(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An interval in Minkovski space is given in spheric coordinates as;

[tex]ds^{2}=c^{2}dt^{2}-dr^{2}-r^{2}d\theta^{2}-r^{2}sin^{2}\theta d\phi^{2}[/tex]

Now I have to find the covariant and contravariant components of the metric tensor.

2. Relevant equations

General expression of a metric tensor is:

[tex]G=g_{\mu\nu}dx^{\mu}dx^{\nu}[/tex] which also equals [tex]ds^{2}[/tex]

3. The attempt at a solution

I have some messy answer written down from the lecture, but cant get it clear.

Seems that I have written down the covariant components of the metric as

[tex]g_{00}=1[/tex]

[tex]g_{11}=-1[/tex]

[tex]g_{22}=-r^{2}[/tex]

[tex]g_{33}=-r^{2}sin^{2}\theta[/tex]

Are these the covariant components that the problem asks me to find? How are they found?

I see a correlation in the Schwarzchild metric

G=[PLAIN]http://rqgravity.net/images/gravitation/Gravitation-92.gif [Broken]

After this it seems that the lecturer has written down that

[tex]g_{\mu\sigma} g^{\mu\rho}=\delta^{\rho}_{\sigma}[/tex]

Is this correct?

So I get a matrix on which all the components on the main diagonal are 1. From there I can derive that the contra-variant components of the metric are simply inverses of the above covariant metric.

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# Homework Help: Special relativity: components of a metric tensor

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