Conserved scalar-field energy in a static spacetime

For a minimally coupled real scalar field with nonnegative squared mass, a future-directed timelike static Killing vector field $K$ defines an energy current from the stress-energy tensor. On a spacelike slice with future unit normal $n$, the energy is $E_{\Sigma }={\int }_{\Sigma }{T}_{ab}{n}^a{K}^{b}d\Sigma$.

In a static slicing with $K=Nn$ and positive lapse function $N$, its integrand is $N[(n^a{\nabla }_a\varphi {)}^2+|D\varphi {|}^2+m^2{\varphi }^2]/2$, hence nonnegative. The Klein-Gordon equation and Killing symmetry give a conserved current. Applying the divergence theorem between slices proves equal energies only when the remaining boundary flux vanishes, for example under appropriate decay or reflecting boundary conditions. Flux through an open boundary must instead appear in the energy balance.