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4D Gravity on a Brane in 5D Minkowski Space

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Gia Dvali, Gregory Gabadadze, Massimo Porrati

High Energy Physics - Theory · Submitted 2 May 2000 · 22 references

Pith's verdict: The founding DGP paper: a clean linearized derivation of a genuinely new mechanism, with the extra scalar mode and the postulated induced term both openly flagged as unresolved.

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arXiv IDhep-th/0005016 (version 2)
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CitationGia Dvali, Gregory Gabadadze, Massimo Porrati. "4D Gravity on a Brane in 5D Minkowski Space." arXiv:hep-th/0005016 (2000). https://pith.science/paper/NTYKMGY4
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The paper

The paper claims

Four-dimensional Newtonian gravity can emerge on a brane floating in an infinite fifth dimension, because a curvature term confined to the brane changes the force law from $1/r$ to $1/r^2$ at a computable crossover distance.

The reading

The paper asks whether ordinary four-dimensional gravity can be recovered on a 3-brane sitting in five-dimensional Minkowski space whose extra dimension is genuinely infinite, neither compactified nor warped. The authors' proposal is that the brane carries its own four-dimensional Ricci scalar term, $M_P^2$ times the integral of $R$ over the worldvolume, in addition to the bulk five-dimensional Einstein-Hilbert term weighted by $M^3$. Because that extra term acts like a delta-function kinetic energy localized at $y=0$, the propagator becomes $1/(M_P^2 p^2 + 2 M^3 p)$, which yields a $1/r$ potential at distances below the crossover scale $r_0 = M_P^2/(2M^3)$ and a $1/r^2$ potential above it. If this works, Newtonian gravity on the brane needs no finite extra dimension at all, and the size of the extra dimension can be truly infinite. The authors then confront the price: the graviton propagator has the tensor structure of a massive four-dimensional graviton, meaning an extra scalar polarization survives and alters predictions such as light bending.

What carries the argument

The load-bearing object is the brane-localized curvature term, a delta-function kinetic operator at $y=0$. In the scalar prototype it turns the bulk equation $M^3 partial_A partial^A Phi + M_P^2 delta(y) partial_mu partial^mu Phi = 0$ into a boundary condition that mixes the four-momentum $p$ with the derivative in the extra dimension; solving it gives the Fourier-space Green function $1/(M_P^2 p^2 + 2 M^3 p)$, whose two terms compete. The ratio of the two Planck masses defines the single crossover length $r_0 = M_P^2/(2M^3)$: below it the $M_P^2 p^2$ term dominates and the brane looks four-dimensional, above it the $2 M^3 p$ term dominates and the brane looks five-dimensional. The same denominator reappears for the graviton, so the entire physical content of the paper is compressed into one scale and one propagator.

Extended reading notes

Core claim

The central claim is that adding a four-dimensional Einstein-Hilbert term $M_P^2$ times the integral of $R$ over a static 3-brane to the five-dimensional bulk action $M^3$ times the integral of $R^{(5)}$ produces the correct four-dimensional Newtonian potential at short distances even though the bulk is flat and extends infinitely. The authors first demonstrate the mechanism with a prototype scalar whose action is $M^3$ times the integral of $(1/2)(1/2)$-type bulk kinetic term plus $M_P^2$ times a delta-function worldvolume kinetic term, solving the Green's function exactly to obtain a potential built from the sine and cosine integrals, $V(r) = -[1/(8p^2 M_P^2 r)]$ times $[sin(r/r_0) Ci(r/r_0) + (1/2) cos(r/r_0)(p - 2 Si(r/r_0))]$, which reduces to $1/r$ for $r$ much less than $r_0$ and to $1/r^2$ for $r$ much greater than $r_0$. They then repeat the analysis for the graviton in harmonic gauge, obtain the propagator denominator $M_P^2 p^2 + 2 M^3 p$, and find the same short- and long-distance crossover. Crucially, the tensor structure they derive is $1/2 eta_{mu alpha} eta_{nu beta} + 1/2 eta_{mu beta} eta_{nu alpha} - 1/3 eta_{mu nu} eta_{alpha beta}$, the massive or five-dimensional form rather than the massless four-dimensional form with coefficient $1/2$. The authors read this as four-dimensional tensor-scalar gravity, and they argue that the coefficient stays at $1/3$ here, unlike in warped constructions, because there is no warp factor and hence no gauge transformation shifting the coefficient to $1/2$.

