[Paper Review] Uniqueness of RS2 type thick branes supported by a scalar field

Sergei T. Abdyrakhmanov, К. А. Бронников+1 more|ArXiv.org|Mar 13, 2005

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper proves the uniqueness of RS2-type thick branes in 5D general relativity, showing they emerge universally as regularized versions of the thin RS2 brane when supported by a scalar field with an alternating-sign potential. The thin brane limit is achieved universally via a one-parameter family of thick models, with the AdS asymptotic and fine-tuned potential condition being universal features, but the stress-energy tensor of test matter diverges at the AdS horizon, undermining physical viability.

ABSTRACT

We study thick brane world models as Z_2-symmetric domain walls supported by a scalar field with an arbitrary potential V(ϕ) in 5D general relativity. Under the global regularity requirement, such configurations (i) have always an AdS asymptotic far from the brane, (ii) are only possible if V(ϕ) has an alternating sign and (iii) V(ϕ) should satisfy a certain fine-tuning type equality. Thus a thick brane with any admissible V(ϕ) is a regularized version of the RS2 brane immersed in the AdS_5 bulk. The thin brane limit is realized in a universal manner by including an arbitrary thick brane model in a one-parameter family, where the parameter "a" is associated with brane thickness; the asymptotic value of V(ϕ) (related to Λ_5, the effective cosmological constant) remains a-independent. The problem of ordinary matter confinement on the brane is discussed for a test scalar field. Its stress-energy tensor is found to diverge at the AdS horizon for both thin and thick branes, making a serious problem for this class of brane world models.

Motivation & Objective

• To establish the uniqueness and universal structure of thick brane solutions in 5D general relativity with scalar field support.
• To resolve the long-standing problem of whether the thin RS2 brane can be consistently derived as a limit of smooth, thick brane models.
• To investigate the physical viability of such models by analyzing the localization of ordinary matter via a test scalar field.
• To determine whether the stress-energy tensor of test matter remains finite on the brane and in the bulk, especially near the AdS horizon.
• To clarify the role of the scalar potential's sign structure and fine-tuning in enabling globally regular, Poincaré-type thick branes.

Proposed method

• Formulates the 5D Einstein-scalar field equations with a Z2-symmetric, static domain wall ansatz for the metric and scalar field depending only on the extra dimension z.
• Applies the harmonic coordinate condition to simplify the Ricci tensor and derive a system of ODEs for the metric function F(z) and scalar field φ(z).
• Imposes global regularity and Z2 symmetry to constrain the potential V(φ), showing that only alternating-sign potentials allow regular solutions with AdS asymptotics.
• Introduces a one-parameter family of thick brane models parameterized by a (related to brane thickness), such that the limit a→0 universally yields the RS2 model.
• Analyzes the Klein-Gordon-Fock equation for a test scalar field χ coupled to the brane field φ, deriving its radial equation and asymptotic behavior.
• Evaluates the stress-energy tensor of the test field and finds its divergence at the AdS horizon due to the z^{-3/4} behavior of the field gradient.

Experimental results

Research questions

• RQ1 Can any regular, thick brane solution in 5D general relativity with a scalar field be consistently embedded in a one-parameter family that includes the thin RS2 brane as a limit?
• RQ2 What conditions on the scalar potential V(φ) are necessary and sufficient for the existence of globally regular, Z2-symmetric thick branes with AdS asymptotics?
• RQ3 Is the thin RS2 brane limit universal across all thick brane models, independent of the potential's shape or parameters?
• RQ4 Does ordinary matter, modeled as a test scalar field, remain localized on the brane, or does its stress-energy tensor diverge in the bulk?
• RQ5 Why does the stress-energy tensor of a test scalar field diverge at the AdS horizon, and is this divergence generic across all such thick brane models?

Key findings

• All globally regular, Z2-symmetric thick branes supported by a scalar field in 5D general relativity necessarily have an anti-de Sitter (AdS) asymptotic behavior at large |z|.
• Such solutions exist only if the scalar potential V(φ) has an alternating sign, and must satisfy a specific fine-tuning condition relating the potential's asymptotic value to the brane tension.
• The thin RS2 brane limit is universally achievable by including any regular thick brane model in a one-parameter family where the parameter a controls brane thickness, with the asymptotic value of V(φ) remaining independent of a.
• The stress-energy tensor of a test scalar field diverges at the AdS horizon due to the z^{-3/4} divergence of the field gradient, rendering the test field approximation invalid in the bulk.
• This divergence is universal and independent of the test field's mass or its coupling to the scalar field φ, indicating a fundamental physical problem for this class of brane world models.
• The origin of the divergence is the repulsive gravitational effect of the AdS bulk, which pulls matter toward the horizon, making the model physically inconsistent for matter confinement.