REVIEW
Brane-world gravity
General Relativity and Quantum Cosmology · Submitted 10 Dec 2003 · 123 references
Pith's verdict: The standard reference for RS brane-world gravity, and it holds up — the only soft spot is that the abstract's 'testable implications' is carried by the conjecture-dependent black-hole bound, not by the solid background results.
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Referee report
The paper
The paper claims
If our visible universe is a three-dimensional surface inside a five-dimensional bulk, then gravity on that surface is a computable deformation of general relativity, with a definite list of corrections that astronomy can test.
The reading
What carries the argument
The machinery is the covariant projection of Shiromizu, Maeda and Sasaki: the Gauss and Codazzi equations relate the four-dimensional curvature of the brane to the projected five-dimensional curvature plus extrinsic-curvature terms, and the Israel-Darmois junction conditions fix the extrinsic curvature at the $mathbb{Z}_2$-symmetric brane in terms of the brane tension and its energy-momentum. Substituting one into the other produces the effective field equations, and the contracted Bianchi identity on the brane becomes a propagation law for $E_{mu u}$ sourced by gradients of $S_{mu u}$. This is what makes the review more than a catalogue of solutions: it isolates exactly which piece of five-dimensional physics a brane-bound observer can reconstruct and which piece is irreducibly out of reach.
Extended reading notes
Core claim
The central claim is that the Randall-Sundrum warped brane-world is not a geometric curiosity but a working effective theory of gravity, with a closed set of equations on the brane and a definite list of corrections to general relativity. Projecting the five-dimensional Einstein equations along the brane gives $G_{mu u} = -Lambda g_{mu u} + kappa^2 T_{mu u} + 6kappa^2lambda^{-1} S_{mu u} - E_{mu u}$, where $S_{mu u}$ is quadratic in $T_{mu u}$ and $E_{mu u}$ is the electric part of the bulk Weyl tensor. The first correction is local and dominates at $ ho gg lambda$; the second is nonlocal, carries the five polarizations of the five-dimensional graviton, and cannot be computed from brane data alone. From this template the review derives the modified Newtonian potential, the modified Friedmann and Raychaudhuri equations, altered shear and vorticity evolution, a tidal-charge black hole metric, and a demonstration that Oppenheimer-Snyder collapse on the brane cannot have a static exterior. The honest core of the claim is that brane-world gravity is predictive where the bulk is simple and silent where it is not: the missing equation for the Weyl anisotropic stress $pi^E_{mu u}$ is what prevents a fully closed four-dimensional treatment of perturbations.
Load-bearing premise
The construction assumes the brane tension is fine-tuned against the negative bulk cosmological constant so that the brane carries zero effective cosmological constant, and the quantitative perturbation and CMB results additionally require a second 'regulator' brane whose backreaction on the visible brane is assumed negligible over the timescales of interest.
Editorial extensions
If this is right
- Newton's law acquires a $1/r^2$ short-distance behaviour below the curvature scale $ell$, so table-top experiments bound $ell lesssim 0.1$ mm and hence the brane tension $lambda > (1\ \mathrm{TeV})^4$ and the fundamental scale $M_5 > 10^5$ TeV.
- The Friedmann equation becomes $H^2 = \frac{kappa^2}{3} ho(1+ ho/2lambda) + m/a^4 - K/a^2$, so the very early universe expands as $a \propto t^{1/3(w+1)}$ instead of $t^{2/3(w+1)}$, and nucleosynthesis caps dark radiation at roughly 5 percent of the radiation density.
- High-energy Hubble damping lets steep potentials inflate, so slow-roll succeeds on the brane where it would fail in general relativity, while the tensor amplitude is enhanced by a factor $F^2 \approx 3H_0/2mu$.
- Oppenheimer-Snyder collapse on the brane cannot have a static exterior, so astrophysical black holes may carry Kaluza-Klein hair; under the AdS/CFT conjecture the same physics shortens evaporation to $(M/M_\odot)^3 (1\ \mathrm{mm}/ell)^2$ years.
- Bianchi universes isotropize at high energy because the $ ho^2$ term outruns the shear term $Sigma^2/a^6$.
Reading between the lines
- The admission that $E_{mu u}$ cannot be reconstructed from brane data is structural rather than technical: a brane-bound observer is always inverting boundary data for a bulk field, the same inversion problem that appears in holographic reconstruction generally, which suggests the review's 'missing equation' is a feature of any such setup rather than a gap specific to this model.
- Because the quadratic correction stiffens the effective equation of state to $w_{\rm tot} \approx 2w+1$, a thermalized early universe above $lambda$ would have behaved like a $w = 5/3$ fluid; the resulting gravitational-wave background would carry a blue tilt that a future detector could in principle separate from standard inflationary signals.
- The tidal-charge solution forces $Q = -2M$ in the small-$r$ limit, making the near-horizon geometry more attractive than Schwarzschild's, which would raise the innermost stable circular orbit frequency and could be checked against X-ray timing if a sufficiently small black hole were found.
- The absence of a normalizable Kaluza-Klein vector zero mode, together with the mass gap $Delta m = 3H_0/2$ on a de Sitter brane, means the lightest bulk gravitational degrees of freedom are inaccessible to any sub-Hubble experiment, which is a large part of why an infinite extra dimension survives laboratory constraints.
