Tensor basis, dual basis, and source-output conversion

Use

Let $S={\mathrm{Sym}}^2(({\mathbb{R}}^4{)}^{\ast })$. Fix a basis $e_I\in S$ and its algebraic dual $e^I\in S^{\ast }$, written as symmetric contravariant tensors:

The algebraic dual $e^I$ is not, in general, the tensor obtained by raising both indices of $e_I$ with $\eta$.

One concrete implementation uses labels $I=(i,j)$, $0\le i\le j\le 3$, with

Thus an off-diagonal basis tensor has two entries equal to one, and its dual has two entries equal to $1/2$.

The tensor Euler derivative is defined by

For propagation we lower this output with the fixed background metric. Its component polynomial is therefore

This is the source-output convention used in $Q$ and $T$ below.

To distinguish it from coordinate variation, let

Then

The ${\mathcal{E}}_I$ are the coordinate Euler components. They are not interchangeable with the propagated components ${\mathcal{J}}^L$. In particular, off-diagonal factors must not be inserted twice.

The linear maps are

Thus

The notations ${\mathsf{A}}^{\mu \nu }{}_L$ and ${\mathsf{b}}_L$, used when contracting a response output, denote evaluation on $e_L$:

Their index placement labels their role in the finite component contraction; it does not instruct an additional multiplication by $\mathbf{G}$.

Fix the constant bilinear map $\mathsf{D}$ through

The general reduction results hold for every fixed such array. Its designation as a particular physical density map remains an input identification.