# Mathematical anatomy of general relativity

I was always told that spacetime in general relativity was a Lorentzian manifold, that is, a Pseudo-Riemannian manifold $$(M, g)$$ with metric signature $$(+, -, -, -)$$ or $$(-, +, +, +)$$ and that that is all the mathematical detail needed to understand the theory.

But in the theory we use covariant derivatives and Christoffel symbols to calculate geodesics and curvature, but both Christoffel symbols and the covariant derivative are representations of a connection (the Levi connection in this case) and a connection (as far as i know) can only be defined in a principal bundle

This would mean we need to lift the notion of spacetime as a manifold to spacetime as a principal $$G$$ bundle. But the question is: What is this bundle? I assume the base space is the manifold and the $$G$$ group is the Lorentz group, but what is the fibre, then? What is the interpretation of spacetime as a bundle? And what structure does this lift yield?

I'm really curious, could someone explain it?

• Isn't this mainly due to physicists rarely aiming for mathematical rigour, and hence happily assume as soon as you have a metric manifold you get the higher order apparatus for free? See this question and the answers: physics.stackexchange.com/q/685429 Commented Jan 25 at 8:53
• @AndersSandberg In this case, it is more that the rigorous mathematical apparatus for Riemannian geometry predates the introduction of more abstract notions like principle bundles. Commented Jan 25 at 9:14
• Exactly, Riemannian geometry (1854) and tensor calculus (Ricci, Levi-Civita 1901) predated general relativity (1915), but the objects where not yet fully interpreted geometrically and fiber bundles were far from existing (1950). It was realized much later that GR talks about a unique type of connection on the frame bundle associated to the tangent bundle. (Weyl in 1918 and 1930 went into that direction but still, fiber bundles where not developed yet). Commented Jan 25 at 12:29
• I thought you only need the tangential and cotangential bundle for that, Commented Jan 25 at 16:36
• You can define an affine connection directly on the tangent bundle as you said (which is what is usually done). However most of the geometrical meaning and depth of the theory of connections appears when you define it at the frame bundle level. It also also allows for generalizations (e.g. on spinors) and allow to build direct analogies with gauge theories. Commented Jan 25 at 21:26

The principal $$G$$ bundle of spacetime is the frame bundle $$LM$$ giving all possible frames of the tangent bundle. It's structure group is $$GL(3,1)$$, which is reduced to $$O(3,1)$$ once a metric is defined. Then the tangent space is defined as an associated bundle to the frame bundle which inherits the connection on $$LM$$.

You can find some more information for example in Nakahara's "geometry topology and physics".

• If one applies that question to the static spherically symmetric spacetime, what is the fiber there, the $S^{2}$ sphere?
– JanG
Commented Jan 25 at 15:29
• No, as long as you consider vector fields, the fiber is always $\mathbb{R}^4$. Commented Jan 25 at 15:39
• Oh, I see, thanks. I wonder if you would mind to answer or to comment my own question on Mathematics site?
– JanG
Commented Jan 25 at 16:04
• With pleasure! I will have a look. Commented Jan 25 at 16:11
• @LéoVacher thanks for your answer, could you please explain how does the $Gl(3,1)$ group reduce to the Lorentz group by defining a metric? Commented Apr 12 at 6:44