Original abstract
The observable universe could be a 1+3-surface (the "brane") embedded in a 1+3+d-dimensional spacetime (the "bulk"), with standard-model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (~ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity "leaks" into the bulk, behaving in a truly 1+3+d-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting testable implications for high-energy astrophysics, black holes and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The observable universe could be a 1+3-surface (the "brane") embedded in a 1+3+d-dimensional spacetime (the "bulk"), with standard-model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (~ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity "leaks" into the bulk, behaving in a truly 1+3+d-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting testable implications for high-energy astrophysics, black holes and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The observable universe could be a 1+3-surface (the "brane") embedded in a 1+3+d-dimensional spacetime (the "bulk"), with standard-model particles and fields trapped on the brane while gravity is free to access the bulk. | background 0.95 | 0.05 | 0.16 | hedged possibility 1.00 |
| s1 | At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (~ TeV) level. | background 0.85 | 0.06 | 0.15 | hedged possibility 1.00 |
| s2 | This revolutionary picture arises in the framework of recent developments in M theory. | background 0.99 | 0.04 | 0.06 | asserted 0.99 |
| s3 | The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. | background 1.00 | 0.03 | 0.04 | asserted 0.71 |
| s4 | General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s5 | At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity "leaks" into the bulk, behaving in a truly 1+3+d-dimensional way. | background 0.55 | 0.08 | 0.20 | asserted 1.00 |
| s6 | This introduces significant changes to gravitational dynamics and perturbations, with interesting testable implications for high-energy astrophysics, black holes and cosmology. | interpretation 0.75 | 0.11 | 0.20 | asserted 0.98 |
| s7 | Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. | background 0.62 | 0.09 | 0.13 | asserted 0.88 |
| s8 | This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models. | contribution 0.99 | 0.46 | 0.11 | asserted 0.95 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:gr-qc/0312059 by the authors.
References
How later papers used it
116 later papers cite it
3 later papers by other authors are listed below: 3 built on it.
Builds on it 3 papers
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Extended some of this paper's results to encompass the Eötvös phenomenological brane case.
Black string corrections in variable tension braneworld scenarios (2012) · 0.98 · paraphrased: the licence does not allow quoting
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Extended some of this paper's results to encompass accrued torsion corrections, obtaining formulas for a Taylor expansion outside a black hole.
Torsion Effects in Braneworld Scenarios (2009) · 1.00 · paraphrased: the licence does not allow quoting
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Extended some of this paper's results to encompass accrued torsion corrections, obtaining formulas for a Taylor expansion outside a black hole.
Possible Generalizations within Braneworld Scenarios: Torsion fields (2010) · 0.89 · paraphrased: the licence does not allow quoting
How the literature uses it
108 arXiv papers cite this in 189 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for high-energy astrophysics, black holes, and cosmology.
Brane-World Gravity · 0.99
Papers citing it · 85 sentences without a confident label
Builds on it 3 papers
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In order to find some typical gravitational signatures of braneworld scenarios with torsion we obtain all the formulæ for a Taylor expansion outside a black hole in Section II, extending some results of Ref. [ref] in order to encompass accrued torsion corrections.
Torsion Effects in Braneworld Scenarios · Introduction · 1.00
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In order to find some typical gravitational signatures of braneworld scenarios with torsion we obtain all the formulæ for a Taylor expansion outside a black hole in Section V, extending some results of Ref. [ref] in order to encompass accrued torsion corrections.
Possible Generalizations within Braneworld Scenarios: Torsion fields · Introduction · 0.89
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Some results [ref] are extended in order to encompass this case.
Black string corrections in variable tension braneworld scenarios · Introduction · 0.98
Uses its method or tool 2 papers
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The presence of the brane action in ([ref]) gives a singular part, which we will take into account by the Israel junction condition [ref] , so now we will consider only the bulk part.
Bouncing cosmology from Kalb-Ramond Braneworld · The 5-D solution · 0.97
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The field equations together with the 5D Einstein and Bianchi equations [ref] are used to compute the bulk metric near the brane, and in particular the black string warped horizon along the extra dimension [ref] .
Regular Bulk Solutions and Black Strings from Dynamical Braneworlds with Variable Tension · Black String Warped Horizon and Variable Brane Tension · 0.96
Compares with it 4 papers
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In particular, Eq.([ref]) — when the torsion, and consequently $A_μν$ defined in Eq.([ref]), goes to zero — is led to the results previously obtained in [ref] .
Torsion Effects in Braneworld Scenarios · Measurable Torsion Effects · 0.89
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Obviously, when the torsion goes to zero, all the results above are led to the ones obtained in [ref] , [ref] , and references therein.
Possible Generalizations within Braneworld Scenarios: Torsion fields · Measurable Torsion Effects · 0.82
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This behaviour has some similarity with steep inflation [ref] , where high energy corrections to the Friedmann equation in RS scenario [ref] permit an inflationary evolution of the brane with potentials too steep to sustain it in the standard 4D case, although the inflationary scenario introduced by Copeland et al in [ref] is supported by an inflaton confined in the brane while in our model inflation on the brane is induced by a dilaton field on the bulk.
Crossing the cosmological constant line in a dilatonic brane-world model with and without curvature corrections · Vacuum solution · 0.95
Cites it as background 79 papers
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This yields the following equations governing the dynamics of gravity on the brane [ref] :
Constraining extra-spatial dimensions with observations of GW170817 · Spacetime with extra dimension: A brief introduction · 0.83
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The coupling coefficient $λ_b$ appearing in the above equation is known as the brane tension and is another characteristic parameter from the extra dimensions [ref] .
Silhouette of M87*: A new window to peek into the world of hidden dimensions · Black hole shadow on the brane · 0.92
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Cosmological evolution on the brane is described by solutions of the non-standard Friedmann equation [ref] .
Cosmological constraints on parameters of one-brane models with extra dimension · Introduction · 0.90
Discussion
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