Would the one accelerating please stand up?

[A.T.]

For cases where a (single) body is accelerated non-uniformly by gravity, there is an equivalent non-gravitational situation, at least in principle.

A single point like object. As soon it's big enough to be deformed by the tidal effects, you cannot reproduce that non-gravitationally.

 

[MikeGomez]

If I understand what is meant by curvature, it has to do with tidal effect and those can be ignored with a body as small as a clock. If I do not understand curvature, further explanation would be greatly appreciated.
--------------------------------------------------------------------------------------

Clock dropped in a hole through the Earth experiment.

Situation A (Gravitational)

Clock frame:
-Clock has 0 proper acceleration.
-Clock has 0 coordinate acceleration.

Earth frame:
-Clock has 0 proper acceleration.
-Clock has non-linear coordinate acceleration.

Situation B (Non-gravitational) Not to be confused with the classic accelerated elevator analogy which implies uniform acceleration. This “elevator” is a non-linearly accelerated spheroid with a hole through the major axis, and it accelerates such that a clock in space (clock goes through the hole of the elevator) sees a relative coordinate acceleration with respect to a position on the elevator which is exactly the same as the Earth situation.

Clock frame:
-Clock has 0 proper acceleration.
-Clock has 0 coordinate acceleration.

Elevator frame:
-Clock has 0 proper acceleration.
-Clock has non-linear coordinate acceleration. The clock has the same (relative) non-linear coordinate acceleration as the case for the hole-in-the-earth situation.

 

[A.T.]

If I understand what is meant by curvature, it has to do with tidal effect and those can be ignored with a body as small as a clock.

Not only the bodies have to be small, but also their trajectory. When two objects oscillate in the hole, and accelerate (coordinate wise) towards each other, that's a tidal effect of curvature too. And it cannot be reproduced in flat space time.

 

[PeterDonis]

I disagree that the equivalence principle should be interpreted as requiring an object’s path to fit into a single local inertial frame.

It isn't a matter of agreement or disagreement. The fact that, in order to have the laws of physics take the same form as they do in flat spacetime, you have to work within a single local inertial frame, is just a fact; it's how the universe works. Whether you want to use the term "equivalence principle" to refer to that fact, or something else, is a matter of terminology, not physics; but since that's the accepted use of the term, you won't get very far trying to make it mean something else.

If I understand what is meant by curvature, it has to do with tidal effect and those can be ignored with a body as small as a clock.

No, they can't. Tidal effects don't just work in the spatial dimensions; they work in the time dimension too.

Here's a simple example: take two spaceships in free fall. One is at some altitude $R$ above the center of the Earth, and is moving at exactly the speed required for a circular orbit about the Earth at that altitude. The second is at a slightly higher altitude $R + \delta R$, and is moving (at some particular instant) at the same speed as the first, so they are at rest relative to each other at that instant. Both spaceships are in free fall.

As time goes on, the two spaceships will not stay at rest relative to each other. (In Newtonian terms, this is because the second spaceship is in an elliptical orbit whose perigee is $R + \delta R$, while the first is in a circular orbit with radius $R$, so the two will move apart as the second gains altitude.) You can make the spaceships as small as you like and this will still happen, because it doesn't depend on the size of the ships, it depends only on the curvature of spacetime, and the effect shows up over time, not over some distance in space at the same instant of time.

This “elevator” is a non-linearly accelerated spheroid with a hole through the major axis, and it accelerates such that a clock in space (clock goes through the hole of the elevator) sees a relative coordinate acceleration with respect to a position on the elevator which is exactly the same as the Earth situation.

Does the spheroid have gravity, or not? I'm assuming not, since spacetime is flat. If it doesn't, what's the point of having it there? Obviously I can make the clock's motion relative to the spheroid match the first clock's motion relative to the Earth, but that's not what the equivalence principle is about. The EP doesn't say that, given any scenario in curved spacetime, we can mock up some scenario in flat spacetime that has the same relative motions. It says that the laws of physics in a local inertial frame in curved spacetime are the same as they would be in a similarly sized patch of flat spacetime. If the spheroid in the second scenario doesn't produce any gravity, it's not affecting any physics (except for the relative motions, which as I've just said, aren't the point), so it's just superfluous.

(That statement about relative motions, btw, is still false. As A.T. said, if you try putting a second free-falling clock in your spheroid scenario, with the same motion relative to the first clock as in the hole in the Earth scenario, it won't work.)

 

[Dale]

There have been a lot of posts here today, so I may be re-hashing stuff that has already been covered.

Ok. I think you are taking the equivalence principle to mean strictly a linearly accelerating elevator and the static gravitational field at the surface of the earth.

I am taking the equivalence principle to have its standard meaning. Applying the standard meaning to this specific scenario requires a uniformly accelerating elevator since only such an elevator would have a uniform accelerometer reading of 1g as would be observed in this specific scenario for the clock at rest on the surface of the earth.

To me it's just semantics to say that there is no "equivalence principle" elevator for the hole-in-the-earth scenario, and therefore the equivalence principle does not apply. The clock falling in the hole and then falling back up has a equivalent inertial counterpart.

