[Paper Review] An Introduction to the Brane World

Abdel Pérez-Lorenzana|ArXiv.org|Jun 24, 2004

advanced mathematical theories3 citations

TL;DR

This paper introduces brane world models as a framework where our 4D universe is a 3-brane embedded in a higher-dimensional bulk space, with gravity propagating in the extra dimensions. It explains how compact or infinite extra dimensions—particularly in the Randall-Sundrum models—can localize gravity on the brane, reproduce four-dimensional gravity, and resolve the hierarchy problem via warped geometry, with key results including the derivation of the effective 4D Einstein equations and the role of the Weyl tensor in non-local bulk effects.

ABSTRACT

The study of the so called brane world models has introduced completely new ways of looking up on standard problems in many areas of theoretical physics. Inspired in the recent developments of string theory, the Brane World picture involves the introduction of new extra dimensions beyond the four we see, which could either be compact or even open (infinite). The sole existence of those new dimensions may have non trivial observable effects in short distance gravity experiments, as well as in our understanding of the cosmology of the early Universe, among many other issues. The goal of the present notes is to provide a short introduction to some basic aspects of the Brane World models. The discussion includes models with flat compact extra dimensions, as well as the so called Randall-Sundrum models.

Motivation & Objective

• To provide a pedagogical introduction to brane world models for researchers new to the field.
• To explain how extra dimensions—compact or infinite—can reconcile gravity with quantum field theory and resolve the hierarchy problem.
• To derive the effective four-dimensional gravity on the brane, including the Newtonian potential and cosmological constant contributions.
• To explore the role of the Weyl tensor and extrinsic curvature in encoding non-local bulk gravitational effects on the brane.
• To present the mathematical and physical foundations of the Randall-Sundrum models with warped compactifications.

Proposed method

• Formal dimensional reduction of a higher-dimensional Einstein-Hilbert action to obtain the effective 4D theory on the brane.
• Use of the Israel junction conditions to derive the induced 4D Einstein equations from the 5D bulk equations with a delta-function source at the brane.
• Derivation of the effective 4D Newton constant in terms of the fundamental 5D Planck scale and brane tension.
• Analysis of the 5D Einstein equations with a cosmological constant in anti-de Sitter (AdS) space to construct the Randall-Sundrum solutions.
• Incorporation of the Weyl tensor component $ E_{MN} $ as a non-local bulk effect that modifies gravity on the brane.
• Inclusion of quadratic energy-momentum contributions $ \pi_{MN} $ from extrinsic curvature to the effective brane stress-energy tensor.

Experimental results

Research questions

• RQ1 How can extra dimensions larger than the Planck scale remain hidden from experiments while still affecting gravity?
• RQ2 What conditions lead to the localization of gravity on a 3-brane in a higher-dimensional bulk?
• RQ3 How is the 4D Newton constant derived from the fundamental 5D Planck scale and brane tension?
• RQ4 What role does the Weyl tensor play in mediating non-local gravitational effects from the bulk to the brane?
• RQ5 How do the Randall-Sundrum models with warped geometry resolve the hierarchy problem compared to the ADD model?

Key findings

• The effective 4D gravity on the brane is governed by the modified Einstein equations: $ G_{(4)MN} = \Lambda_4 \gamma_{MN} + 8\pi G_N t_{MN} + k_*^2 \pi_{MN} - E_{MN} $, where $ E_{MN} $ encodes non-local bulk effects.
• The 4D Newton constant is given by $ 8\pi G_N = \frac{k_*^2 \tau}{6} $, linking the observed gravity strength to the brane tension and fundamental 5D scale.
• The effective cosmological constant on the brane is $ \Lambda_4 = \frac{1}{2}k_*^2\left(\frac{1}{6}k_*^2\tau - \Lambda\right) $, which vanishes under fine-tuning of the brane tension and bulk cosmological constant.
• In the Randall-Sundrum model, the warp factor $ e^{-k|y|} $ localizes zero-mode gravitons on the brane, reproducing 4D gravity at low energies.
• The Weyl tensor component $ E_{MN} $, which arises from the 5D Weyl curvature, contributes to the effective stress-energy and vanishes only in pure AdS bulk.
• The quadratic correction $ \pi_{MN} $, derived from extrinsic curvature, introduces a non-linear, local contribution to the brane stress-energy tensor.