## Friday, November 14, 2008

### General Relativity

As part of my personal campaign to ramp up physics as fast as possible, I determined to study special/general relativity in this winter period, after having finished the OU’s Electromagnetism course in October, and before starting their Quantum Mechanics course this coming February. I therefore bought the middle two ‘blocks’ of the Open University’s “Space, Time and Cosmology” (S357) third-level physics course over the summer (through Amazon) and have now completed them.

Space, Time and Cosmology” has a reputation for being light on maths but heavy on the conceptual foundations. I have found it wonderful for its clarity – particularly General Relativity (GR) – while I wish they had felt able to put a little more maths into it.

Necessary ingredients

• A description of spacetime curvature (the Riemann curvature tensor – with 20 components)

• A description of mass-energy distribution/flow at a point (the energy-momentum tensor Tμν – with 10 components).

• The Ricci curvature tensor Rμν which is a sum of above-mentioned Riemann tensor components. The Ricci tensor is zero in regions without a source term (zero mass-energy) although spacetime may still be curved there. (Nowhere in the universe is truly energy-free).

• The metric tensor, which shows how spacetime intervals are computed within the relevant coordinate systems (gμν– 10 components).

Einstein’s 10 field equations are then (without the cosmological constant):

Rμν- (1/2)gμνR = -8πGTμν

where G is the gravitational constant and R (unsubscripted) is a spacetime function called the curvature scalar.

Given a mass/energy distribution, from the field equations one can compute the metric tensor gμν.

Once we have all the gμν components we can slot them into the metric expression to finally compute geodesics. These are the paths bodies take under gravity, formerly computed using Newton's laws of motion.

This is just as complicated as it sounds, so there was general surprise when Karl Schwarzschild produced an exact solution for the curved spacetime metric around a non-rotating spherically-symmetric mass, in 1916. See here for details.

The Kerr solution, for rotating objects (most notably black holes with angular momentum) was not developed until 1963.

I'm now going to take a brief look at Wheeler's book on GR, which I think is at the same level as the OU text, and then take a look at Sean Carroll's lecture notes on general relativity here.