Can Special Relativity Predict Gravitational Effects?

[snoopies622]

I just had a strange thought:

In section 23 of his popular book, "Relativity - The Special and General Theory", Einstein explains why a clock on the edge of a rotating disk will run more slowly than one at the center, and then says,

"thus on our circular disk, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest)."

In the next paragraph he uses the same kind of argument - applying a prediction of special relativity to the rotating disk - to conclude that,

"the propositions of Euclidean geometry cannot hold exactly on the rotating disk, nor in general in a gravitational field.."

Since an observer on the disk will also notice a Coriolis force (if, say, he has a little Foucault pendulum and sets it in motion) why does it not then follow that every gravitational field is accompanied by a Coriolis acceleration? Are Einstein's arguments in this section simply disingenuous?

 

[Eynstone]

I don't think a constant gravitational field will have any Coriolis acceleration.

 

[Mentz114]

"the propositions of Euclidean geometry cannot hold exactly on the rotating disk, nor in general in a gravitational field."

I think Einstein states two exceptions because they are different. Gravity and rotation are not the same thing, but have a similar consequence.

 

[bcrowell]

The whole argument is just motivation. It isn't meant to be a rigorous proof.

Re the Coriolis force, Einstein is trying to take a particular case that happens to be easy to analyze, then extract features from it that are likely to be of more general significance, rather than those that only apply to that particular case. There are lots of features of the rotating disk that are definitely not true in all cases in GR. Two examples are that the Riemann tensor vanishes, and it's impossible to globally synchronize comoving clocks.

An interesting feature of the rotating disk that has sometimes been argued to be of more general significance is that measuring rods oriented transversely to a gravitational field suffer length contraction. More discussion here, at "Born in 1920": http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.3

I don't think a constant gravitational field will have any Coriolis acceleration.

A complication in discussing this is that there basically isn't any such thing as a constant gravitational field in GR. This is discussed at the link above.

 

[snoopies622]

Thanks, all. It still seems like a bit of verbal slight-of-hand on Einstein's part to me. That is, he seems to be saying, "it happens on a rotating disk, therefore it happens in a gravitational field, too," especially regarding the time-dilation.

 

[bcrowell]

That is, he seems to be saying, "it happens on a rotating disk, therefore it happens in a gravitational field, too," especially regarding the time-dilation.

It's not a proof, it's motivation. Rather than just dumping a bunch of equations on the reader, he's trying to explain to the reader how he himself set out on the beginning of the path to guessing those equations. The rotating disk story doesn't prove that GR is right. To test whether GR was right, it was necessary to do experiments and check it against reality.

 

[snoopies622]

Maybe he should have said something like, "since special relativity predicts that these strange features exist in a rotating frame of reference, and since a rotating frame of reference is one with a non-zero acceleration, and since we are hypothesizing a kind of equivalence between gravitation and acceleration in general, then these strange features may also appear in gravitational fields as well, although at this point in our story, we don't yet know."

 

[Fredrik]

This is a good example of the role of a "principle" in physics. A principle is a loosely stated idea that's supposed to help you guess the structure of a new theory. The equivalence principle is such an idea, and it helped Einstein guess the equation that goes by his name. Of course, once the theory has been found, you can turn the idea expressed by the principle into a precise mathematical statement, but that statement is now a theorem derived from the axioms of the theory.

This is why I don't like how people use the term "Heisenberg's uncertainty principle". The principle is an idea that predates QM, and can be used to guess the structure of the theory. (I don't know the history well enough to say that this is in fact what happened, but I suspect that it is). The result that's derived in every QM book and that's often called the "uncertainty principle" is a theorem, not a principle, so I prefer to call it the "uncertainty theorem" or the "uncertainty relation". (I'm not a fan of the word "uncertainty" either. I think I'd like to call it the "incompatibility theorem" or something like that. The reason I don't is of course that I also want people to understand what I'm talking about).

 

[Passionflower]

A theorem is something which can be proven given a set of mathematical statements. For instance in GR there are many theorems, the theorems are true but only in the context of the theory. I do not see anything similar with the uncertainty principle, as far as I understand it the principle is not mathematically induced but derived from experiments.

 

[George Jones]

A theorem is something which can be proven given a set of mathematical statements. For instance in GR there are many theorems, the theorems are true but only in the context of the theory. I do not see anything similar with the uncertainty principle, as far as I understand it the principle is not mathematically induced but derived from experiments.

Fredrik means exactly the same thing: the uncertainty principle is a theorem proved within the mathematical framework of quantum theory and statistics.

http://en.wikipedia.org/wiki/Uncertainty_principle#Mathematical_derivations

 

[Passionflower]

Fredrik means exactly the same thing: the uncertainty principle is a theorem proved within the mathematical framework of quantum theory and statistics.

http://en.wikipedia.org/wiki/Uncertainty_principle#Mathematical_derivations

I see, I was under the now false impression that the UP was a conclusion from experiment, after this a mathematical framework based on non-commuting operations was used to model this.

I did not realize that the UP is actually a theorem that can be derived from the mathematics of QM theory.

 

[Fredrik]

The term "uncertainty principle" is used both for a loosely stated idea that was motivated by experiments, and for a mathematical theorem. What I'm saying is that I prefer to call the theorem something else (e.g. the "uncertainty theorem"), The way I see it, a principle is supposed to be an idea that helps you guess the structure of a new theory. (The strategy is to only look for theories in which a formal statement of the "principle" can be proved as a theorem).

I included my version of the proof of the theorem in a post a year ago, so I might as well quote myself.

