Einstein gravity extended by a scale covariant scalar field with Bekenstein term and dynamical mass generation

TL;DR

This work develops a relativistic MOND framework based on a single scale-covariant scalar field in integrable Weyl geometry, featuring a non-minimal coupling to the Hilbert term, a cubic Bekenstein-type aquadratic kinetic term, and a second-order mass term activated only for spacelike gradients below MOND scales. In the Milgrom regime the scalar field obeys a covariant deep MOND equation, while outside this regime gravity reverts to Einstein gravity, with the transition controlled by a smooth function of the gradient and MOND scale $a_0$. The Newton–Milgrom limit emerges as a superposition of the baryonic Newton potential and a MONDian scalar-field potential, yielding a total acceleration $a$ that follows a MOND interpolation function; the model also predicts modifications to gravitational lensing, external-field effects, and scalar-field halos that can contribute to cluster masses. Applications to centrally symmetric and axisymmetric galaxy models, along with a Coma-cluster toy model, indicate that the scalar-field halos can substantially reduce the Newtonian missing mass problem, though certain light-deflection predictions and full cosmological implications remain open questions. Overall, the approach offers a coherent, Weyl-geometric route to MOND-like dynamics with explicit energy-momentum contributions from the scalar field, linking galactic phenomenology to a dynamical, non-particle dark-energy/dark-matter–like component, while highlighting remaining theoretical and observational challenges.

Abstract

Under carefully chosen assumptions a single general relativistic scalar field is able to induce MOND-like dynamics in the weak field approximation of the Einstein frame (gauge) and to modify the light cone structure accordingly. This is shown by a Lagrangian model formulated in the framework of integrable Weyl geometry. It contains a Bekenstein-type (``aquadratic'') term and a second order term generating additional mass energy for the scalar field. Both are switched on only if the gradient of the scalar field is spacelike and below a MOND-typical threshold, like in the superfluid model of Berezhiani/Khoury. The mass term induces non-negligible energy and pressures of the scalar field and leads to gravitational light deflection compatible with MOND-ian free fall trajectories. In the weak field (Newton-Milgrom) approximation the Bekenstein term implies a deep MOND equation for the scalar field. In this model the external field effect of the MOND approach has to be reconsidered.

Figures (9)

• Figure 1: Transition function $h(x;\bar{\alpha},\bar{\beta})$ for $\bar{\alpha}=0.1,\, \bar{\beta}=10$.
• Figure 2: Left: Radial acceleration relation $a(y)$ with $y=\frac{a_N}{a_0}$ (where $a_N=a^{(bar)}$) for empirically fitted function $a_{emp}(y)= \tilde{\nu}(y)= y\, (1-e^{-\sqrt{y}} )^{-1}$ (blue) from McGaugh-et-al:2016, compared with the respective relation $a_{\mathtt{sf}} (y)$ of the scalar field model for $\bar{\alpha}=0.1,\, \bar{\beta}=10$, (yellow). The Newtonian assumption $a(y)=y$ is added (green). Right: Radial acceleration relation $a(a_N)$ ($a=g_{tot},\, a_N=g_B$) from Hossenfelder/Mistele:2018, with empirical data (blue) like in fig. \ref{['fig rar']} ($a_0 \approx 10^{-10}\, ms^{-2}$); the red line depicts the relation of Hossenfelder's model ("CEG") and is identical to our \ref{['eq rar WST']} with $h\equiv 1$. Dotted line shows the relation expected in Newton dynamics for baryonic matter only.
• Figure 3: Comparison of the metric coefficients $A(r)$ and $B(r)$ for the numerical relativistic model (blue) and the Schwarzschild solution (yellow) for a typical galaxy ($m \sim 10^{11}M_{\odot}$).
• Figure 4: Left: Radial accelerations in $cms^{-2}$ for galaxies ($M=10^{-5}\, kpc$) for the numerical model $a_{rel}(r)$ (blue), classical MOND $a_M$ (green), and Schwarzschild-Newton dynamics (yellow). Right: Quotient of model accelerations $a_X/a_{rel}$, with $a_{rel}$ the relativistic model, $X$=Newton-Milgrom approximation of relativistic model (blue), $X=$ classical MOND (green), $X=$ Schwarzschild/Newton (yellow).
• Figure 5: Total Newtonian mass equivalent of the scalar field $M_N^{(\phi)}$ (blue) and mass expression of the energy density only $M_e^{(\phi)}$ (yellow) up to coordinate distance $r$, expressed in multiples of the central mass $M$.