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Thread: Einstein-Cartan Theory Primer

  1. #1 Einstein-Cartan Theory Primer 
    Moderator Moderator Markus Hanke's Avatar
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    General Relativity, as successful a theory as it is, suffers from three fundamental problems which restrict its domain of applicability :
    1. It does not incorporate quantum effects
    2. It includes unphysical singularities
    3. 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 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.

    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 parallel transport. 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 does not coincide 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 vectors will coincide ). Here is an illustration of the concept :

    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 curvature. 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 ?

    The answer is yes, we can, and we do it by introducing another fundamental geometrical concept, that of torsion. 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 :

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

    This is the basic idea behind torsion. 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.

    Mathematically, the object which reflects the “mix” of curvature and torsion on a given space-time is called a connection. 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 called Einstein-Cartan theory, and concerns space-times which, in the general case, can have both curvature and torsion. 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 possesses only torsion; this model is called Teleparallelism, 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.

    As it turns out, space-times with torsion and space-times with curvature are not entirely equivalent 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.

    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.


    Last edited by Markus Hanke; March 26th, 2013 at 11:51 AM.
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    Brassica oleracea Strange's Avatar
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    Thanks for that introduction, Markus. How does ECT help with the problem of reconciling GR and QM?

    (I'm sure there will be some people thinking, "hmmm ... torsion ... vortices ... aether ..." )


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    Quote Originally Posted by Strange View Post
    Thanks for that introduction, Markus. How does ECT help with the problem of reconciling GR and QM?
    It doesn't. It is a purely "classical" theory in that it is deterministic, not probabilistic, just like GR is. ECT thus cannot incorporate any quantum effects - sorry, I should have made that clearer.

    (I'm sure there will be some people thinking, "hmmm ... torsion ... vortices ... aether ..." )
    Ha ha ! You are absolutely right - it is just a matter of time
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    Just a few more words on the maths behind ECT.
    A space-time, in the mathematical sense, is a very abstract concept; in order to extract any quantifiable physics from it, we must endow it with three elements :

    1. A coordinate basis , so that we have a set of coordinates to work with
    2. A metric , to define measurements of angles, lengths, areas and volumes in our coordinate basis
    3. A connection , to reflect the underlying geometry of the space-time

    Furthermore it is to be stressed once again that curvature and torsion are encapsulated in the connection defined on the space-time; for any given space-time we are free to choose an infinite number of possible connections, giving different mixes of curvature and torsion. The notion of curvature therefore is a lot less fundamental than it is often made out to be. This means that it is possible to write the physical law, i.e. Einstein-Cartan-Theory for classical gravity, in such a way that it is independent not just under coordinate transformations, but also under multi-valued gauge transformations of the connection. The resulting field equations can be written either in terms of the Ricci tensor, or in terms of the torsion tensor, or a suitable combination of the two. The simplest form is the one using the Ricci tensor - here the field equations become



    This looks deceptively similar to "normal" GR, but it is not the same set of equations ! Firstly, the Ricci tensor is no longer symmetric : , i.e. it contains additional information about torsion. This brings the total number of equations in the system to 16, but trivially solves the troubling issue of spin-orbit coupling. Furthermore, the right hand side contains a new tensor , which is now a canonical stress-energy-momentum tensor; together with the torsion in the Ricci tensor it can be shown that this precludes the formation of space-time singularities. This puts an entirely new twist on black holes and the Big Bang !

    Generally speaking these equations are harder to solve than standard GR, because there is a greater number of them. However, analytically closed solutions do exist, e.g. there is the equivalent to a Schwarzschild solution in a space-time with only torsion.

