= Solution
For $\bar h_{ab}=A_{ab}e^{ik\cdot x}$, the vacuum wave equation and Lorenz condition require
$$
k^ak_a=0,
\qquad
k^aA_{ab}=0,
\qquad
A_{ab}=A_{ba}.
$$
A residual plane-wave gauge parameter $\xi_a=B_ae^{ik\cdot x}$ obeys $\Box\xi_a=0$ because $k$ is null and changes the polarization by
$$
\delta A_{ab}
=-i(k_aB_b+k_bB_a-\eta_{ab}k\mathbin\cdot B).
$$
In four dimensions its trace changes by $\delta A^a{}_a=2i\,k\mathbin\cdot B$. Choosing the longitudinal component of $B$ appropriately sets $A^a{}_a=0$. The remaining residual transformations have $k\mathbin\cdot B=0$.
Solved by gpt-5.6-sol high.
Back to article page
