Mayeul Arminjon

On the equations of electrodynamics in a flat or a curved spacetime and a possible interaction energy

Keywords: Maxwell equations, special relativity, general relativity, alternative theory of gravitation, preferred reference frame PACS: 04.25.Nx, 04.40.-b, 04.50.-h, 98.58.Ay

On the equations of electrodynamics in a flat or a curved spacetime and a possible interaction energy

Mayeul Arminjon

Introduction and summary

The usual approach to classical electrodynamics considers, on one hand, a system of point charges, and on the other hand, the electromagnetic (e.m.) field that both is produced by them and acts upon them [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF][START_REF] Jackson | Classical electrodynamics[END_REF]. The ideal notion of point charge leads to well-known difficulties such as self-force, infinite energy, run-away solutions,... , but actually in the Maxwell equations the charge and current distribution is a continuous one, or at least is preferably so. From the definitions of the "free" densities of electric charge and current [START_REF] Jackson | Classical electrodynamics[END_REF], it certainly follows that the distribution should be continuous for the macroscopic Maxwell equations (the ones that contain the displacement field D and the magnetizing field H in addition to the macroscopic electric field E and the macroscopic magnetic field B). Now, for a linear and isotropic response, the macroscopic equations have the same form as the "microscopic" Maxwell equations (i.e. the most standard ones, with only the microscopic electric field E and the microscopic magnetic field B) [START_REF] Jackson | Classical electrodynamics[END_REF]. Hence, we may and will study the Maxwell equations without D and without H, for a continuous charged medium and its e.m. field. This applies to a macroscopic, practical situation, if one considers then the particular case of a linear and isotropic medium. It applies also to the general situation at a "microscopic but still classical" scale, if one considers the total microscopic charge density and the total microscopic current density, without doing any distinction between "free" and "bound" charges. However, in the case that the source is not considered given, one has to assume that the velocity of the electric current is the velocity of the charged medium, see Eqs. [START_REF] Arminjon | Lorentz-invariant second-order tensors and an irreducible set of matrices[END_REF] and (23) below.

The initial problem that we were investigating in this work is the development of continuum electrodynamics in an alternative, scalar theory of gravitation with a preferred reference frame or "ether", in short the scalar ether theory or SET. An alternative extension of Maxwell's second group to the situation with gravitation, consistent with SET, had been proposed in a previous work [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]. That alternative second group predicted charge nonconservation in a variable gravitational field. However, in a later work [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], it has been found that the charge production/destruction thus predicted seems much too high. It has also been found [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] that the cause for this failure is that in SET one should not assume that the total energy(-momentum-stress) tensor T (whose T 00 component in the preferred frame is the source of the gravitational field [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF]) is the sum of the energy tensors of the charged medium and the e.m. field: T = T chg + T field . The main aim of the present work was therefore to derive explicit equations that should allow one later to calculate the "interaction energy tensor"

T inter := T -T chg -T field (1)
in relevant situations. This is interesting not only for SET in itself, but also because it might be the case that the interaction energy contribute to the "dark matter" [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. To assess T inter , we must precise the constraints that are imposed on it, hence we must investigate in some detail which are the independent equations of continuum electrodynamics and what is their number.

It is instructive and in fact necessary for our purpose to study that problem first for the case of special relativity (SR) and general relativity (GR). This will be the task pursued in Section 2. We shall show there that the notion of differential identity allows one to explain simply why the eight components of the standard Maxwell equations are needed to determine the six components of E and B, when the 4-current is given; and this same notion allows us to show also that the more complete system obtained by adding the equation of motion of the charged continuum is closed. (For that system the 4-current is an unknown.)

The rest of the paper is devoted to the electrodynamics of SET and to the equations for the interaction tensor. In Section 3 we summarize the situation [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF] when the "additivity assumption" (equivalent to T inter = 0 in (1)) is made, and, using the same method as in Sect. 2, we find that also here the equations form a closed system of PDE's. Section 4 briefly exposes the reasons [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] that enforce us to abandon this assumption. Then Sect. 5 studies the constraints that are imposed on T inter . The demand that it be Lorentz-invariant in SR leads us to the simple form (43), depending on a scalar field p. Just one scalar equation is then lacking in the electrodynamics of SET, due to the introduction of just one new unknown p. It follows therefrom that Maxwell's second group cannot be the same in "SET with p" as it is in GR and in the other metric theories of gravitation. In Sect. 6, we consistently close the system of electrodynamics in "SET with p" by adding the charge conservation. We show that the field p is constant (and arguably zero) for a gravitational field that is constant in the preferred reference frame assumed by that theory. Hence, in particular, p is constant, and arguably zero, in SR with this system of equations. Finally, Sect. 7 establishes the explicit equation that determines the field p in a weak and slowly varying gravitational field, Eq. ( 75), and proposes an integration procedure to get the numerical values of p.

Independent equations in standard theory

By "standard theory", we mean the electrodynamics in GR, or possibly in another metric theory of gravity. This includes SR as the case that the spacetime metric is Minkowski's. In any such theory, the dynamical equation verified by the total energy(-momentum-stress) tensor T of matter and nongravitational fields is

T µν ;ν = 0; (2)
and, more generally, there is a rule to go from any equation valid in special relativity in Cartesian coordinates to an equation valid with a general Lorentzian metric in general coordinates: "comma goes to semicolon" [START_REF] Misner | Gravitation, Freeman[END_REF], i.e., partial derivatives have to be replaced by covariant derivatives based on the connection associated with the spacetime metric γ. (This rule becomes ambiguous in cases with derivatives of order larger than one, but one may avoid this in the case of the Maxwell equations: they involve only the e.m. field tensor F , not the e.m. 4-potential A.) That rule does lead to the standard equations for the Maxwell field in a curved spacetime: the first group,

F λµ , ν + F µν , λ + F νλ , µ = F λµ ; ν + F µν ; λ + F νλ ; µ = 0 (3)
(the first equality is an identity due to the antisymmetry of F (F µν = -F νµ ) and to the symmetry of the metric connection), and the second group,1

F µν ;ν = -µ 0 J µ . (4)
(Here J µ (µ = 0, ..., 3) are the components of the e.m. 4-current J and µ 0 is the permeability of free space.) How many independent unknowns do we have in standard theory, and how many independent equations? First, since the first group can be written as

M λµν := F λµ ; ν + F µν ; λ + F νλ ; µ = 0, (5)
and since the l.h.s. is totally antisymmetric, i.e., M λµν = -M µλν = -M λνµ , it is usual to note that (3) contains exactly four linearly-independent equations: for example, M 012 = 0, M 013 = 0, M 023 = 0, M 123 = 0. Also, the four equations in the second group (4) are linearly independent.

Case with given 4-current

Most often in the discussion of the solutions to the Maxwell equations, the 4current J is considered given. Then we have the eight Maxwell equations ( 3)-(4) for the six independent unknowns F µν (0 ≤ µ < ν ≤ 3). It is nevertheless well known that those eight equations are needed, i.e., one cannot consistently remove two of them from the whole. In particular, one cannot consistently remove the two divergence equations div B = 0 and div E = ρ el / 0 from the usual three-vector form of the flat-spacetime Maxwell equations [START_REF] Jiang | The origin of spurious solutions in computational electromagnetics[END_REF][START_REF] Zhou | On independence, completeness of Maxwell's equations and uniqueness theorems in electromagnetics[END_REF][START_REF] Liu | Explanation on overdetermination of Maxwell's equations[END_REF]]. 2Regarding this point, it is observed [START_REF] Jiang | The origin of spurious solutions in computational electromagnetics[END_REF] that the Maxwell equations form two pairs of "div-curl" systems of equations, a div-curl system consisting of four scalar equations for three scalar unknowns u i (i = 1, 2, 3):

div u = f, rot u = s. (6)
Starting from this observation, the following has been shown in a detailed mathematical work [START_REF] Jiang | The origin of spurious solutions in computational electromagnetics[END_REF]: (i) There is a theorem stating uniqueness of the solutions u to the div-curl system, modulo suitable boundary conditions, and provided the data s verifies the compatibility condition div s = 0 [START_REF] Jiang | The origin of spurious solutions in computational electromagnetics[END_REF] (which follows from applying the differential identity div rot u ≡ 0 to Eq. (6) 2 ). (ii) Removing the divergence equation in one div-curl system leads to a system whose solutions are not necessarily solutions of that div-curl system.