    Finally, it is worth mentioning that ECT is a distinct theory, and not the same as GR "with some torsion thrown in". Under some circumstances it produces different physics ( spin-orbit coupling, singularities ), so at least in principle it is possible to physically distinguish these two models. The problem of course is that we cannot directly observe the innards of black holes, and the magnitude of the spin-orbit coupling effect is so small as to be undetectable with current instruments. This might, however, change in the future.
    Last edited by Markus Hanke; March 26th, 2013 at 11:58 AM.
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    Since I know what's bound to hit me, here is a disclaimer - this thread is focused on conveying the general ideas, it is not mathematically rigorous in any way, shape or form. So to all of you mathematicians out there - don't murder me for inconsistencies or lack or rigour. The target audience here is the casual forum reader, the laymen. Corrections and suggestions are always welcome, though.
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    Dang, I should read over this thread sometime when I'm more intelligent. Since I couldn't discern it, I can only assume you made a very nice thread, congratulations.
    "MODERATOR NOTE : We don't entertain trolls here, not even in the trash can. Banned." -Markus Hanke
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    One more important note : Einstein-Cartan theory ( ECT ) is NOT the same as Einstein-Cartan-Evans ( ECE ) theory ! The latter is a unified field model proposed between 2003-2005, but has subsequently shown to be mathematically incorrect. These two are often confused.
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    i found some maths on this [gr-qc/0606062v1] Einstein-Cartan Theory
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    Quote Originally Posted by fiveworlds View Post
    i found some maths on this [gr-qc/0606062v1] Einstein-Cartan Theory
    This is a good article, at least for those of us who are intimately familiar with differential forms
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    Quote Originally Posted by Markus Hanke View Post

    This is a good article, at least for those of us who are intimately familiar with differential forms
    Tell me. Are you "one of those that are "intimately familiar" with the algebra of differential forms?
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    you mean calculus? wouldnt be classifed as algebra nor precalculus just calculus with annoying symbology like rho to make the maths look completely illegible
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    Quote Originally Posted by Guitarist View Post
    Tell me. Are you "one of those that are "intimately familiar" with the algebra of differential forms?
    Absolutely and unfortunately not, Guitarist. You helped me a while back with some basic concepts around forms, but that does not by a long shot make me intimately familiar with them, which is why I avoided any mention of them in my OP. However, while being a very elegant way to formulate ECT, they are not absolutely necessary for the understanding of the theory.
    I did get the basic gist of the article referenced though, if not all the maths involved.
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    Quote Originally Posted by Markus Hanke View Post
    General Relativity, as successful a theory as it is, suffers from three fundamental problems which restrict its domain of applicability :
    1. It does not incorporate quantum effects
    2. It includes unphysical singularities
    3. 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

    .
    Isn't 3. the same as 1.?
    In the information age ignorance is a choice.
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    Quote Originally Posted by Kerling View Post
    Isn't 3. the same as 1.?
    No, number 1 refers more to the fact that GR as a whole is a deterministic theory rather than probabilistic. It is a general statement. Number 3 on the other hand refers to a specific gravitational phenomenon which cannot be modeled using the axioms of GR.
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    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by Kerling View Post
    Isn't 3. the same as 1.?
    No, number 1 refers more to the fact that GR as a whole is a deterministic theory rather than probabilistic. It is a general statement. Number 3 on the other hand refers to a specific gravitational phenomenon which cannot be modeled using the axioms of GR.
    I must honestly say, I have derived spin orbit coupling on various occasions. And I have never ever heard of a method that uses gravity in spin orbit coupling.
    I can imagine that the formalism of GR does not allow a derivation alike that of spin orbit coupling. But spin-orbit coupling is definitely not a gravitational phenomena. It would actually be intrinsically impossible to be one due to symmetries.
    In the information age ignorance is a choice.
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    Quote Originally Posted by Kerling View Post
    I must honestly say, I have derived spin orbit coupling on various occasions. And I have never ever heard of a method that uses gravity in spin orbit coupling.
    I can imagine that the formalism of GR does not allow a derivation alike that of spin orbit coupling. But spin-orbit coupling is definitely not a gravitational phenomena. It would actually be intrinsically impossible to be one due to symmetries.
    To my understanding "spin-orbit coupling" refers in a general sense to any interaction between intrinsic angular momentum and a particle's motion. GR cannot model any such interaction ( although it can handle particles with spin ), since the Einstein tensor is symmetric under exchange of indices, and therefore so is the energy-momentum tensor in the field equations. As you said yourself quite rightly, this precludes any coupling between spin and orbital angular momentum for symmetry reasons. ECT is an attempt to remedy this, because we would expect a theory of gravity to be able to at least describe the classical interaction between spin and orbital angular momentum, if not its quantum counterpart. This was in fact one of the main motivations for Elie Cartan when he developed ECT in the 1920s; see also

