Does the EP Violate in Two Mass Test?

[paw]

We have a large, spherical, mass M in free space away from any other gravitational influence. This so we can claim spacetime is locally flat.

Two small masses, A and B, are positioned, far from M, such that the distance between them is d. Further, A and B are parallel to a tangent on the surface of M.

When masses A and B are released (t=0) they will start to fall toward M.

Q1: Will an observer falling with A and B see the distance (d) between them shrink as a function of t (as they approach M)? That is, are the geodesics followed by A and B parallel or not?

If the answer is that the geodesics are not parallel then the observer should conclude there is a force (or pseudo-force) acting to push A and B closer together.

Q2: If this is true then is it not possible to build a machine which could tell an observer in an 'elevator' whether they are falling in a gravitational field or they are accelerating? Doesn't this violate the strong form of the EP? I would rather think it doesn't but I can't see where my reasoning is wrong.

 

[Chrisc]

paw, the construct you described will provide A and B with evidence of the change in distance d. But this does not test the principle of equivalence quite the way you might think. It is arguably the gravitational attraction of A and B that shortens the distance d. If you think about any test of the principle as having the necessary criteria of showing a discrepancy or distinguishing characteristic in the laws of mechanics instead of the measurable kinematics of an experiment, it will help make this more clear. That does not mean that measurable kinematics are not evidence of the laws but they are not in themselves definitive proof of a change or in-equivalence.

 

[cesiumfrog]

Yes, that method distinguishes a planet's gravity from a rocket's acceleration, but it is not a local measurement (and therefore is irrelevant):
You have relied on the size of your apparatus (distance AB) being "similar" (in terms of your instrument precision) to the size of the region of (possible) space-time curvature (distance AM).

Mathematically, you are employing values of the metric at multiple locations in the manifold, rather than only employing the gradient of the metric at a single location. It's as if you laid down a gigantic circle and measured the ratio of circumference to diameter.

A similar way of cheating is to notice that the electric field around a point charge is different when held near a mass than to when accelerated in empty space, if ignoring that you have waited for the electric field to orbit about the mass (bringing back 'global' information).

(Chrisc, incorrect: recognise weasel-word "arguably" and consider decreasing the test particle masses.)

 

[paw]

Yes, that method distinguishes a planet's gravity from a rocket's acceleration, but it is not a local measurement (and therefore is irrelevant):
You have relied on the size of your apparatus (distance AB) being "similar" (in terms of your instrument precision) to the size of the region of (possible) space-time curvature (distance AM).

Thank you. I should have seen that but didn't. Now that I think of it this scenario is much like the one using tidal effects. I guess I need to remember the 'elevator' needs to be small enough that these effects don't appear.

 

[Chrisc]

(Chrisc, incorrect: recognise weasel-word "arguably" and consider decreasing the test particle masses.)

cesiumfrog, "weasel-word"?
Funny but not much of a critique.
Decrease the test particle masses as much as you want.
Or increase the ratio of the "undefined" distances of A,B to A,M and B,M.
Or increase the rate of acceleration of A, B and the observer toward M.
I think it's called...relativity.

 

[yuiop]

Proof that 2=1

Draw a long horizontal line representing the x axis. Draw 2 vertical rods separated by a distance x. The lower point of each rod lies on the x axis. One rod is twice the length of the other rod. Draw a line connecting the top of the rods. For sufficiently large x (and squinting a little bit) conclude that the line connecting the top of the rods is sufficiently parallel to the x-axis that we can conclude that it IS parallel to the x axis. We have now proved that that both rods have equal length. Therefore 2=1 and any number equals any other number. Now base a whole theory on this aproximation and and assume your theory has sound foundations.

 

[Chrisc]

That's more true than many are willing to admit and few have the courage to face.
All of GR depends on one's willingness to accept its "psychologically natural" path.
The same is true for all of physics.
The cool dude that he was, Einstein had the courage to make that clear from the beginning.

 

[RandallB]

?
How does "squinting a little bit" so you see something different than you measure "prove" anything?

 

[RandallB]

We have a large, spherical, mass M in free space away from any other gravitational influence. This so we can claim spacetime is locally flat.