No it doesn't. There is no way, in flat spacetime, to have one clock with proper acceleration of 1g and to have a second clock fall down and then up all while remaining inertial.

The equivalence principle specifically refers to local experiments. Local is defined to mean that tidal effects are negligible. So saying that there is no "equivalence principle" elevator for the hole-in-the-earth scenario is a way of saying that the hole-in-the-earth scenario is non-local. It cannot be replicated in flat spacetime.

 

[PeterDonis]

There is no way, in flat spacetime, to have one clock with proper acceleration of 1g and to have a second clock fall down and then up all while remaining inertial.

This is a good point that I didn't include in my previous post. The "accelerated spheroid" flat spacetime scenario works if all you want to do is reproduce the relative motion of the clock falling through the hole and the "Earth" (i.e., some large object with a tunnel in it); but it will not reproduce the relative motion of the clock falling through the hole and a clock sitting at rest on the surface of the Earth with a constant 1 g acceleration (the acceleration of a clock in flat spacetime with the same motion relative to the free-falling clock would have to be variable, just like that of the spheroid itself).

 

[MikeGomez]

From Post#63

...When two objects oscillate in the hole, and accelerate (coordinate wise) towards each other, that's a tidal effect of curvature too. And it cannot be reproduced in flat space time.

From Post #64

...two spaceships in free fall. One is at some altitude R above the center of the Earth, and is moving at exactly the speed required for a circular orbit about the Earth at that altitude. The second is at a slightly higher altitude R+δR...

From Post #66

...The "accelerated spheroid" flat spacetime scenario works if all you want to do is reproduce the relative motion of the clock falling through the hole and the "Earth" (i.e., some large object with a tunnel in it); but it will not reproduce the relative motion of the clock falling through the hole and a clock sitting at rest on the surface of the Earth...

I would like to point out that all of these remarks were made after Post# 54 where I indicated in regards to the equivalent hole in the Earth scenario that I was not envisioning a system composed of more than one body.

I agree it would be problematic with more than one body.”

 

[MikeGomez]

It isn't a matter of agreement or disagreement.

It is a matter a agreement or disagreement.

...but since that's the accepted use of the term, you won't get very far trying to make it mean something else.

No, I am not trying to make it mean something else. How would you like it if I were to do to the same, and say that what you are saying is to make the equivalence principle something different than what it is? We do have a disagreement on that subject. Please, if you feel that I have an error in my logic or that I misunderstand something, then simply state your arguments without any hyperbole.

As my argument (and I believe your counter argument) hinges the meaning of the equivalence principle, let us begin there. The equivalence in the equivalence principle concerns the equivalence of gravitational and inertial mass. This has been known since the time of Galileo. What Einstein did was to provide the correct interpretation. Mass is mass. There is no distinction between gravitational mass and inertial mass because they are one and the same. At a fundamental level, the reason that gravitational mass and inertial mass seem to be that same is that they are the same, not because nature conspires to make them appear the same.

Let us forget for a moment the hole in the Earth scenario, and return to the comparison between a body in freefall above the Earth versus a body in freefall in an accelerating elevator. The body above the Earth experiences a gravitational field. What is surprising is that what appears to be a different situation in the case of the body in freefall in the accelerating elevator is actually no different at all. There is a real gravitational field experienced by the body in freefall in the elevator.

Now consider the body in the accelerating elevator on the floor of the elevator. The body experiences a 1g acceleration due to the acceleration of the elevator. Comparing this with situation with a body on the surface of the Earth under the influence of gravity, it might appear that the two situations are different, but that is not the case. The body on the surface of the Earth experiences a real inertial acceleration.

Not only is gravitational mass equivalent to inertial mass, but the gravitational field is equivalent to an inertial field. Einstein presents the comparison of the freefalling body above the Earth with a freefalling body in an elevator to show that the two situations are the same. The reason that he chose a uniformly accelerated elevator is to show that for this example of equivalence, it is an appropriate comparison.

These examples assume the size of the bodies under consideration are small enough that they have negligible internal stress (particle). That is to say that they experience no local tidal effects, but will experience time tidal effects if their trajectory takes them along a path that will do that as has been pointed out.

Einstein could have chosen another situation to describe, such as a particle in a varying gravitational field, in which case an elevator might not be the appropriate non-gravitational counter part, but the equivalence principle would still apply, and at the heart of the equivalence principle is the equivalence between inertial mass and gravitation mass, not any inertial or non-inertial reference frame. What is pertinent to the equivalence principle is that sufficiently small regions in spacetime (such that the effect of gravitation can be neglected) can be considered such that the invariant laws of physics of Special Relativity can be extended to include the non-inertial frames of reference in General Relativity.

Sigh, I'm ready. Tell me how wrong I am.

 

[HiggsBoson1]

Yeah, I realize about the frame of reference part. What I mean is: if someone flew down from outer space, placed a watch on the falling guy, stole it back during mid-fall and brought it to flying spaghetti monster or what have you and flying spaghetti monster looked at the watch, wouldn't he see that the watch read the same time as his (after correcting for the slight acceleration from and to the earth), whereas if he picked up a clock from the Dead Sea he would see the clock show an earlier time than his own (however slight a difference it would be)? Because when the falling guy was in free-fall, he was not effected by gravitational time/length/mass warps (according to spaghetti monster's frame of reference). Or am I way off?

