Matter perturbations such as and obey the following transformation rule
One can construct a number of gauge-invariant quantities unchanged under the transformation (6.20):f (R) gravity one can introduce a scalar field as in Eq. (2.31), so that . From the gauge-invariant quantity (6.31) it is also possible to construct another gauge-invariant quantity for the matter perturbation of perfect fluids:
We note that the tensor perturbation is invariant under the gauge transformation .
We can choose specific gauge conditions to fix the gauge degree of freedom. After fixing a gauge, two scalar variables and are determined accordingly. The Longitudinal gauge corresponds to the gauge choice and , under which and . In this gauge one has and , so that the line element (without vector and tensor perturbations) is given by
The uniform-field gauge corresponds to which fixes . The spatial threading is fixed by choosing either or (up to an integration constant in the former case). For this gauge choice one has . Since the spatial curvature on the constant-time hypersurface is related to via the relation , the quantity is often called the curvature perturbation on the uniform-field hypersurface. We can also choose the gauge condition or .
This work is licensed under a Creative Commons License.