[Paper Review] Towards testing a dark matter candidate that emerges from the scalar ether theory

Mayeul Arminjon|arXiv (Cornell University)|May 22, 2023

Cosmology and Gravitation Theories24 references1 citations

TL;DR

This paper proposes a dark matter candidate emerging from scalar ether theory (SET), where electromagnetic fields in a galaxy's gravitational potential generate an exotic, gravitationally active interaction energy tensor T_inter via a scalar field p. Using homogenization techniques—time, spatial, or spacetime—on the PDE for p, the study shows that spacetime homogenization yields a tractable, macroscopic equation with a non-local homogenized tensor GH, enabling feasible large-scale computation of T_inter as a potential dark matter halo.

ABSTRACT

According to a scalar theory of gravity with a preferred frame, electromagnetism in the presence of a gravitational field implies that there is an additional energy tensor, which might contribute to dark matter. The expression of this tensor is determined by a mere scalar $p$, that depends on the EM field and (for a weak field) on the Newtonian gravitational field. We briefly recall why this tensor arises and how the EM field in a galaxy can be calculated. The data fields that enter the PDE for the scalar field $p$ oscillate very quickly in space and time, as does the EM field. This prevents integration of that PDE at the relevant galactic scale. Therefore, a homogenization of that PDE has to be operated. We discuss in some detail three possible ways of applying the homogenization theory to that PDE: time, space, or spacetime homogenization. The second and third ways may lead to feasible, albeit heavy calculations.

Motivation & Objective

• To investigate whether an interaction energy tensor T_inter, arising from electromagnetism in a weak gravitational field within scalar ether theory (SET), could constitute a viable dark matter candidate.
• To model the interstellar radiation field (ISRF) as an exact solution of the source-free Maxwell equations under axial symmetry, enabling precise computation of EM field components (E, B).
• To address the numerical infeasibility of solving the PDE for the scalar field p due to rapid spatial and temporal oscillations of the EM fields at galactic scales.
• To apply homogenization theory—time, spatial, or spacetime—to derive effective macroscopic equations for p that are computationally tractable at galactic scales.
• To evaluate the feasibility and structure of the resulting homogenized equations, particularly focusing on the non-local nature of the homogenized tensor GH in spacetime homogenization.

Proposed method

• Develops an axisymmetric model of the ISRF as an exact solution to the source-free Maxwell equations, providing (E, B) fields that satisfy the full set of Maxwell equations.
• Derives the PDE for the scalar field p from the interaction energy tensor T_inter = pγμν in SET, with the source term f depending on the Newtonian potential U and its time derivative ∂t(∇U) in the preferred frame.
• Applies three homogenization strategies: time-only, space-only, and spacetime homogenization, each transforming the original PDE into an effective macroscopic equation.
• For spacetime homogenization, uses a two-scale expansion (p = p0(X) + p1(X,Y)) with fast variables Y = (t, x, y, z), leading to a homogenized equation with a tensor GH = ⟨G.(1 + ∇Yχ)⟩ derived from solving a boundary value problem on a spacetime cell Υ.
• Derives the homogenized equation divX⟨q0⟩ + ⟨f⟩ = 0 with ⟨q0⟩ = −GH·∇Xp0, preserving the original PDE structure but with a non-averaged, non-local GH.
• Assesses the computational cost of solving the boundary value problem for GH at each macroscopic point in spacetime, noting it is heavy but potentially feasible in quasi-periodic cases.

Experimental results

Research questions

• RQ1 Can the interaction energy tensor T_inter in scalar ether theory, arising from EM and gravitational fields, account for the observed dark matter distribution in galaxies?
• RQ2 How can the PDE for the scalar field p, which governs T_inter, be effectively solved at galactic scales despite the rapid oscillations of the EM fields?
• RQ3 Which homogenization approach—time, space, or spacetime—yields a numerically feasible and physically meaningful macroscopic equation for p?
• RQ4 What is the structure and behavior of the homogenized tensor GH in spacetime homogenization, and how does it differ from a simple spatial or temporal average?
• RQ5 Does spatial homogenization alone suffice to smooth both spatial and temporal variations of the EM fields, enabling stable integration of the effective PDE at galactic scales?

Key findings

• The interaction energy tensor T_inter = pγμν in scalar ether theory is a viable candidate for dark matter, as it is non-localized, gravitationally active, and not associated with any known velocity-carrying medium.
• The ISRF model produces an exact solution to the source-free Maxwell equations under axial symmetry, yielding a spectral energy density (SED) close to existing models but with significantly enhanced SED on the galactic axis.
• Time homogenization alone fails to resolve the numerical challenge, as the time-averaged fields ¯u and ¯S still exhibit rapid spatial variations at the wavelength scale.
• Spatial homogenization leads to a tractable macroscopic equation for p0 with spatially averaged fields, provided the time variation is also sufficiently smoothed by spatial averaging.
• Spacetime homogenization yields a homogenized equation of the same form as the original PDE, but with a non-local homogenized tensor GH = ⟨G.(1 + ∇Yχ)⟩, which requires solving a boundary value problem on a spacetime cell at each macroscopic point.
• The calculation of GH is computationally intensive, as it depends on the solution χ of a non-trivial spacetime boundary value problem, but remains feasible in quasi-periodic configurations relevant to galactic dynamics.