Load-bearing premise

Everything rests on the brane actually carrying its own four-dimensional curvature term with the right sign and the right size, and the paper offers only speculative origins for that term while explicitly leaving its sign and magnitude to model-dependent details it does not compute.

Editorial extensions

If this is right

  • The gravitational potential between two masses on the brane falls as $1/r$ at short range with a calculable logarithmic correction proportional to $\ln(r/r_0)$, and turns into $1/r^2$ beyond $r_0$, so Newton's law is recovered without compactifying or warping the extra dimension.
  • The crossover scale is fixed by the two Planck masses through $r_0 = M_P^2/(2M^3)$; with $M_P \simeq 10^{19}$ GeV and $M$ in the TeV region, $r_0$ comes out around $10^{15}$ cm, roughly the size of the solar system.
  • Because the graviton propagator carries the coefficient $1/3$ rather than $1/2$, the theory contains an extra scalar polarization that adds an attractive force, and the authors state that this changes light bending in a way that conflicts with observation for a massive graviton of any mass.
  • The same delta-function kinetic-term mechanism localizes scalar and vector fields on a brane in flat higher-dimensional space, and for those spins the phenomenological difficulties that affect the graviton case are absent.

Reading between the lines

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

  • The requirement that $r_0$ exceed the solar-system size by roughly eight orders of magnitude, inferred from Mercury perihelion bounds, pushes the bulk Planck mass a couple of orders of magnitude below the TeV scale, which makes the model harder to embed in a higher-dimensional theory than the original large-extra-dimension picture suggested.
  • The mismatch between the $1/3$ and $1/2$ coefficients is the cleanest observational handle the paper offers: any measurement sensitive to the scalar polarization, such as light deflection by a massive source or precision tests of the equivalence principle, probes the mechanism directly rather than only its distance scaling.
  • The proposed cure for the extra attraction, adding repulsive $U(1)$ gauge interactions to brane matter, would tie the gravitational force law to the gauge charges of the source and could show up as composition-dependent free fall.
  • Because the induced term can be generated by brane matter loops only when conformal invariance is broken, the mechanism predicts a correlation between the matter content of the brane and the size of the crossover scale.
Original abstract

We suggest a mechanism by which four-dimensional Newtonian gravity emerges on a 3-brane in 5D Minkowski space with an infinite size extra dimension. The worldvolume theory gives rise to the correct 4D potential at short distances whereas at large distances the potential is that of a 5D theory. We discuss some phenomenological issues in this framework.

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, 2,895 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.77 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.14 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We suggest a mechanism by which four-dimensional Newtonian gravity emerges on a 3-brane in 5D Minkowski space with an infinite size extra dimension. The worldvolume theory gives rise to the correct 4D potential at short distances whereas at large distances the potential is that of a 5D theory. We discuss some phenomenological issues in this framework.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We suggest a mechanism by which four-dimensional Newtonian gravity emerges on a 3-brane in 5D Minkowski space with an infinite size extra dimension. contribution 1.00 0.73 0.72 hedged possibility 0.91
s1 The worldvolume theory gives rise to the correct 4D potential at short distances whereas at large distances the potential is that of a 5D theory. result 0.99 0.79 0.34 asserted 1.00
s2 We discuss some phenomenological issues in this framework. contribution 0.90 0.73 0.03 unclear 0.82

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.78.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
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

Figures
Figure 1. The one-loop diagram generating the Ricci scalar for 4D graviton. Wave lines denote gravitons, solid lines denote massive scalars/fermions. Vertical short lines on scalar/fermion propagators indicate that they are massive. As a result, the interaction Lagrangian of localized matter with 5D metric fluctua￾tions hAB(x, y) ≡ GAB(x, y) − ηAB, reduces to the following expression: Lint = Z dy hµν (x, y) Tµν (x) δ(y) = h µν (x, 0) Tµν (x) , (4) where the 4D induced metric gµν (x) = ηµν + hµν is defined as in (2). Due to this interaction, a 4D kinetic term can be generated for gµν (x) in…

Figures reproduced from arXiv:hep-th/0005016 by the authors.

References
How later papers used it

768 later papers cite it

Evidence: substantial. 756 of the citing papers share no author with it, and 43 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 1,079 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

Builds on it 1 of 6 papers with this label
Uses its method or tool 1 of 4 papers with this label

  • Used a setup similar to this paper's model to describe a folded space.

