Why is group theory important in General Relativity

Question

I came across the Poincaré group, and most importantly the Lorentz group while studying GR. What is the significance of these groups as well as any other groups used in GR? I mean, why should I care that these set of transformations form groups?

Answer 1

There is a cool vision which I'm recalling from Geroch's Mathematical Physics: in Physics we often encounter structures which happen to satisfy certain axioms and it turns out that often our mathematician friends have been spending quite some time learning about these structures. We can then learn a lot about what we are doing by taking a look at their results. So a first argument could be "because then you can immediately learn a lot about GR by simply taking a quick look at general theorems on groups". For example, there is a unique Poincaré transformation that leaves your coordinate system unchanged. Or the composition of two Poincaré transformations is always a transformation. The fact that Lorentz boosts in different directions do not form a group, for example, can also be read as the fact that if you perform sequential boosts in different directions, you reference systems will have to adapt as to keep the speed of light constant and this requires rotating a little bit (roughly speaking) — see this answer and the ones it refers to for more on this.

More generally, groups are structures that usually arise when we are studying symmetries, where by symmetry I mean "a transformation that leaves the physical laws unchanged". For example, in SR Physics is the same in all inertial frames, which are related to each other by Poincaré transformations. Notice that

1. if I understand a "symmetry" as an action I perform on a physical system, so a function that "eats" my system as it is and "spits" its transformation, the composition of symmetries should be associative, because composition of functions is associative;
2. not changing anything is a symmetry (I can't distinguish the system from itself);
3. if I can't distinguish the transformed system from the original one, I can't distinguish the original one from the transformed, and hence symmetries admit an inverse (just undo the transformation).

That is a group! Hence, whenever we find symmetries in Physics, a group will be there waiting for us, giving mathematical meaning to what it means to be a symmetry. In your particular case, it is encoding the symmetries of Minkowski spacetime (I'm assuming that is the context because it is the spacetime where Poincaré is particularly interesting).

Furthermore, these notions often allows one to get really deep insights about the theory. Let me give a few examples of what I mean.

As a first example, in Particle Physics, one can understand the very notion of what a particle is in terms of irreducible representations of the Poincaré group (roughly speaking, a representation is a way of writing the group in terms of matrices) — see this answer for further discussion. The relevance of group theory comes by means of the notion of representation, which is a structure built upon group theory.

As a second example, considering the sorts of structures which have well-defined transformation properties under the Poincaré group leads one naturally to the notions of tensors and spinors in flat spacetime — see Wald's General Relativity, Chap. 13 for more on that. The relevance of group theory comes once again through representations, which give a precise meaning to "have well-defined transformation properties".

As a third example, the symmetries of the theory are intimately related with its conservation laws by means of Noether's Theorem, meaning, for example, that in a spacetime without a notion of symmetry under time translations (such as homogeneous and isotropic cosmology) you'll have a rough time defining what you mean by the word "energy". The concept of group arises from the notion of symmetry that you need.

One can go on and on forever. The concept of group is definitely of extremely wide usage on contemporary Theoretical Physics, specially when it comes to quantum field theory and general relativity. Nevertheless, I hope this answer might help clarify a bit what are some of the interests.

In summary: group theory is the natural language for dealing with symmetries, and symmetries are quite useful for many purposes. As Sundermeyer puts in his Symmetries in Fundamental Physics, "isn't it surprising that we can do Physics at all"? That's thanks to symmetries, and hence indirectly thanks to group theory.