**Energy
and equations of motion in a tentative theory of gravity with a privileged
reference frame**

** **

*Archives of Mechanics* (Warszawa) **48**,** **No. 1, pp. 25-52 (1996)

** **

M. ARMINJON (GRENOBLE)

**Abstract-** Based on a tentative interpretation of gravity as a
pressure force, a scalar theory of gravity was previously investigated. It
assumes gravitational contraction (dilation) of space (time) standards. In the
static case, the same Newton law as in special relativity was expressed in
terms of these distorted local standards, and was found to imply geodesic
motion. Here, the formulation of motion is reexamined in the most general
situation. A consistent Newton law can still be defined, which accounts for the
time variation of the space metric, but it is not compatible with geodesic
motion for a time-dependent field. The energy of a test particle is defined: it
is constant in the static case. Starting from ‘dust’, a balance equation is
then derived for the energy of matter. If the Newton law is assumed, the field
equation of the theory allows to rewrite this as a true conservation equation,
including the gravitational energy. The latter contains a Newtonian term, plus
the square of the relative rate of the local velocity of gravitation waves (or
that of light), the velocity being expressed in terms of absolute standards.

**1. Introduction**

AN ATTEMPT to deduce a consistent theory of gravity
from the idea of a physically privileged reference frame or ‘ether’ was
previously proposed [1-3]. This work is a further development which is likely
to close the theory. It is well-known that the concept of ether has been
abandoned at the beginning of this century. It is less widely known that, in
the mean time, the objective situation in physics has made it reasonable and
interesting to reexamine this concept. In such a matter, it is necessary for
the writer to appeal to authorities. Thus:

(**i**) Due to the work of BUILDER [6-7],
JANOSSY [11], PROKHOVNIK
[21,23], it has now been proved in detail that special relativity (SR) is,
after all,* fully* compatible with the
assumption of the ether as envisaged by Lorentz and Poincaré, i.e. with the
ether being an inertial frame in which Maxwell’s equations can be written and
relative to which all material objects undergo a ‘true’ Lorentz contraction. This
has been emphasized by physicists as important as BELL [4],
the author of theorems on ‘hidden variables’ in quantum mechanics. In a recent
review on the latter subject, MERMIN [17]
writes: « What Bell’s Theorem did suggest to Bell was *the need to reexamine our understanding of Lorentz invariance* »
(italics ours). It is worth to recall that several basic results of SR,
including the Lorentz transformation and the Lorentz invariance of Maxwell’s
equations, were found by Lorentz and Poincaré, prior to Einstein, *within a theory* *based on ether*. As stated by BELL [4],
the main differences between the approaches by Lorentz (and Poincaré) and by
Einstein are that: (a) « Since it is experimentally impossible to say
which of two uniformly moving systems is *really*
at rest, Einstein declares the notions ‘really resting’ and ‘really moving’ as
meaningless. » (b) « instead of inferring the experience of moving
observers from known and conjectured laws of physics, Einstein starts from the *hypothesis* that the laws will look the
same to all observers in uniform motion. » (italics are Bell’s). It is now
accepted that the Lorentz-Poincaré opposite philosophy leads to the same
physical theory, namely SR.

(**ii**) It has become evident that quantum mechanics leads to attribute
definite physical properties to the so-called ‘vacuum’, and that these
properties do affect our measuring instruments as predicted by quantum
mechanics. As SCIAMA [25] states:
« For example, the electric and magnetic fields in the electromagnetic
vacuum are fluctuating quantities. This leads to a kind of reintroduction of
the ether, since some physical systems interacting with the vacuum can detect
the existence of its fluctuations. However, this ether is Lorentz-invariant, so
there is no contradiction with special relativity. »

(**iii**) Finally, relative velocities of astronomical objects are being
measured with increasing accuracy. We can associate a privileged reference
frame with the ‘cosmic fluid’, the velocity of which is the average velocity of
matter. This reference frame appears clearly in the cosmological models
because, in these simplified models, our universe is assumed homogeneous so
that the average is already done. For example, the Robertson-Walker space-time
metric of an expanding universe distinguishes a particular class of ‘comoving
observers’- that is, comoving with the expansion commonly assumed to explain,
as a Doppler effect, the observed red-shift of the spectra emitted by distant
galaxies. Their set makes a privileged reference frame, as emphasized by PROKHOVNIK [21-22].

Thus,
we have a privileged reference frame in which also the ‘empty’ parts have
physical properties. This may be called an ‘ether’ for short, even though
‘physical vacuum’ would be more appropriate. The use of the old name does not
imply to forget modern physics: but that precisely today's physics allows to
reconsider the assumption of an ‘ether’, provides some justification for
reconsidering also the *possibility*
that, after all, the allegable existence of this ‘ether’ *might manifest itself in certain laws of physics*. Since the
privileged reference frame is kinematically defined from the average motion of
matter at a very large scale, gravitation could be the range of such speculated
manifestations. In other words, it seems allowable to investigate a theory of
gravity with a privileged frame, one which would not necessarily be neutral in
the theory: a non-covariant theory. Some important astronomical facts at the
scale of galaxies, such as the famous rotation curves in the disc galaxies,
have not yet received any undebatable interpretation [19,30]. Hence an
explanation deriving from the effect of motion through an ‘ether’ might be
considered.

However,
Newtonian gravity (NG) gives an excellent description of astronomical motions
at smaller scales, which is even refined by the corrections of general
relativity (GR). Since neither NG nor GR does admit a preferred reference
frame, it is undoubtedly a risky affair to attempt the construction of a theory
of gravity with such a frame. A such attempt has yet been proposed [1-3]. The
aim was to investigate whether the ‘logic of absolute motion’, which furthers
our understanding of *special*
relativity (SR) as compared with the usual ‘space-time logic’ [4,21], can be
extended so as to build a sensible theory of *gravity*. The investigated theory is non-linear, like GR, and
coincides with NG at the lowest approximation, also like GR. As an argument in
favour of giving some attention to this theory, we mention the analytical study
of the gravitational collapse in ‘free fall’ with spherical symmetry, i.e. the
situation analytically studied within GR by OPPENHEIMER &
SNYDER [18]. Since then, it has been shown, notably by PENROSE [20], that this situation contains all essential
features of gravitational collapse for very massive objects in GR. According to
the investigated theory, no singularity occurs during this collapse, neither
for the metric nor even for the energy density, because the implosion would
stop within finite time (for freely falling clocks and for remote clocks as
well) and would be followed by an explosion [3]. On the other hand, this theory
gives exactly the same predictions as GR for the motion of a test particle in
the assumed static gravitation field of a spherical body, for in that case the
Schwarzschild space-time metric of GR is obtained and the Newton law of the
theory implies geodesic motion for any static field. The last result applies to
mass particles [2] and to light-like particles as well [3], and provides a link
between dynamics in NG and in GR, in the sense of MAZILU [16].
In this non-covariant theory, however, a gravitation field is static only if
its source, the mass-energy density in the frame bound to ‘ether’, is
time-independent. Whereas in NG and GR a uniform motion of the massive body can
be eliminated by changing the reference frame, this is not the case here. Thus,
in order to analyse either planetary motion in the solar system or stellar
motion in a galaxy, one might have to reckon with velocities in the range 100-1000
km/s.