Problem is you are not describing a “locally flat” space in GR for gravity. (Note: GR does not derive directly from SR or ‘spacetime”).
A “ large, spherical, mass” does not give a uniform gravitational field. Acceleration does not remain constant but increases at shorter distances from the mass. Thus there is a “tidal effect”. Like the one where the front of the asteroid approaching Jupiter was torn off moving ahead of the rest it, while the back did not increase acceleration quickly enough to keep up; breaking it into several parts.

In order to establish a uniform field in Flat space you can hypothetically design a flat uniform density mass extending out to infinity in all directions. All objects placed at any distance perpendicular to this flat surface mass would accelerate at one fixed rate of acceleration when released in such a “Flat Space”. And yes any two released together they would maintain an unchanging distance of separation and fall along perfectly parallel lines.

 

[yuiop]

?
How does "squinting a little bit" so you see something different than you measure "prove" anything?

Sorry for using sarcasm to make a point. The point I was making is that by taking the paths of two horizontally separated falling particles and making the aproximation that the two nearly parallel paths ARE parallel then you do not rigorously prove anything. The EP principle seems to make this aproximation.

 

[paw]

Problem is you are not describing a “locally flat” space in GR for gravity.

Thanks. I do see the error I introduced in the OP. It was a temproary 'brain lock' from reading too many posts in another thread. : )

 

[yuiop]

Problem is you are not describing a “locally flat” space in GR for gravity. (Note: GR does not derive directly from SR or ‘spacetime”).
A “ large, spherical, mass” does not give a uniform gravitational field. Acceleration does not remain constant but increases at shorter distances from the mass. Thus there is a “tidal effect”. Like the one where the front of the asteroid approaching Jupiter was torn off moving ahead of the rest it, while the back did not increase acceleration quickly enough to keep up; breaking it into several parts.

In order to establish a uniform field in Flat space you can hypothetically design a flat uniform density mass extending out to infinity in all directions. All objects placed at any distance perpendicular to this flat surface mass would accelerate at one fixed rate of acceleration when released in such a “Flat Space”. And yes any two released together they would maintain an unchanging distance of separation and fall along perfectly parallel lines.

There is a serious problem with your description of a uniform gravitational field. Your introduction of a hypothetical flat infinite plane does indeed reproduce the requirement that falling particles fall parallel to each other but it introduces another possibly bigger problem.

You are also right that "All objects placed at any distance perpendicular to this flat surface mass would accelerate at one fixed rate of acceleration when released in such a “Flat Space”.

Now here is the problem:

The Bell's spacships paradox shows that observers stationary in such a gravitational field will see themselves as moving apart spatially over time. For example a tower standing on this infinite plane would appear to be getting larger over time to observer on different floors of the tower. Eventually the tower would break up.

The integral of the gravitational force GM gives a potential of GMR rather than the usual -GM/R. If gravitational time dilation is a function of gravitational potential then the gravitational time dilation factor for such a field would be something like $1/\sqrt{1+2GMR/c^2}$ rather than the usual Schwarzschild $1/\sqrt{1-2GM/(Rc^2)}$.

 

[yuiop]

The point I was trying to make in post #12 is that the measurements made in a rocket undergoing Born rigid acceleration can not be exactly reproduced by a gravitational field, not even a hypothetical gravitational field. The inifinite flat plane gravitational body fixes the parallel falling path problem but it does not reproduce the acceleration profile of acceleration/R experienced by the Born rigid accelerating observers. Any hypothetical gravitational body that reproduces the acceleration/R requirement does not reproduce the parallel falling path requirement. Any equivalance is necessarily an aproximation of the parallel falling paths or of the acceleration versus height profile and this requirement is presumably met by taking an infinitesimally small region and ignoring the small differences. What concerns me is that if General Relativity is based on igonoring small differences in the EP then when GR is applied to extreme situations such as within the singularity of a black hole can be absolutely confident in the predictions of GR and can we be certian those small ignored aproximations do do multiply in magnitude in extreme sitations? I think Einstien had the same concerns and that may explain why Einstein was reluctant to extend GR to the prediction of singularities.

 

[RandallB]

Now here is the problem:

The Bell's spacships paradox shows that observers stationary in such a gravitational field will see themselves as moving apart spatially over time. For example a tower standing on this infinite plane would appear to be getting larger over time to observer on different floors of the tower. Eventually the tower would break up.

