The theory considered interprets gravity as a pressure force. Thus, the scalar gravitational field defines the gravity acceleration field. However, it also determines the relation between the flat “background metric” and a curved “physical metric”. Here we derive the equations of motion of the mass centers of a system of weakly gravitating bodies in the second version of that theory. We use the framework which was built and used for the first version. Namely, we use an asymptotic scheme of post-Newtonian (PN) approximation to derive the local (field) PN equations, and by integration inside the bodies we deduce from those local equations the equations of motion of the mass centers, using also an asymptotic framework for the good separation between the different bodies.

The Lorentz-Poincaré interpretation of special relativity (SR) keeps the classical concepts of separated space and time, at the price of postulating an indetectable preferred inertial frame or “ether”. But SR does not contain gravity. The presence of gravity could make the ether detectable. This is one idea behind the “scalar ether theory of gravitation” (SET), which coincides with SR if the gravity field vanishes, and passes a number of tests. However, the coupling of SET with the Maxwell electromagnetic (EM) field needs to use the theory’s dynamical equation for the energy tensor in a non-trivial way. It cannot be assumed that the energy tensors of the charged matter and the EM field add to give the total energy tensor, source of the gravitational field. Thus, an additional, “interaction” energy tensor T inter has to be postulated. Asking that T inter is Lorentz-invariant in the situation of SR, fixes its form. It depends only on a scalar field p . T inter is an exotic kind of matter and is distributed in the whole space, hence it could contribute to dark matter. For a weak gravitational field, p obeys a first-order partial differential equation (PDE) involving the EM field and the Newtonian potential. However, the EM field varies on the scale of the wavelength, which is extremely small. To get the field p in a galaxy, some averaging has to be done. After several attempts based on the homogenization theory, a simpler way has been found recently: If the macro-averages of p and the EM field vary smoothly, it can be shown that the PDE for p remains valid in the same form with spacetime-averaged fields. The current stage of calculations will also been shown.

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.

Abstract The existing models of the interstellar radiation field (ISRF) do not produce a Maxwell field. Here, the recent model of the ISRF as a Maxwell field is improved by considering separately the different frequencies at the stage of the fitting. Using this improved procedure: (i) It is checked in detail that the model does predict extremely high values of the spectral energy density (SED) on the axis of a galaxy, which however decreases very rapidly when ρ \rho , the distance to the axis, is increased from zero. (ii) The difference between the SED values (with ρ = 1 \rho =1 or 8 kpc 8\hspace{0.33em}{\rm{kpc}} ), as predicted either by this model or by a recent radiation transfer model, is reduced significantly. (iii) The slower decrease of the SED with increasing altitude z z , as compared with the radiation transfer model, is confirmed. We also calculate the evolutions of the SED at large ρ \rho . We interpret these evolutions by determining asymptotic expansions of the SED at large z z , and also ones at large ρ \rho .

In the scalar theory of gravitation with a preferred reference frame, a consistent formulation of electrodynamics in the presence of gravitation needs to introduce an additional energy tensor: the interaction energy tensor. This energy is gravitationally active and might contribute to the dark matter, because it has an exotic character and it is not localized inside matter. In order to check if that energy might form representative dark halos, one has to model the interstellar radiation field in a galaxy as a complete electromagnetic field obeying the Maxwell equations. A model has been built for this purpose, based on assuming axial symmetry and on recent results about axisymmetric Maxwell fields. Its predictions for the variation of the spectral energy density inside our Galaxy are relatively close to those of a recent radiation transfer model, except on the symmetry axis of the Galaxy, where the present model predicts extremely high values of the energy density.

It is shown that in phenomenological models based on plastic spin, the assumed privileged rotating frame is in fact one of constant material symmetry. In polycrystal models, no particular symmetry or privileged frame has to be assumed, but the evolution of symmetry may be conveniently investigated: here both theoretically and in the practical case of simple shear. It is also shown how these models may be operated in a semi-analytical way, giving practical procedures.

Abstract The Maxwell radiation field is an essential physical characteristic of a galaxy. Here, an analytical model is built to simulate that field in an axisymmetric galaxy. This analytical model is based on an explicit representation for axisymmetric source-free Maxwell fields. In a previous work, the general applicability of this representation has been proved. The model is adjusted by fitting to it the sum of spherical radiations emitted by the composing “stars.” The huge ratio distance/wavelength needs to implement a numerical precision better than the quadruple precision. The model passes a validation test based on a spherically symmetric solution. The results for a set of “stars” representative of a disk galaxy indicate that the field is highest near the disk axis, and there the axial component of E {\bf{E}} dominates over the radial one. This work will allow us in the future to check if the interaction energy predicted by an alternative theory of gravitation might be a component of dark matter.

An analytical model for the Maxwell radiation field in an axisymmetric galaxy, proposed previously, is first checked for its predictions of the spatial variation of the spectral energy distributions (SEDs) in our Galaxy. First, the model is summarized. It is now shown how to compute the SED with this model. Then the model is adjusted by asking that the SED predicted at our local position in the Galaxy coincides with the available observations. Finally, the first predictions of the model for the spatial variation of the SED in the Galaxy are compared with those of a radiation transfer model. We find that the two predictions do not differ too much. This indicates that, in a future work, it should be possible with the present model to check if the “interaction energy” predicted by an alternative, scalar theory of gravitation, contributes to the dark matter.

Garay-Avendaño and Zamboni-Rached defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields – and hence, by summation on frequencies, all propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modeled as an exact solution of the free Maxwell equations.

In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation ("SET"), this assumption leads to charge non-conservation, hence an additional, "interaction" energy tensor T inter has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of T inter. Hence, with T inter , at least one additional equation must be provided. This is done by assuming that T inter is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute T inter in a given gravitational plus EM field. T inter may contribute to the dark matter.

