**1. The
Lorentz-Poincaré version of special relativity**

**3.
References quoted in parts 1 and 2
**

**4. ****A naive introduction
to the scalar ether-theory of gravitation*** *

**To a review on
ether theories of gravity, with the scalar theory reviewed in more
detail***
*

The currently accepted theory of gravitation is
Einstein's general relativity (1916),
which arose after a decade of research trying to make gravitation
compatible with relativity. Once the heart of the theory that we now
name special relativity had been found in 1905, it was indeed obvious
that Newton's gravitation was not compatible with this theory. The
first attempt to modify Newton's theory in order to make it compatible
with special relativity was made by Poincaré (1905 , 1906):
he formulated a Lorentz-invariant modification of Newton's attraction
law. Later he found (Poincaré 1908)
that this modification accounted for just one sixth of Mercury's then
unexplained residual advance in perihelion, of 43'' per century.

1. The Lorentz-Poincaré version of special relativity

Poincaré's version of the relativity
principle was
different from Einstein's (1905),
indeed Poincaré's version (1904)
was this: "The laws of physical phenomenons must be the same,
whether for a fixed observer, as also for one dragged in a motion
of uniform translation, so that we do not and cannot have any mean
to discern whether or not we are dragged in a such motion." This
formulation
does not preclude in any way the possibility that there might be an
absolute meaning to the words "fixed" and "moving". Like Lorentz,
Poincaré did not actually reject the concept of an "ether" [see
e.g. Poincaré (1908)]. The
concept of an ether, or physically preferred reference frame, had been
dominating the 19th century physics after the discovery that light
behaved as waves. (It was then considered difficult to think of waves
without a medium to carry them: is it easier now?) Thus the
Lorentz-Poincaré version of special relativity starts from the
"ether", seen as an inertial frame E such that (i) Maxwell's equations
are valid in E (in particular, light propagates at a constant velocity *c*
with respect to E)
and (ii) any material object that moves with respect to E undergoes a
Lorentz contraction. One derives first the Lorentz transformation from
these assumptions, and then the whole of special relativity follows.
This theory (Lorentz 1904 , Poincaré 1905 , Poincaré 1906) has been
thoroughly
investigated by Jánossy ( 1957
, 1965 ), Builder
(1958), Prokhovnik
( 1967 , 1993 ), Pierseaux
(1999) , Brandes (2001), among
others, who all showed that it is
physically equivalent to the textbook version of special relativity.
[The
latter is, of course, derived from the work of Einstein (1905) starting
from (i) the relativity principle and (ii) the invariance of the
velocity
of light.] Indeed both theories lead to the same equations: the Lorentz
transformations for coordinates and for velocities, the relativistic
dynamics with velocity-dependent mass, the transformation of the
electromagnetic field, etc. The crucial equations are already
explicitly written in
Poincaré (1905), the other
ones are in Poincaré (1906).
It is immediate to see this for the Lorentz transformations, and for
the
rest it appears very clearly if one changes Poincaré's notations
and adopts current notations, as was shown by Logunov (1995). Apart from the
historical aspect, it is important for physics research to realize
that the physics of special relativity was found by scientists who
based
their work on the concept of ether. Indeed today's physics makes it
obvious
that vacuum has physical properties.

The Lorentz-Poincaré version sees the key
"relativistic" effects (space contraction and time dilation) as real
physical effects of the motion with respect to the "ether": a moving
ruler is just "really" smaller than a fixed one, a moving clock just
has a "really" greater period than a fixed one. Here, "really" means:
with the simultaneity defined in the preferred inertial frame or ether
E, by using the standard synchronization procedure with light signals
[first proposed by Poincaré (1900),
discussed in more detail in Poincaré
(1904), and rediscovered (?) by Einstein
(1905)]. The reciprocity of these metrical effects (when
interchanging the respective roles of two inertial frames) is then seen
as an illusion due to the necessity of still using the
Poincaré-Einstein synchronization convention in a moving frame:
As long as we don't know the velocity **V** of our inertial frame F
with respect to E, we are obliged to admit that, also in F, light has
the same velocity in both directions **n** and **-n**, for
whatever **n** (that velocity must then be the constant *c*).
What experiments like Michelson-Morley's ask for, and what the rod
contraction and clock slowing say, is merely that the velocity of light
*on a to-and-fro path* is the constant *c*. *In summary:*
according to the Lorentz-Poincaré interpretation of special
relativity, the metrical effects of uniform motion are seen as absolute
effects due to the motion with respect to the preferred inertial frame,
and the relativity of simultaneity is just
an artefact. That artefact leads to the reciprocity of the
"relativistic" effects, which reciprocity is thus itself an artefact.