However, in our opinion, the fact that the div-curl system, or respectively the Maxwell equations, are not overdetermined, is more easily understood from the observation that these systems of PDE's verify definite differential identities. This notion (without its name) is recognized by Liu [START_REF] Liu | Explanation on overdetermination of Maxwell's equations[END_REF] as relevant, see his Definition II, but unfortunately he does not indicate which differential identities do apply to the div-curl system [START_REF] Misner | Gravitation, Freeman[END_REF] or respectively to the Maxwell equations. The identity div rot u ≡ 0, which he mentions, is not a differential identity of the div-curl system [START_REF] Misner | Gravitation, Freeman[END_REF], because rot u = 0 does not belong to the equations of that system (unless s ≡ 0). Also, Eq. ( 7) is obviously not a differential identity, i.e., a partial differential equation that is satisfied by any regular vector function s. What happens is that, if Eq. ( 7) is satisfied by some particular function s, then the following is indeed a differential identity, i.e., it is valid for any regular vector function u:

div (rot u -s) ≡ 0. ( 8
)
Moreover, this is a differential identity of the system (6), i.e., it has the form

n k=1 O k P k u ≡ 0, (9)
where P k u = 0 (k = 1, ..., n, thus n = 4 for the system (6)) are the scalar PDE's of the system, and where O k are scalar differential operators of the first order, or the zero operator. Indeed some among the O k 's, but not all, can be the zero operator, meaning that the corresponding equation P k u = 0 is not involved in the differential identity. (Also, u designates in general the list of unknown functions in the system.) We insist that, by definition, a differential identity like (9) has to apply whether or not any of the equations of the system (P k u = 0) is satisfied. Thus, in view of the scalar identity [START_REF] Zhou | On independence, completeness of Maxwell's equations and uniqueness theorems in electromagnetics[END_REF], it is clear that the system (6) has only three independent equations instead of four. But since it is a differential identity, not an algebraic one, and since that identity involves not merely one but several equations among the scalar PDE's of the system, it is quite clear also that we can't remove any of the scalar PDE's of the system without altering it. In particular, note that (under the validity of the compatibility condition ( 7)), the differential identity ( 8) is already a differential identity of the "curl system" (6) 2 alone, which is hence an underdetermined system: 3 -1 = 2 independent equations for 3 unknowns. It thus becomes obvious that, if one removes the divergence equation ( 6) 1 , one will indeed be able to find "spurious" solutions, i.e., undesired ones.

Consider now the flat-spacetime Maxwell system in vacuo: Eqs. ( 4)-( 5) with commas instead of semicolons (in Cartesian coordinates for the Minkowski metric), and with J µ = 0. As noted by Das [START_REF] Das | The special theory of relativity -a mathematical exposition[END_REF], that system has the two differential identities

S := ε µνρσ M µνρ,σ ≡ 0 (10)
(where ε µνρσ is the signature of the permutation (µνρσ) of {0, ..., 3}), and

F µν ,ν,µ ≡ 0, (11)
which both result from the antisymmetry of the tensor F (after a short algebra, for [START_REF] Das | The special theory of relativity -a mathematical exposition[END_REF]). The first identity is a differential identity of the first Maxwell group ( 5), the second one is a differential identity of the second Maxwell group in vacuo. Thus there are six independent equations for the six unknowns.

Finally, consider the Maxwell system in a general spacetime ( 4)-( 5). In a general coordinate system in a general Lorentzian spacetime, we have similarly with ( 10) and ( 11) the two independent differential identities

e µνρσ M µνρ;σ ≡ 0 (12)
(where e µνρσ is the usual totally antisymmetric tensor that coincides with ε µνρσ in coordinates such that the natural basis is direct and that γ := det(γ µν ) = -1), and

F µν ;ν;µ ≡ 0. ( 13
)
These identities indeed apply (thus whether or not Eqs. ( 4) or ( 5) are verified), because the l.h.s. of either [START_REF] Arminjon | On continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF] or ( 13) is a manifestly invariant scalar, and it is zero (as for [START_REF] Das | The special theory of relativity -a mathematical exposition[END_REF] or [START_REF] Cattaneo | General relativity: relative standard mass, momentum, energy and gravitational field in a general system of reference[END_REF]) in coordinates such that, at the event considered, the Christoffel symbols vanish and the matrix (

γ µν ) = (η µν ) := diag(1, -1, -1, - 1
). The first one, Eq. ( 12), is clearly a differential identity of the first Maxwell group [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF]. The identity [START_REF] Fock | The theory of space, time and gravitation[END_REF] implies that the second Maxwell group (4) has charge conservation as a compatibility condition:

J µ ;µ = 0. ( 14
)
Hence, it is here as with the div-curl system (6): if the compatibility condition ( 14) is satisfied, as it should, then we get from (13) the following differential identity of the second Maxwell group (4):

S := F µν ;ν + µ 0 J µ ;µ ≡ 0. ( 15
)
Thus, here also, we have six independent equations for the six unknowns.

It may be useful to show how the identities ( 12) and ( 15) appear with the usual 3-vector form of the Maxwell equations for a flat spacetime. Using Eqs. (24.13) and (24.14) of Fock [START_REF] Fock | The theory of space, time and gravitation[END_REF], it is easy to check that the l.h.s. of the identity (10) can be rewritten as

S = 6 div rot E + ∂B ∂t - ∂ ∂t (div B) . (16)
In a flat spacetime, the identity S ≡ 0 is thus as well a differential identity of the usual 3-vector form (e.g. Eqs. ( 1) and ( 4) in Ref. [START_REF] Liu | Explanation on overdetermination of Maxwell's equations[END_REF]) of Maxwell's first group. Using µ 0 0 = 1/c 2 , it is also easy to check that, in a flat spacetime, the identity (15) rewrites as the following differential identity of the usual form of Maxwell's second group:

-S = div rot B -µ 0 j + 0 ∂E ∂t + µ 0 0 ∂ ∂t div E - ρ el 0 ≡ 0. ( 17
)
This identity is satisfied provided that the integrability condition of Maxwell's second group: Eq. ( 14) or equivalently ∂ρ el ∂t + div j = 0, is satisfied (here j is the 3-vector with components j i := J i ). Of course, as for S ≡ 0, the identity (17) applies whether or not any of the Maxwell equations is satisfied.

Complete system for a deformable charged medium

Often, in practice, the 4-current density J can be considered given (at least as an approximation), as we just envisaged. This is not the general case, however. A continuous charged medium is subjected to the Lorentz 4-force (density)

f µ := F µ ν J ν , (18)
which tends to modify the (3-)velocity field v of the charged medium (especially if the latter is deformable, as is the case e.g. in magnetohydrodynamics), and hence also to modify the current J as well. Indeed, in any coordinates x µ , the components of the latter 4-vector are defined as ( [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF], Eq. (90.3)):

J µ := ρ el β dx µ dt , (19)
where

β := √ γ 00 , (20)
and t := x 0 /c. The velocity field v (whose value at an event X is the velocity of that infinitesimal volume element of the continuous medium which is at X) is measured with the local standards, and has thus components v i = dx i /dt x where t x is the synchronized local time in the reference fluid F that is considered [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF][START_REF] Cattaneo | General relativity: relative standard mass, momentum, energy and gravitational field in a general system of reference[END_REF][START_REF] Arminjon | On continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]. (Note that the data of one coordinate system (x µ ) automatically defines a unique reference fluid, whose reference world lines are the lines x := (x i ) = Constant [START_REF] Arminjon | On continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF].) Assuming that F admits adapted coordinates that verify the synchronization condition

γ 0i = 0 (21)
(and using any such coordinates), we have

dt x dt = β(t, x), (22)
so that J i = ρ el v i , or j = ρ el v (23)
for the 3-current, a spatial vector field in the reference fluid F. Moreover, a deformable charged medium has state parameters, at least the proper restmass density field ρ * or (equivalently for a barotropic perfect fluid) the pressure field P .