In
the previous work, a Newton law has been defined only in the static case. Since
it then implies geodesic motion, geodesic motion has been assumed in the
general case. The purpose of this paper is to study in greater depth the
formulation of motion in the theory, in connection with the problem of energy.
We first recall (Sect. 2) the basic principles and equations of this theory,
obtaining an alternative (equivalent) form of the field equations. For the
equations of motion (Sect. 3), it is shown that in the general, non-static
case, a consistent Newton law can still be defined, but is incompatible with
the geodesic formulation of motion. Then the energy problem in the theory is
analysed (Sect. 4). As is known, an exact energy can hardly be defined within
GR, since the obtainment of a conservation equation for energy-momentum asks
for a privileged class of reference frames (see e.g. STEPHANI
[28]). After having recalled some aspects of the question of gravitational and
total energy in NG and in GR, it is shown that a local energy balance equation,
including the gravitational energy, is obtained with the studied theory if one
assumes the (extended) Newton law of motion, but not under the geodesic
formulation of motion.

2. **Basic assumptions and equations**

2.1 *Absolute space and physical vacuum*

Since astronomical observations and
the success of Newtonian theory suggest that Galilean space-time is a very
accurate description of our real world-theater, the notion of a physical vacuum
has first to be conciled with *classical
mechanics*. This, however, was not accomplished by ether theorists. For the
ether must be the absolute space, hence rigid- otherwise, one would have *two *independent absolute spaces. But it
also must offer no resistance against motion, and this in Newtonian mechanics
can be true only of a perfect fluid. The proposed answer is that *it is the average motion of the hypothetical
perfect fluid that defines the privileged inertial frame* [1-2],
corresponding to the absolute space of Newtonian theory. Since any motion must
be referred to some space (and time), it may seem that we are lead to a vicious
circle. Thus we first postulate the absolute space or 3-D manifold *M*, which as a working assumption we may
take to have Euclidean structure[1]. And
we postulate the absolute time *t*,
i.e. the world of events *M*^{ 4}
(time and position) is assumed to be the product **R**x*M* [2]. In
Newtonian theory, the natural space metric **g**^{0} on *M* and the absolute time are assumed to
be experimentally accessible, independently of motion and gravitation. In the
studied non-linear theory, the coincidence between ‘absolute’ space (or time)
metric and physical space (or time) measurements is found only in the first
approximation. Then we may define the motion of a continuous medium, relative
to *M*, by its velocity vector field **v**_{0 }= **dx***/dt*. The particular
metric structure we assume for *M*
ensures that the integral of a vector field makes sense and is a vector (note
that it is *not* the case for a general
Riemannian space and thus for GR). Hence, the volume average of the velocity in
any finite domain *W* occupied
by the continuous medium is well defined, say _{}. Now we may precise the assumption according to which
the ‘microscopic ether’, that one which would be a perfect fluid, has in the
average no motion relative to the absolute space *M* : we suppose that the velocity field of the micro-ether, **v**_{0e}, is everywhere defined (it fills the space) and is such
that _{}® 0 as
the size of the regular domain *W* (let us say it is a ball)
infinitely increases, and this independently of the spatial position of * **W*. We
may call *M* the macro-ether.

Since
the micro-ether must fill the space, any kind of matter, made of material
particles, should be actually made of this universal fluid. Thus, material
particles should be local, organized flows in ether, such as vortices which in
a perfect fluid can be everlasting. We note that this concept is compatible
with the lack of complete separability between particles, which is predicted by
quantum mechanics and has received experimental confirmation. It would be an
ambitious program, however, to attempt recovering micro-physics from this
assumption of a ‘*constitutive ether*’
(but not a hopeless program: cf. the hydrodynamic interpretation of
Schrödinger’s wave equation by MADELUNG [15];
more recently, ROMANI [24], WINTERBERG [31-32] and DMITRYIEV [9]
obtained results in the same direction). Anyway, the gravitation theory needs
almost only the macro-ether, which will be physically defined from the average
motion of *matter*, i.e. the average
motions of ether and matter are assumed identical. Recall that the notion of
average used in this definition, is the asymptotic volume average. This means
that no a priori bound needs to limit the size of local inhomogeneities, i.e.
the size of astronomical structures.

2.2 *Scalar field equation and space-time metric*

A perfect fluid can exert only
pressure forces. The gravitation force is tentatively interpreted as the
pressure force or Archimedes' thrust, resulting from the gradient of the
macroscopic pressure *p _{e}* in
the ether. This gives the following expression for the gravity acceleration

(2.1) _{}

where *r** _{e}* is the macroscopic density of the hypothetical fluid,
assumed barotropic; the latter means that

**g **= grad *U*.
Just like for the latter, the macroscopic nature of Eq. (2.1) (the fact that
the involved fields vary significantly over macroscopic distances only) does
not prevent **g** from being felt at the
scale of material particles.

The
field equation for *p _{e}*,
playing the role of Poisson's equation for

(2.2) _{}

with D the Laplace
operator, *r* the density of matter (the density of the conserved
mass in the first, Newtonian approximation; in the non-linear theory, *r * is the mass-energy density in the frame E in which the points of *M* have no motion[3]), *t*** _{x}**
a

(i) Since NG propagates with
infinite speed, it must correspond to the case where the fluid is
incompressible, *r** _{e}* =
Const., and the field equation must become equivalent to Poisson's equation as
the compressibility evanesces.

