General Relativity, as successful a theory as it is, suffers from three fundamental problems which restrict its domain of applicability :In order to overcome the latter two of these problems and thus push the boundaries of the theory’s domain of applicability, several extensions to GR have been proposed over the years; I wish to present one of them here, which was considered by Einstein himself in his later years, and goes by the name “Einstein-Cartan Theory” ( ECT ). I had alluded to this a while back on another thread, but at the time wasn’t ready to discuss this further.

- It does not incorporate quantum effects
- It includes unphysical singularities
- It cannot describe a phenomenon called ‘spin-orbit coupling’, which is the interaction of a particle’s intrinsic angular momentum with its trajectory of motion

In order to get a handle on what ECT is all about, we need to first take a closer look at how we describe the geometric properties of space-time. One of the most important notions in both differential geometry in general and GR and ECT in particular is that of. While a precise and mathematically rigorous treatment of this topic is beyond the scope of this post ( and my own mathematical abilities ), the basic idea is quite simple. Start by visualizing a surface which is intrinsically curved, e.g. the surface of a sphere; pick three points on that surface at random ( let’s call them A, B and C ), then connect these points. Now draw a tangent vector at point A, i.e. a vector which originates at point A and is tangent to the surface at that point - there are infinitely many possible vectors from which we can freely choose, since the exact orientation does not matter so long as it is tangent to the surface. Now do the following : take our vector and transport it along the curve which connects points A and B in such a manner that at every point it remains tangent to the surface. Once you arrive at B, transport the vector to point C, and eventually back to point A, always making sure it remains tangent to the surface. What will we find once we arrive back at the original point A ? We will find that the transported vector doesparallel transportcoincide with the original vector we started with. The degree by which the two vectors differ is a direct measure of how much the geometry of the underlying surface differs from Euclidean geometry ( on a flat Euclidean surface the vectorsnotcoincide ). Here is an illustration of the concept :will

File:Parallel transport.png - Wikipedia, the free encyclopedia

This very simple notion of parallel transport forms the basis of the treatment of non-Euclidean geometry, and thus non-Euclidean space-times.

In Einstein’s General Relativity the parallel transported vector will differ from the original vector because the surface on which this operation is performed possesses intrinsic. The vector’s orientation changes during the transporting procedure because we demand that it must be tangent to the surface, and the surface is not flat, like in the example of the sphere. Treating parallel transport in this way leads us to Riemann geometry and standard General Relativity, which we are all more or less familiar with. But is there another way to look at it ? Can we arrive at the same result ( i.e. the transported vector not coinciding with the original one ) by other means ?curvature

The answer is yes, we can, and we do it by introducing another fundamental geometrical concept, that of. To visualize this, picture holding a piece of string spanned between your two hands. The string is flat and straight. Now twist one end of the string about itself, while keeping the other end fixed. The string is still straight and flat, but now possesses an intrinsic “twist”. If we were to transport a tangent vector along this “twisted” string, its orientation would also “twist” and thus change, even though the string possesses no curvature. Here is an illustration :torsion

File:Torsion along a geodesic.svg - Wikipedia, the free encyclopedia

This is the basic idea behind. It is an intrinsic geometric property just like curvature, and can also be defined using the notion of parallel transport, again just like curvature. Producing the end result of a parallel-transported vector not coinciding with the original vector can thus be done by performing the parallel transport on a surface with curvature ( but no torsion ), on a surface with torsion but no curvature, or by any suitable combination of torsion and curvature. There is thus an infinite number of different ways to produce the same outcome.torsion

Mathematically, the object which reflects the “mix” of curvature and torsion on a given space-time is called a. In the case of standard General Relativity the so-called “Levi-Civita connection” is used; this connection has the property that it is everywhere torsion-free, i.e. space-time under this connection possesses only curvature, but no torsion. This is our normal Riemann space-time. The generalization of GR is now obvious - we can simply permit other connections on our space-time, thus introducing torsion, and see what happens. The resulting theory is calledconnection, and concerns space-times which, in the general case, can have both curvatureEinstein-Cartan theorytorsion. GR is now just a special case of ECT where torsion vanishes everywhere; on the other end of the scale sits a theory where curvature vanishes everywhere, and space-time possessesandtorsion; this model is calledonly, and the space-time is called Weizenboeck space-time. Between these two extremes is an infinite spectrum of models where curvature and torsion are mixed to varying degrees; the fascinating aspect of this is that all of these theories can produce equivalent physics. For example, the gravitational field of the earth can be described via curved space-time in GR, but it can also be completely equivalently described as a flat space-time with torsion under teleparallelism. The resulting physics are equivalent in this case. What this means is that space-time physics are independent of the choice of coordinates, but they are also independent of the choice of connection to some degree. This is an entirely new symmetry, in fact it can be shown that this is a multi-valued gauge symmetry on space-time itself.Teleparallelism

As it turns out, space-times with torsion and space-times with curvature are notequivalent when it comes to the physical outcomes. Coming back to the original problems of GR, it turns out that in space-times with torsion the spin-orbit coupling problem is trivially solved; it also turns out that singularities in black holes no longer occur in ECT, eliminating this problem as well ! It will skip the mathematical details here, as papers treating these results can be easily found on arXiv.entirely

This concludes my little presentation. Note that I have foregone mathematical rigour ( of which I as a non-mathematician am not capable anyway ) in favour of intuitive understanding here, so don’t hang me on the mathematical details.