In the general case, we thus add to the six field unknowns F µν (0 ≤ µ < ν ≤ 3), the field unknowns ρ el and v, plus the other state parameters, say merely the field ρ * for the simplicity of discussion -thus we have five unknowns more, that is 6 + 5 = 11 unknowns. The equation we have in addition to the Maxwell equations ( 4)-( 5) is the dynamical equation of GR for the charged medium subjected to the 4-force (18):

T µν chg ;ν = F µ λ J λ , ( 24
)
where T chg is the energy tensor of the deformable charged medium, depending on v and ρ * . Equation (24) extends a standard equation valid in special relativity (e.g. [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF], Eq. (33.9)), by using the "comma goes to semicolon" rule. It can also be derived from the dynamical equation for the total energy tensor (2), Maxwell's second group (4), and the assumption of additivity of the energy tensors:

T = T chg + T field , (25)
with T field the energy tensor of the e.m. field [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF][START_REF] Fock | The theory of space, time and gravitation[END_REF]:

T µν field := -F µ λ F νλ + 1 4 γ µν F λρ F λρ /µ 0 . (26)
Indeed, by using the identity [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF][START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]

µ 0 T µν field ;ν ≡ -F µ λ F νλ ;ν , (27)
one deduces from the second group (4):

T µν field ;ν = -F µ λ J λ . ( 28
)
Equations ( 2) and (25) then imply immediately (24). Thus, we have five unknowns more than in the case with given 4-current, and merely the four equations (24) more, in addition to (4) and ( 5). 3 However, since in the system [( 4)-( 5), (24)] the current J is unknown as well as is the e.m. field F , Eq. ( 15) is not a differential identity of that system, even though it does have the form (9) (with u denoting the set of all unknowns of the system: now J, F , and ρ * , or equivalently ρ el , v, F , and ρ * ). This is because [START_REF] Arminjon | On the extension of Newton's second law to theories of gravitation in curved space-time[END_REF], as well as the charge conservation [START_REF] Chandrasekhar | Conservation laws in general relativity and in the post-Newtonian approximations[END_REF], apply merely on the solution space of the system. (Both indeed apply there, due to Maxwell's second group (4) and to the identity (13).) Whereas, a differential identity of the system has to be valid for any regular field unknowns J, F , and ρ * . Note also that the identity ( 13) is not a differential identity of the system either, because it does not have the form [START_REF] Liu | Explanation on overdetermination of Maxwell's equations[END_REF]. Thus, for the system [( 4)-( 5), (24)], we have merely the identity ( 12) that counts as a dependence relation. So we have 12 -1 = 11 equations for the 11 unknowns.

Note that the foregoing applies in SR as well as in a metric theory of gravitation. However, in the case with gravitation, strictly speaking, the metric field γ should not be considered given, e.g. because the e.m. energy tensor does contribute to the total energy tensor and hence influences the metric nonlinearly through the Einstein equations (considering GR for definiteness) -even if this influence is very small in usual conditions, e.g. in the solar system. If we account for this, we thus add the ten unknowns γ µν (0 ≤ µ ≤ ν ≤ 3). And we add the ten Einstein equations (of which only six count as independent due to the four differential identities of Bianchi), plus the four equations of the gauge condition which is selected.

Independent equations in SET without interaction tensor

In SET, motion is governed by an extension of the special-relativistic form of Newton's second law to a curved spacetime, written in the preferred reference fluid E assumed by the theory. This extension, which is formulated primarily for a test particle [START_REF] Arminjon | On the extension of Newton's second law to theories of gravitation in curved space-time[END_REF], can be applied to each particle of a dust, that is an ideal continuum made of a myriad of coherently moving test particles. This leads [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF] to the following dynamical equation in the presence of a field of external (3-)force having volume density f with components f i (i = 1, 2, 3):

T µν medium ;ν = b µ (T medium ) + f µ , f 0 := f .v cβ , ( 29
)
where β is defined in Eq. (20), v is the velocity field (with the local time, see after that equation), T medium is the energy-momentum tensor of the continuous medium, and

b 0 (T ) := 1 2 γ 00 g ij,0 T ij , b i (T ) := 1 2 g ij g jk,0 T 0k . ( 30
)
Here g ij are the components of the spatial metric tensor g = g E associated with the spacetime metric γ in the reference fluid E [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF][START_REF] Møller | The theory of relativity[END_REF][START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]. Equation ( 29) is then assumed to be valid for any continuous medium [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF], provided a velocity field can be unambiguously be defined for that medium. In the case that the continuous medium is a charged medium and f is the Lorentz (3-)force, the logic of the theory leads one to define its components f i precisely as the spatial components µ = i in Eq. ( 18) above [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]. Moreover, the f 0 component is thus derived (at least for a dust) to be given by (29) 2 here, and this turns out to be also equal to the µ = 0 component in Eq. ( 18) [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF] although the dynamics is different from that of GR. So for a charged medium the dynamical equation writes:

T µν chg ;ν = b µ (T chg ) + F µ ν J ν . (31)
The dynamical equation for the total energy tensor T is also obtained by induction from what is got for a dust, this time without any non-gravitational external force, and is thus [START_REF] Arminjon | On the possibility of matter creation/destruction in a variable gravitational field[END_REF]:

T µν ;ν = b µ (T ). ( 32
)
It seems natural at first sight to assume the additivity of the energy tensors, Eq. ( 25). It turns out that, together with the dynamical equations for the charged medium and the total energy tensor, Eqs. ( 31) and (32), this determines Maxwell's second group. This is already true for GR, as one can see by reverting the line of reasoning in Eqs. ( 27) and (28) above: starting from the dynamical equations for the charged medium and the total energy tensor in GR, Eqs. ( 24) and (2), and using the additivity (25), we obtain immediately Eq. (28). Equating with the identity (27) gives us

-F µ λ F νλ ;ν = -µ 0 F µ λ J λ , (33)
which, at least for the (generic) case of an invertible matrix (F µ λ ), is equivalent to the second group of GR, Eq. ( 4). For SET, this same argument, with Eqs. (31) and (32) replacing Eqs. ( 24) and (2) of GR, leads first to

T µν field ;ν = b µ (T field ) -F µ ν J ν , (34)
from which one gets by (27) [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]:

F µ λ F λν ;ν = µ 0 b µ (T field ) -F µ λ J λ , (35)
which is the second group got in SET when one assumes the additivity (25).

The foregoing implies that, for SET with the additivity assumption (25), we can take as the system of equations governing the motion of a charged medium and its e.m. field: Maxwell's first group [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF], plus the dynamical equations for the charged medium and the total energy tensor, Eqs. ( 31) and (32) -or, equivalently, Maxwell's first group (5), plus Eqs. (31) and (35). The unknown fields are the same as for GR: F µν (0 ≤ µ < ν ≤ 3), ρ el and v, and ρ * , or equivalently F µν (0 ≤ µ < ν ≤ 3), J, and ρ * -in any case 11 unknowns. And as in GR, we have merely the identity (12) that counts as a dependence relation. So again we have 12 -1 = 11 equations for the 11 unknowns. If we account for the fact that the gravitational field is not given, we add the scalar gravitational field unknown, ψ := -Log β, and we add the scalar flat-spacetime wave equation obeyed by ψ according to that theory [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF].

