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Relativity and gravitational theory Definition

Randall–Sundrum model

Builds on Anti-de Sitter spaceBraneKaluza–Klein theoryEinstein field equations

Definition

Let k>0k > 0 and rc>0r_c > 0 be constants and let ημν\eta_{\mu\nu} be the Minkowski metric on ℝ4\R^4. The Randall–Sundrum model RS1 is five-dimensional gravity with negative cosmological constant on ℝ4×(S1/ℤ2)\R^4 \times (S^1 / \Z_2), where the orbifold S1/ℤ2S^1/\Z_2 is the interval 0≤y≤πrc0 \le y \le \pi r_c. It has two branes with opposite tensions at the fixed points y=0y = 0 and y=πrcy = \pi r_c, and the background metric ds2=e−2k∣y∣ημν dxμdxν+dy2.ds^2 = e^{-2k\abs{y}} \eta_{\mu\nu} \, dx^\mu dx^\nu + dy^2. The RS2 model is the limit rc→∞r_c \to \infty, with one brane at y=0y = 0 and an infinite extra dimension.

Remarks

Between the branes the metric is a slice of five-dimensional anti-de Sitter space of curvature radius 1/k1/k, which solves the bulk Einstein field equations with negative cosmological constant. The jumps of ∂y\partial_y of the warp factor at the branes are matched by the brane tensions, which must be tuned against the bulk cosmological constant. Example: a mass parameter m0m_0 on the brane at y=πrcy = \pi r_c appears to a four-dimensional observer as m=e−kπrcm0m = e^{-k \pi r_c} m_0; with krc≈11k r_c \approx 11 to 1212 this lowers the Planck scale to the weak scale. Non-example: the product metric ημνdxμdxν+dy2\eta_{\mu\nu} dx^\mu dx^\nu + dy^2 on ℝ4×S1\R^4 \times S^1, as in Kaluza–Klein theory, is not warped, so it gives no exponential hierarchy.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Bilinear form, concept, no evidence levelBilinear formconceptBrane, concept, no evidence levelBraneconceptCayley–Klein metric, concept, no evidence levelCayley–Klein metricconceptEinstein field equations, concept, no evidence levelEinstein field equati…conceptEinstein manifold, concept, no evidence levelEinstein manifoldconceptHyperbolic cosine, concept, no evidence levelHyperbolic cosineconceptHyperbolic geometry, concept, no evidence levelHyperbolic geometryconceptIdentity matrix, object, no evidence levelIdentity matrixobjectKaluza–Klein theory, concept, no evidence levelKaluza–Klein theoryconceptMatrix, definition, no evidence levelMatrixdefinitionPoincaré disk model, object, no evidence levelPoincaré disk modelobjectQuadratic form, concept, no evidence levelQuadratic formconceptSectional curvature, concept, no evidence levelSectional curvatureconceptSignature of a quadratic form, concept, no evidence levelSignature of a quadra…conceptUniversal cover, concept, no evidence levelUniversal coverconceptBeltrami–Klein model, example, no evidence levelBeltrami–Klein modelexampleTranspose, definition, no evidence levelTransposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectIndefinite orthogonal group, example, no evidence levelIndefinite orthogonal…exampleHyperboloid model, example, no evidence levelHyperboloid modelexampleAnti-de Sitter space, example, no evidence levelAnti-de Sitter spaceexampleRandall–Sundrum model, definition, no evidence levelRandall–Sundrum modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Bilinear form concept
  2. Brane concept
  3. Cayley–Klein metric concept
  4. Einstein field equations concept
  5. Einstein manifold concept
  6. Hyperbolic cosine concept
  7. Hyperbolic geometry concept
  8. Identity matrix object
  9. Kaluza–Klein theory concept
  10. Matrix definition
  11. Poincaré disk model object
  12. Quadratic form concept
  13. Sectional curvature concept
  14. Signature of a quadratic form concept
  15. Universal cover concept
  16. Beltrami–Klein model example
  17. Transpose definition
  18. Orthogonal matrix definition
  19. Orthogonal group object
  20. Indefinite orthogonal group example
  21. Hyperboloid model example
  22. Anti-de Sitter space example
  23. Randall–Sundrum model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Randall–Sundrum_model
About this page
TagAT9765A1XR
URIm2://AT9765A1XR
Kinddefinition
WritingW0 stub
MSC83E15, 81T30
DefinesRandall–Sundrum model
UsesAnti-de Sitter space, Brane, Kaluza–Klein theory, Einstein field equations
Changed3 November 2026

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