    Einstein
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    You see, this is why I hated Tensors :P
    In the information age ignorance is a choice.
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    Please, could anyone explain to me why the Einstein - Cartan theory is not gauge invariant? What does this actually mean?
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    Please, could anyone explain to me why the Einstein - Cartan theory is not gauge invariant? What does this actually mean?<br><br>
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    Quote Originally Posted by gravity View Post
    Please, could anyone explain to me why the Einstein - Cartan theory is not gauge invariant? What does this actually mean?
    A theory is said to be gauge invariant if its physical content does not change under gauge transformations; i.e. the Lagrangian of the theory remains the same if a gauge transformation is applied to it. The field equations of ECT are



    where T is the torsion tensor, and S is called the canonical spin density tensor of the electromagnetic field :



    It is this latter term which causes the issue, because under the gauge transformation




    the canonical spin density tensor is not invariant ( unlike the l.h.s. of the equation, which is ). The actual maths are anything but trivial, but here is a good summary which employs the same notation as the above :

    http://arxiv.org/pdf/1212.3266v2.pdf
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    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by gravity View Post
    Please, could anyone explain to me why the Einstein - Cartan theory is not gauge invariant? What does this actually mean?
    A theory is said to be gauge invariant if its physical content does not change under gauge transformations; i.e. the Lagrangian of the theory remains the same if a gauge transformation is applied to it. The field equations of ECT are



    where T is the torsion tensor, and S is called the canonical spin density tensor of the electromagnetic field :



    It is this latter term which causes the issue, because under the gauge transformation




    the canonical spin density tensor is not invariant ( unlike the l.h.s. of the equation, which is ). The actual maths are anything but trivial, but here is a good summary which employs the same notation as the above :

    http://arxiv.org/pdf/1212.3266v2.pdf

    thank you very much for your help! But why is it necessary for a theory to be invariant under gauge transformations?
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    Quote Originally Posted by gravity View Post
    thank you very much for your help! But why is it necessary for a theory to be invariant under gauge transformations?
    Gauge invariance is not a necessity, but it is highly desirable since it makes it potentially possible to find a quantum field theory for such a model, i.e. to quantize the theory. The basic idea behind it is that the gauge symmetry groups under which the theory is invariant give us information about the type of gauge bosons which would be the force carriers in the QFT. Also, the presence of gauge symmetries makes analysing the dynamics of a theory easier.

    This is actually a vast and pretty complicated subject - perhaps have a look here :

    Introduction to gauge theory - Wikipedia, the free encyclopedia

    and here

    Gauge theory - Wikipedia, the free encyclopedia

    and also here

    Yang
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  24. #23  
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    Quote Originally Posted by Markus Hanke View Post
    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.
    I think there may be some confusion between torsion and anholonomity here. Anholonomity arises in standard GR when one moves to the orthonormal tetrad (aka vierbein) formalism, or more generally, when one extends the tensor group from the Jacobian matrices to the general linear group of functions (which is not integrable in general). Like the torsion tensor, the object associated with anholonomity is anti-symmetric in the first two indices, but unlike the torsion tensor, the object associated with anholonomity is not a tensor.

    Quote Originally Posted by Markus Hanke View Post
    the exact orientation does not matter so long as it is tangent to the surface
    Parallel transport (of a vector) means that a vector at some given point is parallel to itself at a neighbouring point through which the vector is transported. That it remains tangent to the manifold is part of the intrinsic view but doesn't specifically pertain to parallel transport unless this is being viewed extrinsically.

    Quote Originally Posted by Markus Hanke View Post
    The answer is yes, we can, and we do it by introducing another fundamental geometrical concept, that of torsion. 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.
    In terms of parallel transport, curvature is defined as follows:

    Consider an infinitesimal parallelogram and a vector that is parallel transported around the parallelogram back to its original location. The parallel transported vector will differ from original vector by a doubly-infinitesimal amount that depends on the components of the curvature tensor that align with the parallelogram and also the vector.

    In terms of parallel transport, torsion is defined as follows:

