Taken from Hobson's book:(adsbygoogle = window.adsbygoogle || []).push({});

Metric is given by

[tex]ds^2 = c^2 dt^2 - R^2(t) \left[ d\chi^2 + S^2(\chi) (d\theta^2 + sin^2\theta d\phi^2) \right] [/tex]

Thus, ##g_{00} = c^2, g_{11} = -R^2(t), g_{22} = -R^2(t) S^2(\chi), g_{33} = -R^2(t) S^2(\chi) sin^2 \theta##.

Geodesic equation is given by:

[tex] \dot u_\mu = \frac{1}{2} \left( \partial_\mu g_{v\sigma} \right) u^v u^\sigma [/tex]

The coordinates are given by ##u^0 = \dot t, u^1 = \dot \chi, u^2 = \dot \theta, u^3 = \dot \phi##.

For the temporal component,

[tex]\dot u_0 = \frac{1}{2} (\partial_0 g_{v\sigma})u^v u^\sigma[/tex]

Photons

[tex]u^0u_0 = 0[/tex]

[tex]u^0 g_{00} g^0 = 0 [/tex]

[tex]g_{00}\dot t^2 = 0 [/tex]

[tex]\dot t = 0[/tex]

This doesn't make any sense. For massive particles, ##\dot t = 1##.

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# FRW Metric: time component

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