That sounds right to me.

 

[HiggsBoson1]

It is a matter a agreement or disagreement.
No, I am not trying to make it mean something else. How would you like it if I were to do to the same, and say that what you are saying is to make the equivalence principle something different than what it is? We do have a disagreement on that subject. Please, if you feel that I have an error in my logic or that I misunderstand something, then simply state your arguments without any hyperbole.

As my argument (and I believe your counter argument) hinges the meaning of the equivalence principle, let us begin there. The equivalence in the equivalence principle concerns the equivalence of gravitational and inertial mass. This has been known since the time of Galileo. What Einstein did was to provide the correct interpretation. Mass is mass. There is no distinction between gravitational mass and inertial mass because they are one and the same. At a fundamental level, the reason that gravitational mass and inertial mass seem to be that same is that they are the same, not because nature conspires to make them appear the same.

Let us forget for a moment the hole in the Earth scenario, and return to the comparison between a body in freefall above the Earth versus a body in freefall in an accelerating elevator. The body above the Earth experiences a gravitational field. What is surprising is that what appears to be a different situation in the case of the body in freefall in the accelerating elevator is actually no different at all. There is a real gravitational field experienced by the body in freefall in the elevator.

Now consider the body in the accelerating elevator on the floor of the elevator. The body experiences a 1g acceleration due to the acceleration of the elevator. Comparing this with situation with a body on the surface of the Earth under the influence of gravity, it might appear that the two situations are different, but that is not the case. The body on the surface of the Earth experiences a real inertial acceleration.

Not only is gravitational mass equivalent to inertial mass, but the gravitational field is equivalent to an inertial field. Einstein presents the comparison of the freefalling body above the Earth with a freefalling body in an elevator to show that the two situations are the same. The reason that he chose a uniformly accelerated elevator is to show that for this example of equivalence, it is an appropriate comparison.

These examples assume the size of the bodies under consideration are small enough that they have negligible internal stress (particle). That is to say that they experience no local tidal effects, but will experience time tidal effects if their trajectory takes them along a path that will do that as has been pointed out.

Einstein could have chosen another situation to describe, such as a particle in a varying gravitational field, in which case an elevator might not be the appropriate non-gravitational counter part, but the equivalence principle would still apply, and at the heart of the equivalence principle is the equivalence between inertial mass and gravitation mass, not any inertial or non-inertial reference frame. What is pertinent to the equivalence principle is that sufficiently small regions in spacetime (such that the effect of gravitation can be neglected) can be considered such that the invariant laws of physics of Special Relativity can be extended to include the non-inertial frames of reference in General Relativity.

Sigh, I'm ready. Tell me how wrong I am.

We are all comparable to bodies in an elevator. The earth, orbiting is the elevator, although it's speed is constant. I agree with what you said.

 

[PeterDonis]

At a fundamental level, the reason that gravitational mass and inertial mass seem to be that same is that they are the same, not because nature conspires to make them appear the same.

I agree that this is what General Relativity says. Whether this is "the correct explanation" is still, strictly speaking, an open question, since General Relativity is not a theory of everything. But that's beyond the scope of this discussion.

The body above the Earth experiences a gravitational field.

No, it doesn't. It's in free fall; it feels no force, no acceleration, and no "field".

What is surprising is that what appears to be a different situation in the case of the body in freefall in the accelerating elevator is actually no different at all. There is a real gravitational field experienced by the body in freefall in the elevator.

No, there isn't. The body in free fall in the elevator, just like the one above the earth, is in free fall, feeling no force, no acceleration, and no "field".

Now consider the body in the accelerating elevator on the floor of the elevator. The body experiences a 1g acceleration due to the acceleration of the elevator. Comparing this with situation with a body on the surface of the Earth under the influence of gravity, it might appear that the two situations are different, but that is not the case. The body on the surface of the Earth experiences a real inertial acceleration.

Except for the word "inertial", which does not belong there (if you wanted to add a qualifier to the word "acceleration" here, it should be "proper"), I have no problem with this.

These examples assume the size of the bodies under consideration are small enough that they have negligible internal stress (particle).

No problem here, although I would phrase it that we are assuming the bodies are small enough that we can model them as point particles with no internal structure, since that's a more general statement than just saying they have negligible internal stress.

That is to say that they experience no local tidal effects

Fine.

but will experience time tidal effects if their trajectory takes them along a path that will do that as has been pointed out.

No. "No tidal effects" means "no tidal effects", period. There's no difference between "time tidal effects" and any other tidal effects. (To put it another way, "local" has to mean "local in both space and time".) They're all disallowed if we're going to talk about the equivalence principle. See below.

Einstein could have chosen another situation to describe, such as a particle in a varying gravitational field, in which case an elevator might not be the appropriate non-gravitational counter part, but the equivalence principle would still apply

No, because "varying gravitational field" brings in those "time tidal effects", which aren't allowed. See below.

at the heart of the equivalence principle is the equivalence between inertial mass and gravitation mass, not any inertial or non-inertial reference frame.

