  In this note, we study the first law of entanglement in a boundary conformal
field theory (BCFT) dual to warped AdS cut off by a brane. Exploiting the
symmetry of boundary-centered half-balls in the BCFT, and using Wald's
covariant phase space formalism in the presence of boundaries, we derive
constraints from the first law for a broad range of covariant bulk Lagrangians.
We explicitly evaluate these constraints for Einstein gravity, and find a local
equation on the brane which is precisely the Neumann condition of Takayanagi
[arXiv:1105.5165] at linear order in metric perturbations. This is analogous to
the derivation of Einstein's equations from the first law of entanglement
entropy. This machinery should generalize to give local linearized equations of
motion for higher-derivative bulk gravity with additional fields.