(ii) In the compressible case,
pressure waves must appear (except for static situations where *p _{e}* does not depend on time),
the velocity

*K *= 1/*c _{e}*

(iii) Accounting for SR in the
Lorentz-Poincaré interpretation thoroughly developed by PROKHOVNIK [21,23], the velocity of light, *c*, becomes a limiting speed: so should be also the velocity of the
pressure waves (the ‘sound’ velocity) in the assumed constitutive ether, hence
one must have *c _{e}* =

*p _{e}* =

(iv) In the theory, the equivalence
principle arises naturally, as a correspondence between the absolute metric
effects of motion and gravitation. As a consequence,* the space (time) standards are assumed to be contracted (dilated) in
the gravitation field, in the ratio* *b *= *p _{e}*/

(2.3) (grad *f*)* ^{i}* = (grad

(2.4) D *f* = D_{g}*f* = div** _{g}** grad

Moreover, a local time *t*** _{x}**
appears at any fixed point

(2.5) *dt*_{x}*/dt* = *dt*_{x}_{
}*/dt*_{x}_{0} = *b *= *p _{e}*(

whence the definition of the
derivation with respect to local time, appearing in Eq. (2.2):

(2.6) _{} ,

with *t*_{x}_{0}
= *t* if is **x**_{0} is ‘far enough’, i.e. if *p _{e}*(

The
assumed contraction of the physical space standards (and thus the dilation of
measured distances), with respect to the natural metric on *M*, occurs in the direction of the gravity acceleration **g**. Hence the expression of **g**, the
physical space metric in the frame E, is the simplest
in an ‘isopotential’ coordinate system. This is a space-time coordinate system
(*y** ^{a}*) such
that, at any given time

(2.7) (*g** _{ij}*) =
diag (

(Greek indices will vary from 0 to 3
and Latine ones from 1 to 3). Note that, in the general case where *p _{e}* depends on the time

The
line element of the space-time metric **g**, measuring the
proper time *d**t* along
an element of trajectory, follows straightforwardly from the combination of the
slowing down of the mobile clock due to its absolute motion and to the
gravitation field [2]. If *dl* is the
line element of the space metric **g**, i.e. if *dl*
is the elementary distance covered by the mobile, as measured with rods of the
momentarily coincident observer bound to E, one
has:

(2.8) *ds*^{ 2} = *g _{lm}*

The validity of Eq. (2.8)_{3}
assumes that the coordinates (*x** ^{a}*) are
bound to the frame E. By Eq. (2.7), a diagonal expression is hence
obtained also for

(2.9) (*g _{lm}*) =
diag (

If general coordinates (*z** ^{a}*) are
used, one writes instead of Eq. (2.8)

(2.10)
*dl*^{ 2} = *dt*_{x}^{2}
*g*_{ij}* v _{ }^{i} v_{ }^{j}* =

with *v ^{i}* the components of

_{} if the (*x** ^{a}*) are
bound to E). Of course,

*h** ^{ij}* = -

*dl *’ = (*h*_{ij}^{ }*dz ^{i }dz^{j }*)

2.3 *Expression of the field equation in terms of natural metric and
absolute time*

The expressions of the ‘main’ field
equation (2.2) and the ‘auxiliary’ one (2.1) are in terms of the physical space
metric and local (physical) time, in the frame bound to the absolute space or
macro-ether *M* (it is recalled that
Eq. (2.2) is only valid in this frame E). However, this
physical space-time metric depends on the unknown, i.e. on the field *p _{e}*. In some cases, it is
easier to handle Eq. (2.2) when it is expressed in terms of the natural metric

(2.11) _{}

where (**e*** _{i}*)

(2.12) _{}=(*g*_{00})_{E}

(that (*g*_{00})_{E} =*b*^{ 2} if
one uses the absolute time coordinate *t*,
is due to Eq. (2.8)_{1}). Turning to Eq. (2.2), we have *g* º det (*g** _{ij}*) =

_{}

(cf. Eq. (2.4)), which may be
rewritten as:

(2.13) _{}.

Now, let us assume that our (model
of) universe, in which the studied gravitation field and matter are embedded,
is ‘static at infinity’, which is likely to be an extremely good approximation-
except for cosmological problems. That is, assume that _{} is independent
of the time *t *[4]. In
that case, the term with time derivatives in Eq. (2.2) becomes:

(2.14) _{} .

Combining Eqs. (2.13) and (2.14) and
multiplying by 2*/*(*b*_{}), we rewrite Eq. (2.2) as:

(2.15) _{} .

Thus,
the field equation reduces in the static case to the ordinary Poisson equation,
which is *linear*, and indeed Eqs.
(2.12) and (2.15) are equivalent, for time-independent *f* (and* r*), to the Newtonian equations:
exactly the Newtonian gravity acceleration**
g** will be associated with any given density of mass-energy, *r*(**x**), by Eqs. (2.12) and (2.15). However,
in NG, only the ‘invariable’ (rest) mass is counted in *r* ;
moreover, it remains to precise the definition of *r*, and it turns
out that *r* must depend on the gravitation field, i.e. on *f* itself [Sect. (4.2), point (iv)].
Anyhow, not the same motion will be predicted for test particles if **g** is known, since in the studied theory
Newton's second law is expressed in terms of space and time measurements with
clocks and rods of the local observer in E,
which are affected by the gravitation field. In the static case with spherical
symmetry, for example, Eq (2.15) (plus the requirement that **g** remains bounded as r ®0, *or* the boundary condition *f*=1 at infinity), gives *f *= 1- 2 *Gm/*(*c*^{2}*r*) outside the body (*r* being the radial Euclidean distance,
and with

(2.16) *m* = ò_{M}*r* *dV*^{ 0}_{. }

where *dV*^{ 0} is the volume element of the Euclidean metric).
Then Eq. (2.9), with *a*^{0}_{1
}=1, *a*^{0}_{2} =*r*^{2} and *a*^{0}_{3} = *r*^{2
}sin^{2}*q* in spherical coordinates *r*,*q*,*f*,
leads to Schwarzschild's exterior space-time metric. It is striking that, after
natural account of SR and the equivalence principle, this theory (to which we
have imposed to reduce to NG asymptotically as the ‘ether compressibility’
evanesces), gives exactly and simultaneously the Newtonian attraction field **g** and the Schwarzschild metric in the
spherical static case [2]. The new result is that the Newtonian **g**-field is predicted for any static
situation, whether ‘spherical’ or not. The form (2.15) of the field equation
has important applications also in the non-static case, see Sect. 4.2 (energy
balance).

