The Einstein Equations on the 3-Brane World

Kei-ichi Maeda; Misao Sasaki; Tetsuya Shiromizu

REVIEW

The Einstein Equations on the 3-Brane World

Tetsuya Shiromizu, Kei-ichi Maeda, Misao Sasaki

General Relativity and Quantum Cosmology · Submitted 22 Oct 1999 · 20 references

Pith's verdict: The paper that gave brane-world cosmology its working Einstein equations; the derivation holds, and the RS1 exclusion is stronger than the stress-test allows, though the abstract states it more absolutely than the body does.

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arXiv IDgr-qc/9910076 (version 3)

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CitationKei-ichi Maeda, Misao Sasaki, Tetsuya Shiromizu. "The Einstein Equations on the 3-Brane World." arXiv:gr-qc/9910076 (1999). https://pith.science/paper/PQRHADOP

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This is the paper everyone in brane-world gravity cites for the effective equations, and it earns that. Eq. (17), with the quadratic π_μν and the projected Weyl term E_μν, is derived without any cosmological ansatz: Gauss (1), Codazzi (2), the 5D Einstein equations, the Weyl decomposition (7), Israel (15), Z_2 projection (16). I walked the chain and found no algebraic gap. The identification G_N = κ_5^4 λ / 48π in Eq. (19) is correct, and the appendix is more than an afterthought — Eqs. (A18)–(A20) give the boundary condition and bulk propagation for E_μν, so the system is closed rather than left hanging. That is the part people forget to credit and it is the part that made the formalism usable. The homogeneity result (Eq. 30) follows cleanly from D^ν π_μν = 0 with E_μν = 0, and the paper is upfront that it only holds for exact AdS bulk. On the stress-test: I think it overreaches. In RS1 the visible brane's negative tension is fixed by the construction, not by how one splits S_μν, so the wrong-sign G_N is a real consequence, not an artifact of the λ/τ ambiguity. The ambiguity the authors flag bites on cosmology — an inflaton potential that cannot be cleanly separated from vacuum energy — which is a different regime. That said, 'should be excluded from the physical point of view' in the abstract is a stronger sentence than the body supports, and RS1 was proposed for the hierarchy problem, not as a gravitational theory; a reader should take the sign result and leave the editorializing. Two smaller things: the |E_μν| estimate in Eq. (26) is borrowed heuristics from RS2's Newtonian potential, order-of-magnitude only, and the classifier record claiming no stated limitations is simply wrong — the abstract and the paragraphs after Eqs. (14) and (19) state them plainly. Who should read it: anyone doing brane-world cosmology or modified gravity, and anyone who wants to see a projection formalism done properly. Yes to peer review, and yes, I would cite it.

Referee report

Significance

This is a foundational result: it gives the general low-energy effective gravitational equation for any single-brane model with confined matter, fixes Newton's constant in terms of the brane tension, and isolates the bulk Weyl tensor as the term carrying the five-dimensional gravitational degrees of freedom. The derivation is parameter-free and checkable step by step, and it yields falsifiable statements (wrong-sign G_N for lambda < 0; the homogeneity constraint for a pure-AdS bulk). The authors are candid about the limits of the result: they explicitly flag the ambiguity of the S_munu = -lambda q_munu + tau_munu split after Eq. (14) and after Eq. (19), and they state in the abstract (s4) and the concluding paragraph that the effective equations cease to be valid when the brane cosmological constant is not well-defined. Those acknowledgements are a strength of the presentation and should be weighed in the manuscript's favor.