If A=A^\dagger and B=B^\dagger, then

\Delta A\Delta B\geq\frac{1}{2}|\langle[A,B]\rangle|

where the uncertainty \Delta X of an observable X is defined as

\Delta X=\sqrt{\langle(X-\langle X\rangle)^2\rangle}

Proof:

\frac{1}{2}|\langle[A,B]\rangle| =\frac{1}{2}|(\psi,[A,B]\psi)| =\frac{1}{2}|(B\psi,A\psi)-(A\psi,B\psi)| =|\mbox{Im}(B\psi,A\psi)|

\leq |(B\psi,A\psi)| \leq \|B\psi\|\|A\psi\| =\sqrt{(B\psi,B\psi)}\sqrt{(A\psi,A\psi)}=\sqrt{\langle B^2\rangle}\sqrt{\langle A^2\rangle}

The second inequality is the Cauchy-Schwarz inequality Now replace A and B with A-\langle A\rangle and B-\langle B\rangle respectively. This has no effect on the left-hand side since the expectation values commute with everything, so we get

\frac{1}{2}|\langle[A,B]\rangle| \leq \sqrt{\langle (B-\langle B\rangle)^2\rangle}\sqrt{\langle (A-\langle A\rangle)^2\rangle}=\Delta B\Delta A

That's how I wrote it down in my personal notes, but I should clarify a few things. I'm writing scalar products as (x,y) here. The expectation value \langle X\rangle is defined as (\psi,X\psi), or in bra-ket notation (|\psi\rangle,X|\psi\rangle)=\langle\psi|X|\psi\rangle, so it would be more appropriate to use the notation \langle X\rangle_\psi for the expectation value, and \Delta_\psi X for the uncertainty. We should really be talking about the uncertainty of X in the state \psi.

 

[George Jones]

The term "uncertainty principle" is used both for a loosely stated idea that was motivated by experiments, and for a mathematical theorem.

I am somewhat uncertain about the historical origins of the uncertainty, but I don't think real experiments provided the initial motivation for the uncertainty principle. I think Born's probabilistic interpretation of the wave function motivated Heisenberg to think about Gedanken experiments that led him in a non-rigorous way to his uncertainty principle.

 

[yossell]

Heisenberg was no fan of S's wave mechanics, quite the contrary. I don't think interpretations of S's wave played much of a role in his thinking. Rather, it seems to be a mixture of his positivism plus his analysis of the empirical limitations on accuracy of measurement - such as those found in his analysis of the resolving power of the microscope - that led him to his uncertainty relations.

One difference between Heisenberg's UP and the uncertainty relations of QM: H seems not to have given a general definition of the uncertainty in position and momentum, but would motivate the accuracy of measurement depending on the case that was studied. By contrast, the theorem assumes that uncertainty is to be understood as standard deviation. On a conceptual level, there's room for discussion whether standard deviation is the right analysis of uncertainty - which is not to deny that the formal UP is exceedingly well confirmed. In general, H's discussion of uncertainty - which changed depending on how heavily Bohr was leaning on him - seems to have been a way of helping to understand and comprehend the distinctive nature QM, and its split from classical mechanics.

I'm doubtful that, from a purely logical point of view, the question whether something is taken as an axiom or whether it is derived from a set of axioms, is of great importance - the arrow of logical derivation is not necessarily the arrow of explanation, and I take the use of an axiomatisations to be a way of systematically presenting a theory in a tractable form. It wouldn't bother me if QM were axiomatised directly in terms of time-dependent expectation values, and uncertainty relations between observables, rather than primarily in terms of wave-equations which were then Born-interpreted. Rather, what's I think is significant is the *precision* that is gained from formulating the principle mathematically - though this does involve taking a stance on the meaning of uncertainty and pinning it to a concrete relation between expectation values - and the precise connection that is drawn to other observables.

 

[Troponin]

It's not a proof, it's motivation. Rather than just dumping a bunch of equations on the reader, he's trying to explain to the reader how he himself set out on the beginning of the path to guessing those equations. The rotating disk story doesn't prove that GR is right. To test whether GR was right, it was necessary to do experiments and check it against reality.

Exactly. I wish there were some sort of standard "WARNING: HAND-WAIVING NON-MATHEMATICAL ANALOGY AHEAD" for situations like this.

It seems that the vast majority of posts in this forum are clearing up the confusion that comes from misinterpreting "worded" analogies that were meant to describe the math.

That's more than fine...that's what forums like this are for. But it would be nice if there were some standard preface as a "warning" for when a book is going to try to put the math into "layman's terms." Something like "PROCEED WITH CAUTION, NON-RIGOROUS PROOF AHEAD!"

 

[atyy]

Are there any known theories in which some form of the equivalence principle is obeyed, but yet are non-geometrical?

 

[bcrowell]

Are there any known theories in which some form of the equivalence principle is obeyed, but yet are non-geometrical?

The e.p. guarantees that you *can* choose a geometrical description, but it doesn't force you to. There are non-geometrical ways of describing GR, for example.

 

[snoopies622]

There are lots of features of the rotating disk that are definitely not true in all cases in GR. Two examples are that the Riemann tensor vanishes...

Really? Even though the ratio of a circle's perimeter to its diameter isn't \pi? (At least when the circle has the same center as the disk.)

 

[Fredrik]

Really? Even though the ratio of a circle's perimeter to its diameter isn't \pi?

In SR, a rotating disc is just a bunch of spirals in a 2+1-dimensional spacetime diagram. The metric of spacetime clearly doesn't change just because we choose to study those spirals. If the metric hasn't changed, the curvature hasn't either.

