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 and . As there is now no possibility of confusion we will eliminate the subindex R and write and .
By considering the transformation we will call . But some transformations of the metric tensor do not mean real physical changes. As argued by Weinberg (1972), it is possible to choose . We benefit here from this choice. It is necessary to calculate defined as . The metric tensor must match hence neglecting higher order terms. Therefore , and we have for each component again with .
is equivalent to a three-dimension tensor, as any component containing the time subindex 0 vanishes. When using three-dimension formulae we term . Its trace plays an important role and will simply be called h. We also term .
is the four-velocity of the photon fluid. We have , as with no perturbation. When dealing with three-dimension formulae we term . As we are using comoving coordinates, the unperturbed velocity is .
For quantities of electromagnetic nature, we would have for instance . But as shown in the preceding paragraph the mean quantity is null and therefore we may use instead of . Therefore , , and are perturbed quantities.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998