- #1

- 237

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g

^{cd}g

_{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??

- Thread starter exmarine
- Start date

- #1

- 237

- 6

g

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

WannabeNewton

Science Advisor

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No. ##g^{cd}g_{ac} = \delta^{d}{}{}_{a}## by definition of the inverse of any metric tensor, hence the desired result.(1) Is that correct?

- #3

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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

Nugatory

Mentor

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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.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.

- #5

Matterwave

Science Advisor

Gold Member

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A metric is, by definition, symmetric. :)

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.

- #6

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