# Poincaré's Space Dilemma

1. Jul 1, 2015

### Wes Tausend

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This thread is an offshoot from post #55 on a previous thread called, "Freefall isn't acceleration?".

A.T.,

From good reference, I'm not so sure "constant radius" is that easy to pin down. We may take it upon ourselves to assume that there is no surface movement, no "spatial inflation" change in the general space between and within the atoms comprising earth, but how do we know that for a fact? Afterall, here on PF, students and members alike can expect extraordinary claims to require extraordinary proof. Around 1898, the great mathematician, Poincaré, explored this very principle in his publication, The Relativity of Space, which we may find useful today.

Below, consider the following excerpt from Poincaré's The Relativity of Space in his Science & Method essays:

"...Suppose that in one night all the dimensions of the universe became a thousand times larger. The world will remain similar to itself, if we give the word similitude the meaning it has in the third book of Euclid. Only, what was formerly a metre long will now measure a kilometre, and what was a millimetre long will become a metre. The bed in which I went to sleep and my body itself will have grown in the same proportion. When I awake in the morning what will be my feeling in face of such an astonishing transformation? Well, I shall not notice anything at all. The most exact measures will be incapable of revealing anything of this tremendous change, since the yard-measures I shall use will have varied in exactly the same proportions as the objects I shall attempt to measure. In reality the change only exists for those who argue as if space were absolute..."

From the above, we can logically surmise that Poincaré's "recreational" observation of human cluelessness seems to ring true, if the "jerk of such an abrupt change" did not awaken one (which he left unsaid). Accordingly, even the earth's "new" radius would apparently not yield a clue. That concludes part 1.

Following, in part 2, we can reason our own additional observation:
Our own observation might be the one Poincaré also left unsaid... that perhaps we could not easily detect such an obscure growing dimension even if it were somehow ongoing... as long as a steady growth occurred at uniform motion. Such uniform motion is the same smooth motions of earth traveling around the sun and rotating every 24 hours, none of which are easily noticeable in spite of both occurances together, being a rather complicated combination in the minds eye. The complications of uniform motion have temporarily fooled humans before in geocentricity vs heliocentricity and due care has since been heralded.

But more ominous, further suppose the proposed "growth" of this "fantasy" motion were not uniform? What if the motion were steadily speeding up (accelerating) as it occurred? In that case, we may surmise all living inhabitants of earth would perhaps merely now notice, only an acceleration. Depending on rate, we must reluctantly admit, this "rising ground" phenomena might feel exactly like Einstein's (or even Newton's) gravity. It seems absolutely everything else, the entire universe, would appear exactly the same.

As per Feynman, we can be fooled, but nature cannot. So are we fooled in this case? No, but only because of SR which hadn't been invented in 1898.

As far as I can see, the sole basic item preventing this particular scenario from becoming a legitimate general relative coordinate system is the limiting speed of light which must naturally be included. Matter simply cannot accelerate faster and faster indefinately, and that seems to settle it.

Does anyone else have some other more simple form of basic proof? Agree, disagree?

Wes
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2. Jul 1, 2015

### Ibix

I don't think there's anything stopping you from changing coordinates to ones such that a particular family of freely falling observers are at constant radial coordinate. But it cannot be done globally. Hold two balls one above the other. Drop the higher one and release the lower when the upper one reaches it. Both are free-falling through the same point, but at different velocities. You can't rescale to keep both at a constant distance from the centre of the Earth, which is what I think you are proposing.

3. Jul 1, 2015

### Wes Tausend

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This post is one more offshoot from post #63 on a previous thread called, "Freefall isn't acceleration

A.T.,

I expected a healthy argument and I replied one more time in the former thread out of courtesy to inertiaforce, the OP. So here, I have repeated this post from the other thread to maintain continuity.

I agree. I am merely pointing out the extent of Equivalence by simple observation. Equivalence, along with SR are definately always part of GR. My references to acceleration and motion are Einstein's thought experiment (see post #33) which resulted in Equivalence and therefore GR. Do you have an equally good reference why we cannot refer to such equivalent motion in GR?

I disagree. I see the curvature as the direct result of the bending of light, therefore incorporating SR, also discussed in post #33.

