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.