[Paper Review] Corrections to the Newtonian potential in the two-brane Randall-Sundrum model

Petter Callin|ArXiv.org|Jul 5, 2004

Black Holes and Theoretical Physics3 citations

TL;DR

This paper calculates the Newtonian gravitational potential in the two-brane Randall-Sundrum model, showing that at short distances (r ≪ z_r), the potential matches the one-brane model due to the second brane being effectively invisible, while at long distances (r ≫ z_r), the potential transitions to a four-dimensional form with exponentially suppressed corrections due to the compactified fifth dimension. The key result is that the relevant scale for the correction is the conformal distance z_r ≈ e^{μy_r}/μ, not the physical brane separation y_r.

ABSTRACT

We calculate the Newtonian potential in the two-brane Randall-Sundrum model, emphasizing the effect of the finite distance between the two branes. The result obtained is quite natural: When the distance in the potential is small compared to the brane separation the two-brane model is indistinguishable from the one-brane model, whereas when the distance is large the bulk dimension behaves like an ordinary compact dimension, with an exponentially decreasing correction to the four-dimensional potential. The contribution from the radion is also included, and is found to give only a multiplicative factor of order 1 in the correction.

Motivation & Objective

• To investigate how the finite separation between two branes in the Randall-Sundrum model affects the Newtonian gravitational potential.
• To clarify the role of the conformal distance z_r ≈ e^{μy_r}/μ as the relevant scale for gravitational corrections, rather than the physical brane separation y_r.
• To determine the behavior of the gravitational potential in the limits of short and long distances relative to the brane separation.
• To include the contribution of the radion field to the potential and assess its impact on the correction term.

Proposed method

• Derives the five-dimensional metric with a warped geometry A(y) = e^{-μ|y|} and imposes boundary conditions at y=0 and y=y_r to model two 3-branes.
• Solves the linearized Einstein equations for graviton propagation in the background, using the formalism from Callin (2004) to obtain the spectrum of Kaluza-Klein graviton masses.
• Applies asymptotic expansions of Bessel functions in the limit of small and large μz_r to derive approximate expressions for the graviton mass spectrum.
• Computes the gravitational potential correction Δ by summing over Kaluza-Klein modes, using the wavefunction normalization and coupling strength at the visible brane.
• Evaluates the potential in two regimes: r ≪ z_r (short distance, one-brane-like behavior) and r ≫ z_r (long distance, compact dimension behavior).
• Includes the radion field contribution via a multiplicative factor of order 1 in the correction term, showing it does not alter the qualitative behavior.

Experimental results

Research questions

• RQ1 How does the finite distance between the two branes in the Randall-Sundrum model modify the Newtonian gravitational potential compared to the one-brane case?
• RQ2 What is the physical scale that determines the transition between short-distance (one-brane-like) and long-distance (compact extra dimension) behavior in the potential?
• RQ3 Why is the conformal distance z_r ≈ e^{μy_r}/μ, rather than the physical distance y_r, the relevant scale for gravitational corrections?
• RQ4 What is the quantitative form of the correction to the Newtonian potential in the long-distance regime, and how does it compare to a flat compact extra dimension?
• RQ5 How does the radion field affect the gravitational potential, and does it significantly alter the leading-order correction?

Key findings

• At short distances (r ≪ z_r), the gravitational potential is indistinguishable from that in the one-brane Randall-Sundrum model, with Δ ≃ 2z_r/(πr), matching the 1/r behavior of the one-brane case.
• At long distances (r ≫ z_r), the potential approaches the four-dimensional form with an exponentially decreasing correction: Δ ≃ 2e^{-πr/z_r}, characteristic of a compact extra dimension.
• The transition scale between short- and long-distance behavior is set by the conformal distance z_r ≈ e^{μy_r}/μ, not the physical brane separation y_r, due to the exponential suppression of graviton masses.
• The radion field contributes only a multiplicative factor of order 1 to the correction term, indicating it does not qualitatively alter the potential's behavior.
• In the limit μz_r ≪ 1, the model reproduces the potential of a flat, compact extra dimension with size L = 2z_r, yielding Δ ≃ 2/(e^{πr/z_r} - 1) for r ≫ z_r.
• For r ≪ z_r, the correction Δ ≃ 2z_r/(πr) is consistent with the one-brane result, confirming that the second brane is invisible at short distances.