Major objections (1)

[Abstract (s1) and concluding Summary paragraph] The claim that RS1 'predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view' is stated more absolutely than Eq. (19) establishes. Eq. (19) gives G_N = kappa5^4 lambda / 48 pi, so a negative tension produces a wrong-sign gravitational coupling only when lambda is a well-defined constant that dominates the brane stress-energy in the split of Eq. (14). The authors themselves note after Eq. (14) that this split 'can be ambiguous, particularly in cosmological contexts' and after Eq. (19) that 'it becomes impossible to define Newton's gravitational constant' when the vacuum/matter distinction is ambiguous. The algebra is sound; the issue is that the headline conclusion is unconditional while the supporting step is conditional. Concrete fix: qualify the abstract and the Summary sentence to say that the negative-tension brane yields a wrong-sign induced G_N whenever a constant tension dominates the brane stress-energy, and state explicitly that this is the regime in which the effective description applies. This is a scope/correctness-risk concern, not a challenge to the derivation.

Minor objections (6)

[Text following Eq. (30)] The constraint D^nu pi_munu = (1/6)(rho + P) h_munu D^nu rho = 0 implies partial_i rho = 0 only when rho + P is nonzero; for a cosmological-constant-like fluid (rho + P = 0) the constraint is empty. The abstract's 'generically' (s3) partly covers this, but the sentence after Eq. (30) should state the rho + P != 0 condition explicitly so the homogeneity conclusion is not read as unconditional.
[Eq. (21)] The relation 'D_nu K^nu_mu - D_mu K proportional to D_nu tau^nu_mu = 0' uses a loose proportionality symbol. From Eqs. (10) and (13) the factor is explicit; give it (kappa5^2/2) so the conservation law follows by equality rather than by order-of-magnitude reasoning.
[Eqs. (5) and (9); Appendix A] The tilde notation distinguishing the Riemann-projected quantity E-tilde_munu (Eq. 5) from the Weyl-projected E_munu (Eq. 9) is easy to miss, and Appendix A reuses E-tilde_munu with a different definition in Eq. (A8). State once, near Eq. (5), that the tilde denotes the Riemann-based projection and the untilded symbol the Weyl-based one, and cross-reference Eq. (A10) where the two are related.
[Eq. (26)] The magnitude estimate for E_munu borrows the RS2 Newtonian potential of Eq. (25). State that this is an order-of-magnitude estimate valid in the linearized, short-range regime, so the reader does not read Eq. (26) as a general bound.
[Eq. (11) and surrounding text] The condition a_mu = n^nu nabla_nu n_mu = 0 is a gauge choice on the normal congruence. It would help to say explicitly that the final result (17) is independent of this choice and that Eq. (11) only fixes the coordinate form of the metric in Eq. (12).
[Throughout] The text refers to the 'Codacci equation' (Eq. 2 and surrounding discussion); the standard spelling is Codazzi. The acknowledgements contain 'Postdoctal' and the paragraph after Eq. (17) contains 'unchaged'; these should be corrected.

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The paper

The paper claims

The gravitational equations that a brane-world observer would actually measure are derived, and they rule out the negative-tension brane.

The reading

This paper asks a concrete question: if our four-dimensional world is a thin sheet (a 3-brane) embedded in a five-dimensional spacetime, and all matter except gravity is stuck to that sheet, what gravitational equations do we measure? The authors derive them from the Gauss and Codazzi relations plus Israel's junction conditions, imposing a mirror symmetry across the brane, and get an equation that looks like Einstein's but carries three extra pieces: a net four-dimensional cosmological constant, a term quadratic in the matter stress-energy, and a term inherited from the five-dimensional Weyl tensor. The payoff is a sign argument. Newton's constant comes out proportional to the brane's tension, so a brane with negative tension has the wrong-sign gravitational coupling — an anti-gravity world — which removes the negative-tension brane of the first Randall–Sundrum proposal from consideration while leaving the positive-tension one intact. A second payoff is a constraint: if the bulk is exactly anti-de Sitter, the projected Weyl term vanishes and the four-dimensional Bianchi identities force a perfect-fluid matter distribution on the brane to be spatially homogeneous.

