T2: 1. Geometric Preliminaries and 2. Gravity from Curvature

Question 1

What do round brackets in a lower index indicate?

Answer 1

Symmetrisation of indices: sum of terms with indexes swapped.

Question 2

What do square brackets in a lower index indicate?

Answer 2

Anti-symmetrisation of indices: difference of terms with indexes swapped.

Question 3

What conditions does the weak field limit impose on our model?

Answer 3

The metric differs from flat by small addition g_µν = η_µν + h_µν, where h_µν is small.

Objects move slowly: dx^i/ds «

Question 4

What does static mean for the metric in weak field?

Answer 4

Its time-independent; ∂_t g_µν= 0.

Question 5

State the perfect fluid stress tensor T^µν

Answer 5

T^µν = (ρ + p)U^µU^ν + pg^µν

Question 6

State the Einstein-Hilbert action

Answer 6

S[g] = 1/16πG int(d^4x √−g R)

Question 7

How do we add mass to the EH action?

Answer 7

We require it to be stationary on the EH action so we define G_µν = 8πGT_µν with T_µν = −2/√−g δS_m[g]/δ_gµν(x)

Question 8

How do we add an EM field to the EH action?

Answer 8

Define some field strength tensor with potential Aµ(x):
F_µν(x) = ∇_µA_ν − ∇_νA_µ

Question 9

What is special about the field strength tensor?

Answer 9

We can replace the covariant derivatives with partials; the Christoffels from each cancel.

Question 10

How do we vary the inverse metric?

Answer 10

δM^-1 = -M^-1 δM M^-1