Scalar-field momentum flux ($T^{01}=-{\varphi }_t{\varphi }_x,T^{11}=\frac{1}{2}{\varphi }_t^2+\frac{1}{2}{\varphi }_x^2-V$)

For a real scalar field in one spatial dimension with metric signature $(+,-)$ and potential $V$, the stress-energy conservation law is ${\partial }_t{T}^{01}+{\partial }_x{T}^{11}=0$. The force on the field to the left of a fixed cut is $-T^{11}$ at the cut when the far-left boundary flux vanishes. Finite-energy field configurations also permit this conclusion by spatial cutoffs, without assuming pointwise decay of all derivatives.