Interaction energy tensor in SET

There are three reasons [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] why it turns out to be not satisfying to close the electrodynamics for SET by stating the additivity assumption (25):

(i) The version (35) of Maxwell's second group leads to the non-conservation of charge in a variable gravitational field. (This is seen by using the identity (13) [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF].) It happens moreover that the amounts of electric charge which are thus predicted to be produced or destroyed in a realistic e.m. and gravitational field, seem much too high to be a tenable prediction.

(ii) In general, the energy tensors of the charged medium and the e.m. field are both non-zero. This means that we are in the presence of a mixture. According to the standard theory of mixtures (which has been developed for non-relativistic physics), the effective energy tensor of the mixture is not the sum of the energy tensors of its constituents [START_REF] Müller | A thermodynamic theory of mixtures of fluids[END_REF]. This is not compatible with the additivity assumption (25).

(iii) Equation (34) means that the e.m. field continuum verifies the dynamical equation (29) for a continuous medium with an external force field f field acting on it, with, specifically:

f µ field = -f µ := -f µ chg := -F µ ν J ν . ( 36
)
Using the spatial part of this (µ = 1, 2, 3), and noting respectively v field and v chg the velocity fields of the field continuum and the charged continuum (assuming v field is well defined), the "time" part rewrites as [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]:

f chg . (v field -v chg ) = 0. ( 37
)
However, for a general e.m. field, it is not easy to tell how to define its velocity field v field , be it experimentally or in terms of the energy tensor T field . Anyway, Eq. ( 37) looks problematic for a general e.m. field. For a "null" field (i.e., such that the classical invariants are both zero), the velocity field can be defined naturally in terms of the energy tensor T field [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], and it has modulus v field := (g(v field , v) field ) 1/2 = c. Furthermore, one can then prove [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] that indeed (37) is satisfied (although usually v chg c), but this proof depends heavily on the fact the e.m. field is a null field.

Due to these reasons, we have to abandon the additivity assumption (25). This means that the total tensor T verifying Eq. ( 32) involves another part, say T inter :

T = T chg + T field + T inter . ( 38
)
While introducing that "interaction tensor" T inter , we almost necessarily add new unknowns -unless all state parameters for T inter would be extracted from those of T 1 := T chg and T 2 := T field . 4 Hence, the system [( 5), ( 31), (32)] made of Maxwell's first group and the dynamical equations for the charged medium and for the total energy tensor, cannot be closed any more. So we have to add at least one equation.

Constraints on the interaction tensor

The effect of the gravitational field on the e.m. field can usually be neglected, e.g. because the gravitational force on a charged particle is extremely small as compared with the Lorentz force. When this approximation is done, the e.m. field is as in special relativity (SR), i.e. in a flat Minkowski spacetime.

There is a massive experimental evidence for the predictions deduced from the usual (flat-spacetime) Maxwell equations. Therefore, we ask that the electrodynamics of SET should exactly reduce to that of SR in the absence of gravitation, i.e., when the scalar gravitational field β ≡ 1 so that, accordingly [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF], the physical spacetime metric γ is the Minkowski metric. (See Eq. ( 45) 4 For instance, the standard theory of mixtures makes a definite suggestion to define T in terms of T 1 , T 2 , and the velocity fields v 1 and v 2 : Eq. (2.13) in Ref. [START_REF] Müller | A thermodynamic theory of mixtures of fluids[END_REF]. This suggestion can be extended naturally to SET, by starting from Eq. (39) in Ref. [START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF] (valid for a dust), but this definition for T does not verify Eq. (32). Anyway, an effective energy tensor defined in such a way would not reduce to the sum (25) in special relativity, as is required (see Sect. 5).

below.) In special relativity, the dynamical equation for the charged medium valid in GR, Eq. ( 24), can be derived directly from the Lorentz 4-force (e.g. [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF], Eq. (33.9)), while the dynamical equation for the e.m. field valid in GR, Eq. ( 28), is derived essentially as it was derived hereabove for GR, i.e. from the identity (27) and Maxwell's second group (4) (though in a simpler way for SR, with commas instead of semicolons in Cartesian coordinates; see e.g. Eq. (33.7) in Ref. [START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF]). Hence we have in SR as well as in GR:

T µν chg ;ν + T µν field ;ν = 0. ( 39
)
Again for SR and GR, this is compatible with the additivity assumption (25): of course, the latter plus the dynamical equation ( 2) for the total energy tensor imply (39). Conversely, if we start from the fully general decomposition (38) of the total tensor, then under the validity of Eq. ( 2), we get that (39) is equivalent to

T µν inter ;ν = 0. (40)
We note that Eq. ( 2) does apply in SET for a constant gravitational field: the latter means β ,0 = 0, hence g ij,0 = 0 in (30) (see Eq. (45) below), whence b µ = 0 in (32). Hence, in particular, Eq. ( 2) does apply in SET in the absence of gravitation, i.e., when β ≡ 1. Thus, if we do recover Eq. (39) of SR from SET in the absence of gravitation, then Eq. (40) will apply in that situation. Below, just to check that form of the interaction energy tensor that, we will show independently, is relevant, we therefore assume in advance that Eq. (40) applies to SET in the absence of gravitation, and we will show in Sect. 6 that this is indeed the case with the precise framework that shall be adopted.

In view of the foregoing, it is not a priori obvious that we may impose the condition that the interaction tensor T inter should vanish in SR. However, we may impose that it should be Lorentz-invariant in SR. Any Lorentz-invariant second-order tensor is a scalar multiple of the Minkowski metric tensor, say γ 0 [START_REF] Arminjon | Lorentz-invariant second-order tensors and an irreducible set of matrices[END_REF]. Therefore, T inter should have the form (in Cartesian coordinates for the Minkowski metric, i.e., (γ 0 ) µν = η µν ):

T inter µν = p η µν (SR), (41)
with some scalar field p. This defines of course a Lorentz-invariant tensor field. Note that (41) is equivalent to:

T µ inter ν := η µρ p η ρν = p δ µ ν (SR) (42)
(again in Cartesian coordinates). Now we observe that the definition

T µ inter ν := p δ µ ν , (43)
thus got in Cartesian coordinates in a Minkowski spacetime, is actually generally-covariant: if (43) applies in some coordinates x µ in a general spacetime with metric γ, it still applies after any coordinate change. Therefore, we adopt for the general case the definition (43), which has been got by demanding that T inter be Lorentz-invariant in SR. When gravitation is absent i.e. when the metric turns out to be Minkowski's, we should have Eq. ( 40), and this implies p = Constant, say p ≡ p ∞ . Indeed, if the metric is Minkowski's, Eqs. ( 40) and (41) imply that p ,µ = 0. Thus by requiring only that T inter should be Lorentz-invariant in SR, we get that it is then actually constant. Moreover, it is natural to assume that, very far from any body, the total energy tensor T , as well as T chg and T field , are zero. That assumption implies that the constant p ∞ is zero, and hence that the additivity condition (25) applies in SR. We note also that, with the definition (43), we have just one unknown more (p) in the system of electrodynamics of SET [( 5), (

], see the end of Sect. 3. So we have to find just one scalar equation more. Thus we cannot add the standard Maxwell second group (4), since it would add four independent scalar equations. We have to show this more clearly, because it is an important point.