No there is not a problem in the hypothetical uniform gravitational field.
Assume we design it with a uniform gravitation force of 1G.
A tower standing on that surface would have no more trouble keeping a consistent size without getting larger than the towers we have here on earth. The only difference is the Earth tower will see that things weigh just a bit less up top, not so for the other tower. But both will maintain the same height.

What you refer to is the real problem of trying to employ the equivalence principle to duplicate the uniform field with two rockets; one pulling at the top and the other pushing from the bottom in empty space without the flat plane mass or any other gravity. Now the problem is getting them to start accelerating simultaneously together which is impossible as the problem of simultaneity comes into play.

Of course a method can start them off simultaneously in the initial reference frame. But the acceleration will put them into a some speed relative to the starting point. Which by definition means a new reference frame, which also by definition cannot have seen the two rocket start times as simultaneous in that new frame. Now since the two rockets are using identical acceleration profiles the only way the new reference frame can see the distance between the two rockets as unchanged is if they stared simultaneous. Therefore since we know by the rule of simultaneity they could not have started simultaneously in their new reference frame the distance between the two rockets must be changing.
Thus the tower (aka ‘the string’) must be stretched longer or crushed shorter.

Solving for longer or shorter (break or crush) would depend on establishing one of the two frames as closer to a local preferred frame, but the spaceship problem is unusually not given with a completely defined universe with a CMBR included.

 

[MeJennifer]

No there is not a problem in the hypothetical uniform gravitational field.
Assume we design it with a uniform gravitation force of 1G.
A tower standing on that surface would have no more trouble keeping a consistent size without getting larger than the towers we have here on earth. The only difference is the Earth tower will see that things weigh just a bit less up top, not so for the other tower. But both will maintain the same height.

Of course they remain the same hight in such a uniform gravitational field because both the acceleration and speed with respect to each other is 0. Note that a such a uniform gravitational field with a constant g has zero Riemann curvature and is simply Minkowski spacetime.

 

[yuiop]

..
Of course a method can start them off simultaneously in the initial reference frame. But the acceleration will put them into a some speed relative to the starting point. Which by definition means a new reference frame, which also by definition cannot have seen the two rocket start times as simultaneous in that new frame. Now since the two rockets are using identical acceleration profiles the only way the new reference frame can see the distance between the two rockets as unchanged is if they stared simultaneous. Therefore since we know by the rule of simultaneity they could not have started simultaneously in their new reference frame the distance between the two rockets must be changing.
Thus the tower (aka ‘the string’) must be stretched longer or crushed shorter.
...

In this part of your post you seem to be agreeing with me that height of the tower on the infinite plane gravitational body must change over time if the EP is correct. Now although the proper force of gravity measured at any height by an observer stationary with respect to this field is constant the the gravitational potential increases with increasing height. On the assumption that falling bodies move from a high potential to a lower potential it seems that test particles will fall in the normal direction towards the surface of the gravitational body so there is no apparent anti-gravity at work here. This makes it difficult to give a physical mechanism for why the tower on this body should grow over time because we can not put it down to gravitational force. It seems that we have stumbled across a situation where the equivalence principle demands that the spacetime around such a hypothetical gravitational body would expand over time.

Still, the problem remains that there is no clean and simple hypothetical gravitational body that can accurately duplicate the measurements made inside an rocket with Born rigid acceleration, which is perhaps the most natural form of acceleration, over anything greater than an infinitesimal test volume.

The closest we have come to duplicating the EP, is replicating a rocket undergoing "Bell spaceship type" acceleration, which we know will grow over time and eventually be shredded by tidal forces. This requires a gravitational plane that does not collapse to a point under its own gravitational forces (despite having infinite mass) so the plane must not just be nearly infinite but really infinite to avoid collapse and we would have to accept the rather peculiar expanding spacetime that would be present.

 

[RandallB]

In this part of your post you seem to be agreeing with me that height of the tower on the infinite plane gravitational body must change over time if the EP is correct.

You must not have been reading my post! MeJennifer was able to quote and understand it clearly enough – did you just skim read it? I was clear the only place calculating the height correctly was in the “Two Bell Spaceships” that not even GR (Riemann curvature included) can resolve for the acceleration without gravitation case.