I will say that if we do not allow some argument and a variety of perspectives of observation here, we might as well refer all PF member questions to Wikipedia. Please reply only in the new thread.

Wes
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4. Jul 1, 2015

### Wes Tausend

In Einstein's "elevator" thought experiment (see post #33), the "scientists" may release one ball from a seemingly higher table and another from the seemingly lower table as the first ("higher") ball passes it. Or rather the second ("lower") ball is released during a later period when it has now achieved a greater "inertial starting velocity" from it's respective continuously accelerating table top. The greater difference between the starting velocity of the first ("higher") ball (released at lower earlier acceleration) means it will be hit by the "rising floor" sooner and harder. I think Equivalence holds again in this case.

Wes
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5. Jul 1, 2015

### Wes Tausend

Of course "bending of light doesn't cause curvature". But curvature of light in the elevator first caused Einstein to mathematically treat it that way in his coordinate system. We sometimes mix up the order of geometry and math. One must first imagine the geometry, and only then apply suitable math for proofs, which in this case later turns out to be a mathematical curvature of spacetime... after Einstein has observed that light bends in an accelerating elevator. I believe GR and SR are locked inextricably together in this manner.

Wes
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6. Jul 1, 2015

### stevendaryl

Staff Emeritus
I'm not sure I understand what the disagreement is about. The principle of equivalence is only a local notion. If you confine your experiments to a tiny region where the change in gravity is negligible throughout the region, then the principle of equivalence states that experiments in that region cannot distinguish between being at rest in a gravitational field and accelerating through empty space. So in an elevator sitting on the ground, it will seem just like an elevator accelerating upward in empty space.

But now, if you consider a large region that includes the entire Earth, then it is not equivalent to anything accelerating in empty space. A rule of thumb is this: take two objects, say two apples, and drop them. If their paths remain parallel as they drop, then you might be accelerating through empty space. If the paths diverge, then you are definitely not just accelerating through empty space. Divergence of the paths of freefalling objects is a sign of spacetime curvatures, which is due to the presence of matter and energy.

7. Jul 1, 2015

### Wes Tausend

I think you might mean convergence instead of divergence. Two apples, or balls, will converge over time in real gravity if dropped from a significantly distant point in space. They will not do so if dropped in Einstein's isolated elevator, but rather seem to fall parallel. But in Poincaré's fantasy non-uniform motion universe (see post #1), the balls will again converge as they seem to fall. It takes some thought to realize this. Perhaps reference to another thought experiment will help:

The convergence comes to light in post #59 in this other past thread started by inertiaforce. In my opinion, the ability to visualize the "falling" convergence is served far better by an experimental gravitational model of expanding bubbles in an evacuated bell jar (or aquarium) than by the disgusting old bowling ball sagging a sheet of cloth. I got the "bubble idea" partially from Poincaré, or at least he confirmed my line of thinking. I even mentioned Poincaré later in the thread, but did not elaborate as I have done here. In my opinion, Equivalence is much more equivalent, more global and useful, than we once thought.

Wes
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Last edited: Jul 1, 2015
8. Jul 1, 2015

### WannabeNewton

It can certainly be done globally. Whether or not the resulting constant spatial coordinates can be interpret as having a "radial" component is another story since coordinates do not need to have any intrinsic geometrical interpretation. Furthermore, the resulting coordinates won't be rigid but that doesn't mean such a coordinate system doesn't exist. All I have to do is take the tetrad $e_{\hat{\alpha}}$ attached to the family of observers and apply the exponential map to it. Obviously this will only work if the associated Jacobi field has no caustics.

9. Jul 1, 2015

### Staff: Mentor

But this requirement won't be met in the case under discussion, will it? Suppose we take the family of tetrads attached to observers who are all free-falling towards the center of the Earth from the same altitude. Suppose also that the objects are idealized test objects that do not interact with the Earth's substance. Then their worldlines will all cross at the center of the Earth, and, I believe, this will cause the Jacobi field associated with your construction to have a caustic there, no?

10. Jul 1, 2015

### Staff: Mentor

Um, because we are continually making measurements of the geometry of the Earth, and it stays the same? For millennia we have been sending ships around the oceans, using navigation methods that depend on the geometry of the Earth staying the same; now we have spacecraft constantly orbiting the Earth whose orbits depend on the geometry of the Earth staying the same. If any "spatial inflation" were going on, we'd know it soon enough.