**3. The motion of a test particle: Newton law vs. space-time geodesics**

To state Newton's second law demands
to define the acceleration or rather (since SR must be taken into account) the
time derivative of the momentum. Also because SR must hold at the local scale
(and for sure with physical space-time metric), one has to use the physical,
distorted space and time standards, i.e. the Riemannian space metric **g** and
the local time *t*** _{x}**.

3.1 *The case of a time-independent spatial metric* **g** *(static gravitation field)*

If on a manifold *M* a Riemannian metric is given (thus a *fixed* metric **g**), one has a natural definition of the ‘time’
derivative of a vector **u**(*t*) attached to a ‘trajectory’ i.e. a
differentiable mapping *t* a *X*(*t*)
from an open interval in **R** into *M*. The
components *h** ^{i}* of
the derivative

(3.1) * h** ^{i}* =

with _{}the second-kind Christoffel symbols associated with
metric **g** in coordinates (*x ^{i}*)
and

(3.2) for
any vector **w** in the tangent space *TM _{X}* at point

where point means scalar product **g** and **w’** is the parallel transport (using **g**) of
vector **w** on the trajectory [2];
moreover, the components of the *vector*
*D***u***/Dt* defined in this way, are indeed
given by Eq. (3.1). Note that here *M*
is *also* equipped with the Euclidean
metric **g**^{0 }which
would allow to speak of ‘a vector **v**
in *M *‘ (i.e. not specifically
attached to a point *X*Î*M*), and in fact to identify points in *M* and vectors (once an arbitrary origin
point *O*Î*M* has been selected) [5],
whence our notation **x**Î*M* in previous sections. Thus the Newton law of SR has
been extended to this theory of gravity, in the form:

(3.3)
**F** º **F**_{0} + *m*(*v*) **g** = _{},

(3.4)

where **F**_{0} is the non-gravitational (e.g. electromagnetic)
force, _{} is the
velocity of the test particle with respect to *M*,

*v* = **g**(**v**,**v**)^{1/2}º(*g*_{ij}* v ^{i} v^{j}*)

*m*(*v*)=*m*(0)/(1-*v*^{2}/*c*^{2})^{1/2} is the inertial mass, which is thus
identical to the passive gravitational mass, as is also true in NG. Although
the latter identity is often referred to in GR (under the name of ‘weak
equivalence principle’), it does not make an exact sense in GR, because there
is no Newton law there [27]. The Newton law (3.3) is also defined for* light-like* particles (photons,
neutrinos?), in substituting the energy *e=h**n* (or
rather *e/c*^{2}) for the
inertial mass.

3.2 *Extension to the time-dependent situation*

* *

In the general case, the metric **g** will
yet depend on the time *t* (i.e. its
components *g** _{ij}* in
coordinates bound to E will depend on

(3.4) _{},

where, for a twice covariant
second-order tensor **h** (here_{}) and a vector **u**,
**h****.u** is
the covector with components *h _{ij}
u^{ j}*. And, for a twice contravariant second-order tensor

**g**^{-1}) and a covector **u***, **k****.u*** is
the vector with components *k ^{ij}*

*D*_{1 }**u*** /Dt*. Yet
this time derivative, which accounts for the time variation of the metric, does
not cancel for a vector **u** that is
transported parallel to itself (necessarily with respect to the fixed metric ** _{}**) along the trajectory. The use of Eq. (3.2) for the
definition is hence questionable, since Eq. (3.2) was obtained (in the case of
a fixed metric) under the former requirement, plus the condition that the
Leibniz rule has to be verified. If we provisionally forget the Leibniz rule,
we can therefore define a one-parameter family of vector time-derivatives as
well:

(3.5) _{}

(that *D*_{0 }**u*** /Dt* enters the definition (3.5) with
the coefficient 1 is enforced, since we want to recover Eqs. (3.1) and (3.3)
for a time-independent metric). Now let us come to Leibniz' rule. We obtain
from Eq. (3.5) and the fact that, by construction, *D*_{0}**u***/Dt* obeys the Leibniz rule with the
fixed metric ** _{}**:

_{}

(3.6) =
_{} + 2 *l* _{}.

But, by direct calculation, we also
have:

_{},

(3.7) _{}= _{} + _{}.

By comparing (3.6) and (3.7), we
find that *Leibniz' rule holds (with the
variable metric* **g**_{t}*)* *if and only if **l**=*1/2* in Eq.
(3.5).* Nevertheless, with any value of *l*, we may
associate a particular Newton law, in the following way:

(3.8) **F** º **F**_{0} + *m*(*v*) **g** = _{}, _{}.

Now
is *l*=1/2
really the correct value for the parameter *l*? Another
possible criterion for the choice could be the compatibility of Eq. (3.8) with
the formulation of motion in GR. It has been proved that, *in the static case*, the Newton law (3.3)* implies* that any free test particle follow a geodesic line of the
metric **g**, and this for a mass particle [2] as well as for a
light-like one [3]. It is not difficult (though a bit tedious) to follow the
proof for a mass particle in the static case, the calculation method also
applying to the case where one still has isopotential coordinates bound to E. One
then finds that, already in that particular case, any time dependence of the
field *p _{e}* makes new terms
appear in the geodesic equation for the values 0 and 1 of index

** **

** **

**4. The energy problem in the studied theory of gravitation**

4.1* Some remarks on energy and conservation laws in Newtonian theory and
in general relativity*

(**i**) Is the concept of energy a relative one?

