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

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4. The perturbed Maxwell equations

We will obtain the perturbed equations from Maxwell's, motion, energy and Einstein's field equations. Not all these equations are independent but they are all useful. The general procedure will be as usual. After substituting any generic quantity A by [FORMULA], the original equation is subtracted and second order terms neglected, because we only consider linear perturbations. As an exception, when perturbing Maxwell's equations we will not neglect second order terms, assuming that variations in the metric and in the Faraday tensor are uncorrelated. The reason is that, because the result is the same, and because some conclusions are very important in forthcoming paragraphs, we have preferred to be extremely cautious.

Let us begin with the second set of Maxwell equations:

[EQUATION]

this yields the following perturbed equation

[EQUATION]

Note that the six last terms on the left hand side are second order terms.

In the absence of electric fields, from the [FORMULA], [FORMULA], [FORMULA] and [FORMULA] components, we obtain just [FORMULA]. From [FORMULA]

[EQUATION]

and for [FORMULA] (for instance)

[EQUATION]

From [FORMULA]

[EQUATION]

For [FORMULA] for example

[EQUATION]

The whole set of equations is reduced to

[EQUATION]

[EQUATION]

The first one is familiar. The second one is also familiar but not obvious. It brings to mind the condition of frozen-in magnetic field lines in present cosmic plasmas. What this equation shows is that the magnetic pattern always remains the same, and the magnetic strength is reduced by just the effect of expansion. The original pattern is conserved, even if we are considering small motions of the photon fluid associated with density inhomogeneities. Neither do small metric fluctuations alter the magnetic pattern. This property is also obtained for the linear Newtonian post-Recombination epoch as shown in Appendix B.

From the first set of Maxwell equations

[EQUATION]

with the above procedure and the same change of nomenclature, we obtain

[EQUATION]

The second and the fourth terms are second order terms, which were included in the derivation. One of them vanishes, that is [FORMULA]. The result is

[EQUATION]

[EQUATION]

As [FORMULA] is negligible, we simply obtain that

[EQUATION]

i.e. macroscopic neutrality, a familiar result, and that

[EQUATION]

which simply provides [FORMULA], the electric current, when [FORMULA] has been calculated.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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