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In the wiki page about unsuccessful investigations of Einstein, there is only one sentence describe his unsuccessful of unifying general relativity with electromagnetism:

Einstein spent many years pursuing a unified field theory, and published many papers on the subject, without success.

It has more detail in the Einstein's wiki, but only focus about his being isolation from mainstream physics. However I can't find any info about what we have learnt from his unsuccessful. By learning I mean the physics knowledge, not life lessons like "Never give up" or YOLO. So far the only thing I can tell about that is the concept of unifying physics that Einstein had inspired other physicists.

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    $\begingroup$ The problem of unification of gravitation with other forces is still unsolved. When (and if) it will be solved, one will probably be able to put Einstein's efforts into a perspective, and evaluate his contribution. $\endgroup$ – Alexandre Eremenko Feb 5 '15 at 22:35
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    $\begingroup$ @Ooker: A simple search shows that "Newton" tag should be created first:-) $\endgroup$ – Alexandre Eremenko Feb 6 '15 at 0:26
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    $\begingroup$ There have been successful unified field theories both before and after Einstein's unsuccessful work. Electricity and magnetism were unified by Maxwell in the 19th century. The weak nuclear force was unified with the electromagnetic force ca. 1965. String theory is an example of a (probably unsuccessful) attempt at unifying gravity with other forces. $\endgroup$ – Ben Crowell Feb 6 '15 at 4:22
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What we learned so far belongs to the realm of mathematics, or more generously formal side of theoretical physics, rather than knowledge of nature in the narrow sense. Namely, what effects different unification constructions can produce and account for, and what in them is problematic. Many tricks originally introduced trying to unify gravity and electromagnetism were later used, and continue to be used, in other contexts, some of which have nothing to do with unification.

Unification of gravity and electromagnetism wasn't initiated by Einstein, Herman Weyl pursued it first in 1918-1920, and although Einstein noticed right away that the theory was unphysical what came out of it was the idea of gauge invariance, which became central in the Standard Model. In 1921 Kaluza offered a different model elaborated by Klein in 1926 that introduced the idea of using curled dimensions to unify fields. This idea eventually led to string theory and M-theory with their extra dimensions.

Einstein got involved in 1928, and his main idea was to explore the influence of keeping torsion in play in addition to curvature (in standard general relativity pseudometric connection is used, which by definition has zero torsion). Cartan already suggested a non-zero torsion generalization in 1922, and Einstein attempted to match torsion to the electromagnetic energy tensor. That was unsuccessful, but the Einstein-Cartan theory introduces non-linearity into the Dirac equation which allows to avoid the Big Bang singularity. This feature is currently pursued in cosmology under the name of Big Bounce.

Einstein also went further and replaced the usual metric connections by their direct "opposites", Weitzenböck connections, that have non-zero torsion but zero curvature. This leads to a theory with global parallelism, or teleparallelism, which is equivalently described by a tetrad of vector fields. Again, although the intended goal was not achieved the machinery proved useful elsewhere.

In the theory of gravity itself, for example, Møller showed in 1961 that the tetrad description has some advantages over the metric tensor description, and several theories were proposed that treat gravitational field as a gauge field of the translation group. This approach was later used in loop quantum gravity, which attempts to quantize gravity without unification, ironically. Teleparallelism also came up in two-dimensional non-linear sigma models, where at the infrared limit of renormalization flow curvature disappears but torsion remains. Finally, there is a mathematical analogy between curvature and torsion of spacetime on one hand, and disclinations and dislocations of crystals on the other. Einstein's idea allows one to rewrite curvature effects in terms of torsion, which was fruitful in both areas.

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  • $\begingroup$ Thanks for your answer. In the Avoidance of singularities section, it says that from the Einstein - Cartan theory, we can build a wormhole inside black hole. So I guess the wormhole is one of successful things in unsuccessful things of Einstein, right? And what about the crystals in the last paragraph? Are they the crystals in condense matter physics? $\endgroup$ – Ooker Feb 6 '15 at 3:00
  • $\begingroup$ @Ooker Wormholes "inside" black holes were discovered in standard GR (by Flamm in 1916 and Einstein-Rosen in 1935) with no relation to unified theories, they are unstable though. Only in 2011 Poplawski pointed out that torsion can stabilize them without exotic matter. Yes on condensed matter, see scielo.br/… $\endgroup$ – Conifold Feb 6 '15 at 16:56

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