    At a given point, construct two (different) infinitesimal displacement vectors dx1 and dx2. Parallel transport dx2 along dx1 and dx1 along dx2. When the torsion is zero, the end-points of the two parallel transported vectors will coincide, forming a parallelogram. However, in general (when the torsion is non-zero), the end-points do not coincide and the doubly-infinitesimal displacement between the two end-points depends on the components of the torsion tensor that align with dx1 and dx2.
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    Quote Originally Posted by KJW View Post
    I think there may be some confusion between torsion and anholonomity here. Anholonomity arises in standard GR when one moves to the orthonormal tetrad (aka vierbein) formalism, or more generally, when one extends the tensor group from the Jacobian matrices to the general linear group of functions (which is not integrable in general). Like the torsion tensor, the object associated with anholonomity is anti-symmetric in the first two indices, but unlike the torsion tensor, the object associated with anholonomity is not a tensor.
    While the above is certainly true, this thread is specifically about Einstein-Cartan theory, not standard GR.
    You appear to have good knowledge of the subject, unfortunately few people here will be able to follow or understand what you wrote...
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    Quote Originally Posted by Markus Hanke View Post
    While the above is certainly true, this thread is specifically about Einstein-Cartan theory, not standard GR.
    Yes. The anholonomity object is a part of standard GR as well as natural extensions to it (such as Einstein-Cartan theory). In general, the connection object contains four objects: the partial derivative of the metric tensor, the anholonomity object, the torsion tensor, and the covariant derivative of the metric tensor. Of these, only the latter two are tensors, and these are both zero in standard GR. There is a natural symmetry between the partial derivative of the metric tensor and the anholonomity object (whether or not the two tensors are zero). The two standard formalisms of GR represent the two extreme cases of this symmetry. On the one hand, the metric formalism has arbitrary metric tensor and zero anholonomity, while the orthonormal tetrad formalism has arbitrary anholonomity and Minkowskian metric tensor1.

    It is worth pointing out that Einstein-Cartan theory is a natural generalisation of standard GR in the way torsion emerges as the first integrability condition of the existence of a coordinate transformation between two connections, with curvature emerging as the second integrability condition. What this means is that even if torsion is zero, GR still has to explain why.


    1 The orthonormal tetrad formalism itself might not make this explicit, but the connection between the two can be seen by considering the definition of the orthonormal tetrad in compact form:



    where is the metric tensor of the coordinate system, is the tetrad, and is the Minkowskian matrix that expresses the orthonormality of the tetrad. This expression can be seen as the transformation of the metric tensor from the coordinate system to the anholonomic system where the metric is Minkowskian, with the orthonormal tetrad acting as the transformation matrix. In general, the orthonormal tetrad is non-integrable (unlike the Jacobian matrix associated with a coordinate transformation), and it is this non-integrability that leads to the anholonomity object within the connection object (because the Minkowskian metric is constant, its partial derivative does not appear in the connection object).
    Last edited by KJW; June 27th, 2013 at 01:08 PM.
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    Quote Originally Posted by Kerling View Post
    Quote Originally Posted by Markus Hanke View Post
    Quote Originally Posted by Kerling View Post
    Isn't 3. the same as 1.?
    No, number 1 refers more to the fact that GR as a whole is a deterministic theory rather than probabilistic. It is a general statement. Number 3 on the other hand refers to a specific gravitational phenomenon which cannot be modeled using the axioms of GR.
    I must honestly say, I have derived spin orbit coupling on various occasions. And I have never ever heard of a method that uses gravity in spin orbit coupling.
    I can imagine that the formalism of GR does not allow a derivation alike that of spin orbit coupling. But spin-orbit coupling is definitely not a gravitational phenomena. It would actually be intrinsically impossible to be one due to symmetries.
    I think what was meant was that in the same way that the Einstein tensor represents the gravitation associated with energy-momentum, the asymmetry of the Einstein tensor due to torsion represents the gravitation associated with spin-orbit coupling. While I don't think GR (and its extensions) is just about gravitation, there are those that do, and in a way, I can see their point. Thus, while spin-orbit coupling is not itself a gravitational phenomenon, it is unable to be represented within standard GR, a supposedly gravitational theory that describes the energy-momentum of reality.
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    Quote Originally Posted by Markus Hanke View Post
    While the above is certainly true, this thread is specifically about Einstein-Cartan theory, not standard GR.
    In the context of Einstein-Cartan theory, I think it is important to appreciate where it stands within the fully generalised theory of relativity. I focussed on the object of anholonomity because the algebraic structure of this object closely matches the torsion tensor and therefore may easily be confused with it. Also, it is interesting in its own right. Although it has no physical consequences (all manifestly covariant expressions remain unaltered), it does provide an alternative way to describe physics. It alters the definition of both the curvature tensor and the torsion tensor by the addition of terms which contain the anholonomic object explicitly. However, this generalisation of the description does make the curvature and torsion tensors with respect to a larger group of transformations than the Jacobian matrices (demonstrating that these tensors can still be defined within the context of a larger group of transformations).
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