However, you're talking about a circle in space, not spacetime, and we can at least ask the question if space is curved. To answer that, we have to first decide which 3-dimensional Riemannian manifold to call "space". The obvious choice is to take it to be a subset of spacetime that consists of points that are all assigned the same time coordinate by the rotating coordinate system. But the rotating coordinate system assigns time coordinates to events exactly the same way as the inertial coordinate system in which the center is stationary. So this choice of what to call "space" ensures that space isn't curved. In the 2+1-dimensional spacetime diagram, "space" is just a plane parallel to the xy plane. The metric that's induced on this hypersurface by the Minkowski metric, is just the Euclidean metric. So there's nothing funny about any circles in that plane. They all satisfy circumference=diameter*pi.

The claim that the ratio of the circumference to the diameter isn't pi comes from the fact that if you try to measure the circumference with rulers that are comoving with points on the edge of a rotating disc, these rulers are Lorentz contracted, so you will need more of them to fill up the diameter of the disc than you would when the disc is at rest. However, if you imagine what this would look like in the 2+1-dimensional spacetime diagram, you should see that this procedure doesn't give you the circumference of a circle in any 3-dimensional Riemannian submanifold of spacetime.

Some people want to describe this scenario as circumference/diameter≠pi so desperately that they define a new manifold where there is such a circle. I have no idea why anyone would want to do this (other than a desire to change an incorrect claim into a correct one), but I know that you can do it by taking the world lines of the points on the disc (the spirals in the spacetime diagram) to be the points of a new 3-dimensional manifold, and use the Minkowski metric to define a Riemannian metric on this new manifold. This manifold is curved, but I haven't found a reason to think that this is relevant (and for that reason, I also haven't studied the details).

 

[Austin0]

The claim that the ratio of the circumference to the diameter isn't pi comes from the fact that if you try to measure the circumference with rulers that are comoving with points on the edge of a rotating disc, these rulers are Lorentz contracted, so you will need more of them to fill up the diameter of the disc than you would when the disc is at rest. However, if you imagine what this would look like in the 2+1-dimensional spacetime diagram, you should see that this procedure doesn't give you the circumference of a circle in any 3-dimensional Riemannian submanifold of spacetime.

.

WHy would the rulers be contracted by a different factor than the edge of the disk itself?
The inertial certifugal motion due to rotation perhaps??

 

[Passionflower]

Excellent explanation Fredrik!

 

[Fredrik]

WHy would the rulers be contracted by a different factor than the edge of the disk itself?

I didn't say that they would.

Excellent explanation Fredrik!

Thank you.

 

[snoopies622]

However, if you imagine what this would look like in the 2+1-dimensional spacetime diagram...

Do you mean one that's not rotating and has spirals to indicate physical points on the disk, or one that's rotating along with the disk and describes these points with straight lines?

...you should see that this procedure doesn't give you the circumference of a circle in any 3-dimensional Riemannian submanifold of spacetime.

Isn't it a circle for the person who is at rest with respect to the disk? Of course, if we're considering making simultaneous physical measurements of the circumference and diameter, why bother with the time dimension at all?

 

[bcrowell]

There are lots of features of the rotating disk that are definitely not true in all cases in GR. Two examples are that the Riemann tensor vanishes...

Really? Even though the ratio of a circle's perimeter to its diameter isn't \pi? (At least when the circle has the same center as the disk.)

I was referring to the spacetime Riemann tensor, not the spatial Riemann tensor.

 

[bcrowell]

We had an extremely long thread on the rotating disk back in January: https://www.physicsforums.com/showthread.php?t=367826 My own take on the whole thing is laid out here: http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html (subsection 3.4.4). One point of view advocated in that thread, IIRC, was that it was nonsensical to talk about the spatial metric, so that statements about the spatial curvature were all incorrect. That is a point of view that I have never seen in the published literature. It is easy to find multiple treatments of the noneuclidean spatial metric in the literature by extremely well known relativists, e.g., the textbook treatment by Rindler below.

Below is a list of some good papers on this topic. The Einstein, Dieks, and Egan papers are all freely available online. Amazon will let you peek through a keyhole at the relevant pages of the Rindler book, using their Look Inside feature. The Rizzi and Ruggiero book is absurdly expensive, but I ordered it through interlibrary loan and it had some interesting material in it; it was surprising how many disagreements the various authors had on this topic, which didn't seem to me like a cutting-edge area of relativity.

Einstein, ``The Foundation of the General Theory of Relativity.''
P. Ehrenfest, Gleichf\"ormige Rotation starrer K\"orper und Relativit\"atstheorie, Z. Phys. 10 (1909) 918
Gron, Relativistic description of a rotating disk, Am. J. Phys. 43 (1975) 869
Rizzi and Ruggiero, ed., Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames
Egan, http://www.gregegan.net/SCIENCE/Rings/Rings.html
Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf
Rindler, Relativity: Special, General, and Cosmological, pp. 198-200

 

[Fredrik]

Do you mean one that's not rotating and has spirals to indicate physical points on the disk, or one that's rotating along with the disk and describes these points with straight lines?

The first one. A diagram representing the inertial frame in which the center of the disc is at rest.

Isn't it a circle for the person who is at rest with respect to the disk?

The circumference is a circle in "space" as defined by the rotating coordinate system, if that's what you mean. But a bunch of rulers wouldn't be measuring the proper length of that circle (which is clearly just pi*diameter). Think about how to measure the length of a short segment of the edge with a comoving ruler. Imagine two markings near each other along the edge. As the ruler passes those markings, you need to make simultaneous readings at both markings...simultaneous in the ruler's rest frame, not in the rotating coordinate system. So the ruler measures the proper length of a short curve that's approximately Minkowski orthogonal to the world lines of that segment of the edge. It doesn't measure the proper length of a curve in "space". When you add up the results of all the length measurements to get the "circumference" you're getting something else. You're getting the proper length of a discontinous path in spacetime. You can do the measurements at different times to get a continuous path, but then it's a spiral in the diagram, not a circle.