The concept of energy is among the
most important one in today's physics, especially in classical physics
(classical mechanics of mass points and continuous media, thermodynamics,
classical electromagnetism) and in microphysics as well, but the notable
exception is the gravitation theory (GR): there, this concept can have only an
approximate status since it is not a covariant concept. Already in NG, the
(kinetic plus potential) energy of a mass point, *e *= (*v*^{2}/2 - *U*)*m*,
is not even a Galilean invariant. Indeed, *v*
is obviously not invariant,
whereas the potential *is* a Galilean invariant, since both
Poisson's equation and the merely spatial boundary conditions, e.g. *rU* and *r*^{2}.grad *U*
bounded at infinity, are so (it is recalled that strict NG imposes a spatially
bounded distribution of mass). That *e*
is changed by a constant when changing the inertial frame makes, of course, no
problem in NG since only the variations of *e*
are relevant there. But since NG demands a bounded mass distribution, the
energy in the frame of the mass-center *could*
be preferred on theoretical grounds, and is indeed preferred in actual
analyses: in classical mechanics, be it celestial or terrestrial, one considers
an assumed isolated system and one refers the velocity, and hence the kinetic
energy, to the mass center. Strictly speaking, one should consider an
(approximately) isolated subsystem of the assumed bounded universe and, for
elements of the subsystem, evaluate their velocity in the global mass-center
frame. But this would differ by a *definite*
constant (the velocity of the mass center of the subsystem) from the velocity
in the frame bound to the mass center of the subsystem. Thus in NG (and in the
same way in a good part of classical physics), an absolute concept of energy is
allowable, probably favourable indeed (cf. the definition of temperature in
statistical thermodynamics, as a mean kinetic energy). Apart from this, it
seems that, in non-relativistic quantum mechanics, one would prefer to avoid
discussing the effect, on the energy levels, of changing the reference frame.
It may be that in quantum mechanics also, an absolute concept of energy would
be favourable.

(**ii**) Energy and conservation laws in Newtonian gravity

In addition to this (debatable)
absolute character, a still more important (and undebatable) aspect of the
energy concept is that, in classical physics, it gives rise to local *balance equations* for continuous media,
which lead to global *conservation laws*.
We take the example of NG, which is relevant here, and we assume elastic
behaviour for simplicity (this includes the case of a perfect barotropic
fluid). Thus one has, in NG, the following definition and conservation equation
for the volume density *w* of the total
energy, i.e. the energy of matter (including its potential energy in the
gravitation field), *w _{m}*

(4.1) *w = w _{m} + w_{g}*

(4.2) _{} = 0 ,

with **s** the stress tensor and *P* the
mass density of elastic energy; this is derived from the more usual energy
balance (in which the energy *w _{g}*
does not appear and a source term

(4.3) *E _{m}* +

(4.4) *E _{m}* +

These equations allow a clear
analysis of the energy transfer from matter to gravitation field and, inside
the contribution of matter, from potential to kinetic and internal energy.
Thus, starting from an ‘unbound’ state in which both the (*positive*) gravitational energy and the (negative) potential energy *E _{p}* = ò-

(4.5) _{},

[here **u**Ä**v** is the tensor product, thus

(**u**Ä**v**)* ^{ij}*
=

the vector operator **div** means the Euclidean divergence of a
second-order space tensor **t**, with

(**div t**)* ^{i}* =

in any coordinates deduced from
Cartesian ones by a linear transformation; and **I** or **d** is the identity tensor, *I ^{ij} = *

(4.6) [div_{g}**T**]^{ a}** **º*T ^{ab}*

(**iii**) General relativity

With Eq. (4.6) appears the problem
of energy and conservation laws in GR. As explained by LANDAU & LIFCHITZ [12],
it « does not in general express the conservation law of anything »
(§101), i.e. it cannot be considered as a true conservation equation. The
reason is that no Gauss theorem applies to the divergence of a second-order
tensor in a curved Riemannian space (see also STEPHANI
[28]). In several relevant situations, including that of an asymptoti-cally
flat metric **g** (i.e. a space-time that is Galilean
at infinity), Eq. (4.6) nevertheless implies integral conservation laws. In
order to obtain such laws, one rewrites Eq. (4.6) in the form of a divergence
with respect to a flat space-time metric **g**^{0}, by
adding to **T** a so-called
‘energy-momentum pseudo-tensor of the gravitation field’, **t** (the expression of which is not unique), thus :

(div_{g}_{0}** q**)^{a}_{ }º *q ^{ab}*

(the factor -*g* is
there for reasons which are bound to the form of the field equations in GR).
That the space-time may be equipped with a global flat metric is, of course, a
strong topological assumption; in general, the discussion here is valid in the
domain *W* where the coordinate system (*x** ^{a}*) in Eq. (4.7)

*P** ^{a}* º

This is sometimes interpreted in GR
as the conservation law for energy (*a*=0) and momentum (*a*=1,2,3;
see e.g. LANDAU & LIFCHITZ
[12]). In contrast to Eqs. (4.3)-(4.4), however, the complex expression of
pseudo-tensor **t** in terms of
derivatives of the metric makes it difficult to draw definite conclusions from
Eq. (4.8), regarding the energy transfer. Moreover, it does not seem completely
clear what should be the physically motivated conditions ensuring the
sufficient decrease at infinity for **q**. Thus if one assumes a
‘time-independent far field’, one finds that *P ^{i}*=0 and that

4.2 *The energy conservation in the studied theory with Newton law*

(**i**) The energy of a free test particle, and its time evolution in
non-static situations

As in NG, the energy of a mass point
appears first as a natural conserved quantity in the case of time-independent
gravitation potential. We have in general from Eq. (2.1) with *p _{e}* =

(4.9) **g** = - _{} grad_{g}*b* = grad_{g}*U*,
*U* º -* c*^{2} Log *b* .

By the Newton law (3.8) with purely
gravitational force (**F**_{0}=0),
we have also, using Eq. (3.5):

(4.10) _{}

where, by definition,

(4.11) _{},_{} *v* º **g**_{t} (**v**,**v**)^{1/2} , _{}.

To evaluate the rate of work per
unit rest mass, *c***g.**(**dx***/ds*), with **g** from (4.10), we first observe that

(4.12) _{}

and, as in Eq. (3.7), we find that

(4.13) _{}.

Now, just as in the Lemma in [2] (p.
127), one shows easily with (4.11) that

(4.14) _{}.

Accounting for Eqs. (4.12-14), we
thus obtain with (4.10):

_{}_{} ,

or, using (4.11):

(4.15) _{}+_{}.

Combining Eqs. (4.9) and (4.15), we
get

(4.16) _{}.

In the static case (*b*_{,0}=0,
whence *U*_{,0}=0 and **g**_{,0}=0) we
have thus, coming back to the expression (4.9)_{2} of *U*:

(4.17) _{}.

