T1: 3. Metrics

Question 1

Define a metric

Answer 1

A symmetric, invertible (non-degenerate), type [0,2] vector field g_μν.

Question 2

What does symmetric mean to a metric?

Answer 2

g_μν = g_vμ

Question 3

How many independent components does a (4d) metric have?

Answer 3

10

Question 4

What does invertible (non-degenerate) mean to a metric?

Answer 4

The determinant of the metric g_μν(x^a) as an nxn matrix is not zero; there are distinct eigenvalues.

Question 5

What is the signature of a metric?

Answer 5

The pattern of positive and negative eigenvalues of a metric.

Question 6

How does the metric define the inner product on the space of vectors?

Answer 6

(V,W) ≡ g(V,W) = g_μν V^μ W^ν

Question 7

How does the metric (inverse) relate vectors and covectors (vice versa)

Answer 7

The metric is a map between vectors and covectors; the action of a metric on a vector returns a covector.

Question 8

Generally, how does the metric act on tensors?

Answer 8

The metric is a lowering operator which takes the tensor [r,s] and returns [r-1, s+1] (s, or the number of covector/lower indices increases).

Question 9

Generally, how does the inverse metric act on tensors?

Answer 9

The metric is a raising operator which takes the tensor [r,s] and returns [r+1, s-1] (r, or the number of vector/upper indices increases).