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

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3. The perturbed quantities

As usual, any quantity is substituted by its mean value plus a fluctuating quantity, which in the linear approximation is negligible, i.e. terms including products of two fluctuating quantities are considered second order terms. For instance [FORMULA] and [FORMULA]. As there is now no possibility of confusion we will eliminate the subindex R and write [FORMULA] and [FORMULA].

By considering the transformation [FORMULA] we will call [FORMULA]. But some transformations of the metric tensor do not mean real physical changes. As argued by Weinberg (1972), it is possible to choose [FORMULA]. We benefit here from this choice. It is necessary to calculate [FORMULA] defined as [FORMULA]. The metric tensor must match [FORMULA] hence [FORMULA] neglecting higher order terms. Therefore [FORMULA], and we have for each component [FORMULA] again with [FORMULA].

[FORMULA] is equivalent to a three-dimension tensor, as any component containing the time subindex 0 vanishes. When using three-dimension formulae we term [FORMULA]. Its trace [FORMULA] plays an important role and will simply be called h. We also term [FORMULA].

[FORMULA] is the four-velocity of the photon fluid. We have [FORMULA], as [FORMULA] with no perturbation. When dealing with three-dimension formulae we term [FORMULA]. As we are using comoving coordinates, the unperturbed velocity is [FORMULA].

It is easily calculated that the components of the perturbed affine connection [FORMULA] vanish except


For quantities of electromagnetic nature, we would have for instance [FORMULA]. But as shown in the preceding paragraph the mean quantity is null and therefore we may use [FORMULA] instead of [FORMULA]. Therefore [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are perturbed quantities.

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998