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Tphysicsletters/6981/11/1490/77009901.567tpl/Violation of γ in Brans-Dicke gravity
Wednesday, June 26, 2024 at 1:15:00 PM UTC
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Violation of γ in Brans-Dicke gravity
Hoang Ky Nguyen ∗
Department of Physics, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
Bertrand Chauvineau †
Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, Nice cedex 4, France
Theoretical Physics Letters
2024 ° 026(06) ° 11-10
https://www.wikipt.org/tphysicsletters
DOI: 10.1490/77009901.567tpl
Acknowledgment
BC thanks Antoine Strugarek for helpful correspondences. HKN thanks Mustapha AzregAïnou, Valerio Faraoni, Tiberiu Harko, Viktor Toth, and the participants of the XII Bolyai–Gauss–Lobachevsky Conference (BGL-2024): Non-Euclidean Geometry in Modern Physics and Mathematics (Budapest, May 1-3, 2024) for valuable commentaries
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Abstract
The Brans Class I solution in Brans-Dicke gravity is a staple in the study of gravitational theories beyond General Relativity. Discovered in 1961, it describes the exterior vacuum of a spherical Brans-Dicke star and is characterized by two adjustable parameters. Surprisingly, the relationship between these parameters and the properties of the star has not been rigorously established. In this Proceeding, we bridge this gap by deriving the complete exterior solution of Brans Class I, expressed in terms of the total energy and total pressure of the spherisymmetric gravity source. The solution allows for the exact derivation of all post-Newtonian parameters in Brans-Dicke gravity for far field regions of a spherical source. Particularly for the γ parameter, instead of the conventional result γ PPN = ω+1 ω+2 , we obtain the analytical expression γ exact = ω+1+(ω+2) Θ ω+2+(ω+1) Θ where Θ is the ratio of the total pressure P ∗ ∥ + 2P ∗ ⊥ and total energy E ∗ contained within the mass source. Our non-perturbative γ formula is valid for all field strengths and types of matter comprising the mass source. Consequently, observational constraints on γ thus set joint bounds on ω and Θ, with the latter representing a global characteristic of the mass source. More broadly, our formula highlights the importance of pressure (when Θ ̸= 0) in spherical Brans-Dicke stars, and potentially in stars within other modified theories of gravitation.
Introduction
Brans–Dicke gravity is the second most studied theory of gravitation besides General Relativity. It represents one of the simplest extensions of gravitational theory beyond GR [1]. It is characterized by an additional dynamical scalar field ϕ which, in the original vision of Brans and Dicke in 1961, acts like the inverse of a variable Newton ‘constant’ G. The scalar field has a kinetic term, governed by a (Brans-Dicke) parameter ω in the following gravitation action
In the limit of infinite value for ω, the kinetic term is generally said to be ‘frozen’, rendering Φ being a constant value everywhere. In this limit, if the field ϕ approaches its (non-zero) constant value in the rate O (1/ω), the term ω Φ g µν∂µΦ∂νΦ would approach zero at the rate O (1/ω) and hence become negligible compared with the term Φ R, effectively recovering the classic Einstein–Hilbert action. 1 Together with its introduction [1], Brans also identified four classes of exact solutions in the static spherically symmetric (SSS) setup [2]. The derivation of the Brans solutions was explicitly carried out by Bronnikov in 1973 [4]. Of the four classes, only the Brans Class I is physically meaningful, however. It can recover the Schwarzschild solution in its parameter space. For comparison with observations or experiments, Brans derived the Robertson (or Eddington-RobertsonSchiff) β and γ post-Newtonian (PN) parameters based on his Class I solution
The γ parameter is important as it governs the amount of space-curvature produced by a body at rest and can be directly measured via the detection of light deflection and the Shapiro time delay. The parametrized postNewtonian (PPN) γ formula recovers the result γ GR = 1 known for GR in the limit of infinite ω, in which the BD scalar field becomes constant everywhere. Current bounds using Solar System observations set the magnitude of ω to exceed 40, 000 [6]. We should emphasize that the “conventional” results (2) and (3) were derived under the assumption of zero pressure in the gravity source. It should be noted that these formulae can also be deduced directly from the PPN formalism for the Brans–Dicke action, without resorting to the Brans Class I solution [5, 7]. The PPN derivation relies on two crucial approximations: (i) weak field and (ii) slow motions. Regarding the latter approximation, an often under-emphasized point is that not only must the stars be in slow motion, but the microscopic constituents that comprise the stars must also
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Conclusion
We have derived the exact analytical formulae, (28) and (30), for the PN parameters γ and δ for spherical mass sources in BD gravity. The derivation relies on the integrability of the 00−component of the field equation, rendering it non-perturbative and applicable for any field strength and type of matter constituting the source. The conventional PPN result for BD gravity γ PPN = ω+1 ω+2 lacks dependence on the physical features of the mass source. In the light of our exact results, the γ PPN should be regarded as an approximation for stars in modified gravity under low-pressure conditions. Our findings expose the limitations of the PPN formalism, particularly in scenarios characterized by high star pressure. It is reasonable to expect that the role of pressure may extend to other modified theories of gravitation.
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References
[1] C. H. Brans and R. Dicke, Mach’s Principle and a Relativistic Theory of Gravitation, Phys. Rev. 124, 925 (1961)
[2] C. H. Brans, Mach’s Principle and a relativistic theory of gravitation II, Phys. Rev. 125, 2194 (1962)
[3] H. K. Nguyen and B. Chauvineau, O(1/ √ ω) anomaly in Brans-Dicke gravity with trace-carrying matter, arXiv:2402.14076 [gr-qc]
[4] K. A. Bronnikov, Scalar-tensor theory and scalar charge, Acta Phys. Polon. B 4, 251 (1973), Link to pdf
[5] C. M. Will, Theory and Experiment in Gravitational Physics, second edition, Cambridge University Press, Cambridge, 2018
[6] C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev. Relativ. 17, 4 (2014), doi.org/10.12942/lrr-2014-4
[7] S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972
[8] H. K. Nguyen and B. Chauvineau, An optimal gauge for Tolman-Oppenheimer-Volkoff equation in Brans-Dicke gravity (in preparation)
[9] B. Chauvineau and H. K. Nguyen, The complete exterior spacetime of spherical Brans-Dicke stars, Phys. Lett. B 855, 138803 (2024), arXiv:2404.13887 [gr-qc]
[10] H. K. Nguyen and B. Chauvineau, Impact of Star Pressure on γ in Modified Gravity beyond Post-Newtonian Approach, arXiv:2404.00094 [gr-qc]
[11] J. C. Baez and E. F. Bunn, The Meaning of Einstein’s Equation, Amer. Jour. Phys. 73, 644 (2005), arXiv:grqc/0103044
[12] J. Ehlers, I. Ozsvath, E. L. Schucking, and Y. Shang, Pressure as a Source of Gravity, Phys. Rev. D 72, 124003 (2005), arXiv:gr-qc/0510041
Abstract
Introduction
Conclusion
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