  We consider a perturbation of the Einstein gravity with the Neumann boundary
condition, which is regarded as an end of the world brane (ETW brane) of the
AdS/BCFT correspondence, in the AdSd+1 spaectime where d is larger than 2. We
obtain the mode expansion of the perturbations explicitly for the tensionless
ETW brane case. We also show that the energy-momentum tensor in a d-dimensional
BCFT should satisfy nontrivial constraints other than the ones for the boundary
conformal symmetry if the BCFT can couple to a d-dimensional gravity with a
specific boundary condition, which can be the Neumann or the conformally
Dirichlet boundary conditions. We find these constraints are indeed satisfied
for the free scalar BCFT. For the BCFT in the AdS/BCFT, we find that the BCFT
can couple to the gravity with the Neumann boundary condition for the
tensionless brane, but the BCFT can couple to the gravity with the conformally
Dirichlet boundary condition for the nonzero tension brane by using holographic
relations.