11. Jul 1, 2015

### Wes Tausend

Peter,

First thank you for your kind remark about no-controversy-worry in the other thread. I really do want to remain in compliance here on PF, yet resolve some long held, deep questions as well as help other PF members see further.

You queried, "Um, because we are continually making measurements of the geometry of the Earth, and it stays the same?" That is precisely the "dilema" crux of Poincaré's Space Dilema, thus the title. Let me repeat his observation (from post #1) in his own words with appropriate emphasis added. Poincaré wrote:

"...Suppose that in one night all the dimensions of the universe became a thousand times larger. The world will remain similar to itself, if we give the word similitude the meaning it has in the third book of Euclid. Only, what was formerly a metre long will now measure a kilometre, and what was a millimetre long will become a metre. The bed in which I went to sleep and my body itself will have grown in the same proportion. When I awake in the morning what will be my feeling in face of such an astonishing transformation? Well, I shall not notice anything at all. The most exact measures will be incapable of revealing anything of this tremendous change, since the yard-measures I shall use will have varied in exactly the same proportions as the objects I shall attempt to measure. In reality the change only exists for those who argue as if space were absolute..."

I took the liberty to further observe that if any "spatial inflation" continuously occurred in uniform motion that we would likely still neither measure nor "notice anything at all". If non-uniform motion, such as a form of global Equivalence took place, we would not casually measure any difference either. However we would notice the acceleration. It would seemingly be identical to gravity, as well it should considering our thought experiment about comprehensive Equivalence.

My take on all this is that Poincaré is arguing that a steady size of space (or the radius of earth etc) can not properly be considered Absolute any more than Absolute Rest is assured. I don't see global Equivalence as a problem, but a wonderful opportunity to expand the academic reach of Einstein's two Relativities in a simple, dynamic, geometric model.

No one must make such a choice by todays standards. But if you had to make a choice into the future, would you insist space is Absolute, or would you acknowledge that all coordinate systems are more likely relative? We are reminded to consider such related concepts as Big Bangs, singularities and more recently, cosmological inflation.

To me, it seems everything should obey the Copernican principle. Recently, the principle has been generalized to the relativistic concept that humans are not privileged observers of the universe. Our atoms, consisting of space and mass, have their own small amount of built-in gravity/Equivalence. We observe with this built-in handicap.

The only obstacle to increasingly speeding floors/surfaces seems to be the limiting speed of light (SR), but I believe this, too, can be easily dealth with using simple, common observation, to allow Equivalence to still mathematically function as needed and keep GR and SR intact. I have yet to directly ask about this and it is a daunting task.

Wes
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12. Jul 1, 2015

### A.T.

Is there a published quantitative theory based on this wonderful model?

13. Jul 1, 2015

### Staff: Mentor

My take on it is that the concept of "size of space" as Poincare and you are using it here is physically meaningless. What matters is the relative sizes of different objects. If I have a standard 1-meter ruler, the standard ruler defines what "1 meter" means. If somebody claims that the universe just increased in size by a factor of a thousand, so that the standard ruler is now "1 kilometer" in length, my response is that such a claim is meaningless, because "1 meter" or "1 kilometer" is not defined by somebody's abstract claim but by some well-defined physical thing--in this case, the length of the standard ruler.

14. Jul 1, 2015

### Mentz114

I cannot follow Poincaré at all. If something happens that no-one can ever detect - why are we even talking about it. My friend Ginger the twelve-foot hedgehog agrees completely with me.

I mean how does 'something happen' that cannot be detected. It is a contradiction. If nothing can be detected then it is pretty safe to assume that nothing happened.

Hang on ... call from Ginger.

15. Jul 1, 2015

### A.T.

Reminds me of the hollow Earth idea, based on spherical inversion:
https://en.wikipedia.org/wiki/Hollow_Earth#Concave_hollow_Earths

Purportedly verifiable hypotheses of a "concave hollow Earth" need to be distinguished from a thought experiment which defines a coordinatetransformation such that the interior of the Earth becomes "exterior" and the exterior becomes "interior". (For example, in spherical coordinates, let radius r go to R²/r where R is the Earth's radius.) The transformation entails corresponding changes to the forms of physical laws. This is not a hypothesis but an illustration of the fact that any description of the physical world can be equivalently expressed in more than one way.[46]

16. Jul 1, 2015

### Mentz114

Yes that's where Ginger goes on holiday.

