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I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I know that the starting point for Einstein's considerations was the equivalence principle. Nevertheless I can't make the connection from there, to why gravity is the curvature of spacetime. Any help or reference to where this is explained would be much appreciated.

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See Riemann. I think he was the one who elaborated the idea of 4th dimension and introduced the idea that forces were just "crumpling" of space in higher dimensions. I read this in "Hyperspace" by Michio Kaku. – Yashbhatt May 26 '14 at 16:47

Perhaps more of a comment that was too long, rather than an answer.


It seems a historical account for the motivation and construction of a derivation of the field equations,

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G \, T_{\mu\nu}$$

is probably what is required to answer your question. In 1913, Einstein published the entwurf equations which were covariant under general linear transformations; it was only much later did Einstein return to general covariance, and in fact had rejected such constructs in the past...

A historical account of every dead end Einstein faced, and how he eventually arrived at the Einstein field equations is a long story, too long for a physics S.E. answer. As such, I redirect you to Untying the Knot: How Einstein Found His Way Back to Field Equations Discarded in the Zurich Notebook, which is freely available here. If I recall correctly, the truth is Einstein wrote the correct field equations, but unknowning of their correctness, discarded them in his 'Zurich' notebook. The answer to your question (from Einstein's original perspective) almost surely lies within the text; in addition see http://arxiv.org/abs/1201.5353.

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Thanks for your answer. I understand how one gets from the idea gravity = curvature in spacetime, to the Einstein field equations. Nevertheless I'm still unsure were this idea comes from in the first place. Can this be motivated in any way? Most books I know take this as the starting point or as a given assumption without further motivation. I'm less interested in Einsteins historical struggles and would love a more modern perspective on how to get to this insight. Any further idea besides: "Because it works", would be much appreciated. – JakobH May 27 '14 at 9:42
    
I think it really started with Minkowski space, where Minkowski was able to create a 4-dimensional spacetime geometry that would allow the Lorentz transformations as "rotations" in the geometric model he had created. At first Einstein viewed it as a mathematical tool, but Einstein relized that spacetime was the physical reality and that the idea could be expanded to all reference frames and developed General Relativity with the non-Euclidean (curved) framework of spacetime. – Peter R 2 days ago

First of all gravity is not the spacetime curvature, instead it is the geodesics assumed by a physical entity as it moves along the curved spacetime. What do I mean by geodesics? A geodesic is simply the shortest path a particle assumes in some paramitized curve. To understand the abovementioned think of a heavy ball on top of your bed or fabric, what would happen if marbles were placed near it? The genius of Einstein arises from his claim that gravity arises because of the curvature of spacetime, and not because of some mystic force field as Newton believed. Thus objects can assume intricate paramitized curves (Orbit of mercury), instead of the simple elliptic paramitized curve.

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"think of a heavy ball on top of your bed or fabric, what would happen if marbles were placed near it?" Sure, they would roll down toward the heavy ball. But only because there is the force of gravity "underneath" - if there was only curved fabric and no gravity the marbles would not move at all. – bright magus May 27 '14 at 15:37
    
Yes on Earth it would seem so, but think about in outer space. Einstein commensurated that fabric experiment with what happens in the heavens, instead of having heavenly bodies moving due to some mystic force they might as well do so due to the curvature of spacetime. – user34793 May 28 '14 at 8:37
    
I don't understand. He actually conducted an experiment in "heavens"??? Still, there is no answer to the question: What makes things move? Mystic curvature? – bright magus May 28 '14 at 9:29
    
He performed at thought experiment...Einstein thought of geometry of spacetime dictating the motion of heavenly bodies,primarily because there wasn't any apparent force that could perfectly explain why objects attract. Newton's gravity theory was problematic. Newton described the force applied to heavenly objects and not neccessarily what governs the force, Einstein's quest was to explain this...In short,according to Einstein it is the geometry of spacetime that manupilates the Newton's Gravitational force. – user34793 May 29 '14 at 14:29
    
I'm perfectly aware that Einstein wanted to replace the force of gravity with curvature. But still, as I showed, curvature itself does not produce movement. Movement due to curvature alone is as magical as Newton's force at a distance. – bright magus May 29 '14 at 14:54

Since you mention the following in one of your comments

I'm less interested in Einsteins historical struggles and would love a more modern perspective on how to get to this insight.

