# Geodesics: Stationary Property & Dirac

• exmarine

#### exmarine

In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks)

gcdgac(dva/ds) becomes (dvd/ds)

I seems to me that that only works if the metric matrix is diagonal.
(1) Is that correct?
(2) If so, that doesn't seem to be a legitimate limitation on the property of geodesics??

(1) Is that correct?

No. ##g^{cd}g_{ac} = \delta^{d}{}{}_{a}## by definition of the inverse of any metric tensor, hence the desired result.

Ah yes, I see that now. Thanks.

PS. Must the metric matrix always be symmetric? For example, a skew-symmetric matrix can produce the same ds^2 interval as a diagonal. Just curious.

Must the metric matrix always be symmetric? For example, a skew-symmetric matrix can produce the same ds^2 interval as a diagonal. Just curious.

It gives you the same ##ds^2## as some diagonal matrix, but it won't work for calculating the inner product of two different vectors.

Ah yes, I see that now. Thanks.

PS. Must the metric matrix always be symmetric? For example, a skew-symmetric matrix can produce the same ds^2 interval as a diagonal. Just curious.

A metric is, by definition, symmetric. :)

it won't work for calculating the inner product of two different vectors.

To expand on this just a bit, the metric must be symmetric because the inner product is commutative; ##g(a, b) = g(b, a)## for any two vectors ##a## and ##b##. If you write this out in components, you get that ##g## must be a symmetric matrix.