What carries the argument

The engine is the pair of embedding identities — the Gauss equation relating the brane's Riemann tensor to the bulk Riemann tensor plus extrinsic curvature, and the Codazzi equation relating the divergence of the extrinsic curvature to the bulk Ricci tensor contracted with the normal — combined with Israel's junction condition across the brane and a $Z_2$ reflection symmetry that makes the extrinsic curvature on the two sides equal and opposite. That mirror symmetry is what turns a jump condition into an algebraic formula for $K_{mu nu}$ in terms of the brane stress-energy, and it is that formula which, fed back into the Gauss equation, produces the effective four-dimensional Einstein equations. The final equation's structure is worth naming: $8pi G_N tau_{mu nu}$ is ordinary matter, $kappa_5^4 pi_{mu nu}$ is a high-energy correction quadratic in the stress-energy, and $E_{mu nu}$ is the imprint of the bulk gravitational field, traceless and constrained in divergence but not in transverse-traceless part.

Extended reading notes

Core claim

Starting from the Gauss equation, the Codazzi equation, and the decomposition of the five-dimensional Riemann tensor into Weyl, Ricci, and scalar parts, the authors obtain the induced four-dimensional Einstein tensor on a hypersurface without assuming any symmetry. They then specialise to the brane-world setting: a five-dimensional stress-energy tensor of the form $T_{ab} = -Lambda g_{ab} + S_{ab} delta(chi)$ with $S_{mu nu} = -lambda q_{mu nu} + tau_{mu nu}$, Israel's junction condition $[K_{mu nu}] = -kappa_5^2 (S_{mu nu} - q_{mu nu} S/3)$, and $Z_2$ symmetry about the brane. The junction condition then fixes the extrinsic curvature algebraically, $K_{mu nu} = -kappa_5^2(S_{mu nu} - q_{mu nu}S/3)/2$, and substituting this back yields the central result: $^{(4)}G_{mu nu} = -Lambda_4 q_{mu nu} + 8 pi G_N tau_{mu nu} + kappa_5^4 pi_{mu nu} - E_{mu nu}$, with $Lambda_4 = kappa_5^2(Lambda + kappa_5^2 lambda^2/6)/2$, $G_N = kappa_5^4 lambda / 48 pi$, $pi_{mu nu}$ a specific quadratic expression in $tau$, and $E_{mu nu}$ the electric part of the five-dimensional Weyl tensor evaluated at the brane. Because the sign of $G_N$ tracks the sign of $lambda$, the negative-tension brane of the first Randall–Sundrum scenario is an anti-gravity world and the authors reject it, while the positive-tension second scenario gives ordinary attraction. The same equation, combined with the contracted Bianchi identity, gives $D^mu E_{mu nu} = kappa_5^4[tau^{alpha beta}(D_nu tau_{alpha beta} - D_beta tau_{nu alpha}) + (tau_{mu nu} - q_{mu nu}tau) D^mu tau/3]/4$, so the Weyl term is not free but sourced by matter; setting $E_{mu nu} = 0$ for an exactly anti-de Sitter bulk and taking a perfect fluid forces $partial_i rho = 0$, i.e. spatial homogeneity.

Load-bearing premise

Everything rests on splitting the brane's stress-energy into a constant vacuum tension plus ordinary matter, $S_{mu nu} = -lambda q_{mu nu} + tau_{mu nu}$, with $lambda$ a well-defined number — and the authors themselves note this split is ambiguous, particularly in cosmology, and that Newton's constant becomes undefinable when the distinction between vacuum and matter energy blurs.

