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Astron. Astrophys. 326, 13-22 (1997)

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5. The perturbed equation of motion-energy

We start with the equation of motion

[EQUATION]

where

[EQUATION]

The electromagnetic contribution is as in Eq. (11) but now taking into account [FORMULA]. We now obtain

[EQUATION]

which yields

[EQUATION]

which yields for the 0th component

[EQUATION]

which is the equation of energy conservation. For the i th component:

[EQUATION]

However, it is possible to show that [FORMULA]. This comes from the well known property of the tensor metric [FORMULA], and therefore [FORMULA] and [FORMULA], which becomes in our case [FORMULA] where [FORMULA] now corresponds to the Robertson-Walker metric. We have

[EQUATION]

hence

[EQUATION]

and

[EQUATION]

With this simplification and some algebra, we have for the i th component

[EQUATION]

which is the equation of motion.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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