Radion and Holographic Brane Gravity

The low energy effective theory for the Randall-Sundrum two brane system is investigated with an emphasis on the role of the non-linear radion in the brane world. The equations of motion in the bulk is solved using the low energy expansion method. This allows us, through the junction conditions, to deduce the effective equations of motion for the gravity on the brane. It is shown that the gravity on the brane world is described by the quasi-scalar-tensor gravity with a specific coupling function $\omega(\Psi) = 3\Psi /2(1-\Psi) $ on the positive tension brane and $\omega(\Phi) = -3\Phi /2(1+\Phi) $ on the negative tension brane. In contrast to the usual scalar-tensor gravity, the quasi-scalar-tensor gravity couples with two kinds of matter, namely, the matter on the positive tension brane and that on the negative tension brane with the different effective gravitational coupling constants. In particular, the radion disguised as scalar fields $\Psi$ and $\Phi$ couples with the sum of the tracepart of the energy momentum tensors of matter on both branes. In the course of the derivation, it has been revealed that the radion converts the non-local Einstein's theory with the generalized dark energy to the local quasi-scalar-tensor gravity. Moreover, we have obtained the effective action by substituting the bulk solution into the original action. It is also shown that the quasi-scalar-tensor gravity works as holograms at the low energy in the sense that the bulk geometry can be reconstructed from the solution of the quasi-scalar-tensor gravity.