I hereby unashamedly ignore history, and offer instead a quick plausibility argument.

Let's start with the equivalence principle which, loosely speaking, says that a (sufficiently small) system freely falling in a gravitational field is indistinguishable from the same system floating inertially in empty space.

Consider now a satellite freely orbiting the earth in a 2-D plane. Now use the vertical direction of the picture to embody time to form a spacetime representation, and in this picture plot the trajectories of the center of mass of each body.

Again roughly speaking, and using our eyes' geometry, you would draw the earth's world-line as a straight vertical line and the satellite's as a helix winding its way around this center line, right?

Now we try to reconcile this picture with the equivalence principle which, again, seems to suggest that each of these lines must be 'straight', because each object is really floating inertially.

They certainly don't look straight, but what if straightness means something non-Euclidean in spacetime? And that's where it would make sense to vary the geometrical stucture of spacetime in such a way that both trajectories really are geodesics of that non-conventional metrical structure.

Does that feel compelling enough to you?

Again, we are not pretending Einstein did this of course. Nor are we pretending that we could reconstruct general relativity easily from this argument. We are just making the connection plausible to a modern reader, rather than making it appear magically out of the blue.

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In "The Foundation of the General Theory of Relativity" Albert Einstein conducted two thought experiments - with a lift and with a spinning disk - where he compared gravity and acceleration. He considered a path of light as seen by an accelerated and a non-accelerated observer. As a result of his reasoning he concluded (at the end of the section 2):

"The path of a ray of light with respect to K' must in general be curvilinear, if with respect to K light is propagated in a straight line with a definite constant velocity."

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Thanks for the Link. This goes exactly in the right direction. We can get the effect of gravity by coordinate transformations and equally transform the effects of gravity away, because gravity acts on all objects equally. Nevertheless I still can't make the final connection why this implies that gravity is a effect of curved spacetime. – JakobH May 27 '14 at 9:50
    
Well, I gave you official explanation. Now, the problem is that Einstein wrote the above sentence about a body located in a field of force (gravitational) comparing it to a body experiencing actual movement due to acceleration. If you are on a merry-go-round and throw a stone, you will see the path of the stone as curvilinear, but person on the ground will see it as straight. However, the person on the merry-go-round not only feels acceleration, but he is also moving due to it. This would be equivalent to a person in free fall in gravitational field (accelerating) and not stationary. – bright magus May 27 '14 at 10:32
    
So, Einstein simply produced a derivation of an equation proving what he claims, but there are a lot of weird things in there. For instance, you can read further that "For infinitely small four-dimensional regions the theory of relativity in the restricted sense is appropriate, if the coordinates are suitably chosen." This means that at the limit SR applies, and therefore gravitation does not exist at the limit. If so, than how can you derive equations for gravitation at the limit, if it is said not to exist there? You should read the whole thing yourself and arrive at your own opinion. – bright magus May 27 '14 at 10:34
    
N.B. I'm encouraging you to read the original text, because I think you are not likely to obtain a clear answer to you question here. As you see, no-one else even tried to give you answer going down to the origins of Einstein's concept here. My experience is people (PhDs even) excel in complicated maths, but they don't care much about basics (that's why you often get this "because it works" answer). And lots of "why" questions are being dismissed as "non-physical". – bright magus May 27 '14 at 11:47

The first motivation was perhaps the incompatibility of Newton's law with special relativity, which eliminated the idea of gravity being a force, and rather the geometric property of space and time, whose mechanism was curvature. Then, like @Yashbhatt said, it was the study of tensor analysis on curved manifolds initiated by Ricci and Riemann which led to the so called 'trampoline' analogy, which was perhaps the genesis of general relativity, and of course, it took Einstein's genius to realise that.

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