I Is Einstein's elevator different for gravity?

[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?

 

[PeterDonis]

We only detect acceleration because parts of the system are acted on by different amounts right?

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.

 

[FactChecker]

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?

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.

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.

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.

 

[PeterDonis]

A free-fall situation in gravity would feel the same as unaccelerated linear motion.

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).

 

[csullens]

What other force is like gravity?

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.

 

[csullens]

No. We detect acceleration because it causes stresses in objects that would otherwise not be there. Even if you act on every atom in the object the same way, the stresses are still there because the equilibrium state of the object is different when it is accelerated than when it is in free fall.

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.

 

[PeterDonis]

the particles looking around at each other wouldn't notice that they were being accelerated at all

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.

 

[csullens]

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.

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?

 

[PeterDonis]

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?

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.

 

[PeterDonis]

How could the clocks decide to move differently when the only difference between the particles is their position in space?

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.

 

[csullens]

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.

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.

 

[PeterDonis]

Wouldn't such time dilation be due to the different curvature of space time in the different locations though?

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.

 

[csullens]

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.

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 = m_inertiala = m_{gravitational}g would not reduce to a = g, but rather a = m_{gravitational}g/m_inertial and the equivalence principle would be out the window.

 

[PeterDonis]

I'm talking about the case of free-fall

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.

By Einstein's own thought experiment, the clocks in various places inside the elevator should all agree

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 started by saying that there is no measurable difference between free fall in a gravitational field and floating freely in deep space right?

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.

Then he investigated carefully the implications of such a claim. From that he deduced things about clocks changing speed

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.

and massive bodies bending space-time,

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.

it seems that the only thing that makes gravity have this property that Einstein so carefully explored, is that gravity accelerates all masses equally.

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.

the only requirement for all masses to accelerate by the same amount is that the force applied is proportional to the mass

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.)

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.

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.

It seems to me that this is the equivalence which must be at the heart of GM.

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.

 

[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.

 

[FactChecker]

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).

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.

 

[csullens]

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.

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?

 

[Nugatory]

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.

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.

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?

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.

 

[FactChecker]

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?

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.

 

[PeterDonis]

what was it about the elevator thought experiment that prompted Einstein to develop GR

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.

At the time he had essentially the old notions of Newtonian gravity still

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.

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

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.

 

[csullens]

It's not just that fact - there are many other differences

When you look at the force law for Newtonian gravity and the force law for the electric force, they look almost identical. The similarity is striking to anyone. The differences are, it seems, the sign, and M vs Q. The point that no one seems to want grant me is the essential importance of the M in the gravitational force law being the same m that is in the inertial force law F = ma. If the gravitational force law used a charge that was not proportional to mass, but rather just was just another basic property of particles, the gravitational charge, and if you kept everything else the same, this would mean gravity would suddenly be much more like the electric force, and the equivalence principle would no longer hold.

 

[Nugatory]

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?

Yes. The easiest way to see this is to imagine that the particles are strobe lights flashing once per second (according to someone sitting right next to them in the elevator). A person sitting at the bottom of the elevator receives one flash per second from the clock next to him. How many flashes per second does he receive from a clock at the top of the elevator? A person up on the ceiling of the elevator also receives one flash per second from the clock next to him. How many flashes per second does he receive from a clock on the floor? Before you answer, consider that although the distance between floor and ceiling is constant, the distance (as observed by a non-accelerating observer watching the elevator) traveled by each individual flash of light is not.

 

[csullens]

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.

I can grant that you guys understand this material better than me. And even as I try to be humble and acknowledge that, I think there is one part of this that you have just not got quite right. And that is that any force which accelerates all objects, all particles, all atoms, equally and in the same direction, would adhere to this equivalence principle. That's really the only pre-requisite. For any force for which that would be true, the equivalence principle would hold, and any force that failed to be true, it wouldn't hold. Maybe I'm wrong about that, but its hard for me to see how.