All you pointing out is how this version of the example shows that the “Two Bell Spaceships” paradox is still a paradox.
And IMO what shows “Two Bell Spaceships” it to still be a true paradox is not so much those that consider it an unresolved paradox, as it is that those that claim to have “definitively” resolved it have done so with different and conflicting concluding results depending on the assumptions made.

 

[yuiop]

You must not have been reading my post! MeJennifer was able to quote and understand it clearly enough – did you just skim read it? I was clear the only place calculating the height correctly was in the “Two Bell Spaceships” that not even GR (Riemann curvature included) can resolve for the acceleration without gravitation case.

All you pointing out is how this version of the example shows that the “Two Bell Spaceships” paradox is still a paradox.
And IMO what shows “Two Bell Spaceships” it to still be a true paradox is not so much those that consider it an unresolved paradox, as it is that those that claim to have “definitively” resolved it have done so with different and conflicting concluding results depending on the assumptions made.

I wondered why you appeared to contradicting yourself in post#14 but I will agree that I can see that you are being self consistent now if you are working on the mistaken assumption that the Bell's spaceships paradox is a unresolved true paradox. There are no unresolved true paradoxes in Special Relativity!

General Relativity is built on Special Relativity. If General Relativity disagrees with or contradicts Special Relativity on any issue then it General Relativity that must be questioned first. Saying that Special Relativity has unresolved true paradoxes is the same as saying Special Relativity is flawed and it automatically follows that General Relativity is flawed. You really should re-examine your position on this. It came as a surprise that you do not agree with Bell's conclusion in his spaceships paradox, because you have not posted anything in the thread dedicated to that issue.

 

[MeJennifer]

General Relativity is built on Special Relativity. If General Relativity disagrees with or contradicts Special Relativity on any issue then it General Relativity that must be questioned first.

I disagree with that notion. For instance in spacetimes where the cosmological constant is not equal to zero the laws of special relativivty do not apply but must be changed accordingly.

I would rather say that general relativity is built on Newtonian gravity but with relativistic physics.

 

[RandallB]

I wondered why you appeared to contradicting yourself in post#14 but I will agree that I can see that you are being self consistent now if you are working on the mistaken assumption that the Bell's spaceships paradox is a unresolved true paradox. There are no unresolved true paradoxes in Special Relativity!

Do you think a Gravitational vs. Acceleration issue could have anything to do with SR?
SR is limited to fixed speed refrance frames; on its own it cannot resolve such a paradox, thus resolving or not resolving in cannot speak to SR.
Can you explain why you think an unresolved Bell spaceships paradox should reflect on SR?

General Relativity is built on Special Relativity.

You can say that in the Theory of Relativity that SR came before GR, but that does not mean GR was built and derived from SR which it was not. Were it that simple it would not have taken 8 to 12 years to build GR.

Love to see a Ref: to something that attempts to derive GR from SR if you think you know one.

It came as a surprise that you do not agree with Bell's conclusion in his spaceships paradox, because you have not posted anything in the thread dedicated to that issue.

I’m confused – you brought up the EP (Equivalence Principle) issue with an incorrect description of Uniform Gravity, Bell spaceships didn’t apply till you brought up the “tower example”

You seem to think the Bell spaceships paradox is conclusively resolved. Why?

You agree that a stationary tower on Earth would behave the same as a tower on the surface of the hypothetical flat surface. I.E. height does not change over time and a string tied from top to bottom does not break or gains slack; Likewise, for the identical string attached to two rockets next to the tower maintaining stationary positions due to trust equivalent to gravity.
I assume you agree YES unless you say NO and explain.

Remove the mass and gravity and the string attached to the rockets still feels the same gravity as they go into motion from that starting position.
Now the Bell spaceships paradox says one of three things must happen to the string: 1) It Breaks 2) It Stays Taut or 3) Goes Slack between the two rockets.
Correct me if I’m wrong, but didn’t Bell end up with the string breaking?
You tell me; what In Your Opinion does that mean for EP?

IMO (Especially, since I’ve also seen arguments for a Slack String) I do not consider Bell spaceships a “resolved” Paradox. At best it may be worth questioning EP further elsewhere becuase of the pardaox here, but for me nowhere near enough to reject the Equivalence Principle based on this paradox alone.

Also IMO “You really should re-examine your position on” all these issues.