Editorial extensions

If this is right

Newton's constant on the brane is fixed by the brane tension, $G_N = kappa_5^4 lambda/48pi$, so a brane with negative tension has a gravitational coupling of the wrong sign and cannot host ordinary attractive gravity.
The first Randall–Sundrum scenario, whose visible brane carries negative tension, is excluded on physical grounds; the second, positive-tension scenario survives and recovers four-dimensional Einstein gravity at low energies.
In a strictly anti-de Sitter bulk, the vanishing of the projected Weyl tensor plus the four-dimensional Bianchi identity forces a perfect fluid on the brane to be spatially homogeneous, so inhomogeneous matter requires the bulk to deviate from exact anti-de Sitter.
The quadratic term $pi_{mu nu}$ grows relative to the linear matter term as the square of the energy scale, so it matters in the early universe and is negligible today.
The effective equations stop being meaningful in any era where the split between vacuum energy and ordinary matter on the brane is ill-defined, such as when a scalar field's potential energy dominates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's sign argument turns brane tension into a selection principle rather than a free parameter, which suggests that any brane-world model builder should treat the sign of the tension as a consistency check before doing cosmology.
Because $E_{mu nu}$ carries the transverse-traceless part that is not fixed by brane matter, brane observers are in principle able to infer something about the bulk geometry from gravitational-wave data alone; the bulk is not sealed off from brane measurement.
The acknowledged ambiguity in separating $lambda$ from $tau_{mu nu}$ is sharpest precisely during inflation, so the framework's most-wanted application and its weakest point coincide; a treatment that absorbs the scalar potential into an effective, time-dependent tension would be the natural next step.
The homogeneity constraint in the exact anti-de Sitter case suggests a diagnostic: any brane-world cosmology with structure must come with a non-zero Weyl contribution, so the size of that contribution is a bookkeeping entry for how much bulk gravity the model is secretly importing.

Original abstract

We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with $Z_2$ symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. {}From our general argument we conclude that the first Randall & Sundrum-type theory (RS1) [hep-ph/9905221] predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view. Their second-type theory (RS2) [hep-th/9906064] where the brane has the positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti-de Sitter, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by the potential energy of the scalar field.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,220 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.84 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.82 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with $Z_2$ symmetry. We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. {}From our general argument we conclude that the first Randall & Sundrum-type theory (RS1) [hep-ph/9905221] predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view. Their second-type theory (RS2) [hep-th/9906064] where the brane has the positive tension provides the correct signature of gravity. In this latter case, if the bulk spacetime is exactly anti-de Sitter, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities. By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor. In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by the potential energy of the scalar field.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We carefully investigate the gravitational equations of the brane world, in which all the matter forces except gravity are confined on the 3-brane in a 5-dimensional spacetime with $Z_2$ symmetry.	contribution 0.70	0.88	0.04	asserted 0.91
s1	We derive the effective gravitational equations on the brane, which reduce to the conventional Einstein equations in the low energy limit. {}From our general argument we conclude that the first Randall & Sundrum-type theory (RS1) [hep-ph/9905221] predicts that the brane with the negative tension is an anti-gravity world and hence should be excluded from the physical point of view.	result 0.75	0.90	0.55	asserted 0.99
s2	Their second-type theory (RS2) [hep-th/9906064] where the brane has the positive tension provides the correct signature of gravity.	result 0.78	0.43	0.13	asserted 1.00
s3	In this latter case, if the bulk spacetime is exactly anti-de Sitter, generically the matter on the brane is required to be spatially homogeneous because of the Bianchi identities.	result 0.95	0.68	0.07	asserted 1.00
s4	By allowing deviations from anti-de Sitter in the bulk, the situation will be relaxed and the Bianchi identities give just the relation between the Weyl tensor and the energy momentum tensor.	result 0.86	0.75	0.08	asserted 0.94
s5	In the present brane world scenario, the effective Einstein equations cease to be valid during an era when the cosmological constant on the brane is not well-defined, such as in the case of the matter dominated by the potential energy of the scalar field.	limitation 0.56	0.74	0.10	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (20)

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L. Randall and R. Sundrum, hep-th/9906064

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C. Csaki, M. Graesser, C. Kold and J. Terning, hep-ph/99 06513

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hep-th 2026-10 conditional novelty 5.0 of 10

In DGP braneworlds, black hole entropy equals holographic entanglement entropy precisely when the brane extrinsic curvature satisfies a condition that is automatic for static black holes, and the same condition govern...
Long-lived ringing of Bronnikov--Kim brane-world black holes and wormholes
gr-qc 2026-09 conditional novelty 5.0 of 10