 

[PeterDonis]

any force which accelerates all objects, all particles, all atoms, equally and in the same direction, would adhere to this equivalence principle

And would therefore not be a force, by the same argument by which Einstein concluded that gravity is not a force: its effects could be exactly duplicated in an environment where, by hypothesis, this "force" was entirely absent, and all we had was an accelerating rocket.

 

[csullens]

Well, then I suppose we have come full circle because that is precisely the point I made early on and which you claimed I was incorrect about. But now you seem to agree with it.

 

[Nugatory]

When you look at the force law for Newtonian gravity and the force law for the electric force, they look almost identical. The similarity is striking to anyone. The differences are, it seems, the sign, and M vs Q.

Yes, but there's only so far that that you can push that analogy. For example...
- The gravitational force law works if the particles are moving, whereas the electric force law does not. That's why Coulomb's law is part of electrostatics, and was a major embarrassment for classical physics which predicts that planetary orbits around the sun would be stable but electron orbits around the nucleus would not.
- If an object of mass M in a gravitational field splits into two smaller pieces, their trajectories will be very different than those of the parts of an object with charge Q moving an electric field and split into two.
- The internal stresses in your hypothetical uniformly charged object are different from the internal stresses produced by gravity.

 

[PeterDonis]

that is precisely the point I made early on

You have never, as far as I can tell, agreed with the statement that the type of "force" you describe would not be a force at all. You have insisted on viewing it as "a force that acts on everything equally". That is not the same as saying it's not a force at all.

Furthermore, I pointed out in an earlier post an obvious difference between the two cases, from the standpoint of GR: a non-gravitational force that "acts on everything equally" would still produce nonzero proper acceleration, which even if we assume will not be detectable by any stresses or strains in the object acted on (because every single atom in the object is effectively subjected to Born rigid acceleration, as I described earlier), would still be detectable by its time dilation effect. There is no such effect on an object free-falling under gravity.

So I still don't see how you are saying the same thing I am saying.

 

[Nugatory]

And that is that any force which accelerates all objects, all particles, all atoms, equally and in the same direction, would adhere to this equivalence principle.

Yes, and the only forces other than Newtonian gravity that have that property are the "fictitious" forces produced by acceleration. Thus, we consider the possibility that Newtonian gravity should be considered one of these forces as well, and that takes us to general relativity and curved spacetime.

 

[csullens]

You have never, as far as I can tell, agreed with the statement that the type of "force" you describe would not be a force at all. You have insisted on viewing it as "a force that acts on everything equally". That is not the same as saying it's not a force at all.

Furthermore, I pointed out in an earlier post an obvious difference between the two cases, from the standpoint of GR: a non-gravitational force that "acts on everything equally" would still produce nonzero proper acceleration, which even if we assume will not be detectable by any stresses or strains in the object acted on (because every single atom in the object is effectively subjected to Born rigid acceleration, as I described earlier), would still be detectable by its time dilation effect. There is no such effect on an object free-falling under gravity.

So I still don't see how you are saying the same thing I am saying.

So on the one hand a force which acts equally on everything is really "not a force at all," but on the other hand it will produce a nonzero proper acceleration and time dilation? If it wasn't really a force could it still do those things?

 

[Nugatory]

So on the one hand a force which acts equally on everything is really "not a force at all," but on the other hand it will produce a nonzero proper acceleration and time dilation? If it wasn't really a force could it still do those things?

Yes. (For the time dilation, that is. Gravity does not cause any proper acceleration; an object in free-fall has zero proper acceleration).

A few posts back I suggested that you work out gravitational time dilation for yourself by considering the behavior of clocks on the floor and ceiling of the elevator, that emit a flash of light every time they tick. Have you tried that yet?

 

[csullens]

Yes, and the only forces other than Newtonian gravity that have that property are the "fictitious" forces produced by acceleration. Thus, we consider the possibility that Newtonian gravity should be considered one of these forces as well, and that takes us to general relativity and curved spacetime.