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top of this page (Lorentz-Poincaré relativity and a scalar
theory of gravitation)

2. Why is the Lorentz-Poincaré version of special
relativity important for the theory of gravitation?

If all of physics is really Lorentz-invariant, we
shall never be able to measure the "absolute" velocity **V**, so
that the difference between the "true" simultaneity (that defined in
the preferred inertial frame E) and the "artificial" ones (those
defined in the other inertial frames F) will remain at the metaphysical
level. In that case, one may indeed qualify the concept of ether as
"superfluous", as Einstein (1905) put
it. His own starting point, from the relativity principle, is then to
be preferred. Moreover, the full equivalence between all inertial
frames, obtained in that case, leads to give a physical meaning to the
concept of space-time, which was Minkowski's
(1907) assumption. [In his "Rendiconti" paper, Poincaré had already
introduced the concept of the space-time, as a 4-dimensional space with
coordinates *x*, *y*, *z*, *ict *; but this
looked more like a (very useful and clever) mathematical tool in his
paper.] This means that past, present and future are relative notions,
and that time travels are at least theoretically possible. It also
enforces to consider this mixture of space and time as the true arena
for physical theories. In particular, it may lead (though not in a
compelling way) to Einstein's general relativity (GR), according to
which gravitation is the curvature of space-time, and in which free
test particles follow the geodesics of a pseudo-Riemannian metric on
space-time. [If general relativity is correct, then, according to Bonnor (2002), time travels are indeed
quite plausible.] In my opinion, this is no physical explanation of
gravity. Moreover, GR's invariance under arbitrary coordinate changes
is antinomic to quantum theory, in which, in particular, the choice of
the *time* coordinate cannot be arbitrary. This leads to the
longstanding difficulties with quantum gravity, which the mainstream
researchers want to overcome by going into string- and M-theories with
their many-dimensional manifolds. One should be allowed to explore
simpler possibilities.

There is indeed another possibility, which seems to have been little explored before: that not all of physics is subjected to the relativity principle, and that the force which contradicts relativity is precisely gravitation. (Of course, the conflict with relativity has to be small in usual conditions.) This possibility exists only in the Lorentz-Poincaré version of special relativity (SR), that sees SR as a consequence of the Lorentz contraction: it obviously doesn't exist if the reason for SR is the universal validity of the relativity principle, as in Einstein's version. Why should gravitation be the range of a such violation? Simply because SR really doesn't include gravitation. After all, even in GR, the relativity principle of SR does not hold true.

The scalar theory investigates just this
possibility. It
does have a physical explanation for gravity, it is much simpler than
GR,
yet it seems to agree with observational facts. Rigorous work has been
done,
and further work is in progress, to check this in celestial mechanics,
and
the results of this work are, in my opinion, of importance also for GR.

3. References quoted in parts 1 and 2

I am grateful to the Institutions that implemented the online versions of the original works, and to Emili Bifet, who provided the links to those online versions.

W. B. Bonnor (2002) Communication at
the Second British Gravity Meeting, Queen Mary University of London,
June 10/11, 2002; http://arxiv.org/abs/gr-qc/0211051.

J. Brandes (2001) *Die
Relativistischen Paradoxien und Thesen zu Raum und Zeit*, Verlag
Relativistischer Interpretationen, Karlsbad.

G. Builder (1958) *Austr. J. Phys.*,
**11** , 279-297 and 457-480.

A. Einstein (1905)* Ann. der Phys.*, (4) **17**,
891-921 (Eingegangen 30. Juni 1905).

A. Einstein (1916) *Ann. der Phys.*,
(4) **49**, 769-822.

L. Jánossy (1957) *Usp. Fiz.
Nauk*,* ***62**, N°1, 149-181.

L. Jánossy (1965) *Acta
Phys. Polon.*,* ***27**, 61-87.

A. A. Logunov (1995) *On the
Articles by Henri Poincaré "On the Dynamics of the Electron"*
(English translation G. Pontecorvo), Publishing Department of the Joint
Institute for Nuclear Research, Dubna. (1st Russian edition 1984.) This
monograph includes an English translation of Poincaré 1905 and
1906.

H. A. Lorentz (1904) *Proc.
Acad. Sci. Amsterdam*, **6**, 809- . German translation: H. A.
Lorentz, A. Einstein, H. Minkowski, *"das
Relativitaetsprinzip"*, Teubner, Leipzig u. Berlin, 1913, pp. 6-26.

H. Minkowski (1907) *Nachr.
Kgl. Ges. d. Wiss. zu Goettingen, Math.-phys. Kl.,* Sitzung vom 21.
Dezember 1907.

Y. Pierseaux (1999) *La
"Structure Fine" de la Relativité Restreinte* (L’Harmattan,
Paris).

H. Poincaré (1900) *Arch.
Néerland. Sci. Ex. Nat.*, (2)* ***5**, 252-278
(1900). Reprinted in *Oeuvres Complètes, tome IX*
(Gauthier-Villars, Paris, 1954), pp. 464-488.