Massive scalar quasinormal modes of Bronnikov-Kim brane-world black holes and wormholes become increasingly long-lived as the field mass grows while keeping a nonzero oscillation frequency, in contrast to the Casadio-...
Extreme mass-ratio inspirals and extra dimensions: Insights from modified Teukolsky framework
gr-qc 2025-07 conditional novelty 5.0 of 10

A modified Teukolsky equation and the Dudley-Finley approximation give nearly the same LISA detectability bound for the braneworld tidal charge, with MTE mismatches growing faster for high-eccentricity EMRIs.
Geodesic dynamics in brane-de Sitter wormholes
gr-qc 2025-04 conditional novelty 5.0 of 10

For an asymptotically de Sitter brane wormhole, the throat is shown to be an unstable fixed point and photon sphere, with a claimed Bogdanov-Takens bifurcation for radial null geodesics.
Echoes from the braneworld black holes
gr-qc 2020-01 conditional novelty 5.0 of 10

Brane-localized black holes with negative tidal charge should produce gravitational-wave echoes whose time delay grows with the charge, offering a potential probe of extra dimensions.
Helical phase inflation and its observational constraints
hep-ph 2019-08 conditional novelty 5.0 of 10

Helical phase inflation, using the phase of a complex scalar as the inflaton, fits Planck 2018 and BICEP2 data and is claimed to show alpha-attractor behavior, with a brane version that reduces field excursions.
Avoiding the string swampland in single-field inflation: Excited initial states
hep-th 2018-09 conditional novelty 5.0 of 10

An excited initial state for tensor perturbations can suppress the tensor-to-scalar ratio enough to reconcile steep single-field inflation with the swampland criteria.
Cosmology on a Brane in Minkowski Bulk
hep-th 2000-10 accept novelty 5.0 of 10

A brane with an intrinsic curvature term in a Minkowski bulk has two cosmological branches; one produces late-time accelerated expansion sourced by the brane's own curvature, with no brane tension and no dark energy.
Cosmological Evolution of the Randall-Sundrum II Model with Running Vacuum: A Special Class of Solutions
gr-qc 2026-09 accept novelty 4.0 of 10

A hybrid RS-II braneworld and running vacuum model fails to produce natural inflation without extreme fine-tuning or unnatural parameters.
Compact stars in a large-tension braneworld: mildly negative Weyl coupling consistent with NICER and gravitational-wave data
gr-qc 2026-07 conditional novelty 4.0 of 10

A large-tension braneworld with α_U ≈ −0.15 fits NICER and GW170817 data for SLy, raising M_max to ~2.30 M_⊙ and R_1.4 to ~13.3 km relative to GR.
Two-scalar-field $f(R)$ Thick Branes, Gravitational Resonances and Quasinormal Modes
hep-th 2026-06 unverdicted novelty 4.0 of 10

In ghost-free two-scalar f(R) thick branes, internal structure produces no narrow real-axis tensor resonances; quasinormal modes are broad with quality factors 0.9-1.9.
Investigating the interplay of the braneworld gravity and the plasma environment on the black hole shadow
gr-qc 2025-12 unverdicted novelty 4.0 of 10

Negative tidal charge enlarges the shadow of rotating braneworld black holes while inhomogeneous plasma shrinks it and homogeneous plasma enlarges it; EHT data limits q to roughly -1.15 to 0.45 for M87* in the low-den...
Scalar-gravitational quasinormal modes and echoes in a five dimensional thick brane
gr-qc 2025-08 conditional novelty 4.0 of 10

Scalar perturbations of a split five-dimensional thick brane yield long-lived quasinormal modes and position-dependent echo waveforms: clean periodic echoes on a sub-brane and modulated echoes between sub-branes.
Wormholes from beyond
gr-qc 2024-12 conditional novelty 4.0 of 10

Using the Nakas-Kanti top-down approach, the author embeds the Morris-Thorne and Molina-Neves wormholes in a regular five-dimensional anti-de Sitter bulk, so the wormhole on the brane is supported entirely by bulk fields.
Wormholes without exotic matter: quasinormal modes, echoes and shadows
gr-qc 2021-07 conditional novelty 4.0 of 10

