Radion and holographic brane gravity

Abstract

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 a 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 a quasi-scalar-tensor theory with a specific coupling function ω(Ψ) = 3Ψ/2(1 − Ψ) on the positive tension brane and ω(Φ) = −3Φ/2(1 + Φ) on the negative tension brane, where Ψ and Φ are non-linear realizations of the radion on the positive and negative tension branes, respectively. In contrast to the usual scalar-tensor gravity, the quasi-scalar-tensor gravity couples with two kinds of matter, namely, the matters on both positive and negative tension branes, with different effective gravitational coupling constants. In particular, the radion disguised as the scalar fields Ψ and Φ couples with the sum of the traces of the energy momentum tensor on both branes. In the course of the derivation, it has been revealed that the radion plays an essential role to convert the non-local Einstein gravity with the generalized dark radiation to the local quasi-scalar-tensor gravity. For completeness, we also derive the effective action for our theory by substituting the bulk solution into the original action. It is also shown that the quasi-scalartensor 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.

Low Energy Effective Action for Dilatonic Braneworld —A Formalism for Inflationary Braneworld—

“…It turns out that χ µν and C becomes zero when two branes get separated infinitely. However, we should be careful to treat this limit, because next order corrections might diverge in this limit as is suggested by analysis of the linearized gravity [14].…”

“…In this section, we begin with the single-brane system. Since we know the effective 4-dimensional equations hold irrespective of the existence of other branes [14], the analysis of the single-brane system is sufficient to derive the effective action for the two-brane system as we see in the next section.…”

“…Then, following the above procedure, we can derive the effective low energy action with corrections coming from Kaluza-Klein modes. Let us explain this more concretely (see [11] for detailed calculations and discussions).…”

“…In the first Randall-Sundrum (RS1) model, the two flat 3-branes are embedded in AdS 5 and the brane tensions given by ⊕ σ = 6/(κ 2 ℓ) and ⊖ σ = −6/(κ 2 ℓ). The point is the fact that the equations of motion on each brane take the same form if we use the induced metric on each brane [14] (see also [19] for other approaches). The effective Einstein equations on each positive (⊕) and negative (⊖) tension brane at low-energies yield…”

“…Because both approaches are complementary and useful, their mutual relations should be understood. This was done by us through the explicit construction of the low energy effective action [11]. It would be beneficial if we could generalize these results to the dilatonic braneworld.…”