    How to form a wormhole (2020) · 0.89 · paraphrased: the licence does not allow quoting

Uses its data or values 1 paper
Compares with it 3 of 31 papers with this label
How the literature uses it

660 arXiv papers cite this in 948 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 · 380 sentences without a confident label

Contests it 2 papers
Builds on it 5 papers
Uses its method or tool 4 papers
  • To describe a folded space we will use a setup similar to the so-called DGP model [ref] .

    How to form a wormhole · A 4D brane embedded in a 5D bulk and attractive force between two masses · 0.89

  • In other words, we consider a Randall-Sundrum type setup with a 4d brane in a 5d ambient spacetime [ref] .

    Entanglement islands in higher dimensions · Introduction · 0.92

  • We model the subleading term in the effective action of the source as a Dvali-Gabadadze-Porrati (DGP) term [ref] , i.e. induced gravity on the bulk source, and, while the multiplicative constants cannot be determined, we observe that the remaining on-shell action has a parametric scaling that is consistent with that of $δ_12 _12^K$.

    Bubbling wormholes and matrix models · Free energy and on-shell action · 0.85

Uses its data or values 1 paper
  • One can use the approximate solution for the growth rate for the different gravity model specified by $γ_i$, f_i ≡d lng_id lna ≈Ω_m^γ_i(z) One can integrate the above equation to obtain $g_i(z)/g_i(z_0) = exp[∫_-ln[1+z]^0 Ω_m^γ_i(z') d ln(1+z') ]$ where we use $γ_DGP ≃11/16$ [ref] , $γ_GR ≃0.555$ [ref] , and $γ_f(R) ≃0.41$ [ref] .

    Measuring bias from unbiased observable · General Relativity vs Modified Gravity · 0.97

Compares with it 30 papers
  • So in this regime, the potential energy is approximately the same as the one found in the original DGP setup [ref] .[ This is not surprising, since for two point sources separated at a large distance $r$, their bulk separation becomes less important and the system with two branes behaves like a system with one brane provided that $r_c 2 r_c$.] Since we are only interested in the regime $r ≪r_c$, the above expression reduces to [equation]

    Modification of the laws of gravity in the DGP model by the presence of a second DGP brane · Potential energy in the 5-d description · 0.98

  • On the other hand, in [ref] they find that with their analysis of the redshift space power spectrum for the braneworld model of gravity by Dvali, Gabadadze and Porrati (DGP) [ref] , negligible model bias is found.

    Theoretical Accuracy in Cosmological Growth Estimation · Introduction · 0.82

  • The DGP model is characterized by an induced curvature term on the brane, and (in its original form) is symmetric around the brane, which is tensionless and embedded in 5D Minkowski space [ref] : [equation] The brane cosmological equations from this action are given by [ref] ρ= 6 M_4^2 H^2 ∓12 M_5^3 H

    The Cosmology of Asymmetric Brane Modified Gravity · Asymmetric Braneworld Models · 0.91

Cites it as support 2 papers
  • Hints towards this statement are, e.g., Ref. [ref] where it is shown that the Dvali-Gabadadze-Porrati gravity [ref] violates Birkhoff's theorem; or the impossibility of designing Einstein-Straus models with $f(R)$ theories [ref] .

    Cosmic backreaction and Gauss's law · Discrete cosmology with a modified theory of gravity · 0.99

  • This is realized explicitly in the DGP model where the massless graviton mode is non-normalizable [ref] .].

    Causality in Curved Spacetimes: The Speed of Light & Gravity · Graviton Dispersion Relation · 0.96

Cites it as background 437 papers
  • The extra dimensions can be strongly warped, or flat, and the brane itself can have various terms in its effective action: [equation] where $Λ$ is a possible bulk cosmological constant, $σ$ is the brane tension (branes typically have an energy momentum proportional to the induced metric), $K$ is the extrinsic curvature of the brane, which is the Gibbons-Hawking term present when we regard the brane as a boundary to the bulk, and $ℛ$ is the intrinsic Ricci curvature of the brane, and represents the DGP [ref] term.

    The three burials of Melquiades DGP · Overview · 0.97

  • This model is known as the Dvali-Gabadadze-Porrati (DGP) braneworld model [ref] .

    Softly higher-derivative massive gravity · Propagator in a softly higher-derivative massive gravity · 0.80

  • One of the most promising scenarios for modifications of GR in a relativistic and general covariant framework is the Dvali-Gabadadze-Porrati (DGP) model [ref] .

    Higher Curvature Brane Corrections to the DGP Model · Introduction · 0.85

Discussion

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