Wormholes that mimic near-extremal Reissner-Nordstrom black holes in their ringdown and shadow still produce detectable echoes, computed here for scalar, electromagnetic and Dirac perturbations.
Primordial Gravitational Wave Signals in Modified Cosmologies
gr-qc 2020-08 conditional novelty 4.0 of 10

Scalar-tensor and braneworld modifications of the early expansion rate leave detectable imprints on the primordial gravitational wave spectrum, with scalar-tensor boosts visible to high-frequency interferometers and b...
Silhouette of M87*: A new window to peek into the world of hidden dimensions
gr-qc 2019-09 conditional novelty 4.0 of 10

The M87* shadow diameter, combined with stellar-dynamical or EHT mass estimates and an assumed spin of 0.5, is better fit by a braneworld black hole with negative tidal charge than by a Kerr black hole, though the pre...
Probing the era of reheating for reconstructed inflationary potential in the RS II braneworld
astro-ph.CO 2019-08 reject novelty 4.0 of 10

The paper derives constraints on the reheating temperature and the five-dimensional Planck mass in RS II braneworld inflation using a reconstructed attractor-like inflaton potential.
Astrophysical environment around a black hole in the braneworld and its optical signatures
hep-th 2026-05 unverdicted novelty 3.0 of 10

Braneworld quadratic and nonlocal corrections weaken gravity in anisotropic Einstein-cluster environments around black holes, blocking horizon formation and shifting Einstein-ring and shadow radii in ways that may con...
Unification of inflation and dark energy {\it \`a la} quintessential inflation
gr-qc 2014-10 conditional novelty 3.0 of 10

A pedagogical review of quintessential inflation, surveying two classes of single-field models that join early-universe inflation to late-time dark energy, together with their instant-preheating and relic gravitationa...
Brane-world gravity
gr-qc 2003-12 accept novelty 2.0 of 10

A comprehensive 2003 survey of Randall-Sundrum brane-world gravity, covering the covariant projection formalism, brane cosmology, the black hole problem, and cosmological perturbation theory, with an explicit list of ...
TASI Lectures on Extra Dimensions and Branes
hep-ph 2004-04 accept novelty 1.0 of 10

A pedagogical survey of brane and extra-dimension model building, covering large flat dimensions, split fermions, gaugino mediation, orbifold symmetry breaking, and the Randall-Sundrum warped solution with its gravito...
Linear Dilaton in Cosmology and Particle Physics
hep-th 2026-08

How later papers used it

240 later papers cite it

Evidence: substantial. 234 of the citing papers share no author with it, and 47 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 428 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

18 later papers by other authors are listed below: 1 built on it and 17 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 of 3 papers with this label

Found the (4+n)-dimensional generalization of this paper's equations using its junction conditions.

Kaluza-Klein brane cosmology with a bulk scalar field (2011) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 18 of 39 papers with this label · 1 by the same authors

Used the effective gravitational field equations derived by this paper.

Wormholes in de Sitter branes (2012) · 1.00 · paraphrased: the licence does not allow quoting
Involved the gravity on the brane using the method provided in this paper, with the brane evolution determined by the effective Einstein field equation on the brane and the Israel junction condition together.

Brane universe and holography in spacetime of charged AdS dilaton black hole (2019) · 1.00 · paraphrased: the licence does not allow quoting
Wrote down the effective gravitational equations on the brane by means of the covariant decomposition developed by this paper.

Anisotropic fluid from nonlocal tidal effects (2014) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's recipe to find the brane-world stress tensor satisfying the equation of state ρ = -p = const. < 0, giving an AdS-type brane metric.

Negative $\Lambda$ induced by accelerated motion (2014) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's formalism, employing the Gauss-Codazzi equations and Israel-Darmois junction conditions to project the five-dimensional curvature along the brane.

Cosmological accretion onto braneworld black holes: a relativistic treatment (2026) · 1.00 · paraphrased: the licence does not allow quoting
Obtained the effective Einstein equation on the brane following the approach introduced by this paper.