Once an interaction tensor is introduced through Eq. (38), we may a priori postulate the standard Maxwell second group (4) in addition to the system of electrodynamics of SET [( 5), (31), (32)] [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. This leads to (Eq. (104) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]):

T µν inter ;ν -b µ (T inter ) = b µ (T field ). (44)
Let us compute the l.h.s. (this will be used also in the next section). Until the end of this section, and again in Sect. 7, we will use coordinates x µ that are adapted to the preferred reference fluid E and, more specifically, we will assume that the spatial coordinates are Cartesian for the Euclidean spatial metric g 0 assumed in the theory (g 0 is time-independent in any coordinates that are adapted to the preferred frame E), and x 0 = cT with T the preferred time of the theory. In such coordinates, the curved "physical" spacetime metric γ is by assumption [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF]:

ds 2 = γ µν dx µ dx ν = β 2 (dx 0 ) 2 -g ij dx i dx j = β 2 (dx 0 ) 2 -β -2 dx i dx i . ( 45
)
This implies that, in such coordinates, we have

γ := det (γ µν ) = -β -4 . ( 46
)
With the help of an identity for T ν µ ;ν ([1], Eq. (86.11)) and the definition (30) of b µ , we obtain thus (for whatever symmetric tensor T µν ):

T ν 0 ;ν -b 0 (T ) ≡ β 2 T 00 ,0 + T 0j ,j -ββ ,0 T 00 , (47)
T ν i ;ν -b i (T ) ≡ β 2 T ν i β 2 ,ν -β -3 β ,0 T 0i -β -3 β ,i T jj -ββ ,i T 00 . ( 48
)
With the definition just adopted (43) for T inter , we get from this: (26) and (30).) Hence, Eqs. ( 49) and (50) imply that it cannot in general exist a scalar field p that verify the four equations (52): already the three spatial equations (µ = i = 1, 2, 3) imply that rot (b i ) = 0, which is not true in general.

δ 0 = p ,0 -3 p β ,0 β -1 , (49)
δ i = p ,i , (50)
Thus, postulating the validity of the standard Maxwell second group (4) would demand to have a more general form than (43) for the interaction tensor T inter , which instead should depend on four scalar fields. In that case, the interaction tensor would not be Lorentz-invariant in SR any more. It is hard to say in advance whether or not it would then be possible to get the constancy of T inter in SR -which applies to the single-scalar form (43) as we will show in the next section. of Sect. 3, the fact that the scalar gravitational field β is also unknown can be accounted for by adding the scalar flat-spacetime wave equation for ψ := -Log β.) Now consider the case of a constant gravitational field, which includes the situation without any gravitational field. In that case, the general dynamical equation of GR [START_REF] Jackson | Classical electrodynamics[END_REF] applies to SET as noted after Eq. ( 40), so that we get from the general decomposition (38) and the definition (43) of the interaction tensor:

T µν ;ν = 0 = T µν field ;ν + T µν chg ;ν + T µν inter ;ν = T µν field ;ν + T µν chg ;ν + p ,ν γ µν . (58)
Hence, the system [( 5), ( 31), ( 55), (57)] is solved by p = Constant := p ∞ and with the fields F , J, ρ * being the solution of the system [( 4)-( 5), (24)] for the given time-independent metric and for the relevant boundary conditions. Therefore, we get that Eq. ( 40) is true in a constant gravitational field. In particular, Eq. ( 40) is true when the latter vanishes -as we provisionally assumed after postulating the form (43) of the interaction energy tensor, to check that form. Moreover, as noted after Eq. ( 43), we may assume that the constant p ∞ is zero, hence the additivity (25) of the energy tensors T chg and T field does apply in a constant gravitational field.

If the e.m. field F and the gravitational field β are considered given, then the scalar field p, and hence the interaction energy tensor, may in principle be calculated by solving the equation for charge conservation, Eq. (57). Of course F and β are coupled with the other fields that include precisely p.

7 First approximation of the scalar field in a weak gravitational field

In this section we will obtain equations that should allow one, in a future work, to assess numerically the scalar field p and the interaction energy in a given e.m. field and in a given weak and slowly varying gravitational field.

To this aim, we may use the results of the asymptotic framework developed in some detail in Sects. 4 and 5 of Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. The only change with respect to that former work is the additional term -G µν δ ν (p) in Eq. (55), as compared with Eq. ( 22) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], and correspondingly the additional equation (57).

and the like for T 1 [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. This can be then calculated explicitly: as given by Eqs. ( 53) and (54) of Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF],

ρ = c -3 e i ∂ T U ,i 1 + O c -2 , (67)
where

e i = 1 2 c µ 0 (B 1 E 1 +B 2 E 2 +B 3 E 3 ) × ×   c 2 B 1 3 + B 1 B 2 2 + B 1 B 3 2 + B 1 E 1 2 -B 1 E 2 2 -B 1 E 3 2 + 2 B 2 E 1 E 2 + 2 B 3 E 1 E 3 c 2 B 2 3 + B 2 B 3 2 + B 2 B 1 2 + B 2 E 2 2 -B 2 E 3 2 -B 2 E 1 2 + 2 B 3 E 2 E 3 + 2 B 1 E 2 E 1 c 2 B 3 3 + B 3 B 1 2 + B 3 B 2 2 + B 3 E 3 2 -B 3 E 1 2 -B 3 E 2 2 + 2 B 1 E 3 E 1 + 2 B 2 E 3 E 2   .
(68) In Eq. ( 68), E i := E i and B i := B i are the components of the first approximations of the electric and magnetic fields in the frame E, in coordinates of the class specified after Eq. (44). I.e., E i and B i are extracted (Note 2) from the first approximation F 1 of the e.m. field tensor F , Eq. (66), that obeys the flat-spacetime Maxwell equations [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. Inserting (63) and ( 67) into (57) using u

µ ;µ = (u µ √ -γ) ,µ / √ -γ, with √ -γ = 1 + O(c -2
) owing to (46) and (61), gives us:

(G µν 1 (p 1 ) ,ν ) ,µ = G µν 1 ,µ (p 1 ) ,ν = c -3 e i ∂ T U ,i 1 + O c -2 , (69)
where p 1 := c 2n-5 0 p is the first approximation of p. The first equality is due to the antisymmetry of G µν and G µν 1 . The matrix G := (G µν 1 ) is given explicitly by Eq. (50) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], which can be rewritten as

G = -c E.B H, H :=     0 B 1 B 2 B 3 -B 1 0 -E 3 c E 2 c -B 2 E 3 c 0 -E 1 c -B 3 -E 2 c E 1 c 0     . (70)
It is easy to check that Maxwell's (flat-spacetime) first group, verified by F 1 , implies that

H µν ,µ = 0. ( 71
)
It follows from this and (70) that we have explicitly in (69):

(k ν ) := G µν 1 ,µ = (72) 1 (E.B) 2 ×     -B i c (E.B) ,i E 3 (E.B) ,2 -E 2 (E.B) ,3 + B 1 c (E.B) ,0 E 1 (E.B) ,3 -E 3 (E.B) ,1 + B 2 c (E.B) ,0 E 2 (E.B) ,1 -E 1 (E.B) ,2 + B 3 c (E.B) ,0     , i.e., k 0 = -c (E.B) 2 B.(∇(E.B)), (73)
(k i ) = 1 (E.B) 2 ∂ (E.B) ∂T B -E ∧ (∇(E.B)) . (74)
Equation ( 69) can be rewritten in the form

∂ T p 1 + u j ∂ j p 1 = S, (75)
where

S := c -2 (e i ∂ T U ) ,i k 0 (76)
(no confusion can occur with the sum S in Subsect. 2.1), and

u j := c k j k 0 . (77)
We assume k 0 = 0 in Eq. (72), i.e. B.(∇(E.B)) = 0. Note that E.B = 0 is required since Sect. 6. Remind that here the first-approximation fields E and B are involved, and they obey the flat-spacetime Maxwell equations. Equation ( 75) is an advection equation with a given source S for the unknown field p 1 . This is a hyperbolic PDE whose characteristic curves are the integral curves of the vector field u := (u j ). That is, on the curve C(T 0 , x 0 ) defined by dx dT = u(T, x), x(T 0 ) = x 0 ,

we have from (75):

dp 1 dT = ∂p 1 ∂T + ∂p 1 ∂x j dx j dT = S(T, x). (79)
We note that the field u is given, Eq. (77), i.e., it does not depend on the unknown field p 1 . Therefore, the integral lines (78) are given, too, hence the characteristic curves do not cross. Thus, the solution p 1 is got uniquely by integrating (79):

p 1 (T, x(T )) -p 1 (T 0 , x 0 ) = T T 0 S(t, x(t)) dt, ( 80
)
where T → x(T ) is the solution of (78). If at time T 0 the position x 0 in the frame E is enough distant from material bodies, one may assume that p 1 (T 0 , x 0 ) = 0.