Agreed. See below.

What is pertinent to the equivalence principle is that sufficiently small regions in spacetime (such that the effect of gravitation can be neglected) can be considered such that the invariant laws of physics of Special Relativity can be extended to include the non-inertial frames of reference in General Relativity.

Wrong, even by your own standards, since you just said frames were not what was important.

The equivalence principle is the statement that, in a sufficiently small region of spacetime (such that tidal effects can be neglected--all of them, including "time tidal effects"), the laws of physics take their special relativistic forms. Since those laws can be written in such a way as not to require or assume any frames at all, inertial or non-inertial, the EP does not say or need to say anything about frames. Frames are a convenience for computation. And since the equivalence between inertial and gravitational mass is necessary for the laws in a small enough region of spacetime to take their SR forms, that equivalence can indeed be viewed as part of the EP.

The reason all tidal effects ("time" or otherwise) are disallowed is that the laws of SR do not allow them, so if the laws of physics in a small enough region of spacetime are going to take their SR forms, the region must be small enough for tidal effects--all of them--to be negligible. It has to be all of them because "region of spacetime" means "region of spacetime"; its size in time is limited just like its size in space. Otherwise we would see effects that do not match the SR laws (like two free-falling objects starting out at rest relative to each other but not staying at rest relative to each other).

Tell me how wrong I am.

Done. See above.

 

[A.T.]

I was not envisioning a system composed of more than one body.

Reproducing motion in special cases has nothing to with a general principle in the sense of the standard EP, where all local experiments give the same results. It's not really a generalization of the standard EP.

 

[MikeGomez]

No, it doesn't. It's in free fall; it feels no force, no acceleration, and no "field".

Considering that every particle is an extension of a field which extends to infinity, how could we possibly conceive of it otherwise?

All of the particles composing the body in freefall are an extension of the field, while at the same time (being that they consist of mass) possesses some measure of inertia. The particles composing the elevator are an extension of the field and possesses inertia as well. So what can be said about the inertial relationship between a particle of the body in freefall and a particle of the elevator? Well, what is meant by “inertia” here is momentum. Momentum has the identical numeric value as velocity per unit of mass. However position and velocity can only be measured relatively, so it comes as no surprise that inertia (momentum) is relative as well, and it is invariant in the sense that the relative inertia between two particles is the same whether viewed as from one particle or the other.

The inertia of a particle determines its change in position from one instant to another. The relative change in position between two particles is determined by the relative inertia of each. Due to relativity, we can choose the make measurements either from the body or from the floor and we shall be justified either way. At the same time, that also says that we can not for certain make a judgment as to the absoluteness of position or velocity of one or the other. As far as the particles are concerned, they have inertia relative to the inertia of other particles, and (due to that) they change position relative to the position of other particles. That is the situation for the particles of the body in freefall above the elevator and for the particles composing the elevator as well.

Concerning the particles of the body in freefall above the Earth and for the particles composing the earth, the situation is identical. The particles all have some inertia, but that is meaningful to say only in a relative manner. Again, as in the other case, the relative inertia of the particles will advance their relative position from one instant to another towards (or away from) each other.

In neither case do the particles posses any different form of inertia or another, which can be said to be based on whether the system is viewed as gravitational or flat-space.

The equivalence principle must hold good no matter what view of gravity/inertia you take. If you prefer the stress energy tensor side of the EFE equation (ie bodies do not follow the path of least resistance, they follow the path of greatest energy) then you must be sure that this applies to the flat-space scenario as well as all others. If you prefer the QM view of gravity as an interaction, then you must check that this applies to the accelerating elevator scenario and all others. If you prefer to view things as in QLG, you need to insure that space in quantized in the accelerating elevator scenario as well as all others. Any scheme which purports to involve particle with qualities of inertia which are distinguishable in this manner (source of gravity versus source of acceleration) is either in error or a candidate for disproving the equivalence principle.

As an example, look at the exquisite symmetry between the Unruh temperature equation and the Hawking black-hole temperature equation. If, due to the scrutiny of some brilliant scientist, one of these were to be proven false, necessarily so then would the other.

 

[MikeGomez]

Reproducing motion in special cases has nothing to with a general principle in the sense of the standard EP, where all local experiments give the same results. It's not really a generalization of the standard EP.

But I am not trying to reproduce motion in special cases, and I am sorry if what I have posted indicates that. My point is quite the contrary. I believe that in every sense, motion at a fundamental level is identical in every situation.

 

[Dale]

What is pertinent to the equivalence principle is that sufficiently small regions in spacetime (such that the effect of gravitation can be neglected) can be considered

And this excludes the hole in the Earth scenario. It is not sufficiently small.

 

[MikeGomez]

Reproducing motion in special cases has nothing to with a general principle in the sense of the standard EP, where all local experiments give the same results. It's not really a generalization of the standard EP.

Ok, I see what you mean. It is in regards to my comments in post #54 about only doing the experiment with one body at a time.