Self-accelerating the normal DGP branch (2009) · 1.00 · paraphrased: the licence does not allow quoting
Wrote down the effective gravitational equations on the brane by means of the covariant decomposition developed by this paper.

Regular Schwarzschild-like spacetime embedded in a five dimensional bulk (2021) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's formalism, in which gravitational phenomena in a brane immersed in a five-dimensional bulk are described by effective four-dimensional Einstein equations.

Black holes and wormholes in AdS branes (2010) · 0.99 · paraphrased: the licence does not allow quoting
Following the approach suggested by this paper, wrote the effective four-dimensional gravitational field equations in a vacuum Randall-Sundrum brane.

Electromagnetic perturbations in new brane world scenarios (2016) · 0.99 · paraphrased: the licence does not allow quoting
Summarized the 5D action and equations in the formalism developed by this paper.

Holography for the Trace Anomaly Action (2023) · 0.98 · paraphrased: the licence does not allow quoting
Following many brane-world scenarios including this paper, wrote the brane energy-momentum tensor with brane tension λ and matter tensor τ_μν.

Wormholes from beyond (2024) · 0.83 · paraphrased: the licence does not allow quoting
Used the equations above, which were used to develop a covariant analysis of the weak field in this paper, to develop a Taylor expansion of the metric along the extra dimension predicting the black string behavior.

Black string corrections in variable tension braneworld scenarios (2012) · 0.82 · paraphrased: the licence does not allow quoting
The effective 4D Einstein equation seen by a brane observer is computed from the 5D Einstein equation by projecting on the brane via the Gauss equation together with the Israel junction condition [ref] .

Cosmological Dark Matter from a Bulk Black Hole (2022) · The 5D scalar-gravity system · 0.95
Following this paper, wrote 5-dimensional Einstein equations with a bulk cosmological constant and brane energy-momentum as source.

A New Fate of a Warped 5D FRW Model with a U(1) Scalar Gauge Field (2015) · 0.89 · paraphrased: the licence does not allow quoting
Using this paper's procedure, gave the field equations on the brane with brane tension λ, energy momentum tensor τ_μν, π_μν, and the electric part of the Weyl tensor.

Brane-$f(R)$ gravity and dark matter (2012) · 0.81 · paraphrased: the licence does not allow quoting
Used the Israel junction conditions and the Gauss-Codazzi equations, as employed by this paper, in obtaining the field equations on the brane.

Anisotropy in Born-Infeld brane cosmology (2011) · 0.96 · paraphrased: the licence does not allow quoting
Following this paper's derivation, wrote the induced Einstein equations on the brane with a 4-dimensional cosmological constant, κ, S_μν, ℱ_μν, and ξ_μν.

Relativistic Images in Randall-Sundrum II Braneworld Lensing (2009) · 0.87 · paraphrased: the licence does not allow quoting

By the same authors

Used this paper's equation to analyze the Kaluza-Klein cosmology.

Kaluza-Klein braneworld cosmology with static internal dimensions (2007) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

219 arXiv papers cite this in 394 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 259 sentences without a confident label

Builds on it 3 papers

Using this junction conditions, we find the $(4+n)$-dimensional generalization of the Shiromizu-Maeda-Sasaki equations [ref] ,

Kaluza-Klein brane cosmology with a bulk scalar field · Action and field equations · 0.98
Building on the framework of Refs. [ref] , one can incorporate matter on the brane—here, an Einstein cluster—into the effective four-dimensional field equations [ref] and investigate its impact on both the matter distribution and the resulting spacetime geometry.

Astrophysical environment around a black hole in the braneworld and its optical signatures · Introduction · 0.97
This approach has been previously used [ref] - [ref] in brane scenarios and now we apply it to the case of brane vectors.

Brane Vector Dynamics from Embedding Geometry · Embedding geometry and Einstein equation on brane · 0.89

Uses its method or tool 37 papers

Within the braneworld scenario, the field equations induced on the brane are derived via an elegant geometric approach developed in Ref. [ref] .