But the expression of the (internal
plus kinetic) energy in SR is

(4.18) *e _{pm} *=

hence we have get the result that
the total energy of the test particle, including its potential energy in the
gravitation field, is simply

(4.19) *e _{m}* =

As in NG, this total energy is lower
than the ‘pure’ energy *e _{pm}*,
in other words the gravitational (potential) energy of

(4.20) _{}+ _{},

(4.21)
_{} + *b g*_{v
}_{} , or _{} + _{}.

For a light-like particle (photon)
one defines *e _{pm}=h*

(4.22) _{}.

This is the same as Eq. (4.21) with *l*=1.

In
summary, if one assumes a Newton law (3.8), then the balance equation (4.21)_{2}
is derived for the total energy *e _{m}*
of any free test particle (mass point or light-like particle), which includes
its potential energy in the gravitation field. Unless

(**ii**) The balance equation of energy for dust

We first consider dust, since we
have an expression of the energy balance for a free test particle [Eq. (4.21)].
Dust is a continuum made of non-interacting mass particles, each of which
conserves its rest mass, so that we may use both Eq. (4.21) and the continuity
equation for the density of rest mass, expressing the conservation of the rest mass.
The latter means that the rest mass *d**m*_{0 }contained
within any given ‘substantial’ domain *dw* of the continuum
is constant, and is thus a statement that can be expressed in terms of any
consistent space and time metric (in so far as the very notion of rest mass is
taken for granted). If we follow the motion of the continuum from the
privileged frame, we can define its velocity **v**_{0} with the absolute time, and we can use the volume
measure *dV*^{ 0} associated
with the natural metric **g**^{0} on *M*, thus

**v**_{0} = **dx***/dt*,*
dV*^{ 0} =_{}*dx*^{1} *dx*^{2}* dx*^{3}.

We then have the usual continuity
equation:

(4.23) _{}, *r*_{00 }º*d**m*_{0}*/ **d**V*^{ 0}, div_{0 }º_{} ,

which leads to the following
expression for a ‘substantial’ derivative:

(4.24) _{}_{0}_{}

(where the scalar product is of
course **g**^{0}; but it is true that

**g**(grad_{g}*y*, **w**)= *y*_{,i}* w ^{i}*

for any metric **g** and
in any coordinates). We apply this to *y* = *bg** _{v}* . According to
Eqs. (4.18) and (4.19),

*d **e _{m}* =

hence *e** _{m}* is the volume density of the energy

(4.25) _{}_{}.

It
is necessary to make contact with the mass tensor **T**, which may be defined for a perfect barotropic fluid as in SR
(and as in GR also), thus (cf. FOCK
[10]):

(4.26) *T*^{ l}* _{m}* = (

*m** º *r**(1 +*P */* c*^{2}), *r** =* r**(*p*),
*P *º _{} ,

(4.27) *u** ^{l}* º

and the same for the *T*^{ lm}^{ }(or *T** _{lm}*)
components, though with

*u*^{l}* u** ^{m}* (or

(4.28) *d**V*_{E} = *d**V*_{C}_{ }*/**g** _{v }*,

(note that the observer in E uses the ‘true’ simultaneity, hence he finds
the same relation *d**V*_{E} = *d**V*_{C}_{ }*/**g** _{v }*if he simply
remembers that the measuring rods of the observer in C are
‘truly’ Lorentz-contracted in direction

(4.29) *d**V*^{ 0} = *b d**V*_{E}_{ }, *r*_{00} = *r*_{0}*/b* = *g*_{v}*r***/b* .

Now, from Eqs. (4.26) and (4.27), we
have for dust (*p*=0, *P*=0):

* **r** *g*_{v}^{2 }= (*T*^{ 0}_{0})_{E}_{ }, *r** *g*_{v}^{2 }*v*_{0}^{i}*/c* = (*T*^{ i}_{0})_{E} (*x*^{0}=*ct*),

whatever space coordinates bound to E are used; indeed, one has always *g*_{0i }=0 in such coordi-nates
[see Eq. (2.8)], hence *u*^{0 }*u*_{0 }= *g*_{00}(*u*^{0})^{2 }=*g*_{v}^{2}. By
(4.29), we have *r** *g*_{v}^{2 }=* r*_{00 }*g*_{v}*b* = *e** _{m}* , hence we may
rewrite the balance equation (4.25) as:

(4.30)
_{}_{}

where *x*^{0}=*ct*, the
space coordinates are bound to E, and the space-time metric **g**^{0} is
defined by

(*ds*^{0})^{2}=(*dx*^{0})^{2}-(*dl*^{ 0})^{2 }

with *dl*^{ 0} the line element of the natural space metric **g**^{0}; the
first identity is valid in any
coordinates linearly bound to Galilean coordinates of the metric **g**^{0} [if (*M*, **g**^{0}) is Euclidean.
Except for the first identity, Eq. (4.30) is yet valid also if (*M*, **g**^{0}) is a space with
constant curvature].

Equation
(4.30) [or (4.25)] still contains the two source terms on the right-hand side.
If it transforms into a true conservation equation, this must include the
energy of the gravitation field itself. As
in NG, we have to use the field equation in order to replace the energy
density by a combination of derivatives of the potential [here *f*º*b*^{ 2}, with
*b**=p _{e}/p_{e}*

(4.31) _{}_{}

[where the scalar square **g**^{2} is in terms of the
natural metric, **g**^{2}=**g**^{0}(**g**,**g**)].
This strongly recalls the ‘pure gravitational’ terms in the Newtonian Eq.
(4.2), though with an additional term (*f*_{,0}*/f*)^{2}. Now we observe that the
first source term on the right of Eq. (4.30) can be written as

(4.32) ** _{}**.

Hence, if we interpret the
mass-energy density *r* as *T*^{ 00}, and if we demand that Leibniz’ rule applies to the
definition of a vector time derivative, i.e. *l*=1/2 in Eqs.
(3.5) and (3.8), then we obtain for dust, from Eqs. (4.30-32), the following *conservation equation* *for the energy*:

(4.33) _{}.

As will be seen, the interpretation
of *r* as *T*^{ 00} means that the
gravitation field definitely reinforces itself and is thus very plausible.
Other possible interpretations: *r* = *T*^{ 0}_{0} or *r* = *T*^{ }_{00}, certainly do
not allow to rewrite the energy balance (4.30) as a true conservation equation
if *l*=1/2
[because one then adds to Eq. (4.33) the term

*f*_{,0}(*T*^{ 00}- *r*)/2,
which is clearly not a 4-divergence]. Finally, if a different value is assumed
for *l*, not
only Leibniz’ rule fails to apply, but the mixed source term [the last term in
Eq. (4.30)] makes it more than unlikely that a conservation equation could be
obtained for the energy. Thus *we assume*
*l*=1/2 *and* *r* = *T*^{ 00} *from now on*.

