Brane-Localized Gravity

The following sections are included:

• Foreword

• Preface

• Contents

While there is currently no experimental evidence at all that supports the notion that our universe might possess any spacetime dimensions beyond the four established ones, it is a rather remarkable fact that we actually know of no compelling reason as to why the number of spacetime dimensions should actually be four. Certainly, a universe with four dimensions has some very special properties. For instance, it is only in four dimensions that the trilinear fermion, anti-fermion, gauge boson coupling constant is dimensionless, one of the key features which leads to the renormalizability of the strong, electromagnetic and weak interaction gauge theories. Nonetheless, it is equally possible to formulate such theories in spacetimes with dimension higher than four — they just would not be as easy to deal with — and even studying these theories in spacetimes with dimension lower than four has often been found to be extremely instructive. Consequently, it is legitimate to consider the possible existence of additional dimensions beyond four, with a view to either learning something theoretical about them which proves informative for 4-dimensional theories, or to being able to perform some experiment in four dimensions which might reveal their presence. What makes this enterprise so interesting is that the very entering into the issue of spacetime dimensions immediately entails the involvement of gravity, to thereby provide a possible road to a unification of gravity with the other fundamental forces.

The following sections are included:

• The Einstein tensor of pure Ads₅

• Introducing the brane

• Gravitational Gauss's law in the context of the brane world

• Compactified brane world with two branes

• Removing the second brane

The following sections are included:

• Topology of anti-de Sitter space

• Coordinate system choices for anti-de Sitter space

• “PoincarB patch” coordinates

• Other coordinate sectionings of anti-de Sitter space

The following sections are included:

• Gravitational versus purely geometric embeddings

• Gauss embedding formula

• Evaluation of the induced Einstein tensor in a specific case

The following sections are included:

• Junction conditions in the Newtonian and cases

• General Israel junction conditions

The following sections are included:

• Covariant analysis of an embedded Newtonian sheet of matter

• The embedded sheet Riemann tensor and its weak gravity limit

• Analogy to the static Einstein universe

The following sections are included:

• General structure of the Einstein equations in the RW case

• Static RW brane in an empty bulk

• Static RW brane in a bulk with a cosmological constant

The following sections are included:

• Embedding of a de Sitter brane in Ads₅

• Embedding of an anti-de Sitter brane in Ads₅

The following sections are included:

• Embedding an RW brane in a bulk with a cosmological constant

• Embedding an RW brane in an AdS₅ bulk

• Difference between embedding RW and Schwarsschild branes

The following sections are included:

• Embedding a k > 0 RW brane in M(4,2)

• Embedding a k > 0 RW brane and AdS₅ bulk in M(4,2)

• Embedding a k > 0 de Sitter brane and AdS₅ bulk in M(4,2)

• Analog to the Rindler horizon

• Embedding a k < 0 RW brane and AdS₅ bulk in M(4,2)

• Embedding a k = 0 RW brane and AdS₅ bulk in M(4,2)

The following sections are included:

• Null geodesics of the k > 0 RW brane world

• Null geodesics of the k = 0 and k < 0 RW brane worlds

• Alternate derivation of the k = 0 null geodesics

• Variation of the metric coefficients along a bulk null geodesic

The following sections are included:

• Derivation of the generalized Einstein equations on the brane

• Some exact solutions to the modified Friedmann equation

• Localization of gravity around a Robertson-Walker brane

The following sections are included:

• Structure of the brane-world background geometries

• General formulation of the gravitational fluctuation equations

• Gauge invariance considerations

• Imposition of the axial gauge in the brane world

• Gravitational fluctuation equations in the axial gauge

The following sections are included:

• NT sector fluctuations

• TT sector fluctuations

• Physical status of the NT and TT sector fluctuations

• Comparison with unembedded M₄ fluctuations

The following sections are included:

• Gauge invariance in the dS₄ and AdS₄ brane worlds

• NT/TT decomposition of the fluctuation equations

• Comparison with unembedded AdS₄ fluctuations

The following sections are included:

• Orthogonality measure for TT modes

• Energy of gravitational fluctuations around flat spacetime

• First order total gravitational energy as a surface term

• Second order total gravitational energy as a surface term

• Second order gauge transformation properties of the energy

• Gauge invariant conservation condition for the total energy

• Origin of the gauge invariant energy conservation condition

• Abbott and Deser treatment of the curved space case

• The needed curved space Killing vectors

• The needed brane-world energy-momentum tensor

• total energy, normalization and completeness relations

• Construction of the propagator via completeness

The following sections are included:

• mode basis

• Mode propagation in and long distance gravity

• Causality issues in the brane world

• Causality in flat spacetime

• Upper half plane determination of the propagator

• Lower half plane determination of the propagator

• Evaluation of the cut discontinuities of the propagator

• Alternate propagators in :

• Some exact solutions to the fluctuation equations

• Connection to RW perturbations on the brane

The following sections are included:

• mode basis

• Completeness test for normalizable basis modes

• Completeness test for non-normalizable basis modes

• Upper half plane determination of the propagator

• Lower half plane determination of the propagator

• Negative signature modes in the propagator

• Temporal evolution of localized square steps in

• Some exact solutions to the fluctuation equations

The following sections are included:

• mode basis

• Normalizability criterion for : modes

• Discrete normalizable mode basis for

• Completeness test for normalizable basis modes

• Normalizable basis mode propagator

• Completeness test for non-normalizable basis modes

• Some exact solutions to the fluctuation equations

• Constructing the causal : propagator

• Causal propagation in the brane world

The following sections are included:

• Equivalent brane-world Schrödinger equation

• Instructive approximate brane-world Schrödinger equation

The following sections are included:

• mode basis

• Causality in the brane world

The following sections are included:

• Kink generated M₄ domain wall

• Soliton generated M₄ domain wall

• dS₄ domain wall

The following sections are included:

• Appendices
  • Gauge Invariance of the Fluctuation Equation
  • Perturbative Weyl Tensor
  • Transverse and Transverse-Traceless Projection
    • Transverse projection for vector fields
    • Transverse projection for flat space tensor fields
    • Transverse-traceless projection for flat space tensor fields
    • Transverse-traceless projection for curved space tensor fields
  • Fluctuation Energy-Momentum Tensor
    • General second order Einstein-Hilbert action
    • Second order fluctuation energy-momentum tensor in AdS₅
    • Conservation of the second order energy-momentum tensor
    • Brane-world second order energy-momentum tensor
  • Retarded Propagators for Pure AdS and Pure dS Spacetimes
    • Retarded propagator for pure AdS spacetimes
    • Retarded propagator for pure dS spacetimes
    • Retarded propagator for pure M₄ spacetime
  • Embedding M(2,1) Branes in AdS₄
    • and brane-world background geometries
    • Fluctuations in the and brane worlds
    • brane-world mode basis
    • Causality in the brane world
    • Causality in the brane world

• Bibliography

• Author Index

• Subject Index