Of course, if we're considering making simultaneous physical measurements of the circumference and diameter, why bother with the time dimension at all?

Because a length measurement requires two simultaneous readings.

 

[bcrowell]

WHy would the rulers be contracted by a different factor than the edge of the disk itself?
The inertial certifugal motion due to rotation perhaps??

They're not. They're contracted by the same factor as the edge of the disk.

Of course, if we're considering making simultaneous physical measurements of the circumference and diameter, why bother with the time dimension at all?

It's not quite as simple as you might think to get rid of the time dimension.

For instance, if you set out to measure the disk using rulers, you might think that that would eliminate time as a factor. But real rulers are dynamical objects that stretch and bend due to centrifugal and Coriolis forces, and relativity places absolute limits on the rigidity of materials, so you can't just say you want to make them infinitely rigid. There is a notion of Born rigidity, but it doesn't have all the properties you'd like in a rigid object (e.g., a Born-rigid object with finite area can't have a nonzero angular acceleration), and in order to make something act Born-rigid, you have to decide on a notion of simultaneity (which is frame-dependent), and then arrange to have forces applied simultaneously to various points on the object in order to keep it from deforming.

A more straightforward way to define length measurement is using radar, and this does bring in time implicitly. If you define length measurements using radar, and carry them out on the rotating disk, then you can ultimately arrive at a description of its noneuclidean spatial geometry in which time has been eliminated. This is, for example, a valid physical interpretation of the spatial metric discussed in the link and references I gave in #25.

 

[Fredrik]

One point of view advocated in that thread, IIRC, was that it was nonsensical to talk about the spatial metric, so that statements about the spatial curvature were all incorrect.

If you're referring to what I said in that thread, I wouldn't use the words "nonsensical" and "wrong". "Pointless" and "irrelevant" would be more accurate. Maybe there is a good reason do those fancy definitions and complicated calculations, but I still haven't seen one. The fancy stuff is certainly not needed to get the main point of the problem, which is that a solid disc can't be given a spin without forcefully stretching the material.

 

[Mike_Fontenot]

"the propositions of Euclidean geometry cannot hold exactly on the rotating disk, nor in general in a gravitational field.."

Since an observer on the disk will also notice a Coriolis force (if, say, he has a little Foucault pendulum and sets it in motion) why does it not then follow that every gravitational field is accompanied by a Coriolis acceleration? Are Einstein's arguments in this section simply disingenuous?

No, they were not disingenuous. In fact, that example was (I believe) the spark that put Einstein on the trail of GR...he basically knew THEN what GR needed to be like (because of his strong belief in the general equivalence principle)...but it took him 10 years to learn what kind of mathematics was required to give him what he already knew he needed. It (differential geometry) had already been worked out by some mathematicians, who (I imagine) had no idea what practical application their mathematics had (and probably would have been offended by the question itself!).

The gravitational field that is equivalent to the rotating disk SR problem is a very specific, and a very odd, gravitational field. It is a field that is cylindically symmetric, directed radially outward, and whose intensity grows linearly with the distance from the central spatial point...probably doesn't happen very often with mass distributions in our universe!

Mike Fontenot

 

[bcrowell]

The gravitational field that is equivalent to the rotating disk SR problem is a very specific, and a very odd, gravitational field. It is a field that is cylindically symmetric, directed radially outward, and whose intensity grows linearly with the distance from the central spatial point...probably doesn't happen very often with mass distributions in our universe!

It happens all the time in our universe. It happens anywhere that spacetime is flat -- provided that you describe that flat spacetime in rotating coordinates!

 

[atyy]

I think reading Einstein's writings are more history of science these days - I think this particular sentence is borderline meaningful - but his talk about general covariance was just completely wrong. Anyway, it is fun to see if we can make these vague statements correct. How about this interpretation?

Acceleration is like gravity.
An accelerated frame is not globally inertial, so gravity presumably also creates a global noninertial frame.
Except that acceleration is only fake gravity, so you can always recover a global inertial frame with acceleration, but never with gravity.

 

[bcrowell]

I think reading Einstein's writings are more history of science these days - I think this particular sentence is borderline meaningful

Which sentence are you referring to?

- but his talk about general covariance was just completely wrong.

Why?

 

[atyy]

Which sentence are you referring to?

The one about rotation and curved space and gravity and curved spacetime.

Why?

All theories can be written in a generally covariant form, so general covariance is not terribly meaningful. Einstein meant "no prior geometry", but I think that realization had to wait for Anderson.

 

[bcrowell]

All theories can be written in a generally covariant form, so general covariance is not terribly meaningful. Einstein meant "no prior geometry", but I think that realization had to wait for Anderson.

I see. I guess that's basically the argument made in this paper? http://arxiv.org/abs/gr-qc/0603087

Although I might agree in principle, in reality general covariance seems to be very, very hard for most people to accept. Even smart people who have studied a lot of GR often tie themselves up in horrible knots because they keep thinking that coordinates have some direct physical interpretation. For instance, the end of this paper

Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf

has a long list of statements by authoritative people (including Feynman) about how you can never receive a photon from a galaxy that's receding from you at >c. All those people made that mistake, but it's not a mistake you'd make if you'd truly accepted coordinate-independence.

 

[Austin0]

The claim that the ratio of the circumference to the diameter isn't pi comes from the fact that if you try to measure the circumference with rulers that are comoving with points on the edge of a rotating disc, these rulers are Lorentz contracted, so you will need more of them to fill up the diameter of the disc than you would when the disc is at rest. However, if you imagine what this would look like in the 2+1-dimensional spacetime diagram, you should see that this procedure doesn't give you the circumference of a circle in any 3-dimensional Riemannian submanifold of spacetime.