Thank you. I remarked earlier that one seeming reason for this is that the "gravitational charge" from Newtonian gravity cancels with the inertial mass from F=ma leaving g = a on the surface of the Earth, showing that all objects will accelerate in the same way. Is this not a key feature which allowed Einstein to consider the equivalence principle for gravity, but not for other forces?

 

[csullens]

Yes. In fact, only a fictitious force can do that.

A few posts back I suggested that you work out gravitational time dilation for yourself by considering the behavior of clocks on the floor and ceiling of the elevator, that emit a flash of light every time they tick. Have you tried that yet?

If the elevator is in free fall, wouldn't that violate the equivalence principle if I should show that clocks were ticking off at different rates at different locations in my elevator?

 

[PeterDonis]

The gravitational force law works if the particles are moving

No, it doesn't; at least, not if we take relativity into account. This is one of the key problems that led Einstein to believe that Newtonian gravity is incompatible with relativity as it stands. (Note that electrodynamics does not have this problem: it is compatible with relativity as it stands.)

The issue is that the Newtonian force law has the square of the distance in the denominator. But distance in which frame? Distance is frame-dependent in relativity. So the force law as it stands cannot be relativistically covariant; it has to be rewritten.

What's more, the obvious way to rewrite it--introducing a finite propagation speed (the speed of light) by direct analogy with electrodynamics--doesn't work (unlike the case of electrodynamics), because it predicts aberration: the gravitational force no longer points directly at the position "now" of the source, due to propagation delay. (There are some complications here that I won't go into; what I've just said is a basic heuristic.) In the case of electrodynamics, the aberration predicted by the theory exactly matches what is observed. However, in the case of gravity, aberration is not observed--at least, not with the accuracy available to 19th and early 20th century physics. (Strictly speaking, this is not quite correct, since the precession of Mercury's perihelion is actually an observational effect of gravitational aberration. But nobody understood that in 1905/1907, and in any case the observed aberration in this case is still far smaller than the modified Newtonian theory I am describing predicts.)

 

[Nugatory]

Thank you. I remarked earlier that one seeming reason for this is that the "gravitational charge" from Newtonian gravity cancels with the inertial mass from F=ma leaving g = a on the surface of the Earth, showing that all objects will accelerate in the same way. Is this not a key feature which allowed Einstein to consider the equivalence principle for gravity, but not for other forces?

It's not a key feature, it's the essential problem. It is so totally amazing that the gravitational mass (the $m$ in $F=Gm_1m_2/r^2$) should be equal to the inertial mass (the $m$ in $F=ma$) that we should be predisposed to reject any theory that just assumes that it is so. The point of the elevator thought experiment is that it allows us to drop that assumption.

 

[csullens]

It's not a key feature, it's the essential problem. It is so totally amazing that the gravitational mass (the $m$ in $F=Gm_1m_2/r^2$) should be equal to the inertial mass (the $m$ in $F=ma$) that we should be predisposed to reject any theory that just assumes that it is so. The point of the elevator thought experiment is that it allows us to drop that assumption.

Thank you for that. Exactly what I'm trying to drive at! Why is it that I get the distinct impression that the two of you agree with each other completely, and yet somehow one manages to support the basic idea I'm presenting, while the other does not. LOL.

 

[Nugatory]

If the elevator is in free fall, wouldn't that violate the equivalence principle if I should show that clocks were ticking off at different rates at different locations in my elevator?

If the elevator is in free fall (and is small enough that tidal effects are negligible - that's the "local" word again - as is the case in all equivalence principle scenarios) the clocks will not tick at different rates. If the elevator is accelerating, then it's not in free fall and the clocks will tick at different rates.

But you shouldn't take my word for it. It is a really good exercise to try calculating the relative tick rates for the two clocks in the acceleration and no-acceleration cases for yourself.