There was a discussion involving clocks all over the place, flying down a whole in the earth, flying around in orbit around the planet, or the surface or whatnot, and a comment was made regarding the inability for a dropped clock to fall down and then back up in flatspace and I disputed that.

BTW, for me a distinction should not be made in the first place. I wouldn’t consider the Earth as possessing a gravitational field with any more consideration that as possessing a bunch of inertia due to its configuration of mass (energy). The Earth being only a big giant collection of particles, and the accumulated total energy of which creates a giant gravitation/inertial field which influences other matter in the vicinity such as the clocks. But for the benefit of people who view the situation in a gravitational manner (everyone else?) I suggested a configuration in the so called flat-space arena which could mimic that effect.

Away, my bumbling first attempt to present an example situation in the other arena seemed to not be able to accommodate more that one body at a time, and so I stated in post #54 that this might be problematic and we should only consider one body at a time. Since then I may be able to devise a better example which would accommodate more than one clock, however the conversation has shifted to a discussion regarding the equivalence principle, and I think its best try and get that more or less resolved before continuing with the hole in the Earth saga.

 

[PeterDonis]

Considering that every particle is an extension of a field which extends to infinity

Not in General Relativity. In GR a "particle" is just an idealized point-mass. Strictly speaking, it should have a small enough mass that its gravitational effects are negligible--the full technical term is "test particle"--and it should have no "charges" linked to other fields (e.g., no electric charge). If you want to discuss how things are under some other theory than GR, you should start a separate thread; there's no point in talking past each other.

(Even in quantum field theory, which is where the idea of a "particle" always requiring a field that is everywhere, the way you phrase this isn't quite right--it should be "every particle is an excitation of an underlying quantum field which is present everywhere in spacetime".)

what can be said about the inertial relationship between a particle of the body in freefall and a particle of the elevator? Well, what is meant by “inertia” here is momentum.

No, inertia is not momentum. They are different concepts. You are either confused about physical terminology (note that this terminology is not restricted to GR, what I said is true in Newtonian physics as well), or you are talking about your own personal theory rather than mainstream physics. In either case, this discussion appears to me to be going off the rails.

The inertia of a particle determines its change in position from one instant to another. The relative change in position between two particles is determined by the relative inertia of each.

This could be a garbled way of stating that F = ma, but I'm not sure, because you aren't including force at all. You also appear to be focusing on coordinate acceleration, which is a mistake: the key concept for purposes of the equivalence principle is proper acceleration.

The equivalence principle must hold good no matter what view of gravity/inertia you take.

Sure, nobody is disputing that. The EP is an experimental fact, so of course any theory must account for it.

 

[PeterDonis]

a comment was made regarding the inability for a dropped clock to fall down and then back up in flatspace and I disputed that.

The comment you responded to was that there is no way to have a dropped clock fall down and then back up in flat spacetime (not flat "space") in a uniformly accelerated elevator. You pointed out (correctly) that if we allow the elevator's acceleration to vary with time, we can make the dropped clock follow a "fall down and back up" trajectory in flat spacetime. (Indeed, we can make the clock follow any trajectory we wish by adjusting the elevator's acceleration profile appropriately).

You then went on to make another claim which is not correct. You claimed that this "variably accelerating elevator" scenario being able to reproduce the "fall down and then back up" trajectory of the clock falling through the hole in the Earth is an example of the equivalence principle. It isn't. It has already been pointed out to you, multiple times, that tidal effects are different between the two scenarios and that this is observable even with measurements restricted to inside the elevator only. (I said in a previous post that this is because the experiment covers a long enough period of time for such effects to become observable; but actually, it only takes time for them to become observable if we assume the elevator is small, and that's not really correct--see below.)

However, there is also another reason the two situations aren't equivalent, which has also been pointed out (though not as many times): the proper acceleration of the elevator in the flat spacetime version is different from the proper acceleration of the Earth (or any observer at rest with respect to the Earth) in the hole in the Earth version. The EP does not cover scenarios where the two "elevators" have different proper accelerations.

Finally, there is a point which hasn't been commented on explicitly, but which is worth bringing up (I referred to it briefly above): in order for a clock to follow the "fall down and then back up" scenario in flat spacetime while remaining inside the elevator, the elevator has to be large enough to contain the clock's entire trajectory. That would mean an elevator the size of the Earth, in order to correctly reproduce the distances and times involved in the hole in the Earth scenario. But, again, an "elevator" the size of the Earth in the hole in the Earth scenario is way too large to be a local inertial frame--tidal effects are easily observable, so it's clearly not the same as a similarly sized elevator in flat spacetime, regardless of proper acceleration or the clock's trajectory or anything else.

 

[MikeGomez]

Nevermind the "multiple times" bs. If you think I'm too dense to figure something then just give up on me.

You (and others) spend a lot of time responding, and you have no idea how much I appreciate that, but if you are frustrated I'd rather not carry on.

 

[MikeGomez]

So far my attempts at explaining my understanding of the equivalence principle have been a bit long winded, so here is a short one.

