# Stationary property of geodesics - Dirac

1. Jan 23, 2015

### 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??

2. Jan 23, 2015

### WannabeNewton

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

3. Jan 24, 2015

### exmarine

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.

4. Jan 24, 2015

### Staff: Mentor

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.

5. Jan 24, 2015

### Matterwave

A metric is, by definition, symmetric. :)

6. Jan 29, 2015

### Staff: Mentor

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.