H. Poincaré (1904) *Bull.
Sci. Math.*, (2) **28**, 317-, November 1904. Reprinted in *La Valeur de la Science*
(Flammarion, Paris, 1905), chapters 7,8 and 9. English translation:
*Bull.
Amer. Math. Soc.* **37** (2000), 25 - 38. (Several
books of Poincaré may be accessed on line - in French - at
http://www.ac-nancy-metz.fr/enseign/philo/textesph/default.htm#poinca).

H. Poincaré (1905) *C.-R. Acad. Sci. Paris *(Séance du 5
Juin 1905) **140**, 1504-1508. (The online version is due to
the Bibliothèque Nationale de
France.) Reprinted in *Oeuvres Complètes, tome IX*
(Gauthier-Villars, Paris, 1954), pp. 489-493.

H. Poincaré (1906) *Rendiconti
Circ. Matemat. Palermo *(Adunanza del 23 Luglio 1905),** 21**,
129-176. Reprinted in *Oeuvres Complètes, tome IX*
(Gauthier-Villars, Paris, 1954), pp. 494-550, and also in *La Mécanique Nouvelle*
(Gauthier-Villars, Paris, 1924), pp. 18-76. (The online version is
due to the Bibliothèque Nationale de France.) Partial English
translation available at

http://www.univ-nancy2.fr/poincare/bhp/pdf/hp1906rpen.pdf

H. Poincaré (1908) *Rev.
Gén. Sci. Pures et Appl.*, **19**, 386-402. Reprinted in *Oeuvres
Complètes, tome IX* (Gauthier-Villars, Paris, 1954), pp.
551-586.

S. J. Prokhovnik (1967)* The
Logic of Special Relativity* (Cambridge University Press, Cambridge,
U.K.).

S. J. Prokhovnik (1993) *Z.
Naturforsch.*, **48a**, 925-931.

To the
top of this page (Lorentz-Poincaré relativity and a scalar
theory of gravitation)

4.** **A naive introduction to the scalar ether-theory of
gravitation

*b* ) *Archimedes'
thrust
in the micro-ether*

*c *) *The
micro-ether as the universal fluid. The velocity of light and the
velocity
of "sound"*

*d *) *The
macroscopic nature of the fields *p_{e}*
and *rho_e

*e* ) *The
macro-ether* *or preferred reference frame of the theory*

* f* ) *The
limiting case of Newtonian gravity** and the effect of
the compressibility of the fluid*

* g *)
*Provisional conclusion and acknowledgment*

The basic idea of this theory is that gravity could be just
Archimedes' thrust exerted on matter at the scale of elementary
particles by a perfect continuous fluid, which I call the micro-ether
(you will see at point *e* ) why
"micro"). The fluid is assumed continuous at any scale. It means that
its fluid nature is not the resultant of a microscopic substructure
made of a myriad of molecules that move in all directions, but is
intrinsic. In contrast, the liquids or gases that we know are made of
molecules. The pressure exerted by them may thus be interpreted as
resulting from shocks of those tiny particles. I.e., we now know that
it just turns out to be so for the fluids we meet every day: water,
air, wine, etc. However, mind that the molecular nature of these fluids
is known since hardly more than a century. During several centuries, it
was considered that all of these fluids are continuous, and that did
not make any problem in interpreting their mechanical behaviour, which
is the behaviour of interest here. I mean that from the *mechanical*
point of view, fluids might be continuous as well. For sure, the
intuitive notion of a fluid refers to something continuous, may be it
is the most obvious representation of what "continuous" means. What
makes the molecular structure of usual fluids necessary is their *thermal*
behaviour. An usual fluid may be cold or hot, i.e., it may take energy
from neighbouring bodies or bring energy to them, and we know that this
is due to the random motion of the constitutive molecules: the
temperature of the fluid is
related to the mean kinetic energy per particle. By contrast, we may
infer
that a truly continuous fluid has no temperature and does not allow any
heat exchange, hence it must be a perfect fluid, with merely two state
variables: pressure and density, which are connected together by a
one-to-one relationship -- the barotropic state equation.

