Membrane paradigm in large D dimensions

With Professor Shiraz Minwalla and collaborators at TIFR Mumbai, I studied the black hole membrane paradigm in large D dimensions. Previous studies in the spectrum of quasinormal modes around the Schwarzschild black hole, obtained from 1/D linearized expansion, reveal the presence of a decoupled sector of gravitational dynamics at low frequencies. We have determined the effective description of these slow modes at non linear level by explicitly constructing the relevant manifold of slow solutions in gravity, order by order in an expansion in 1/D.

We define a membrane region near the horizon of the black hole and find the equation of motion of the codimension one surface in terms of the one forms n (the normal to the surface), u (the velocity field) and ds (the exterior derivative of the big sphere radius which preserves a rotational symmetry). These equations are

U_{\perp}.K.U_{\perp}+n_S(n_S^2-1)/S=0\\ {\cal P}_a^b (U_{\perp}.\nabla u_b)=0,


U_\perp= U- (U.n) n, ~~~U= dS+ n_S^2(dS-u_\mu dx^\mu),

K is the extrinsic curvature of the space and {\cal P}_a^b is a projector to the subspace perpendicular to the space spanned by the three one forms. Please study for the details.