Conclusion

The main results obtained in this paper are the following ones:

1) The structure of classical electrodynamics based on the standard Maxwell equations of special relativity or general relativity has been discussed and it has been shown by using the notion of differential identity that the number of independent scalar PDE's is the same as the number of unknown fields. This applies to both the case with given 4-current and the more general case where one takes into account the equation of motion of the charged medium, the 4-current then belonging to the unknowns. Of course the result for the first case is well known (e.g. [START_REF] Jiang | The origin of spurious solutions in computational electromagnetics[END_REF][START_REF] Zhou | On independence, completeness of Maxwell's equations and uniqueness theorems in electromagnetics[END_REF]), but the explanation by considering the differential identities ( 12) and ( 15) is more straightforward and we could not find it in the literature. For the more general case also, our discussion is based on differential identities: ( 12) is still valid but [START_REF] Arminjon | On the extension of Newton's second law to theories of gravitation in curved space-time[END_REF] holds now only on the solution space; that discussion too seems new.

2) In the investigated theory of gravity ("SET"), with the additivity assumption (25), electrodynamics consists of the system [( 5), (31), (32)], i.e., Maxwell's first group and the dynamical equations for the charged medium and for the total energy tensor. Using the same method as for SR and GR, it has been shown that this is also a closed system of PDE's. While introducing the "interaction energy tensor" T inter by switching to (38), one necessarily introduces new unknowns, so that the foregoing system is not closed any more and one needs new equations. We imposed on T inter that, in SR, it should be a Lorentz-invariant tensor. This determines that in the general case it has the form T µ inter ν := p δ µ ν , with p a scalar field (which, we showed, is constant and even zero in SR). Thus, only one additional equation is needed and this can consistently be imposed to be the charge conservation.

3) Considering a weak and slowly varying gravitational field, we derived the equation that determines the scalar field p, whose knowledge is equivalent to that of the interaction energy tensor: Eq. (75). We indicated how one may in principle compute that field in a given EM field and in a given weak and slowly varying gravitational field, Eq. (80). The interaction energy is gravitationally active, because its density T 00 inter = p γ 00 contributes to the total energy density of matter and non-gravitational fields, T 00 . According to SET [START_REF] Arminjon | Space isotropy and weak equivalence principle in a scalar theory of gravity[END_REF], T 00 is the source of the gravitational field. The interaction energy is not especially localized inside matter, and it has to be present in space as soon as there is matter that is electromagnetically active. It could thus be counted as "dark matter". To learn more, it will be necessary to have recourse to a numerical work.

  whereδ µ := δ µ (p) := T ν inter µ ;ν -b µ (T inter ).(51)Equation (44) is equivalent toδ µ (p) = b µ (T field ) (µ = 0, ..., 3). (52)Clearly, for a general e.m. field, b µ (T field ) can be essentially any 4-vector field. (This can be checked by using the explicit forms of T field and b µ , Eqs.

In this paper, as in the foregoing[START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], we are using the SI units and the (+ ---) signature. (In Refs.[START_REF] Landau | The classical theory of fields, 3rd English edition[END_REF][START_REF] Fock | The theory of space, time and gravitation[END_REF][START_REF] Arminjon | Continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF], the Gauss units were used.) Greek indices go from 0 to 3, Latin ones from 1 to 3. Indices are raised or lowered using the spacetime metric γ.

Here ρ el is the volume density of the electric charge and 0 is the permittivity of the vacuum. With the SI units and the (+ ---) signature, the relation between the field tensor F and the electric and magnetic fields E and B in Cartesian coordinates in a Minkowski spacetime is given on the Wikipedia page "Electromagnetic tensor", or by Eq. (47) of Ref.[START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF].

At least if one assumes a perfect fluid, the mass conservation can be exactly deduced from (24), in nearly the same way as one deduces it from (2)[START_REF] Chandrasekhar | Conservation laws in general relativity and in the post-Newtonian approximations[END_REF] in the case without the e.m. field and current (in both cases, mass conservation applies iff the fluid is isentropic). Hence it is not an additional equation.

The remainder in this expansion was incorrectly written as O c -5 in Eqs. (45) and (54) of Ref.[START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. In fact it is O c n-7 while the main term is ord c n-5 , Eq. (41) of Ref.[START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]. The important point is precisely that the remainder is O c -2 times the main term.

Acknowledgement: I am grateful to Jerzy Kijowski for a discussion on the motion of the sources. Also, a referee asked for more explanation about why the interaction energy could contribute to dark matter.

Charge conservation and determination of the interaction tensor

With the general decomposition (38), the dynamical equation (32) for the total energy tensor in SET is equivalent to:

By using the identity (27) for T µν field ;ν , the dynamical equation (31) for the charged medium, and the definition (51), this rewrites as

If the matrix (F µ λ ) is invertible, which is the generic situation and is equivalent to E.B = 0 [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], this can still be rewritten as

where (G µ ν ) is the inverse matrix of matrix (F µ ν ). By using the identity (13), we get from this:

Apart from the identities (27) and ( 13) (which are valid independently of any physical theory), the validity of Eqs. ( 55) and (56) depends only on the validity of Eqs. ( 32) and (31), accounting for the general decomposition (38) with the interaction energy tensor (43). In Sect. 5, we showed that, to close the system of electrodynamics of SET [( 5), (31), (32)] in the presence of the interaction energy tensor (43), we need just one scalar equation more. Therefore, it suggests itself to close the system by adding the conservation of charge J µ ;µ = 0, i.e., in view of (56), by adding the following equation:

Thus we have the system [( 5), (31), (32), (57)], or equivalently the system [( 5), (31), (55), (57)], which has the differential identity [START_REF] Arminjon | On continuum dynamics and the electromagnetic field in the scalar ether theory of gravitation[END_REF]. So we have 13 -1 = 12 equations for 12 unknowns F µν (0 ≤ µ < ν ≤ 3), J, ρ * , and p. It is natural to expect that the system [( 5), (31), (55), (57)] has a unique solution when suitable boundary conditions are imposed, just as has the system of electrodynamics of a metric theory, [( 4)-( 5), ( 24)]. (As at the end Therefore, the analysis done in that work applies almost without any change.

In view of the additional equation ( 57), we must require that δ ν (p) be of the same order as is b ν (T field ) in the gravitational weak-field parameter λ, with λ = c -2 in specific λ-dependent units of mass and time. From Eqs. (34) and (36) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF], we have for the order of b ν :

where

is the expansion of the e.m. field tensor. The gravitational field is expanded as (Eq. (28) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]):

where U is the Newtonian gravitational potential. Using this and setting

we get from (49) and (50) that δ µ = ord(c q ), hence our requirement is satisfied iff q = 2n -5 so that

Let us define

as in Eq. (23) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] -but now ρ = J µ ;µ , unlike in the latter work. With this definition, everything in Sects. 4 and 5 of Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF] remains valid and we have in particular (Eq. (45) in Ref. [START_REF] Arminjon | Charge conservation in a gravitational field in the scalar ether theory[END_REF]):

where G 1 µν and T 1 µν are the first approximations of G µν and T µν , i.e.,


Philippe Giraudeau
email: philippe.giraudeau@inria.fr

Design and Evaluation of a Mixed-Reality Interface for Collaborative Learning at School

Keywords: Mixed-Reality, Tangible Interaction, Spatial Augmented Reality, Instructional Design, Collaborative Learning

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

Since at least two decades, schools have introduced computers in classrooms to allow children to discover the digital world and learn how to use it. It was at the beginning a way to get use to these new interfaces instead of using them as pedagogical tools. The shift started about fifteen years ago while school books began to be sold with compact discs, providing the first digital resources for teachers and pupils. More recently, schools started to use tablets instead of traditional computers that propose a similar access to digital information, but in a more convenient way.

