# What are these perturbations perturbations to?

In CMB papers, I often find that "perturbations" can be decomposed into scalar, vector, and tensor perturbations, with equations looking something like $$\Pi_{ab} = \Pi_{ab}^{(S)} + \Pi_{ab}^{(V)} + \Pi_{ab}^{(T)}.$$ These are then referred to as density fluctuations, vorticity, and gravitational waves. I understand that they are just defined based on their rotational properties and can have various sources. However, I've struggled to find what these are perturbations to.

The fact that scalar perturbations are density fluctuations make me think they are perturbations to the stress-energy tensor. But if tensor perturbations are gravitational waves, that sounds like perturbations to the metric. I imagine that the use of the perturbation is either $$g_{ab} = g_{ab,0} + \Pi_{ab}$$ or $$T_{ab} = T_{ab,0} + \Pi_{ab}.$$ It seems unintuitive to me that it would be possible to add perturbations to the metric to perturbations to the stress-energy tensor.

• The revised question boils down to what is meant by the notation $\Pi$. I believe this typically denotes perturbations to the stress energy tensor but this can depend on the particular source you are looking at. Note that even if $\Pi$ does refer to perturbations in the stress energy tensor it is still possible for metric perturbations to appear depending on how things are defined. For example often the energy density and pressure are defined in terms of $T^\mu_\nu$, so $T_{\mu\nu}$ can have metric perturbation factors for lowering an index. Jan 9, 2017 at 15:10

• How can you add together perturbations of different things? For example, I've seen equations like $\Pi_{ab) = \Pi_{ab)^{(S)} + \Pi_{ab)^{(V)} + \Pi_{ab)^{(T)}$ Jan 8, 2017 at 16:20
• Think of it like a matrix equation. The matrix $\Pi$ can be written as a sum of three pieces, each of which transforms in a nice way under spatial rotations. (Note this means $\Pi$ itself does not transform nicely). A simpler example of the same kind of thing is that you can decompose any matrix $M$ into a piece that's symmetric (under transposing) and a piece that's antisymmetric, $M=M_s+M_a$, where $M_s^T=M_s$ and $M_a^T=-M_a$. Jan 9, 2017 at 13:22
• Right. I understand that much. But if $\Pi^S$ is a perturbation to the stress-energy tensor and $\Pi^T$ is a perturbation to the metric, you can't just add those. Jan 9, 2017 at 14:21
• I don't know exactly what you mean by $\Pi$. But just think of Einstein's equations. When you perturb them you will have metric and matter perturbations in the same equation. You can then decompose the components of the equation into S,V, and T--each of those pieces will have metric and matter perturbations. Since everything is linear you could further split up, say, the S equation into a sum of different species (metric S, photon S,...). A similar kind of thing happens for the conservation of stress energy equations. Jan 9, 2017 at 14:55