Geodesics: Stationary Property & Dirac

[exmarine]

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

g^cdg_ac(dv^a/ds) becomes (dv^d/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??

 

[WannabeNewton]

(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.

 

[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.

 

[Nugatory]

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.

 

[Matterwave]

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. :)

 

[PeterDonis]

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.