Nonetheless, we still use a single kind of device, equipped with (touch) screens, mice and keyboards and all the current computers rely on the well-know WIMP (Windows, Icon, Menu, Pointer) paradigm. "This generic interface provides us a window, through which we need to jump in order to reach the digital realm." [START_REF] Sol | One Reality Augmenting the human experience through the combination of physical and digital worlds[END_REF]. Such interfaces have shown great benefits for web browsing, text editing, and so on. However, they are limited as soon as collaborative interactions or hands-on activities are required [START_REF] Marshall | Do tangible interfaces enhance learning[END_REF]. This is particularly true for learning applications [START_REF] Stanton | The effects of multiple mice on children's talk and interaction[END_REF].

During this PhD, we will focus on the design of interfaces that go beyond the traditional and wide WIMP (Window, Icon, Menu, Pointer) paradigm and explore approaches that favor rich 3D interactions, giving more importance to physical manipulations. We want to enable children to collaborate around a unique space by anchoring the collaborative learning experience with the digital, in the real world, allowing the computer to disappear.

From these objectives, two mains research questions appear :

-What are the requirements to build a space where digital and physical world coexist ?

-How this kind of hybrid space can support the learning in collaboration ?

Learning, in collaboration

Learning is a cognitive process that permit to acquire new knowledge or modify existing knowledge. It can be impacted by numerous factors as psychological or social. As stated by S. Dehaene in his class at Le Collège de France, the field of cognitive neuroscience has identified at least three factors that can tune the speed and learning facility at school. 1) Attention : The fact that children are focused on the right things to learn without being disturbed by other factors.

2) Active learning or active engagement (intrinsic motivation): The fact that children are engaged in their activity. For that, we have to maximize curiosity and active prediction.

3) Feedbacks, error signals and rewards (extrinsic motivation). The comparison between the prediction and the outcome is a key role in the learning process.

Additionally to those factors, the Instructional Design (ID) approach aims to distinguish the impact of different aspects that influence the cognitive load required during a learning activity [START_REF] Sweller | Instructional Design in Technical Areas[END_REF]. The first one is the content to learn, called the intrinsic cognitive load. The extrinsic load refers to the manner in which information is presented to a learner. Finally, the germane load is the way information is processed and automated.

One of our objectives in this thesis will be to reduce extrinsic cognitive load during manipulation on our interfaces, to favor collaborative learning (CL).

CL is a situation of collaboration where the common goal is to learn or attempt to learn something together [START_REF] Dillenbourg | What do you mean by ' collaborative learning '? Collaborative learning Cognitive and computational approaches[END_REF]. All the mechanisms dedicated to collaboration such as negotiation, sharing of meanings, coordination, are then applied to the purpose of learning. CL is often used for learning complex notions that are difficult to learn alone and where the cognition of pairs allow to reduce the amount of working memory needed to complete the task.

The e-TAC project

This PhD is part of an interdisciplinary project called e-TAC that aims to design tangible and augmented collaborative learning environments for children at school. e-TAC works also towards promoting social relationship, knowledge sharing and conceptualization of new concepts. This project gathers together researchers in Human-Computer Interaction (HCI), education science and ID. Furthermore the goal is not only to design that kind of interfaces but also to create adapted pedagogical content by working with practitioners like teachers, inspectors of the national French education as well as pupils.

MIXING THE REALITY WITH THE DIGITAL

During this PhD project, we take sides to base our interfaces on Tangible User Interface (TUI) and Spatial Augmented Reality (SAR).

Manipulating the digital with physical interaction

TUIs are interfaces in which physical objects and their surfaces are enhanced through embedded computation. Thus, the physical world becomes an interface that can connect objects and the environment with the digital world [START_REF] Shaer | Tangible User Interfaces: Past, Present, and Future Directions[END_REF]. Since TUIs are becoming in HCI a common way to create interfaces that go beyond the use of screens, keyboards and mice. TUIs originally come from the special issue "Back to the Real World" (1993) [START_REF] Wellner | Computer-augmented environments: back to the real world[END_REF], which argued that computers and virtual reality take humans away from their "natural environment". Later, Ishii et al [START_REF] Ishii | Tangible bits: towards seamless interfaces between people, bits and atoms[END_REF], presented a more comprehensive approach with Tangible Bits and with first prototypes like Urp [START_REF] Underkoffler | Urp: a luminous-tangible workbench for urban planning and design[END_REF].

Bringing the digital onto the real world

Exposing digital information onto the real world without using screens is relatively new for HCI. To create such augmentation, we make use of SAR, which is a specific kind of augmented reality (AR) that proposes to project onto the real world digital contents [START_REF] Raskar | The office of the future: A unified approach to image-based modeling and spatially immersive displays[END_REF] instead of see-through devices as usually done in AR. The digital desk [START_REF] Wellner | Computer-augmented environments: back to the real world[END_REF] is one of the first interfaces to take advantage of the projection to augment the environment. The main advantage of using SAR is the possibility to augment different surfaces like horizontal surfaces (i.e. [START_REF] Furió | HOBIT: Hybrid Optical Bench for Innovative Teaching[END_REF][START_REF] Jordà | The reacTable: exploring the synergy between live music performance and tabletop tangible interfaces[END_REF]) and objects surfaces that can be moved inside the projector's frustum.

Benefits of mixed-reality

Mixed-reality based on projection and tangible interactions comes with some benefits, in particular for education purposes. First of all, Jacob [START_REF] Robert | Reality-based interaction: a framework for post-WIMP interfaces[END_REF] proposed the notion of reality-based interaction (RBI) as a unifying framework that ties together a large subset of these interaction styles including augmented reality and tangible interaction. Jacob et al, identified four types of interaction that all styles have in common: Naïve Physics, Body Awareness and skills, Environment Awareness and Social Awareness.

Researchers suggest that physical interactions (Naïve physics) could reduce the requirement of abstraction and ease the learning, thanks to the more intuitive way to interact with the interface and its content [START_REF] Marshall | Do tangible interfaces enhance learning[END_REF]. Tangible components of MR interfaces could also promote active learning and hands-on engagement [START_REF] Marshall | Do tangible interfaces enhance learning[END_REF][START_REF] Zuckerman | Extending tangible interfaces for education: digital montessori-inspired manipulatives[END_REF]. This allows the learning of abstract concepts with concrete representations [START_REF] Fleck | Making tangible the intangible: Hybridization of the real and the virtual to enhance learning of abstract phenomena[END_REF] and in particular thanks to the 3D representations of manipulated elements. MR interfaces allow users to try more solutions with physical manipulation and allow them to spend more time exploring when performing problem solving tasks [START_REF] Do-Lenh | Task Performance vs. Learning Outcomes: A Study of a Tangible User Interface in the Classroom[END_REF][START_REF] Schneider | Benefits of a tangible interface for collaborative learning and interaction[END_REF].

Tangible interactions take place in the external (physical, real) world that acts as a support for our cognition by loading and storing information as described by the extended cognition [START_REF] Sutton | The psychology of memory, extended cognition, and socially distributed remembering[END_REF].

In addition of the external world, other people or artefacts can also share and store information regarding an activity (i.e. problemsolving), as stated by the distributed cognition theory [START_REF] Donald A Norman | Cognitive artifacts. Designing interaction: Psychology at the human-computer interface[END_REF]. MR based on projection can also support collaboration by displaying and spatializing digital contents onto the real world (Environment Awareness). Those contents become manipulable by multiple users, a negotiation space favoring collaboration through speech and mutual gazes (Social Awareness) [START_REF] Dillenbourg | What do you mean by ' collaborative learning '? Collaborative learning Cognitive and computational approaches[END_REF][START_REF] Schneider | Using Mobile Eye-Trackers to Unpack the Perceptual Benefits of a Tangible User Interface for Collaborative Learning[END_REF].