(**iii**) Extension of the energy conservation equation to general
matter behaviour

For general matter behaviour, the
constitutive material particles interact with each other and do not necessarily
conserve their rest mass. Still, other conservation laws of microphysics hold
true in rather general situations, in particular that for the baryon number
which was proposed by CHANDRASEKHAR [8]
as a substitute for the mass conservation. However, it is considered here that
no conservation law is more fundamental than that for energy. Equation (4.33)
applies in the absence of gravitation, i.e. in SR, with **T** the mass tensor of any kind of matter and field {incidentally,
matter and non-gravitational fields would be of similar nature, i.e. all these
would be microscopic flows in ether, according to the concept of constitutive
ether (cf. ROMANI [24])}. In the presence of gravity, Eq. (4.33) has
been derived, for dust, from the assumed conservation of the rest mass. We
therefore postulate that Eq. (4.33) holds true with **T** the (mixed) mass tensor of any kind of matter and
non-gravitational field, in the presence of gravitation. Since the total
energy, not the rest mass, is thereby conserved, this postulate contains the
possibility that (depending on the constitutive equation) *matter can be created or destroyed by its interaction with a variable
gravitation field.*

* *

(**iv**) Some local and global consequences of the conservation equation

The density of pure gravitational
energy (or rather of its mass equivalent), with respect to the natural volume
measure *dV*^{ 0}, appears
clearly in Eq. (4.33); it is

(4.34) _{},

where _{}º *dl*^{ 0}*/dt* = *cf* [cf. Eqs. (2.5) and
(2.7)] is the ‘absolute’ velocity of light (and that of waves of the pressure *p _{e }*, i.e. gravitation waves)
in the direction

(4.35) *T*^{ 0}_{0}=*e** _{m}* ( =

which also is relative to *dV*^{ 0}, and which enters the
balance equation (4.33), includes its ‘potential’ energy in the gravitation
field [cf. Eqs. (4.18-19)]. Let us examine, in the static case, what follows
from assuming that *r* on the r.h.s. of the field equation
(2.15) is

*T*^{ 00 }=*T*^{ 0}_{0}*/b*^{ 2} (as
is imposed if one admits that a conservation equation must exist for energy).
Since the density of the ‘pure’ energy of matter, i.e. without its potential
energy, is

*e** _{pm}* =

*r *= *e*_{pm}*/b*. Due to the ‘Poisson equation’
(2.15) with *f*_{,0}=0, Eq.
(2.16) determines in the general static case the ‘active mass’ *m*, giving the expression (2.9) of the
space-time metric: at large distance from the massive body (which is unique,
since any other body would fall, and which is at rest in the ether), we have
approximate spherical symmetry, hence

(4.36) *f *= 1- 2 *Gm/*(*c*^{2}*r*) + *o*(1*/r*)

(*r*
being the Euclidean distance from the body), with

(4.37) *m* = ò_{body} *r* *dV*^{ 0}
= ò_{body} (*e*_{pm}*/b*)* **dV*^{ 0}

This means that the density *e** _{pm}* of the ‘pure’
(internal plus kinetic) energy is reinforced, as regards its contribution to
the ‘active gravitational’ mass

*r *= *T*^{ 00},

we would assume

*r *=*T*^{ 0}_{0} or *r *=*T*^{
}_{00},

we would get *e*_{pm}*b* or *e*_{pm}*b*^{ 3}
in the place of *e*_{pm}*/b* in Eq. (4.37), i.e. the gravitation field
would have a *weakening* effect on
itself. But the notion of active mass, especially its comparison with the
passive gravitational mass [which is also the inertial mass, and still
coincides with the pure energy of matter, cf. the Newton law (3.8) and Eqs.
(4.18-19)], implies a linearity of the acceleration field **g** with respect to the amount of matter. This amount is best
represented by the pure energy of matter, thus its density *e** _{pm}* does not
coincide with

*r *=* e*_{pm}*/b*], hence the same is true for the
vector **g** [Eq. (2.12)]. One therefore
cannot isolate the gravitation force exerted by a subdomain *W* on a
mass point, i.e. the contribution of *W* to the field **g**. Hence, an actio-reactio principle cannot even be defined for the
gravitation force, even in the static case. In summary, the studied theory
implies (at least in the static case) that the gravitation field really
reinforces itself, and this forbids to define the active gravitational mass for
a subdomain of the system. As to the global active mass, it may be defined, at
least in the static case, and it is then greater than the sum of the pure
energy of matter, for the same reason.

In
the more general situation of an isolated, but not necessarily static matter
distribution, embedded in *Euclidean*
space *M*, the integration of Eq.
(4.33) over the whole space and its transformation is possible, under the
sufficient condition that (*f*_{,0})^{2}
is *o*(1/*r*^{3}) and **g** is *O*(1*/r*^{2})
at large *r* (with *r* the current Euclidean distance from some point in the group of
bodies, and using the fact that *f* »1 at
large *r*. In assuming that the
behaviour at large *r* is as in NG, one
indeed would expect this decrease of **g**
like 1*/r*^{2}; actually, this
is not the general case, due to the gravitation waves). Using Gauss’ theorem on
the sphere *r=R* and making *R* tend towards +¥, we
obtain then

(4.38) *W _{m}* +

which is the same as Eq. (4.3),
though here *e** _{m}* and

(4.39) _{}_{}.

At large *r*, we have Eq. (4.36) plus **g
**» - (*G m /* *r*^{2}) **e*** _{r}* (with

*W _{g}* º ò

Since *e*_{m}* =** r** f* = *e*_{pm}*b* (= *T*^{ 0}_{0}), we can thus
rewrite the conserved energy, in the static situation (which may be the case
for the initial, as well as for the final state), as

(4.40)
*W *= _{}ò_{body} *e*_{m}*dV*^{ 0} + _{}ò_{body} _{} *dV*^{ 0}.