So

I see I misread you and read circumference in both places. But I am still a bit confused; the rulers would be contracted when aligned to measure the cicumference but wouldn't they be their proper rest length when aligned with the diameter ?? Or am I not understanding what you mean by fill up the diameter of the disk?

 

[atyy]

I see. I guess that's basically the argument made in this paper? http://arxiv.org/abs/gr-qc/0603087

Yes. Just to clarify what I meant, I would say GR is distinguished by general covariance of the field that is varied in the action, although not by general covariance of the equations of motion.

Although I might agree in principle, in reality general covariance seems to be very, very hard for most people to accept. Even smart people who have studied a lot of GR often tie themselves up in horrible knots because they keep thinking that coordinates have some direct physical interpretation. For instance, the end of this paper

Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf

has a long list of statements by authoritative people (including Feynman) about how you can never receive a photon from a galaxy that's receding from you at >c. All those people made that mistake, but it's not a mistake you'd make if you'd truly accepted coordinate-independence.

Well, Feynman even made mistakes with Gauss's law in his celebrated Lectures on Physics (http://www.feynmanlectures.info/flp_errata.html, see Thorne's preface at the bottom), which I am sure he understood most of the time, so I think the list is entertaining, but I wouldn't read much more into it. Actually, the fact that Feynman got some bits of classical electrodynamics wrong based on the equivalence principle is further reason not to take Einstein's motivating ideas (general covariance, EP) for GR too seriously - modern textbooks such as MTW, Rindler and Carroll essentially concur - d'Inverno is the odd one out, I think.

 

[Fredrik]

the rulers would be contracted when aligned to measure the cicumference but wouldn't they be their proper rest length when aligned with the diameter ??

That's right.

 

[DrGreg]

Some people want to describe this scenario as circumference/diameter≠pi so desperately that they define a new manifold where there is such a circle. I have no idea why anyone would want to do this (other than a desire to change an incorrect claim into a correct one), but I know that you can do it by taking the world lines of the points on the disc (the spirals in the spacetime diagram) to be the points of a new 3-dimensional manifold, and use the Minkowski metric to define a Riemannian metric on this new manifold. This manifold is curved, but I haven't found a reason to think that this is relevant (and for that reason, I also haven't studied the details).

This is one of the rare occasions I have to disagree with Fredrik, not on the technical details but on his opinion that this is irrelevant. The manifold being referred to here (technically a "quotient manifold" I believe), represents the intuitive notion of a geometry being represented by a rigid grid of rulers glued together.

For instance, if you set out to measure the disk using rulers, you might think that that would eliminate time as a factor. But real rulers are dynamical objects that stretch and bend due to centrifugal and Coriolis forces, and relativity places absolute limits on the rigidity of materials, so you can't just say you want to make them infinitely rigid. There is a notion of Born rigidity, but it doesn't have all the properties you'd like in a rigid object (e.g., a Born-rigid object with finite area can't have a nonzero angular acceleration), and in order to make something act Born-rigid, you have to decide on a notion of simultaneity (which is frame-dependent), and then arrange to have forces applied simultaneously to various points on the object in order to keep it from deforming.

Imagine a giant wheel of rulers, one set of rulers as radial spokes and another set connected to form concentric circles. If you glued the rulers together at rest and then tried to rotate the wheel, it would fall apart as the circumferential rulers would contract and the radial rulers would not. Nevertheless, you could imagine gluing the structure together while it was already spinning. It might be a severe engineering challenge to achieve this in practice, but we can envisage it happening without any laws of physics being broken. Once constructed, the spinning structure is perfectly stable so long as it continues to rotate with constant angular velocity and we don't try to move any pieces to a new relative location (provided the centripetal forces are not large enough for the structure to fall apart). This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

Of course, if we're considering making simultaneous physical measurements of the circumference and diameter, why bother with the time dimension at all?

Because a length measurement requires two simultaneous readings.

In general that's true, but it's not true when the points being measured are permanently at rest in your coordinate system. In that case you can take your measurements any time you like. So when you measure your circumference, it doesn't matter whether each infinitesimal measurement you make is synchronised to any other measurement or not. You just start at one worldline, and work your way round the circle until you get back the the worldline you started at, taking as much time as you want. The length measurement you make does not need to be associated with a curve in space-time. It is a space measurement, not a spacetime measurement. In particular there's no need to identify a spacelike surface within which to make measurements (and in the case of rotating coordinates, you can't find any such surface orthogonal to the "at rest" spiralling worldlines).

 

[Fredrik]

This is one of the rare occasions I have to disagree with Fredrik, not on the technical details but on his opinion that this is irrelevant. The manifold being referred to here (technically a "quotient manifold" I believe), represents the intuitive notion of a geometry being represented by a rigid grid of rulers glued together.

I don't mind when you disagree with me, because I usually learn something new when you do. I think I expressed myself a bit too strongly in #19. #28 is a better representation of what I've actually been thinking:

Maybe there is a good reason do those fancy definitions and complicated calculations, but I still haven't seen one.

I actually started writing an addition to #19, where I said that the fancy stuff might be useful if we're trying to develop an understanding of non-local measurements, perhaps with the intention of replacing the axioms of SR and GR (which refer to local measurements) with axioms that refer to measurements of a more general kind. But I never finished it, because I felt that it might just lead to 20 questions about what I meant, and then I'd have to spend too much time explaining it.

If this approach does lead to a more general notion of measurement, and especially if it gives us what we need to replace the length measurement axiom of GR with a prettier one, I might just have to study those details.

Imagine a giant wheel of rulers,
...you could imagine gluing the structure together while it was already spinning.
It might be a severe engineering challenge to achieve this in practice,

I'm not convinced that it's just an engineering challenge, because we have to consider the elastic deformation of the rulers (and spokes) under centrifugal forces. Or are we just assuming that they're rigid in some specific sense?

