  We consider black-brane spacetimes that have at least one spatial translation
Killing field that is tangent to the brane. A new parameter, the tension of a
spacetime, is defined. The tension parameter is associated with spatial
translations in much the same way that the ADM mass is associated with the time
translation Killing field. In this work, we explore the implications of the
spatial translation symmetry for small perturbations around a background black
brane. For static charged black branes we derive a law which relates the
tension perturbation to the surface gravity times the change in the the horizon
area, plus terms that involve variations in the charges and currents. We find
that as a black brane evaporates the tension decreases. We also give a simple
derivation of a first law for black brane spacetimes. These constructions hold
when the background stress-energy is governed by a Hamiltonian, and the results
include arbitrary perturbative stress-energy sources.