(I apologise for my levity and I'll cease and desist from now).

17. Jul 1, 2015

### Ibix

Isn't that kind of the point? My reading of the OP was that was trying to find a way to regard any free-falling observer as being at a constant radial coordinate. If your scheme doesn't have an identifiable radial coordinate, it's not the one I think he's looking for.

His response to my post suggests I may be mis-interpreting him.

18. Jul 1, 2015

### Ibix

I think I misunderstood you. I thought you were seeking a system in which all freely-falling observers are at rest while the floor accelerates up at them. Instead you seem to be looking for one in which all freely-falling observers are moving at constant coordinate speed while the floor accelerates up at them.

Do I understand you right? If I do, is it possible to describe both orbitting bodies and radially falling bodies as having constant coordinate speed in this scheme?

19. Jul 2, 2015

### harrylin

I don't think that he meant a change that leads to physical phenomena; quite the contrary I would say that he was imagining an expansion of space that has no noticible effects.
It's obvious that you got your inspiration from the video in the other thread and for completeness I'll cite here below* the full warning of Einstein against such ideas here below.

Suppose that the earth expands to double its radius. Then the distance Earth-moon must also expand to double the distance in order for us to not notice it. But then the distance Moon-Sun should also double at the same rate. If I'm not mistaken, this implies a linear acceleration of the universe outward from the Earth that replaces the Earth's gravitational acceleration. Physically it makes no sense to assume that the Earth is by chance the center of the universe in such a way. Even more, the Sun has a different gravitational acceleration so that it has to expand at a different rate, and the space around it as well, including the Earth. And the same for other planets. [Edit:] And this does not even take into account the required different gravitational accelerations of bodies at different distances from the Earth. It seems obvious to me that such contradictory requirements cannot be fulfilled. Perhaps that is also what others have in mind.

* From our consideration of the accelerated chest we see that a general theory of relativity must yield important results on the laws of gravitation. In point of fact, the systematic pursuit of the general idea of relativity has supplied the laws satisfied by the gravitational field. Before proceeding farther, however, I must warn the reader against a misconception suggested by these considerations. A gravitational field exists for the man in the chest, despite the fact that there was no such field for the co-ordinate system first chosen. Now we might easily suppose that the existence of a gravitational field is always only an apparent one. We might also think that, regardless of the kind of gravitational field which may be present, we could always choose another reference-body such that no gravitational field exists with reference to it. This is by no means true for all gravitational fields, but only for those of quite special form. It is, for instance, impossible to choose a body of reference such that, as judged from it, the gravitational field of the earth (in its entirety) vanishes.[emphasis mine] - https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_II

Last edited: Jul 2, 2015
20. Jul 2, 2015

### Wes Tausend

Yes, GR is based on Equivalence. I make no other claim other than observation of it's usefulness.

One might feel that simple Equivalence only applies locally, but on the other hand GR itself is a global application derived from it. If the current coordinate math allows a global application, then we might agree somehow there exists an irrefutable pictoral geometry since the accepted math is based in said geometry. In essence, Einstein appears to take the bent (curved) path of the accelerating elevator light beam and apply it as a curved space (spacetime) around the mass of the globe. I agree that this geometry may be difficult to envision, but there is no reason we might not attempt to see further using perhaps untried but simple, logical observations. Good science is also better understood by attempts and occasional observations in error. And, inspired by Poincaré, that is all I'm doing... observing as best I can.

Equivalence is a solid, grand principle and like many here, I see the universe-machine is built of geometry, math and staid philosophical principles. Einstein's principle is solid as a tree and we have already harvested the fruit from the bottom branches. If some other curious primate were to possibly observe more fruit higher up, well we must thoroughly investigate. Perhaps the higher fruit is spoiled, perhaps not, but we can climb, look closer and qualitive questions can and should be asked.

First, as a group, we must all make the same complete observation or it won't mean much. Overall IMO, this thread seems to be proceeding satisfactorily. The concept certainly does not come quickly.

Wes
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