This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

I'm trying to come up with a sentence that describes how it's relevant, something like the axiom I use to describe how clocks are relevant: "A clock measures the proper time of the curve in spacetime that represents its motion". What would the corresponding statement be for this wheel? How about this?

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q whose projection onto spacetime is a timelike geodesic. The wheel measures proper lengths in Q of lines through p and circles around p".

In general that's true, but it's not true when the points being measured are permanently at rest in your coordinate system. In that case you can take your measurements any time you like. So when you measure your circumference, it doesn't matter whether each infinitesimal measurement you make is synchronised to any other measurement or not.

OK, so you're thinking of rulers attached to the disc in a very complicated way, rather than inertial rulers momentarily comoving with a small segment of the edge. The thing is, they give us the same results as inertial comoving rulers, and each of those rulers involve two simultaneous readings, and it's clear that they're measuring the length of a discontinous curve in spacetime, or the length of a continuous spacelike spiral.

I don't know what constraints you would impose on the attached rulers to prevent them from deforming under all the forces involved, or rather to get them to deform in exactly the right way. I also don't know how you would justify the claim that what you're measuring is "the circumference".

The length measurement you make does not need to be associated with a curve in space-time.

Then why would we call it a "length" measurement? Because there exists a manifold that has nothing to do with what we previously called "space", in which there is a circle with that circumference? I think we need a much better motivation than that.

It is a space measurement, not a spacetime measurement.

But the result is also not the proper length of any (relevant) continuous curve in space, so it doesn't make sense to call it a space measurement either...

...until we have defined "space" to mean something very different from what it meant to us before we started working on this problem!

This is what really bugs me about these claims (when they're made in books). The authors seem to start out thinking that we can measure the circumference of a disc in a certain way, and then when they realize that this is just wrong, they redefine the meaning of the words to make their first claim right. It's like saying that an obviously discontinuous function is continuous...and then go "oops...uh...because every subset of the real numbers is open". I don't doubt that some of them have a better understanding of the issues than that, but they don't always show it.

I find it very strange that some authors try to slip in a redefinition of the term "space" without even mentioning that it is a redefinition.

The claim (not made by you) that I find the most bizarre is that the quotient manifold is space in the rotating coordinate system. It clearly doesn't have anything to do with that coordinate system, or any other.

 

[DrGreg]

I'm not convinced that it's just an engineering challenge, because we have to consider the elastic deformation of the rulers (and spokes) under centrifugal forces. Or are we just assuming that they're rigid in some specific sense?

If you like, we can wait until the rotating structure has settled down to an equilibrium state, i.e. all points are relatively at rest in the rotating frame, and then calibrate it (i.e. engrave the markings on the rulers) with the help of some comoving inertial rulers. Or, even better, by using local (infinitesimal) radar.

How about this?

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q whose projection onto spacetime is a timelike geodesic. The wheel measures proper lengths in Q of lines through p and circles around p".

Points in Q do not correspond to timelike geodesics in spacetime. They correspond to spiral worldlines (at rest in the rotating coordinates) which are timelike but certainly not geodesic. I don't know whether that was just a typo from you, or whether you have misunderstood the quotient construction.

And although I talked of radial and tangential rulers, in principle you could have rulers in any direction you like (again assembled and radar-calibrated while already rotating).

OK, so you're thinking of rulers attached to the disc in a very complicated way, rather than inertial rulers momentarily comoving with a small segment of the edge. The thing is, they give us the same results as inertial comoving rulers, and each of those rulers involve two simultaneous readings, and it's clear that they're measuring the length of a discontinous curve in spacetime, or the length of a continuous spacelike spiral.

Any distance is a "sum of infinitesimal distances" (to use somewhat imprecise language, but you know what I mean) and each individual infinitesimal distance coincides with a comoving inertial measurement, and the two ends of that infinitesimal measurement are indeed simultaneous in the comoving frame (i.e. the infinitesimal segment in spacetime is orthogonal to the spiral worldlines at that point). But when you stitch all these measurements together, it doesn't matter whether whether they are synchronised to each other. You could, if you want, synchronise them to each other and find that when you complete the circle (reaching the same spiral worldline where you started), the two ends aren't simultaneous. It doesn't matter. When you project each infinitesimal segment onto the quotient manifold, the time gets lost and you get the same projected segment no matter how each segment is synced to its neighbours.

Then why would we call it a "length" measurement? Because there exists a manifold that has nothing to do with what we previously called "space", in which there is a circle with that circumference?

I'm not sure why you have such difficulty in accepting this as "space". It takes a bit of getting used to (and non-mathematical souls who haven't previously come across other examples of "quotient spaces" in other maths contexts may have more difficulty) but to my mind this seems to be the correct definition of space. It just happens to be isomorphic to a spacetime surface of simultaneity in other cases such as Minkowski coords or Rindler coords where there is no torsion.

But the result is also not the proper length of any (relevant) continuous curve in space, so it doesn't make sense to call it a space measurement either...

It is the length of a continuous curve in the quotient manifold, a circle in fact. It's just not the length of a closed curve in spacetime.

The claim (not made by you) that I find the most bizarre is that the quotient manifold is space in the rotating coordinate system. It clearly doesn't have anything to do with that coordinate system, or any other.

I don't really understand why you think that last sentence is true.

 

[bcrowell]

For instance, if you set out to measure the disk using rulers, you might think that that would eliminate time as a factor. But real rulers are dynamical objects that stretch and bend due to centrifugal and Coriolis forces, and relativity places absolute limits on the rigidity of materials, so you can't just say you want to make them infinitely rigid. There is a notion of Born rigidity, but it doesn't have all the properties you'd like in a rigid object (e.g., a Born-rigid object with finite area can't have a nonzero angular acceleration), and in order to make something act Born-rigid, you have to decide on a notion of simultaneity (which is frame-dependent), and then arrange to have forces applied simultaneously to various points on the object in order to keep it from deforming.

