Would the one accelerating please stand up?

[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.

 

[Dale]

If you want a mainstream reference, look in any GR textbook; my personal favorite is Misner, Thorne, and Wheeler

And mine is Carroll's lecture notes

http://preposterousuniverse.com/grnotes/

This material is the beginning of chapter 4

 

[PeterDonis]

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.

And just to clarify my request for a mainstream reference, the key thing that needs a reference is not the term "relative inertia" by itself, but whatever it is that you are describing by those words that you think is common knowledge. From your previous posts where you describe what you mean by that term, it doesn't look to me like any mainstream concept that I'm aware of. But if you have a mainstream reference and can provide it, then we can get a better idea of what you are trying to describe.

 

[MikeGomez]

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".

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".

A gravitational field exists in all four of these cases…

1: A gravitational field exists for the body in freefall above the earth.

2: A gravitational field exists for the body at the surface of the earth.

3: A gravitational field exists for the body in freefall above the floor in the accelerating elevator scenario.

4: A gravitational field exists for the body at the surface of the floor in the accelerating elevator scenario.

"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.

Sure, everyone knows that GR is not a theory of everything, but an intellect such as Einstein makes one of the most profound statements in the history of science, and even a century later no one (and I am sure many have tried) has been able to disprove it. Yet you say that this idea in such doubt by mainstream science today that it is invalid to use in a modern day discussion of the equivalence principle?

 

[MikeGomez]

And just to clarify my request for a mainstream reference, the key thing that needs a reference is not the term "relative inertia" by itself, but whatever it is that you are describing by those words that you think is common knowledge. From your previous posts where you describe what you mean by that term, it doesn't look to me like any mainstream concept that I'm aware of. But if you have a mainstream reference and can provide it, then we can get a better idea of what you are trying to describe.

What I am trying to describe is the relative relationships between.the body in freefall and the body at the surface for both scenarios of the the planet and accelerating elevator.

Relative relationships exist for position, velocity, momentum (not inertia), gravitational attraction, and all physical aspects of what is under consideration here. The two common scenarios of the body at the planet and the body at the elevator do not apply for every physics example regarding gravitation and acceleration, but the equivalence principle does (when understood in the correct way and when applied in the correct way).

I am trying to establish the relative relationships for the bodies for the scenarios that everyone is familiar with. That is essential because once a relational understanding is established for the scenarios that everyone is familiar with, that can be extended to create appropriate scenarios for the other cases that we are discussing such as the "body falling down a hole in the earth" scenario.

 

[Dale]

The two common scenarios of the body at the planet and the body at the elevator do not apply for every physics example regarding gravitation and acceleration, but the equivalence principle does (when understood in the correct way and when applied in the correct way)

This is not correct. The equivalence principle explicitly restricts itself to local experiments. So it explicitly does not apply to non-local examples. The hole in the Earth scenario is such an example.

 

[PeterDonis]

A gravitational field exists in all four of these cases…

1: A gravitational field exists for the body in freefall above the earth.

2: A gravitational field exists for the body at the surface of the earth.

3: A gravitational field exists for the body in freefall above the floor in the accelerating elevator scenario.

4: A gravitational field exists for the body at the surface of the floor in the accelerating elevator scenario.

This is not a standard use of the term "gravitational field". (Admittedly, there is not a single "standard" use of that term; but your usage does not match either of the standard usages that I'm aware of, as I'll show below.) So you are going to have to define specifically what you mean by "gravitational field". By "specifically", I mean "in terms of the actual math of GR".

In what is probably the most common standard usage of "gravitational field", it refers to particular connection coefficients in a chosen coordinate chart. So before you can even say whether it "exists" in any of your examples, you need to specify what coordinate chart you are using. In the most natural coordinate chart for each of your cases, there is no gravitational field in cases #1 and #3, because the most natural coordinate chart is a local inertial frame in which the free-falling bodies in those cases are at rest, and in any local inertial frame the connection coefficients are zero, so there is no "gravitational field". In cases #2 and #4, the most natural coordinate chart is a non-inertial chart in which the accelerated bodies in those cases are at rest; in that chart, the appropriate connection coefficients are nonzero, so a "gravitational field" does exist in those cases. So in this usage, you are correct for cases #2 and #4 but wrong for cases #1 and #3.

The other fairly common usage of the term "gravitational field" is as a synonym for "spacetime curvature", i.e., tidal gravity. In that usage, a gravitational field exists in cases #1 and #2, but not in cases #3 and #4.

 

[PeterDonis]

Yet you say that this idea in such doubt by mainstream science today that it is invalid to use in a modern day discussion of the equivalence principle?

I said no such thing. Read what I posted again, carefully.

 

[PeterDonis]

What I am trying to describe is the relative relationships between.the body in freefall and the body at the surface for both scenarios of the the planet and accelerating elevator.

The correct term to describe that is "coordinate acceleration", not "relative inertia". In coordinates in which either body is at rest, the other body is accelerating. The other relationships all follow from that one--at least, the relationships for position, velocity, and momentum (and energy) do. I'm not sure what you mean by a "relative relationship" with reference to "gravitational attraction". (This has been a recurrent issue in this discussion--you insist on using your own idiosyncratic terminology instead of learning the standard language in which these things are described. That makes communication difficult, since the rest of us know the standard language and are using it.)

once a relational understanding is established for the scenarios that everyone is familiar with, that can be extended to create appropriate scenarios for the other cases that we are discussing such as the "body falling down a hole in the earth" scenario.

