The actual response on the plane

Fix a real scalar solution with Schwartz Cauchy data, and $f\in {C_c}^{\infty }((t_0,T)\times {\mathbb{R}}^3)$. Let $u_f=-G_{A}f$. On the plane, $D(k)=0$ and $k_{0\mu }=0$. The response is therefore

The truncated causal support of $u_f$ is compact, so this integral is well defined even though $k$ need not be spacelike compact.

With $v_1=e,v_2=r$, define the actual coefficients

The raw-current derivation remains valid with $V=\Theta (t-t_0)u_f$, even for Schwartz scalar data. Its surface coefficient is (eq:surface) translated to $t_0$. That surface term pairs to zero with a spatial-TT tensor: its nontrace part has a temporal index and its remaining part is a spatial trace. This observation is not yet a conserved-current construction.