 

[PeterDonis]

on the one hand a force which acts equally on everything is really "not a force at all," but on the other hand it will produce a nonzero proper acceleration and time dilation?

No. I was not describing two different ways of looking at the same "force". I was describing two different possibilities that you appear to me to be conflating:

(1) A "force" that does not produce any proper acceleration at all. This is "really not a force at all", by the same argument that Einstein used to show that gravity is not a force: we can duplicate its effects in an accelerating rocket where, by hypothesis, the "force" is entirely absent.

(2) A force that does produce proper acceleration, but acts on every atom in an object in exactly the same way, so that the distances between the atoms remain the same (Born rigid acceleration), and therefore no stresses or strains appear. An example of this would be a hypothetical object consisting of, for example, two electrons connected by a massless spring, oriented parallel to a constant electric field (for example, somewhere between the plates of a very, very large capacitor). Both electrons would be accelerated equally, so the spring would remain unstretched; but they would still be under nonzero proper acceleration and you could not duplicate the effects in an accelerating rocket where no electric field was present, because there would be time dilation between the two electrons (see below for more on that).

If the elevator is in free fall,

Then the time dilation effect is not present, as I said in an earlier post. There has to be nonzero proper acceleration.

More precisely: if we have two objects, both in free fall (zero proper acceleration), at a constant separation, there will be no time dilation between them. This applies to two objects at rest in a freely falling elevator, or two objects dropped inside a rocket that is accelerating because its engines are firing.

But if we have two objects, both under nonzero proper acceleration, then there will be time dilation between them if they are at different heights (i.e., different positions along the direction of the proper acceleration). This applies to two objects at rest at the top and bottom of an accelerating rocket, or two electrons oriented parallel to a constant electric field as in the huge capacitor I described above.

 

[Nugatory]

No, it doesn't; at least, not if we take relativity into account.

Yes, you're right. It only works if we don't take relativity into account, which is another way of concluding that the naive analogy between Newtonian gravitation and classical E&M can't be made to work.

 

[csullens]

This applies to two objects at rest at the top and bottom of an accelerating rocket,

I admit, I have no intuitive understanding of why two objects at opposite ends of an accelerating rocket, would experience relative time dilation. I will need to look into that.

 

[csullens]

If the elevator is in free fall (and is small enough that tidal effects are negligible - that's the "local" word again - as is the case in all equivalence principle scenarios) the clocks will not tick at different rates. If the elevator is accelerating, then it's not in free fall and the clocks will tick at different rates.

But you shouldn't take my word for it. It is a really good exercise to try calculating the relative tick rates for the two clocks in the acceleration and no-acceleration cases for yourself.

I'm afraid I don't have the tools to solve that problem in such a way that I would get a different answer for the rocket than I would get for the free falling elevator. Which tells me that I guess I don't know how to approach that problem correctly.

 

[PeterDonis]

I have no intuitive understanding of why two objects at opposite ends of an accelerating rocket, would experience relative time dilation. I will need to look into that.

The pitt.edu article I linked to earlier has a brief discussion of it, under "Gravitational Slowing of Clocks".

 

[pixel]

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.

I don't know the historical sequence, but the "happiest thought" does imply relativistic notions of gravity. A freely falling elevator is an inertial frame of reference. So if you aim a beam of light across the elevator, it will travel horizontally. But to an observer outside the elevator, the beam is falling along with the elevator and so follows a curved path. Thus, the "happiest thought" implies the bending of light in a gravitational field.

 

[PeterDonis]

to an observer outside the elevator

In other words, to an observer who has nonzero proper acceleration.

the "happiest thought" implies the bending of light in a gravitational field

For observers who have nonzero proper acceleration, i.e., who are at rest in the field, yes. But not for observers who are freely falling in the field. That's why Einstein had to take the next step, to looking at accelerated observers, to start obtaining results about the behavior of gravitational fields.