(4.41)

It is smaller than the active mass *m* [Eq.(4.37)], and the latter does not
need to remain the same in any couple of initial and final states, both being
assumed static. Furthermore, and in contrast with NG [see the discussion after
Eq. (4.4)], the structure of Eq. (4.40) does not seem to imply that the pure
energy of matter,

ò_{matter} *e*_{pm}*dV*^{ 0} ,

is necessarily increased as the pure gravitational energy *W _{g}* is increased, i.e. in the
case of gravitational concentration of matter.

**5. Conclusion**

Although less immediately than in
the case of a static gravitation field, a consistent Newton law can still be
defined in the general situation within the present theory (and in fact in any
theory which, in a reference frame, provides us with a space metric and a time
metric). It is consistent in that it is based on a true vector derivative of
the momentum, obeying Leibniz’ rule with the variable metric **g** (here
**g** is the physical space metric in the privileged frame
bound to ether); moreover, this time derivative coincides with the usual
absolute derivative in the case of a time-independent metric **g**, i.e.
in the static case. The requirement that Leibniz’ rule must hold true permits
to select the relevant vector derivative, i.e. *l*=1/2 in Eq.
(3.5), from a one-parameter family of candidates. Among the possibilities which
are thereby eliminated, the value *l*=1 is likely to correspond to
Einstein's motion along space-time geodesics; anyhow,* l*=1
does correspond to geodesic motion in the important case where ‘isopotential’
coordinates bound to ether still exist (see the definition near Eq. (2.7); this
case includes spherical symmetry around a point bound to ether). A general
Newton law is important for the status of the theory, since it means that the
acceleration field **g** [Eq. (2.1)]
keeps a direct physical meaning in the most general case. Thus the above result
implies a crucial departure of the theory from the ‘logic of space-time’. The
latter is the commonly accepted *interpretation*
of SR, even though the ‘logic of absolute motion’ can be vindicated as well in
SR [4, 6-7, 11, 21-23]. The logic of space-time is, however, *essential* in GR, and even also in its
various formulations or modifications as a field theory in flat space-time
(including the theory proposed by LOGUNOV *et al.* [14]).

It
seems therefore natural to hesitate before taking the new direction at this
bifurcation in the studied theory. But another argument leads to the same
choice, and this is not quite a detail: if one wants to have a true
conservation equation for energy in this theory, one also must assume the
Newton law (with Leibniz’ rule, thus* l*=1/2) and hence forget space-time
geodesics. One then indeed obtains for energy a local balance equation without
source term, and this is in terms of a flat space-time metric. The
gravitational energy is thereby unambiguously defined (in contrast with GR).
Just like in NG, one must recognize that one part of gravitational energy is
embedded in matter, as its (negative) potential energy in the gravitation
field, while the other (positive) part is present in the whole space and may be
called the pure gravitational energy. The expression of the latter in the
present theory [Eq. (4.34)] contains just the Newtonian expression, plus a term
bound to the fact that any time-dependence of the gravitation field implies a
time-dependence of the local space and time standards: according to the studied
theory, such variation demands an energy supply. This feature of the theory
will be worth to discuss in connection with the question of gravitation waves.
When the local balance equation can be integrated (which is not the case if
gravitation waves are filling the space), it implies the conservation of the
global energy [Eq. (4.38)], which is the sum of the energy of matter (including
its negative potential energy) and the pure gravitational energy. The
gravitation field does reinforce itself in a direct sense [Eq. (4.37)].

According
to the present ether theory, gravitation would have some rather concrete
aspects. But it is once again emphasized that the theory is non-covariant,
which is a risk as regards its application to celestial mechanics. One thus has
to study what would be the effect, say on the motion of a planet considered as
a test particle, of a *uniform* motion
of the attracting body. This effect will be at most in *u*^{2}*/c*^{2}
with *u* the corresponding constant
velocity. It is therefore expected that the perturbation of the Newtonian
analysis is small enough so that the theory, at least, is not *worse* than NG for mechanics of the solar
system, except perhaps for time scales beyond the ‘horizon of predictibility’
implied by chaotic behaviour of N-bodies problems. It might happen, however,
that (due to the effect of the velocity *u*),
the theory could fail to account for such very small effects as the advance of
perihelion of Mercury, i.e. it could fail to improve NG in that respect (the
motion of the perihelion is very sensitive to almost any kind of perturbation).
As to the gravitational red shift, the deflection of light rays and the delay
of radar signals, it seems plausible that the effect of the velocity *u* does not change significantly their
magnitude, which would thus be correctly predicted. The writer has already
verified this point as regards the red shift.

ACKNOWLEDGEMENTS

I am very grateful to Profs. P.
Guélin and E. Soós for discussions which helped me to realize that a consistent
energy concept is even more necessary in a non-covariant theory like the
present one.

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Laboratoire
‘Sols, Solides, Structures’ [associated with the Centre National de la
Recherche Scientifique], Institut de Mécanique de Grenoble, B.P. 53 X, F-38041
Grenoble cedex, France.

[1]
However, the whole theory can be written if *M*
is more generally assumed to be equivalent (isometric) to either of Euclidean
space **E**^{3}, (hyper)sphere **S**^{3}, or Lobatchevsky space **L**^{3}, i.e. *M* is a space with constant curvature, be
it nil, positive or negative: in either case, *M* is equipped with a natural metric **g**^{0} which admits a 6
parameter group of isometries.

[2]
Formally speaking, one should start with *M*^{
4} and admit privileged space and time projections: *M*^{ 4} ® *M* and *M*^{
4} ® **R **making *M*^{
4} isomorphic to **R**x*M* in a canonical way. This is the ‘Newton-Lorentz
universe’ considered by Soós [26].

[3] A (generally deformable) frame F can be defined as a time-dependent diffeomorphism *y** _{t}* of

[4] This assumption is in fact less
restrictive than that of an ‘insular matter distribution embedded in a Galilean
space-time’, which is commonly set in GR (e.g. FOCK [10], LANDAU & LIFCHITZ [12]). In particular, does not actually need to be
reached somewhere, even asymptotically [3], and thus can be constant even if
the matter distribution has unbounded support.

[5] In the case where (*M*,**g**^{0}) would
be instead a space with constant curvature, one also might identify vectors in *TM _{X } *and in

[6] A sufficient condition is: *q ^{a}*

[7] Express the Newton law in terms of *D*_{1/2} (*e _{pm}*