Dynamical Foundations of the Brane Induced Gravity

We present a comprehensive formalism to derive precise expressions for the induced gravity of the braneworld, assuming the dynamics of the Dirac-Nambu-Goto type. The quantum fluctuations of the brane at short distances give rise to divergences, which should be cutoff at the scale of the inverse thickness of the brane. It turns out that the induced-metric formula is converted into an Einstein-like equation via the quantum effects. We determine the coefficients of the induced cosmological and gravity terms, as well as those of the terms including the extrinsic curvature and the normal connection gauge field. The latter is the characteristic of the brane induced gravity theory, distinguished from ordinary none-brane induced gravity.