# Dirac Talks: Introduction to mathematical structure of general relativity

I'll give a brief review of the mathematical structures

underlying General Relativity, at each step giving a physical motivation

for invoking the structure. My perspective will be from the standpoint

of asking what kind of global structure we can reasonably infer from the

local observations we're limited to making. In particular, I'll begin

with the definition of a smooth manifold and discuss the construction of

its tangent space and the notion of a connection. I'll introduce tensors

and why special relativity leads us to expect that a manifold model for

the universe should come equipped with a metric tensor. Finally, I'll

introduce the Levi-Civita connection and use it to talk about the

decomposition of a general connection into its metric compatibility and

torsion tensors, demonstrating how torsion can change geodesics when

metric compatibility is fixed. This will set the stage for Ben's

discussion, which will culminate in a computation of how light redshift

would be different if one takes the geometric motivation of dark matter

through the addition of torsion seriously.