1:
The equivalence principle means that a body at the surface of the Earth or in freefall (small body with negligible local tidal effects) experiences equivalent (more or less exact) physical effects as does a body which is far removed from gravitation does in an (proper) accelerated container, either in freefall or on the floor of that container (aka elevator). An accelerometer the surface of the Earth reading 1g does not know whether it is really at the surface of the Earth or in an accelerated elevator out in space.

But the equivalence principle doesn’t only mean that.

2:
It also means that an accelerometer in a 0.5g elevator on Earth will not be able to determine if it is really at the surface of the Earth or in an accelerated elevator in space at 1.5g. This case covers all (infinite number of) of combinations of…

static_gravitation_on_earth + uniform_acceleration_on_earth = some_uniform_accelleration_in_space

And equivalence principle also means…

3:
static_gravitation_on_earth + non_uniform_acceleration_on_earth = some_non_uniform_accelleration_in_space

And equivalence principle also means…

4:
variable_gravitation_on_earth + uniform_acceleration_on_earth = some_non_uniform_accelleration_in_space

And equivalence principle also means…

5:
variable_gravitation_on_earth + non_uniform_acceleration_on_earth = some_non_uniform_accelleration_in_space

Can we agree on this definition of the equivalence principle?

 

[PeterDonis]

The equivalence principle means that a body at the surface of the Earth or in freefall (small body with negligible local tidal effects) experiences equivalent (more or less exact) physical effects as does a body which is far removed from gravitation does in an (proper) accelerated container, either in freefall or on the floor of that container (aka elevator). An accelerometer the surface of the Earth reading 1g does not know whether it is really at the surface of the Earth or in an accelerated elevator out in space.

This is true if you add the qualifier "over a short enough period of time". Otherwise those "time tidal effects" come into play and the body will be able to tell whether it is in an accelerating rocket in flat spacetime or at rest in a gravitational field in curved spacetime.

It also means that an accelerometer in a 0.5g elevator on Earth will not be able to determine if it is really at the surface of the Earth or in an accelerated elevator in space at 1.5g.

I think you mean "if it is really moving upward from the surface of the Earth as an elevator with an additional 0.5g upward thrust over its weight would move", correct? An elevator with 0.5g applied in addition to its weight at the surface of the Earth will not stay at rest on the surface.

That said, this is not quite the same as the first scenario, because in this scenario, the downward coordinate acceleration of a single free-falling body inside the elevator will vary with time in the elevator in the Earth's gravitational field, whereas it won't in the elevator accelerating in flat spacetime. So "time tidal effects" will show the difference between the two cases even with only a single free-falling body. Restricting the experiment to a short enough period of time eliminates this, and with that restriction, yes, this is a valid application of the EP. But the definition of "short enough period of time" in this case will be more restrictive than in the first case.

static_gravitation_on_earth + non_uniform_acceleration_on_earth = some_non_uniform_accelleration_in_space

In this case, the non-uniformity of the acceleration makes the "short enough period of time" restriction even more stringent. I'm pretty sure it makes it unrealizable, which is why I said in previous posts that the EP does not allow non-uniform acceleration. But you're welcome to run some numbers if you want to try to pin down exactly how short a period of time is needed to eliminate all "time tidal effects" in the case of non-uniform acceleration. Similar comments apply to the other two non-uniform cases.

Can we agree on this definition of the equivalence principle?

Not as you state it. With qualifiers, for the uniform case, yes. For the non-uniform case, no. See above.

 

[PeterDonis]

those "time tidal effects" come into play and the body will be able to tell whether it is in an accelerating rocket in flat spacetime or at rest in a gravitational field in curved spacetime.

I should probably expand on this a bit. Remember that the EP doesn't just apply to a 1 g accelerating rocket vs. a person standing at rest on the surface of the Earth. It applies to all possible scenarios in any curved spacetime in which an observer "at rest" in some gravitational field experiences a 1 g proper acceleration. All of those scenarios, in a sufficiently small region of spacetime, must look the same as the flat spacetime case.

Also, the same applies to any proper acceleration, not just 1 g. So, for example, it applies to an observer at rest at the surface of a neutron star, compared to an accelerating rocket with the same proper acceleration (trillions of g's, if I've done my back of the envelope math right). And it applies to an observer "hovering" just above the horizon of a black hole. So all of these cases have to be taken into account.

 

[MikeGomez]

This is true if you add the qualifier "over a short enough period of time". Otherwise those "time tidal effects" come into play and the body will be able to tell whether it is in an accelerating rocket in flat spacetime or at rest in a gravitational field in curved spacetime.

Yes, I did mean over a short period of time. For an extended period of time the body will not feel internal stress type tidal effects because we stipulate that it is very small, however it does experience time tidal effects due to its path in a changing gravitational field.

For the body in freefall above the Earth for an extended period of time, we can integrate over the difference in gravitational potential between the start position and end position to find the time tidal effect. Is that correct? If so, then it might appear that the situation for the body in freefall in the accelerating elevator does not experience the same time tidal effect. The body appears to be in freefall and not accelerating, so we might be tempted to integrate over zero, since the change in potential in this case is zero.

