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Nordstrom's scalar theory of gravity and the equivalence principle
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TL;DR
Deruelle shows that Nordström’s scalar theory of gravity, like General Relativity, can satisfy the strong equivalence principle beyond post-Newtonian order, with gravitational mass equal to inertial mass when \(F(\Phi)=\Phi\). The paper supports this via explicit examples and a proposed Nordström-Katz superpotential, while noting the general proof remains incomplete.
Key findings
- Nordström’s theory is observationally ruled out: it gives PPN parameter \(\gamma=-1\), perihelion advance \(\Delta\omega = -\frac{1+a_2}{6}\Delta\omega_{\rm GR}\), and \(\beta=\frac{1+a_2}{2}\).
- For a two-body system at lowest order, the paper derives \(M_g/M_{\rm in}=1+\eta E_g/(Mc^2)\) with \(\eta=2a_2\), so \(M_g=M_{\rm in}\) if \(a_2=0\).
- For a constant-density perfect-fluid star, it finds \(M_g/M_{\rm in}=1+\eta E_g/(Mc^2)\) with \(\eta=2a_2\) and \(E_g=-3GM^2/(5R)\); a Mathematica check shows \(M_{\rm in}=M_g\) exactly if \(F(\Phi)=\Phi\).
- In the Jordan-frame superpotential approach, the inertial mass is \(M_{\rm in}=\frac{M_g}{3}(1-\alpha)\); fixing \(\alpha=-2\) from the Einstein-frame constant-density result gives \(M_{\rm in}=M_g\).
- Appendix result: for a radiating binary, the leading energy-loss term is \(1/6\) the general relativistic value, with an orbit-averaged expression given in the paper.
Methods
- Theoretical study of Nordström’s scalar gravity in both the Einstein frame (scalar field on flat spacetime) and Jordan frame (conformally flat metric).
- Uses analytic calculations for two point-like particles at lowest order and for a static, spherically symmetric constant-density perfect-fluid star, including a numerical/symbolic check of junction conditions.
- Constructs a Katz-type superpotential and conserved charge for Nordström’s Jordan-frame action, and proposes a Palatini-like action for future work.
Limitations
- The paper states that its argument that the strong equivalence principle always holds in Nordström’s final theory is “a bit weak.”
- The Jordan-frame Lagrange multiplier \(\lambda^{\mu\nu\rho\sigma}\) has ten independent components while the traceless equations give only nine, so the system is underdetermined and the inertial mass is not fully fixed; the constant \(\alpha\) is fixed by returning to the Einstein-frame constant-density example.
- The explicit star calculation uses the “unrealistic but simple” constant-density equation of state; more realistic barotropic fluids are cited as studied numerically elsewhere but are not used to prove the general case.
- The proposed Palatini-like action and construction of a “Nordström-Katz” superpotential are left to further work.
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In the paper
“However, λ µνρσ , having the symmetries of the Weyl tensor, possesses ten independent components, whereas (5), being traceless, has only nine components.The system of equations ( 5) for λ µνρσ is therefore undetermined.”
§ Nordström's scalar theory of gravity and the equivalence principle > VIII. A JORDAN ACTION FOR NORDSTR ÖM'S GRAVITY
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In the paper
“We have therefore shown that the strong equivalence principle holds exactly (in this particular case of a constant density “star”) in Nordström's “final” theory [1] [2] where F (Φ) = Φ.”
§ Nordström's scalar theory of gravity and the equivalence principle > VI. THE EXAMPLE OF A CONSTANT DENSITY "STAR"
What the sources say
boostedSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“lim r→∞ r 2 D σ λ σ 0r0 = α GM g c 2 (9.18) with α an arbitrary constant.”
§ Nordström's scalar theory of gravity and the equivalence principle > B. Application to Nordström's theory
plainstates an association- Asserted: passes — The authors put it forward as true.
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In the paper
“ds 2 = (1 + Φ) 2 [-dt 2 + dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 )] with Φ = - GM g c 2 r . (9.15)”
§ Nordström's scalar theory of gravity and the equivalence principle > B. Application to Nordström's theory
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In the paper
“At lowest order the motion is coulombian, ˙ V = qq ′ M N /mm ′ R 2 so that, for a 2 = 0 : Mg = M + qq ′ /2Rc 2 , which, again, is its inertial mass.”
§ Nordström's scalar theory of gravity and the equivalence principle > X. CONCLUSION
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In the citing paper
“A key feature is that the scalar gravity theory respects the Strong Equivalence Principle.”
From Small vacuum energy from small equivalence violation in scalar gravity (2017) · cites this paper as [14]
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In the paper
§ Nordström's scalar theory of gravity and the equivalence principle > V. THE EXAMPLE OF THE TWO-BODY MOTION AT LOWEST ORDER
“where Φ is given by (3), that is, at lowest order, by Φ = Gm c 2 r”
§ Nordström's scalar theory of gravity and the equivalence principle > V. THE EXAMPLE OF THE TWO-BODY MOTION AT LOWEST ORDER
“Φ(X µ ) = - Gm c 2 r R + O(G 2 ) (5.3)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Nordström's scalar theory of gravity and the equivalence principle > VI. THE EXAMPLE OF A CONSTANT DENSITY "STAR"
“If M in = M g or equivalently, -GM g /c 2 R = Φ R , then we must find that, for all Φ R :”
§ Nordström's scalar theory of gravity and the equivalence principle > VI. THE EXAMPLE OF A CONSTANT DENSITY "STAR"
“Φ R ≡ -GMg c 2 R an integration constant”
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Metadata
- Authors
- Deruelle, N.
- Affiliations
- Centre National de la Recherche Scientifique (FR)
- Corresponding author
- Deruelle, Nathalie
- Venue
- General Relativity and Gravitation
- Published
- 2011-08-07
- Volume
- 43
- Issue
- 12
- Pages
- 3337-3354
- ISSN
- 0001-7701 (ISSN-L)
- 1572-9532
- arXiv
- 1104.4608
- OpenAlex
- W2169707543
- Cited by
- 59
- References
- 25
- Citations by year
- 2025: 15 · 2024: 6 · 2023: 3 · 2021: 2 · 2020: 6 · 2019: 2 · 2018: 6 · 2017: 3 · 2016: 4 · 2015: 5 · 2014: 4 · 2013: 3
- FWCI
- 0.89
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- General Relativity and Gravitation (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- equivalence principle, strong equivalence principle, superpotential, gravitational mass, inertial mass, post-Newtonian corrections, scalar field gravity, flat spacetime
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