Mass radius relation of compact stars in the braneworld · Tolman-Oppenheimer-Volkoff Equations in the braneworld · 0.87
Following Nakas and Kanti [ref] , the total energy-momentum tensor could be written as [equation] where the delta $δ(y)$ asks for a junction condition, and the brane energy-momentum tensor, following many brane-world scenarios [ref] , is given by [equation] The constant $λ$ is the brane tension, which is some sort of vacuum energy on the brane, and the tensor $τ_μν$ represents all matter fields or sources exclusively on the brane.

Wormholes from beyond · The brane equations · 0.83
We used the effective gravitational field equations derived by Shiromizu, Maeda, and Sasaki [ref] .

Wormholes in de Sitter branes · Introduction · 1.00

Compares with it 8 papers

The main result of this paper is the existence of the trace part of the brane energy-momentum tensor in the modified Einstein equations on the brane, which is a modification of the SMS effective equation [ref] .

Modified Gravitational Equations on Braneworld with Lorentz Invariant Violation · Conclusions · 0.80
For low energy density, $GL^2ρ≪1$ and $L^2 C/a^4≪1$, in the approximation beyond the leading order, cf. Eq.([ref]), this equation reads[We assume that the dark radiation term is redshifted for growing $a$ faster than matter term and expand to the second order in $ρ$, but the first order in $C$.] [equation] and coincides with the modified Friedmann equation in the Randall-Sundrum model [ref] [equation] where $σ=3/8πG_5L=3/4πGL^2$ is the Randal-Sundrum brane tension and $C$ is the braneworld constant of motion [ref] .

Anomaly Driven Cosmology: Big Boost Scenario and AdS/CFT Correspondence · AdS/CFT and Randall-Sundrum braneworld setup · 0.91
For low energy density, $GL^2ρ≪1$ and $L^2 C/a^4≪1$, in the approximation beyond the leading order, cf. Eq.([ref]), the modified Friedmann equation coincides with the modified Friedmann equation in the Randall-Sundrum model [ref] [equation] where $σ=3/8πG_5L=3/4πGL^2$ is the Randall-Sundrum brane tension and $C$ is the braneworld constant of motion [ref] .[We assume that the dark radiation term is redshifted, as $a$ grows, faster than the matter term and expand to the second order in $ρ$, but the first order in $C$.] Note that the thermal radiation on the brane (of non-Casimir energy nature) is equivalent to the mass of the bulk black hole associated with this constant.

Density matrix of the Universe reloaded: origin of inflation and cosmological acceleration · Dual description via the AdS/CFT correspondence · 0.91

Cites it as support 1 paper

However, this problem can be avoided if our universe is confined on the positive tension brane [ref] .

Gravity localization and mass hierarchy in scalar-tensor branes · I. Introduction · 0.98

Cites it as background 74 papers

The effective 3+1 dimensional Einstein field equations on the brane for the Randall and Sundrum model were derived by Shiromizu, Maeda, and Sasaki [ref] .

Gravitational Geons on the Brane · Introduction · 0.93
In the 5D warped space models involving a non-compact extra dimension, the gravitational field equations in the bulk can be expressed in the form [ref] [equation] where the fundamental 5D Planck mass $M_P$ enters via $k^2=8π/M_P^3$, $λ$ is the brane tension, and $Λ$ is the bulk cosmological constant.

Quasiperiodic oscillations in a strong gravitational field around neutron stars testing braneworld models · Gravitational field equations on the brane and the exterior spacetime to the uniform density configurations · 0.90
According to the general formulation of SMS braneworld scenario [ref] , our $4D$ universe (the brane) is embedded locally and isometrically in a codimension-1 ambient (the bulk) space, in which the standard model fields are confined to the brane, while gravity propagates in all spatial dimensions.

Intrinsic quantum dynamics of particles in brane gravity · Newtonian dynamics of particles · 0.95

Discussion

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