But that is why I spent a bunch of time describing relative inertia. In order to find the equivalent time tidal effect for the case of the body in freefall above the accelerating elevator, the elevator would now need to be accelerating non-uniformly, and we would integrate over that changing potential between the body and the floor of the elevator.

You may feel that the body per se has experienced no time tidal effects, but that is an incorrect way to view the situation. It is that is to ignore the relative relation between the two bodies.

 

[PeterDonis]

For the body in freefall above the Earth for an extended period of time, we can integrate over the difference in gravitational potential between the start position and end position to find the time tidal effect. Is that correct?

No. Tidal gravity is manifested by nearby free-falling objects that start out at rest relative to each other, not staying at rest relative to each other. For example, two bodies that start out at rest at slightly different altitudes, and free-fall in the gravitational field of the Earth, will not stay at rest relative to each other (in Newtonian terms, this is because the one that starts slightly lower will accelerate downward, in the coordinate sense, slightly faster, and so will move away from the one that starts out slightly higher). It should be obvious that there is no way to duplicate this effect in flat spacetime.

In the case of a non-uniform acceleration in flat spacetime, there is another effect that could be described as a "time tidal effect"; I referred to it in my previous post. It is the fact that the coordinate acceleration of a freely falling object, relative to the elevator, will change with time. However, you still can't use this to duplicate tidal gravity in the gravitational field of the Earth, as I described it in the previous paragraph; two freely falling objects that start out at rest relative to each other will stay at rest relative to each other (and they will both have coordinate accelerations relative to the elevator that vary with time in exactly the same way, whereas in a non-uniformly accelerating elevator in the gravitational field of the Earth, they wouldn't).

The rest of your post is based on your incorrect understanding of what tidal gravity is; but you should also be aware that your concept of "relative inertia" is not, to the best of my knowledge, a mainstream scientific concept, or even a speculative one. If you think it is, you will need to give a reference. If not, it's a personal theory and is not a permitted topic of discussion in this forum.

 

[MikeGomez]

The comment you responded to was that there is no way to have a dropped clock fall down and then back up in flat spacetime (not flat "space") in a uniformly accelerated elevator. You pointed out (correctly) that if we allow the elevator's acceleration to vary with time, we can make the dropped clock follow a "fall down and back up" trajectory in flat spacetime. (Indeed, we can make the clock follow any trajectory we wish by adjusting the elevator's acceleration profile appropriately).

You then went on to make another claim which is not correct.

What are you tryng to say? That I am so incorrect that even when I say something correct, immediately after that I am making another claim that is incorrect?

You claimed that this "variably accelerating elevator" scenario being able to reproduce the "fall down and then back up" trajectory of the clock falling through the hole in the Earth is an example of the equivalence principle. It isn't.

Poor wording. It’s not an example of the equivalence principle. I mean that the proper invocation of the equivalence principle applies here as it can be applied in all cases.

Finally, there is a point which hasn't been commented on explicitly, but which is worth bringing up (I referred to it briefly above): in order for a clock to follow the "fall down and then back up" scenario in flat spacetime while remaining inside the elevator, the elevator has to be large enough to contain the clock's entire trajectory. That would mean an elevator the size of the Earth, in order to correctly reproduce the distances and times involved in the hole in the Earth scenario. But, again, an "elevator" the size of the Earth in the hole in the Earth scenario is way too large to be a local inertial frame--tidal effects are easily observable, so it's clearly not the same as a similarly sized elevator in flat spacetime, regardless of proper acceleration or the clock's trajectory or anything else.

Yes, the elevator would have to be large, but the density of material that it is constructed from could be of such light material (or even perforated) such that ththe gravitation due to its mass would be negligible.

 

[MikeGomez]

The rest of your post is based on your incorrect understanding of what tidal gravity is; but you should also be aware that your concept of "relative inertia" is not, to the best of my knowledge, a mainstream scientific concept, or even a speculative one. If you think it is, you will need to give a reference. If not, it's a personal theory and is not a permitted topic of discussion in this forum.

Peter,

I would like to request that you please remove this comment, reason being that I do abide by the rules of the forum. The rules are reasonable, and I especially like the parts about not putting down other members and treating members with respect even if you don’t agree with them.

In general I attempt to be careful about the way I word things around here. For example if I were to attempt to make an analogy using the term flying spaghetti monster, my point would be derailed because members here would tell me that flying spaghetti monsters were not physical and therefore such and such could never happen. Using the term “relative inertia” is not a “theory” of mine, just English words trying to explain my point of view of the equivalence principle that are either common knowledge or at least things that I thought were common knowledge, not something new. I don’t have any theories, only ideas about the way the world works just like everyone else, some correct and some incorrect.

Position is relative, as is velocity, acceleration, gravitation, size, color, weight, charge, time, and the list goes on and on. Do you really think that using the word relative associated with the concept of inertia seems outside of the mainstream? You think that is crackpottery? Ok, then fine, then I’ll refrain from doing that anymore. In addition you pointed out that I don’t know the meaning of tidal gravity. Now the obvious thought that comes to mind is that when I do understand it, I might feel inclined to describe that as relative also. Well then, if the phrase relative tidal gravity were to be declared taboo, I will honor that as well and not use that phrase either.

