Notes for 'Advanced General Relativity' (Warwick postgraduate course)

Pseudo-Riemannian space

element of a vector space V can be expanded in terms of a basis

For a vector bundle the basis of sections does not exist [probably no one will understand]

p-form valued sections

g_jk,l = Γ_jkl + Γ_kjl

g_ij are not observable like the potentials

g is a transverse wave

Fun - what is 'teleparalellism'?

exercise. GR two-body problem

solution to ‘teleparallelism’ exercise

acceptable solution

- parallel vectors at different points of space

full marks

- h is symmetric and rank n - 1

probably should not try to figure out in full detail

another acceptable solution

g^ij R_ij = R

what is the relationship between R and Λ

R is the Ricci scalar

It is not very difficult to understand the Einstein field equation, but writing them in a form in which they can be manipulated is very hard

How did Hilbert discover the field equations?

think before looking at the answer

g is the Lorentzian metric and $\delta g$ is any symmetric (0, 2)-tensor that vanishes outside a compact set. This allows us to define the variation of the action as 

so we have

we get

[ when Hilbert told Einstein about this, he explained to Hilbert what the christoffel symbols are ]

The power of the Jacobian under transformation is called the weight of the tensor (density)

In order to perform the variation of the action, we need a lagrangian which is a scalar

√g has weight -1

d⁴ x has weight 1

g^{μ ν} has weight -2

so, if

ℒ = - 1/16πG g^{μ ν} R_{μ ν}

(this is 16 instead of 8 because we did not include the speed of light)

-1/16πG ∫ d⁴x √g ( R_{μ ν}  - 1/2 R g_{μ ν} ) 𝛿g^{μ ν}

variation of  ∫ ( L - 1/2κ R ) √g d⁴ x = 0

gives the field equations

Hilbert misunderstood the meaning of integration by parts

what are the Bianchi identities?

you just say that the curl of R^i_{jklm} = 0