Tangible and augmented interfaces for education

Among the previous works that have been done on MR for education, we can cite Hobit for Hybrid Optical Bench for Innovative Teaching [START_REF] Furió | HOBIT: Hybrid Optical Bench for Innovative Teaching[END_REF]. It is an augmented tabletop that proposes to enhance the learning of optics with an augmented tabletop that reproduces practicals at university. This interface benefits from both physical manipulation and pedagogical supports that are embedded within the experimentation. The Tinker Lamp is an augmented tabletop interface for training logistics assistants [START_REF] Jermann | Tinkering or sketching: Apprentices' use of tangibles and drawings to solve design problems[END_REF][START_REF] Zufferey | TinkerSheets: using paper forms to control and visualize tangible simulations[END_REF]. This system uses paper sheets as input. Papers are also used as output surface in order to fit the projection size. It allows students to create and shape warehouse with 3D shelves.

Still for the purpose of learning new concepts, Do-lenh et al. , [START_REF] Do-Lenh | TinkerLamp 2.0: designing and evaluating orchestration technologies for the classroom[END_REF], built an application for The Tinker Lamp, allowing users to build concept maps on a synaptic transmission. Learners could for instance create links between two concepts by placing pieces of paper side by side and delete the same link by placing two fingers onto the link.

FIRST RESULTS

The following section summarizes our initial results presented in [START_REF] Giraudeau | Towards a Mixed-Reality Interface for Mind-Mapping[END_REF]. (See the paper for more details).

In this work, we have explored an approach based on the hybridization of physical and digital content for mind-mapping activities at schools. We have designed a MR interface called Reality-Map.

Introduction

Mind-mapping is a technique for fast idea generation and is often used in planning, critical thinking, studying, note taking and problem solving. Buzan et al [START_REF] Buzan | The concept map book[END_REF], showed that mind-mapping make ease categorization and learning.

With Reality-Map, users can spatially organize the studied content by manipulating augmented papers and plastic-coated cards allowing users to write with whiteboard markers on cards and embed them within the mind map (see Figure 1). Users can also create (draw) and remove links with a specific tracked pen. These links are highlighted straight lines projected between two objects.

Implementation

We implemented Reality-Map using a projector-camera (procam) system to display digital information on the table and tracked which and where were pieces of paper and objects on the table. We calibrated the procam system using a manual technic (see Appendix A.2.2 Manual techniques [START_REF] Sol | One Reality Augmenting the human experience through the combination of physical and digital worlds[END_REF]). The hardware side consists of an Optitrack's Trio composed of three infrared cameras and a videoprojector placed above the user for the augmentation of the horizontal surface. The software side is composed of Optitrack Motive, that handles input from Optitrack Trio and then send information via UDP to the second software, Unity 3D. Unity3D is used to create the graphical UI (documents and links). The 3D calibration allowed to project the digital information to the right placed, creating the illusion that digital contents are attached to their corresponded objects.

Pilot study

We carried out a pilot study to evaluate Reality-Map by comparing it with a WIMP interface called Draw.io. The assumption of this study was that mixed-reality interfaces based on tangible interactions and projection are not only suitable for providing reality-based interactions but may also favor the building of knowledge for learning complex notions. For this pilot-study, we chose basic brain functions as the targeted knowledge. Our hypotheses were : H1: Learning scores will be better in the tangible group compared to control group.

H2: Participants will better recall knowledge learned during the experiment after 5 days with Reality-Map.

Experimental design

For this between-subjects experiment, we randomly split participants in two groups. A "Control group" had to use the WIMP interface to create the mind-map and the "Tangible group", which had to do the same task but with Reality-map.

The procedure was conducted as follows. First, participants answered pretest regarding their knowledge about the brain. Then, depending on the group, participants used the WIMP software Draw.io or Reality-Map. During 10 min, participants discovered and manipulated documents about brain areas. Then, the task consisted in the creation of a mind-map on the theme: "Brain structures and their cognitive processes". This task had to be done in 10 min. Finally, participants had to fill a post-test questionnaire regarding what they learned during the session. Five days after the experiment, participants were invited to answer a second post-test questionnaires.

Results

Analyses (a two-way mixed ANOVA) showed that participants scored significantly higher in the post-test that they did in the pretest,(F=5,256; p<.05) for the two groups, however the tangible group was significantly better than the control group.

CURRENT WORK

From the literature and previous experiments realized during this PhD, we started to develop a second major prototype that will be deployed in a pilot classroom. The current interface is supporting the same features as Reality-Map and aims to go beyond. Most of the work done yet concerns the technology that supports the projection. Indeed, SAR compels to calibrate the 3D environment by getting intrinsic (i.e. lens distortion, size, optical center), extrinsic (position) parameters of each optical element in order to project digital media at the right position and without distortion. Reality-Map used Optitrack camera and software, which are very precise but expensive and thence cannot be deployed in classrooms. In parallel to the creation of this hybrid environment, we are also designing TUIs that will provide to users a way of manipulating digital content.

Interaction techniques developed for this interface will respect ID principles and learning theories that have been presented previously. We also want to promote the curiosity and exploration possibilities of children by not constraining them too much and provide an environment that let them discover [START_REF] Bonawitz | The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery[END_REF]. Finally, the interface will be built to support collaborative interactions. Some actions will be allowed only when there is more than one user around the table. On the contrary, other actions will be permitted only once at a time to create a bottleneck that will force negotiation and argumentation.

FUTURE WORK 5.1 Focus group and elicitation sessions

With the prototype that is currently under development, we will enter in a second phase where future users will be directly involved in the process of development. Two focus groups will be organized. The first one will be held with teachers and practitioners from the French national education. This focus group will let them see and give feedback on the first concrete MR prototype.

A second series of focus groups will be organized with pupils. The goal of these sessions will be to elicit feedback on how they perceive the interface and the developed interaction techniques. Furthermore, letting children interact spontaneously with an interface without giving them instructions can be really interesting by observing how they interact with the interface. Such observations can highlight some aspects of our interface that we did not through about.

For instance, during the first elicitation session, some children try by themselves to zoom on the picture that was projected on the table. Several focus groups will be regularly organized until a consensus between researchers and practitioners is found.

Finally, we will also focus on the usability and acceptability of our interfaces, which are the main current approaches used in HCI to evaluate interfaces.

Evaluation

We will formally evaluate interfaces in both control and ecological environment. Alongside of general questions of usability and acceptability and more generally, the user experience, we would like to answer these following questions:

1. Will the learning performances be the same with a mixedreality interface than with robust interactive tabletops based on multi-touch screen ?

2. Are mixed-reality interface good for all types of content ? 3. Do we have the same representations of the content we manipulate on post-WIMP interfaces compared to more traditional WIMP interfaces?

We choose these questions based on the literature and previous experiences with MR interface for education. Indeed, for the first question, researchers point out the fact that using AR interfaces could improve the learning compared to more traditional tools as non-augmented environment and interactive tabletops based on multi-touch screen. These interfaces are robust but expensive and/or do not allow object surface augmentation.

The second one is directly questioning the generalization learning capabilities of such interfaces.

The latter question will probably not be answered during the three years of this PhD because of its deep complexity. Nonetheless, exploring how the brain learns during manipulation of documents whether on a WIMP or post-WIMP is crucial for designing hybrid environments that are fully dedicated to learning.

Figure 1 :

 1 Figure 1: Reality-Map is dedicated to mind-mapping activities by combining real and digital content an unique hybrid space.

ACKNOWLEDGEMENT

I would like to thank Martin Hachet, Inria senior researcher and supervisor of this PhD. I would also like to thank Marie Gonzalez for her work on tangible tokens. The e-TAC project is financially supported by the French government in the e-FRAN call for projects within the framework of the Program of Investments for the Future (PIA) operated by Caisse des Dépôts et Consignation.