Dalespam says that the equivalence principle does not apply to the hole in the Earth scenario. I say that he is incorrect, and that the meaning of equivalence has to do with gravitational and inertial equivalence on a broad scale and as such covers every scenario. What I do not say is that this is a personal theory of his.

 

[Dale]

Dalespam says that the equivalence principle does not apply to the hole in the Earth scenario. I say that he is incorrect, and that the meaning of equivalence has to do with gravitational and inertial equivalence on a broad scale and as such covers every scenario. What I do not say is that this is a personal theory of his.

It does not apply. There are three different equivalence principles (taken from http://en.wikipedia.org/wiki/Equivalence_principle):

Strong: The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

Einstein: The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.

Weak: The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception.

All three refer to "local" which means that the experiment is conducted in such a small region of space and brief duration of time that tidal effects are negligible. This is simply not the case in the hole-in-the-earth scenario. The scenario involves two clocks, one at the surface of the Earth and one that falls through a hole in the Earth and back up again. The time that this requires and the distance from one side of the Earth to the other are both sufficiently large that the experiment is non-local and tidal effects are non-negligible.

In the hole-in-the-earth scenario the following are observed:
1) the distance between the two clocks starts at 0, increases, and then decreases back to 0 (as measured by radar)
2) the relative velocity between the two clocks starts at 0 when they are co-located, increases away from each other, decreases away from each other, increases towards each other, and decreases towards each other back to 0 (as measured by Doppler radar).
3) the proper acceleration of the first clock is always 0 (as measured by an accelerometer on the first clock)
4) the proper acceleration of the second clock is always g in the direction away from the first clock (as measured by an accelerometer on the second clock)
5) the proper time accumulated on the second clock is greater than the proper time of the first clock (as measured by the two clocks themselves)

These 5 results are impossible to replicate in flat spacetime. Even if you are using only the weak equivalence principle this scenario fails to qualify. The effects of motion in the hole-in-the-earth scenario are, in fact, distinguishable from an accelerated observer in flat spacetime, therefore the scenario is not local.

 

[PeterDonis]

It’s not an example of the equivalence principle. I mean that the proper invocation of the equivalence principle applies here

The "proper invocation" of the EP in the "clock in the hole" scenario eliminates basically the entire scenario; a single local inertial frame can only cover a portion of the clock's worldline so small that the variability of the "gravitational field", which is the whole point of the scenario, is not detectable. So while your statement that "the proper invocation of the equivalence principle applies here" is technically true, I fail to see the point of making it. If all you've been trying to say is that a single local inertial frame can cover a basically infinitesimal portion of the clock in the hole scenario, why have you continued to object when we point out exactly that fact?

the elevator would have to be large, but the density of material that it is constructed from could be of such light material (or even perforated) such that the gravitation due to its mass would be negligible.

You're missing the point. The elevator's gravity can be made negligible in the flat spacetime case (it would have to be for spacetime to be flat), but the Earth's gravity cannot be made negligible in the clock in the hole case. The Earth's gravity completely changes the relationship (in comparison with the flat spacetime case) between the proper acceleration experienced by the "elevator" (the Earth in the clock in the hole case) at a given point on the clock's worldline, and the relative velocity of the clock and the "elevator" (the Earth) at the same point. You can reproduce the profile of relative velocity in flat spacetime with a variably accelerating elevator, but you cannot, with the same variable acceleration of the elevator that you need to reproduce the profile of relative velocity, also reproduce the profile of proper acceleration of each point on the Earth as the clock passes it. And the proper acceleration of the piece of the Earth that the clock is passing at any given point is a locally measurable quantity. So if the EP applied to an "elevator" the size of the Earth in this scenario, it would be easy to tell the "elevator" with the real Earth moving in it from the elevator variably accelerating in flat spacetime, by the different proper acceleration profiles--meaning the EP would be violated.

 

[PeterDonis]

Do you really think that using the word relative associated with the concept of inertia seems outside of the mainstream?

Yes. If you disagree, please provide a mainstream reference. That is one of the rules of PF.

 

[PeterDonis]

Dalespam says that the equivalence principle does not apply to the hole in the Earth scenario. I say that he is incorrect, and that the meaning of equivalence has to do with gravitational and inertial equivalence on a broad scale and as such covers every scenario. What I do not say is that this is a personal theory of his.

That's good, because it isn't. Dalespam's statement about the EP follows trivially from the facts that the EP only holds within a single local inertial frame, and that a single local inertial frame cannot cover more than a basically infinitesimal portion of the clock in the hole's worldline. That is a fundamental part of the mainstream concept of the EP in GR. If you want a mainstream reference, look in any GR textbook; my personal favorite is Misner, Thorne, and Wheeler. (Section 1.6 talks about the basic meaning of "spacetime curvature" and is probably the best place to start, but there are discussions of the EP in its various forms in a number of places in the book.)

So by the rules of PF, DaleSpam's statement is not a personal theory. But unless you can provide a mainstream reference, your statement about "relative inertia" is.