Imagine a giant wheel of rulers, one set of rulers as radial spokes and another set connected to form concentric circles. If you glued the rulers together at rest and then tried to rotate the wheel, it would fall apart as the circumferential rulers would contract and the radial rulers would not. Nevertheless, you could imagine gluing the structure together while it was already spinning. It might be a severe engineering challenge to achieve this in practice, but we can envisage it happening without any laws of physics being broken. Once constructed, the spinning structure is perfectly stable so long as it continues to rotate with constant angular velocity and we don't try to move any pieces to a new relative location (provided the centripetal forces are not large enough for the structure to fall apart). This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

I don't see anything to disagree with here. Is this in reply to the text you quoted from my #27? I think what I said and what you said are consistent with one another. I agree with you that the rotating spatial metric is well defined, interesting, and non-Euclidean.

 

[DrGreg]

I don't see anything to disagree with here. Is this in reply to the text you quoted from my #27? I think what I said and what you said are consistent with one another. I agree with you that the rotating spatial metric is well defined, interesting, and non-Euclidean.

I wasn't disagreeing with you, I was quoting this as it seemed relevant to the point I was making in my post, and I wanted to address the issues of rigidity etc. My post was really aimed at Fredrik, Austin0 & snoopies622. Sorry for any confusion.

 

[jason12345]

I just had a strange thought:

In section 23 of his popular book, "Relativity - The Special and General Theory", Einstein explains why a clock on the edge of a rotating disk will run more slowly than one at the center, and then says,

"thus on our circular disk, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest)."

In the next paragraph he uses the same kind of argument - applying a prediction of special relativity to the rotating disk - to conclude that,

"the propositions of Euclidean geometry cannot hold exactly on the rotating disk, nor in general in a gravitational field.."

Since an observer on the disk will also notice a Coriolis force (if, say, he has a little Foucault pendulum and sets it in motion) why does it not then follow that every gravitational field is accompanied by a Coriolis acceleration? Are Einstein's arguments in this section simply disingenuous?

Maybe you should do a search on the The Ehrenfest Paradox as well, since the above is related:

http://en.wikipedia.org/wiki/Ehrenfest_paradox

I find it curious that Einstein jumps to the conclusion that the rods around the periphery of the circle are Lorentz contracted even though they're being accelerated. Worse still, all the points of the rotating frame have unique comoving frames and so there is no comoving frame for an infinitesimal rod on the circumference. Of course, since I know little about GR, the error must lie with my lack of understaning ;)

 

[Fredrik]

If you like, we can wait until the rotating structure has settled down to an equilibrium state, i.e. all points are relatively at rest in the rotating frame, and then calibrate it (i.e. engrave the markings on the rulers) with the help of some comoving inertial rulers. Or, even better, by using local (infinitesimal) radar.

OK, that makes sense.

Points in Q do not correspond to timelike geodesics in spacetime. They correspond to spiral worldlines (at rest in the rotating coordinates) which are timelike but certainly not geodesic. I don't know whether that was just a typo from you, or whether you have misunderstood the quotient construction.

It wasn't a typo, and I don't think I have misunderstood it. Actually, it looks like you just misread what I said. There is exactly one point in the quotient manifold that corresponds to a timelike geodesic. That's the point I called p. The only thing I want to change right away about what I said is the word "projection". (To be consistent with standard terminology, e.g. in the context of fiber bundles, it's the map that takes a point in spacetime to the spiral in spacetime that it belongs to, i.e. to a point in Q, that should be called a projection). So let's replace that with "corresponds to".

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q that corresponds to a timelike geodesic in spacetime. The wheel measures proper lengths in Q of lines through p and circles around p".

If we do what you suggested in the first quote above, and engrave distance markings on e.g. a rotating solid disc, at locations determined by radar, we might be able to use it to measure the proper length of other curves in the quotient manifold. I'm saying "might", because I still haven't thought about how to define the metric on Q.

Any distance is a "sum of infinitesimal distances" (to use somewhat imprecise language, but you know what I mean) and each individual infinitesimal distance coincides with a comoving inertial measurement, and the two ends of that infinitesimal measurement are indeed simultaneous in the comoving frame (i.e. the infinitesimal segment in spacetime is orthogonal to the spiral worldlines at that point). But when you stitch all these measurements together, it doesn't matter whether whether they are synchronised to each other. You could, if you want, synchronise them to each other and find that when you complete the circle (reaching the same spiral worldline where you started), the two ends aren't simultaneous. It doesn't matter. When you project each infinitesimal segment onto the quotient manifold, the time gets lost and you get the same projected segment no matter how each segment is synced to its neighbours.

Agreed. But this isn't relevant until we have redefined "space" to mean a quotient manifold defined by a congruence instead of a submanifold defined by a coordinate system.

I'm not sure why you have such difficulty in accepting this as "space". It takes a bit of getting used to (and non-mathematical souls who haven't previously come across other examples of "quotient spaces" in other maths contexts may have more difficulty) but to my mind this seems to be the correct definition of space. It just happens to be isomorphic to a spacetime surface of simultaneity in other cases such as Minkowski coords or Rindler coords where there is no torsion.

My strongest objection isn't against the claim that it makes sense to call this "space". It's against the suggestion that this simply is space, and no redefinition is necessary, even though the term already means something else.

I think I understand what you said about how the distance markings we engrave on the disc will assign lengths to curves in Q in agreement with the metric on Q, but I don't know if I can be convinced to think that this is a good enough reason to redefine the word "space" to not be a submanifold of spacetime.

