# Deducing some GR from SR?

1. Aug 11, 2010

### 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?

2. Aug 11, 2010

### Eynstone

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

3. Aug 11, 2010

### 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.

4. Aug 11, 2010

### bcrowell

Staff Emeritus
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 [Broken]

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.

Last edited by a moderator: May 4, 2017
5. Aug 11, 2010

### 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.

6. Aug 12, 2010

### bcrowell

Staff Emeritus
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.

7. Aug 12, 2010

### 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."

8. Aug 12, 2010

### Fredrik

Staff Emeritus
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).

9. Aug 12, 2010

### 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.

10. Aug 12, 2010

### George Jones

Staff Emeritus
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

11. Aug 12, 2010

### Passionflower

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.

12. Aug 12, 2010

### Fredrik

Staff Emeritus
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.

13. Aug 12, 2010

### George Jones

Staff Emeritus
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.

14. Aug 12, 2010

### 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.

15. Aug 12, 2010

### Troponin

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!"

16. Aug 12, 2010

### atyy

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

17. Aug 12, 2010

### bcrowell

Staff Emeritus
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.

18. Aug 12, 2010

### snoopies622

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.)

19. Aug 13, 2010

### Fredrik

Staff Emeritus
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).

Last edited: Aug 13, 2010
20. Aug 13, 2010

### Austin0

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??