal force. However, to model the non-local (tidal) effects of gravity as a fictitious force we have to introduce curved spacetime. For example, two people standing on opposite sides of the earth experience acceleration at 1g in opposite directions - yet are not moving apart. That can't happen with elevators in flat spacetime, but it can in curved spacetime. So yes, the equivalence principle leads to thinking in terms of curved spacetime and not the other way around.

19. Apr 8, 2017

### FactChecker

I guess if it behaved exactly like gravity, then it would be indistinguishable from gravity. All the effects and conclusions would be the same. And we would call it gravity. If it looks like a duck and swims like a duck and sounds like a duck, then maybe it is a duck.

PS. You can not separate the thought experiment from the profound conclusions. Einstein would not have used the thought experiment if it didn't lead somewhere. He probably had a lot of other thoughts that led to dead ends.

20. Apr 8, 2017

### Staff: Mentor

Einstein's way of stating what he called "the happiest thought" of his life was this: "if a person falls freely, he will not feel his own weight." Note that the words "acceleration" and "force" do not appear in this at all.

No, he didn't. He already knew that Newtonian gravity was inconsistent with special relativity and was convinced that it was Newtonian gravity that was wrong, not SR. In fact, he was in the process of writing a review article that would discuss this issue when the "happiest thought" came to him.

It's true that this conclusion follows from Newtonian gravity, because Newtonian gravity includes as an assumption that inertial mass (the $m$ in $F = ma$) and gravitational mass (the $m$ in $F = GMm/r^2$) are equal. But this assumption in Newtonian gravity has no basis other than the fact that it is needed to make the theory match experiment; it doesn't follow from any other assumptions of the theory. And of course Einstein knew all this before the "happiest thought" came to him. So this, in itself, can't be what the "happiest thought" told him; it was already known to him and every other physicist who dealt with gravity.

What did the "happiest thought" tell him? It told him that, to understand how to make a relativistic theory of gravity, just looking at free fall wasn't enough, precisely because in free fall, there is no gravity. It's not felt at all. So in order to investigate gravity, we have to look at a scenario where it is felt--or at least where something is felt that has some relationship to gravity. That is what led Einstein to the next step: comparing an accelerating rocket to a room sitting at rest on the surface of the Earth. It was the equivalence between those two cases that started him on the road to GR.

Now, when considering the accelerating case, in flat spacetime, there is no gravity present, by hypothesis. Yet to a person inside the rocket, there is certainly an apparent "gravitational field": if he releases a rock, it falls (in coordinates in which the rockets are fixed) with an acceleration that is independent of its mass, just as if the rocket were sitting at rest in the gravitational field of a massive body like the Earth. This is what made Einstein realize that gravity can't be a force; the effects that we normally attribute to "the force of gravity" can be duplicated (locally) in a scenario where we have carefully excluded gravity altogether. And this also neatly explains the equivalence of "inertial mass" and "gravitational mass", which has no explanation in Newtonian gravity: there is no such thing as "gravitational mass" in the Newtonian sense because gravity is not a force at all; all things "fall" in a gravitational field with the same (coordinate) acceleration because the "field" is not being produced by a force, it's being produced by the proper acceleration of the observer.

For a good online presentation of Einstein's pathway of thought to GR, see here:

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/general_relativity_pathway/

A good printed reference is Kip Thorne's Black Holes and Time Warps; one of the early chapters discusses the subject in some detail.