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

I don't really understand why you think that last sentence is true.

Perhaps that statement was too strong. I meant that the quotient manifold is defined by the congruence, not by the coordinate system. The coordinate system defines something completely different as "space". The only relationship between the rotating coordinate system and the quotient manifold is that the congruence that defines the quotient manifold is easier to describe in those coordinates than any other.

 

[DrGreg]

It wasn't a typo, and I don't think I have misunderstood it. Actually, it looks like you just misread what I said. There is exactly one point in the quotient manifold that corresponds to a timelike geodesic. That's the point I called p.

Oops, my mistake. I get it now.

If we do what you suggested in the first quote above, and engrave distance markings on e.g. a rotating solid disc, at locations determined by radar, we might be able to use it to measure the proper length of other curves in the quotient manifold. I'm saying "might", because I still haven't thought about how to define the metric on Q.

Have you seen bcrowell's own website that he linked to in an earlier post? http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4

I already talked about how to measure each infinitesimal "ds" between two spirals in a direction orthogonal to them. That's more-or-less it.

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

It only makes sense with a field associated with a stationary coordinate system, so that each pair of timelike worldlines are a constant distance from each other e.g. as determined by two-way doppler shift. (I'm still a beginner in this area but I think this all means that stationary coordinates are characterised by time-independent spacetime metric components or equivalently that your vector field X must be a Killing vector.)

If you have a copy of Rindler's Relativity: Sp, Gen & Cosm, it's covered there in Chap 9 of the 2nd edition.

(Haven't time to say more now, it is way past bedtime in my part of the world.)

 

[bcrowell]

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

It only makes sense with a field associated with a stationary coordinate system, so that each pair of timelike worldlines are a constant distance from each other e.g. as determined by two-way doppler shift. (I'm still a beginner in this area but I think this all means that stationary coordinates are characterised by time-independent spacetime metric components or equivalently that your vector field X must be a Killing vector.)

If you have a copy of Rindler's Relativity: Sp, Gen & Cosm, it's covered there in Chap 9 of the 2nd edition.

My own current understanding of this is written up here: http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.2 The distinction I make there between stationary spacetimes with only one (timelike) Killing vector and those with additional symmetries may be one that can be eliminated, but for me at my current point of understanding it seems helpful. If there's only one Killing vector, which is timelike, then points in space have permanent identities that are defined in a natural way, and defining a spatial metric becomes a pretty natural thing to do.

I found Rindler's treatment helpful, but a little opaque in places.

 

[Fredrik]

It only makes sense with a field associated with a stationary coordinate system,

Minkowski spacetime is a stationary spacetime no matter what curves we choose to consider. Perhaps what you have in mind is that there's a coordinate system in which all points that belong to the image of the same curve in the congruence have the same spatial coordinates? We can use my almost arbitrary vector field X to define such a coordinate system, e.g. like this: Choose some inertial frame x. For each event p, follow the integral curve of X through p to the event q, where it intersects the t=0 plane. Assign spatial coordinates (x¹(q),x²(q),x³(q)) to p.

I don't know if this can be used for anything interesting. I'm just saying that (almost) every congruence seems to be associated with such a coordinate system.

For me to feel that it's appropriate to call something "space", I think it should at least give meaning to "space, at time t". A quotient manifold doesn't seem to do that.

 

[DrGreg]

Perhaps what you have in mind is that there's a coordinate system in which all points that belong to the image of the same curve in the congruence have the same spatial coordinates?

That is indeed what I meant (and why I said "stationary coordinates" rather than "stationary spacetime"). Your vector field needs to be a timelike non-vanishing Killing vector field for the coordinates to be "stationary", i.e. a constant distance from each other over time.

For me to feel that it's appropriate to call something "space", I think it should at least give meaning to "space, at time t". A quotient manifold doesn't seem to do that.

But space applies to all possible times. You are thinking of a bundle of spaces, one for each time, but I'm thinking of a single space.

 

[DrGreg]

For each event p, follow the integral curve of X through p to the event q, where it intersects the t=0 plane. Assign spatial coordinates (x¹(q),x²(q),x³(q)) to p.

You can use that to define the coordinates, but what I'm saying is you don't want to use the t=0 plane to measure distance. The distance between nearby points in the plane does not necessarily equal the orthogonal distance between the curves, so it doesn't represent distance as measured by a comoving inertial observer (a tangent line to the curve).

 

[Fredrik]

You are thinking of a bundle of spaces, one for each time, but I'm thinking of a single space.

Yes, but Minkowski spacetime is a bundle of spaces, and it's quite useful. I still don't really get why we would want a single space.

You can use that to define the coordinates, but what I'm saying is you don't want to use the t=0 plane to measure distance. The distance between nearby points in the plane does not necessarily equal the orthogonal distance between the curves, so it doesn't represent distance as measured by a comoving inertial observer (a tangent line to the curve).

Good point.

I've been thinking some more about what sort of congruence of curves that can define a quotient manifold in a useful way, and I think I get it. I just don't have an elegant way of saying it yet. I'm thinking that if we consider two small neighborhoods of two points on the same curve in the congruence, they should "look approximately the same" in their comoving inertial frames. The approximation must become exact in the limit where the size of the region goes to zero, and this must hold for any two points on any single curve in the congruence. To "look approximately the same" here means that any other curve that passes through the two neighborhoods must be at approximately the same spatial coordinates in the two comoving inertial frames.

(There must be a better way of saying that).

This seems to be the requirement that must be met for the "object" whose motion is represented by the congruence of curves to be useful as a generalized "ruler", because it allows us to calibrate it once and for all with local radar measurements, and engrave it with distance markings (as you described above) that will not "move around relative to each other".