Abstract The standard interpretations of special relativity (Einstein–Minkowski) and general relativity (GR) lead to a drastically changed notion of time: the eternalism or block universe theory. This has strong consequences for our thinking about time and for the development of new fundamental theories. It is therefore important to check this thoroughly. The Lorentz–Poincaré interpretation, which sees the relativistic effects as following from a “true” Lorentz contraction of all objects in their motion through the ether, uses a conservative concept of time and is in the absence of gravitation indistinguishable from the standard interpretation; but there exists currently no accepted gravitation theory for it. The scalar ether theory of gravitation is a candidate for such a theory; it is presented and discussed. The equations of motion for a test particle are derived; the case of a uniformly moving massive body is discussed and then specialized to the case of spherical symmetry. Formulas for the acceleration of test particles are given in the preferred frame of the ether and in the rest frame of the massive body that moves with velocity V with respect to the ether. When the body rests in the ether (V = 0), the acceleration is up to order c−2 identical to GR. The acceleration of a test particle for V ≠ 0 is given; this makes it possible to fit observations in celestial mechanics to ephemerides with V as a free parameter. The current status of such fits (although to ephemerides and not to observations) is presented and discussed.

According to the "gravitationally-modified" Maxwell equations that were proposed for an alternative scalar theory with an "ether", electric charge would not be conserved in a time-dependent gravitational field. We define an asymptotic expansion scheme for the electromagnetic field in a weak gravitational field. This allows us to assess the amounts of charge production or destruction which are thus predicted. These amounts seem high enough to discard that version of the gravitationally-modified Maxwell equations. We show that this failure is due to the former assumption of additivity of the energy tensors: an "interaction energy tensor" has to be added. Then the standard Maxwell equations in a curved spacetime become compatible with that scalar theory, and they predict charge conservation.

In this paper the independent equations of continuum electrodynamics and their quantity are investigated, beginning with the standard equations used in special and general relativity. Using differential identities it is checked that there are as many independent equations as there are unknowns, for the case with given sources as well as for the general case where the motion of the charged medium producing the field is unknown. This problem is then discussed in an alternative theory of gravity with a preferred reference frame, in order to constrain an additional, "interaction" energy tensor that has to be postulated in this theory, and that would be present also outside usual matter. In order that the interaction tensor be Lorentz-invariant in special relativity, it has to depend only on a scalar field p. Since the system of electrodynamics is closed in the absence of the interaction tensor, just one scalar equation more is needed to close it again in the presence of p. That equation is taken to be the equation for charge conservation. Finally, the equations that allow the determination of field p are derived in a given weak gravitational field and in a given electromagnetic field.

We prove that, up to multiplication by a scalar, the Minkowski metric tensor is the only second-order tensor that is Lorentz-invariant.To prove this, we show that a specific set of three 4 × 4 matrices, made of two rotation matrices plus a Lorentz boost, is irreducible.

Two questions are investigated by looking successively at classical mechanics, special relativity, and relativistic gravity: first, how is space related with spacetime? Our answer is that each given reference fluid, that is a congruence of reference trajectories, defines a physical space. The points of that space are formally defined to be the world lines of the congruence. The second question is: does the geometric structure of the spacetime determine the physics, in particular, does it determine its relativistic or preferred-frame character? We find that it does not.

A modification of the Maxwell equations due to the presence of a gravitational field was formerly proposed for a scalar theory with a preferred reference frame. With this modification, the electric charge is not conserved. The aim of the present work was to numerically assess the amount of charge production or destruction. We propose an asymptotic scheme for the electromagnetic field in a weak and slowly varying gravitational field. This scheme is valid independently of the theory and the “gravitationally-modified” Maxwell equations. Then we apply this scheme to plane waves and to a group of Hertzian dipoles in the scalar ether theory. The predicted amounts of charge production/destruction discard the formerly proposed gravitationally-modified Maxwell equations. The theoretical reason for that is the assumption that the total energy tensor is the sum of the energy tensor of the medium producing the electromagnetic (e.m.) field and the e.m. energy tensor. This means that an additional, “interaction” tensor has to be present. With this assumption, the standard Maxwell equations in a curved spacetime, which predict charge conservation, are compatible with the investigated theory. We find that the interaction energy might contribute to the dark matter.

A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].

An alternative, scalar theory of gravitation has been proposed, based on a mechanism/interpretation of gravity as being a pressure force: Archimedes’ thrust. In it, the gravitational field affects the physical standards of space and time, but motion is governed by an extension of the relativistic form of Newton’s second law. This implies Einstein’s geodesic motion for free particles only in a constant gravitational field. In this work, equations governing the dynamics of a continuous medium subjected to gravitational and non-gravitational forces are derived. Then, the case where the non-gravitational force is the Lorentz force is investigated. The gravitational modification of Maxwell’s equations is obtained under the requirement that a charged continuous medium, subjected to the Lorentz force, obeys the equation derived for continuum dynamics under external forces. These Maxwell equations are shown to be consistent with the dynamics of a “free” photon, and thus with the geometrical optics of this theory. However, these equations do not imply local charge conservation, except for a constant gravitational field.

We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame and that, however, they can be given a rigorous meaning. Then, we review the definitions of the canonical and Hilbert energy-momentum tensors from a Lagrangian through the principle of stationary action in general space-time. Using relatively elementary mathematics, we prove precise results regarding the definition of the Hilbert tensor field, its uniqueness, and its tensoriality. We recall the meaning of its covariant conservation equation. We end with a proof of uniqueness of the energy density and flux, when both depend polynomially on the fields.