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The Lie derivative measures the difference between two paths in the timespace manifold, and hence the commutator bracket occurs naturally, as explained in the presentation What is a Tensor? Lesson 21: The Lie Derivative by XylyXylyX.

The question is at what point in GR does the special linear algebra, understood as the group of invertible traceless matrices with the commutator bracket, make its appearance and why?

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  • $\begingroup$ Out of curiosity, is your question (about the $sl$ algebra) spurred by something you read/watched? $\endgroup$ – Qmechanic Jan 5 at 22:29
  • $\begingroup$ @Qmechanic Just the Xy...yX lectures on GR. $\endgroup$ – Antoni Parellada Jan 6 at 0:09
  • $\begingroup$ Does he talk about $sl$? $\endgroup$ – Qmechanic Jan 6 at 16:08
  • $\begingroup$ @Qmechanic No, he does not, but it was that lecture that prompted me to ask this question. $\endgroup$ – Antoni Parellada Jan 6 at 16:09
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at what point in GR does the special linear algebra, understood as the group of invertible traceless matrices with the commutator bracket, make its appearance and why?

Typically, Lie groups appear in GR when one wishes to define and work with the Killing vector fields of some spacetime. Clearly, the Lie derivative defines the Killing field, and you can see here how the Killing form plays a basic role in the theory of Lie algebras (also see the last paragraph). In GR, a non-degenerate Killing form induces a Levi-Civita connection on the Lie group (in this case the Poincare group), and the Levi-Civita connection is important since it is the torsion-free metric connection.

But there is seemingly always newer interest in how the Lie algebra itself may be used, for instance in this article they use the underlying Lie algebra of minkowski spacetime to provide a boost for non-inertial frames providing for a completely dynamical framework.

In the case of the Minkowski spacetime (Lorentz manifold with Minkowski metric), the Lorentz Group is the symmetry group of the manifold (the group of isometries of the metric) and is itself a Lie group. However, the corresponding Lorentz Lie algebra is isomorphic to the special linear group, see here (p. 9) and here. Thus, the special linear group underlies all of special relativity and general relativity. Here is a very nice discussion of why we use the complexification of the special linear group to achieve the Lorentz group.

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Fundamental physical fields $\phi^a(x^\mu)$ are representations of the Lorentz group $SO(3,1)$ at every point $x^\mu$. This is true both in special as well as general relativity, only in GR we need to define the action of the group with respect to some physical frame at every point (i.e., generally not a coordinate frame).

The group $SL(2,\mathbb{C})$ is the universal cover of the connected part of Lorentz group and its point-to-point representations correspond to possible fundamental fields on a manifold. Its most obvious representation, $2\times 2$ matrices with determinant 1 acting on 2-dimensional complex vectors corresponds to Weyl spinors, which are fields of massless particles with spin 1/2. If we require parity invariance, we have to directly sum two copies of this representation, and we end up with Dirac spinors, which correspond to massive particles with spin 1/2, such as electrons.

As to the Lie algebra of $SL(2,\mathbb{C})$, it is straight-forwardly related to $so(3,1)$. The generators of the Lorentz group can be shown to be related to "internal angular momentum" of the field, also known as "spin". Most importantly, the value of the Casimir element of $sl(2,\mathbb{C})$ in the given representation is known as the total spin magnitude of the given fundamental field. This number is, after all, what we mean when we say that a particle/field has a certain value of spin.

In other words, the group $SL(2,\mathbb{C})$ and its Lie algebra are intimately connected to fundamental fields "filling" the GR space-time. However, understanding the properties of such fields in the given space-time is often informative for seemingly disconnected investigations. Perhaps the most famous one in the field of General relativity is Witten's proof of the Positive mass theorem using spinor techniques.

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