For some relationships, yes, but not for all. There are relationships in curved spacetime that cannot be duplicated in flat spacetime. We have been over this already.

 

[PeterDonis]

the equivalence principle does (when understood in the correct way and when applied in the correct way).

We have given you two mainstream references now that describe what "the correct way" is. Before making further statements about what you think "the correct way" is, I strongly advise you to read at least one of those references (the Carroll lecture notes that DaleSpam linked to are easier because they're online and free) and take some time to think over what it is telling you.

 

[Dale]

@MikeGomez FYI, it is considered very poor form on PF to not provide a reference when asked, even if the concept seems to you to be completely standard or obvious. Such references provide valuable learning material, clarify the point being made, and serve to ensure that the content of PF remains consistent with the professional scientific community. Please take all such requests seriously.

 

[MikeGomez]

@MikeGomez FYI, it is considered very poor form on PF to not provide a reference when asked, even if the concept seems to you to be completely standard or obvious. Such references provide valuable learning material, clarify the point being made, and serve to ensure that the content of PF remains consistent with the professional scientific community. Please take all such requests seriously.

Sorry if I'm taking too long to reply guys. I am extremely hard pressed for time here.

My reference is "Relativity" by Albert Einstein, last printed in 1952.

Thank you both for the references you have provided for me. I have the link that you have provided and at first glance it looks great. I'll download the MTW also. I've seen plenty of references to that one so it so it must be good as well. Please be a little patient and I will get back.

 

[MikeGomez]

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.

We need to produce a gravitational field in flat spacetime for the clock in the hole, which is uniform (uniform in the sense that it does not create proper acceleration). I do not agree about the worldlines and that has to with the ongoing issue we have about the meaning of the equivalence principle.

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.

Once again issues regarding equivalence principle.

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.

Einstein EP or stong EP.

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.

Number 1 looks fine, although we shouldn't need to measure the distances by radar.

Number 3 is fine. The proper acceleration for the hole in the Earth clock must remain zero by definition of the problem.

Number 5 is the difference in time in the Earth case which must be reproduced in the flat spacetime scenario, so that is fine.

Numbers 2 & 4 are the problem, and yet again (no surprise), this is due to what we think about the meaning of the equivalence principle.

Please be a little patient, I am under serious time constraints here.

Thank you.

 

[harrylin]

A gravitational field exists in all four of these cases…

That depends... I can easily see how that phrasing resulted in some disagreement! I have the impression that - as all too often - this discussion has deteriorated onto a rather useless discussion about words. Perhaps you will agree with the following refinement of your statement, as follows.

Coordinate systems can be chosen by means of which according to Einstein's GR a gravitational field appears to exist or may be held to exist in all four of these cases:

1: The case of a body in freefall above the earth.

2: The case of a body at the surface of the earth.

3: The case of a body in freefall above the floor in the accelerating elevator scenario.

4: The case of a body at the surface of the floor in the accelerating elevator scenario.

[..]
My reference is "Relativity" by Albert Einstein, last printed in 1952 [..].

That reference may suffice indeed. Here's my attempt on specifying appropriate references for 1-4 as reformulated by myself, taken from https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_II:

1. Check - using any reference system fixed to the Earth:
"The action of the Earth on the stone takes place indirectly. The Earth produces in its surrounding a gravitational field, which acts on the stone and produces its motion of fall."
- OK.
2. Check - using any reference system fixed to the Earth: I did not find an appropriate reference in that discussion.
However, a body at the surface of the Earth is surely not an issue.
3. Check - using any reference system fixed to the accelerating chest:
"the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment. Relying on his knowledge of the gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time."
- OK.
4. Check - using any reference system fixed to the accelerating chest:
"the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time."
- OK.

 

[A.T.]

We need to produce a gravitational field in flat spacetime for the clock in the hole, which is uniform (uniform in the sense that it does not create proper acceleration).

That is not what uniform gravitational field usually means. Uniform gravitational field usually means that it produces no tidal effects. Free falling bodies (point masses) always have zero proper acceleration, regardless whether the gravitational field is uniform or not.

The proper acceleration for the hole in the Earth clock must remain zero by definition of the problem.

The proper accelerations of all involved objects must be reproduced, in order to have an equivalence of physical laws between the two cases.

 

[PeterDonis]

(uniform in the sense that it does not create proper acceleration)

A gravitational field never "creates" proper acceleration; objects moving solely under gravity are always in free fall, feeling zero proper acceleration. This is true for the Earth's gravitational field (or any field) just as much as for a "uniform" gravitational field. Any object that feels proper acceleration, feels it because some non-gravitational force is acting on it.

As A.T. said, the usual way of distinguishing a "uniform" gravitational field (i.e., one produced by acceleration in flat spacetime) from a "real" gravitational field (i.e., one produced by a massive body in curved spacetime) is by tidal effects; the latter has them, the former does not. However, as I pointed out before, in both cases the presence of the "gravitational field" in this sense depends on a particular choice of reference frame: you have to pick a frame in which the object feeling proper acceleration is at rest (i.e., the object at rest in the "elevator", the object at rest on the Earth's surface). If you pick a frame in which a freely falling object is at rest, the "gravitational field" vanishes.

I do not agree about the worldlines and that has to with the ongoing issue we have about the meaning of the equivalence principle.

We should defer further discussion of that until you've had time to look at references, since you said you were doing that.