Multiple D0-branes in Weakly Curved Backgrounds

We investigate further our recent proposal for the form of the matrix theory action in weak background fields. Using Seiberg's scaling argument we relate the matrix theory action to a low-energy system of many D0-branes in an arbitrary but weak NS-NS and R-R background. The resulting multiple D0-brane action agrees with the known Born-Infeld action in the case of a single brane and gives an explicit formulation of many additional terms which appear in the multiple brane action. The linear coupling to an arbitrary background metric satisfies the nontrivial consistency condition suggested by Douglas that the masses of off-diagonal fields are given by the geodesic distance between the corresponding pair of D0-branes. This agreement arises from combinatorial factors which depend upon the symmetrized trace ordering prescription found earlier for higher moments of the matrix theory stress-energy tensor. We study the effect of a weak background metric on two graviton interactions and find that our formalism agrees with the results expected from supergravity. The results presented here can be T-dualized to give explicit formulae for the operators in any D-brane world-volume theory which couple linearly to bulk gravitational fields and their derivatives.