I Did Born Rigidity Lead Einstein to Conclude Spacetime is Curved?

[luro1964]

I'll set out by saying that I have no real formal training in physics or maths.
However, I have been keen to try to understand what exactly convinced Einstein that spacetime must be curved. As I understand it, the bending of star light was already explained by Newtonian physics, although of course the calculations didn't match experimental evidence.
So, my reading suggests it may have Born rigidity. I believe Einstein assumed that geometry for the rigid rotating (accelerating) disk is non-euclidean and that he therefore needed to adopt non-euclidian geometry for GR.
Am I on the right track?

 

[A.T.]

However, I have been keen to try to understand what exactly convinced Einstein that spacetime must be curved.

That's a history of science question, which assumes there was one single thing that convinced him, instead of multiple considerations.

As I understand it, the bending of star light was already explained by Newtonian physics, although of course the calculations didn't match experimental evidence.

Einstein also had to reconcile the bending of light with the constancy of light speed he postulated in SR.

So, my reading suggests it may have Born rigidity. I believe Einstein assumed that geometry for the rigid rotating (accelerating) disk is non-euclidean and that he therefore needed to adopt non-euclidian geometry for GR.

That was likely just one of multiple things that lead him to this idea.

 

[Ibix]

As I understand it, the bending of star light was already explained by Newtonian physics,

That's a rather dubious claim, which depends on exactly how one pretends massless objects moving at light speed fit in to Newtonian physics. You certainly can do a "Newtonian" calculation where you get light deflected by half the relativistic figure, but it's a deeply suspect model. I believe Eddington actually did something a bit more sophisticated (neglecting spatial curvature, perhaps?) which is often described as Newtonian because it gives the same answer as the dubious model, but I may have that wrong.

I would think that the equivalence principle, supporting the realisation that inertial paths are actually the "curved" freefall paths, was a critical part of his thinking. It eliminates the possibility of global inertial frames in the presence of gravity, which I don't think can be reconciled with flat spacetime.

But for an actual answer to the question as asked I think you need a ouija board. And Einstein's resurrected spirit might not even be able to answer the question. Do you always know exactly how you came to some conclusion?

 

[Dale]

I have been keen to try to understand what exactly convinced Einstein that spacetime must be curved

As an historical question about what specifically convinced Einstein himself, you would have to look to historical writings where he may have given hints about his thought process. This sort of thing is more likely to be found in personal letters to colleagues than in his peer reviewed papers.

However, it follows directly from the equivalence principle. The equivalence principle tells you that free fall motion is inertial motion. Inertial motion is represented by a straight line in spacetime, per Newton’s first law. Two free-falling objects can collide twice, meaning that two straight lines in spacetime can intersect twice. In flat Euclidean geometry that is not possible, but it is possible in curved geometry. So the geometry of spacetime must be curved.

 

[PeterDonis]

my reading suggests it may have Born rigidity. I believe Einstein assumed that geometry for the rigid rotating (accelerating) disk is non-euclidean and that he therefore needed to adopt non-euclidian geometry for GR.

Logically, this is an obvious non sequitur, because the Born rigid rotating disk can be described entirely using flat spacetime. The "non-euclidean geometry" of the disk is a particular mathematical construction based on the flat spacetime model. It does not in any way show curvature of spacetime.

Historically, I don't know whether a heuristic argument along these lines was significant for Einstein. But I doubt that it would have been, because there is a much better argument having nothing to do with the rotating disk that easily leads to the idea that gravitation involves spacetime curvature: namely, the argument that @Dale gave in post #4.

 

[Sagittarius A-Star]

I believe Einstein assumed that geometry for the rigid rotating (accelerating) disk is non-euclidean and that he therefore needed to adopt non-euclidian geometry for GR.
Am I on the right track?

Yes. Einstein wrote in 1922:

Einstein and general relativity
The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity.[4] He referred to it in several publications in 1912, 1916, 1917, 1922 and drew the insight from it, that the geometry of the disc becomes non-Euclidean for a co-rotating observer. Einstein wrote (1922):[5]

66ff: Imagine a circle drawn about the origin in the x'y' plane of K' and a diameter of this circle. Imagine, further, that we have given a large number of rigid rods, all equal to each other. We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to K'. If U is the number of these rods along the periphery, D the number along the diameter, then, if K' does not rotate relatively to K, we shall have $U/D=\pi$ . But if K' rotates we get a different result. Suppose that at a definite time t of K we determine the ends of all the rods. With respect to K all the rods upon the periphery experience the Lorentz contraction, but the rods upon the diameter do not experience this contraction (along their lengths!). It therefore follows that $U/D>\pi$ .

It therefore follows that the laws of configuration of rigid bodies with respect to K' do not agree with the laws of configuration of rigid bodies that are in accordance with Euclidean geometry. If, further, we place two similar clocks (rotating with K'), one upon the periphery, and the other at the centre of the circle, then, judged from K, the clock on the periphery will go slower than the clock at the centre. The same thing must take place, judged from K' if we define time with respect to K' in a not wholly unnatural way, that is, in such a way that the laws with respect to K' depend explicitly upon the time. Space and time, therefore, cannot be defined with respect to K' as they were in the special theory of relativity with respect to inertial systems. But, according to the principle of equivalence, K' is also to be considered as a system at rest, with respect to which there is a gravitational field (field of centrifugal force, and force of Coriolis). We therefore arrive at the result: the gravitational field influences and even determines the metrical laws of the space-time continuum. If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean.

Source:
https://en.wikipedia.org/wiki/Ehrenfest_paradox