*
b* )

Let us assume that the void between the molecules of ordinary
matter, or rather (since molecules are composite objects) between the
elementary particles: electrons, quarks, etc., is not empty in fact,
but is filled by a such continuous fluid. Because a such fluid should
be a perfect
fluid, the motion of the particles is not braked by the presence of
this fluid (this is well-known in fluid mechanics as "d'Alembert's
paradox",
but it is not a paradox actually). At least, this is so for a uniform
motion. As to accelerated motion of a particle, it does feel the fluid,
in a way that might give insights on the relativistic mass increase
with
velocity -- but that is another story. Also, the particles are not
point
particles (a point particle does not make any sense, it seems to me),
instead each has a finite size. In that case, *if there is a gradient
of the fluid pressure, say more pressure upwards, then any
particle
will be pushed downwards, *with a

*
c *)

We know from accelerators experiments that existing particles,
even the elementary ones (electrons, photons, quarks, ...), can be
transformed into other ones, e.g. electron-positron annihilation giving
rise to
photons. We also know from quantum mechanics that there is some
fuzziness
in their position and velocity, and that there is some indiscernability
between identical particles. These facts would be easier to understand
if actually all particles were just local organizations of the
hypothetical
fluid which I assumed above to be between them. Thus, an
elementary
particle would be a kind of localized flow in the micro-ether,
something
like a vortex. (A vortex may be everlasting in a perfect fluid;
however,
note that particle physics says that there are much more unstable
particles
or "resonances" than stable ones: this also is easy to understand with
this
assumption.) In other words, *the micro-ether would be the
universal, "constitutive" fluid, of which any matter should be made. *If
we assume this, it becomes just natural that the mass density inside a
particle is the same (or nearly so) for all particles -- the
more so, if this hypothetical universal fluid has a very low
compressibility: the common density inside the
particles would then be just the nearly uniform density *rho_e* in the universal fluid. The mathematical form of
the relation between the (field of) ether pressure *p** _{e}*
and the gravity acceleration

**g** = `-` (grad *p** _{e}*)/

Actually, relativity suggests strongly that the
compressibility of the micro-ether should be *K* = 1*/c*^{2}
with *c* the velocity of light, which is indeed an extremely
small compressibility. The latter equality means exactly that the
"velocity of sound" (the propagation speed of the pressure waves) in
this fluid, say *c _{e}*, is equal to

* d *)
*The macroscopic nature of the fields *p_{e}*
and *rho_e

Gravitation is a macroscopic force, in the sense that the
gravity acceleration **g** varies significantly only over
macroscopic distances, and due to the presence of massive, macroscopic
bodies (although **g** is felt by small particles). This is related
to the fact that gravity
is a very small force as compared with the other forces
(electromagnetic
and nuclear), which enormously dominate at small distances. According
to
the concept of the micro-ether, gravity is indeed a kind of *correction*:
the fields *p** _{e}* and

* e* ) *The
macro-ether* *or preferred reference frame of the theory*

The ether should account for the inertial frames which are
revealed by Newton's mechanics. The inertial frames of classical
mechanics lead to Newton's concept of the absolute space: as is
well-known, there is just one inertial frame, up to a uniform
translation. (Rotation, in particular, is detectable by experiments
such as Foucault's, but it is also the case for any kind of *deformation*
of the selected reference frame with respect to an inertial frame [A8].) For Newton,
this could be true only if one of the inertial frames had a privileged
status. Thus the ether should define that preferred inertial frame, for
otherwise we would have two independent privileged spaces: that of
Newton's theory, plus the ether. The universal fluid which I postulate
(following Romani), and which I name micro-ether, is assumed to have a
complex microscopic motion, involving in particular those motions which
should define the elementary particles of matter. Hence the
micro-ether, whose motion thus includes that of any kind of matter,
cannot define an inertial frame.

However, the average motion of matter at a very large scale
defines the best approximation of an inertial frame. Note, for
instance, that the astrometrical reference frames are kinematically
defined from distant "astronomical pointers". These are material
objects, including very distant quasars which have no detectable
motion, and also including usual stars, which are closer objects, whose
motion is indeed averaged. And those kinematical reference frames are
excellent dynamical reference frames, i.e., they are very-well adapted
to write and solve the equations of celestial mechanics -- in other
words, they are good inertial frames. (The plural is to account for the
successive improvements of the astrometrical reference frames.) Now, in
the
line of the concept of the constitutive micro-ether or universal fluid,
it
is natural to assume that the average motion of the micro-ether
coincides with the average motion of matter. Therefore, the average
motion of the micro-ether
should define an inertial frame, which I call the macro-ether. It is in
that reference frame that the equations of the theory, most of which
are
indeed merely space-covariant, are assumed to be true.

* f* ) *The
limiting case of Newtonian gravity** and the effect of
the compressibility of the fluid*

Thus, gravitation would result from the
macroscopic part of the pressure gradient in the micro-ether.
We expect
then that, if that fluid is (macroscopically) compressible, a
disturbance
in the (macroscopic) ether pressure should propagate with the "sound"
velocity,

*c*_{e} = [*dp*_{e}/d(*rho_e*)]^{1/2}.
(2)

On the other hand, we know that Newton's gravitation propagates instantaneously. Therefore, it should correspond to the limiting case of an incompressible fluid. The latter should be an excellent approximation in many cases, because Newton's gravity is already very accurate in certain situations (in particular at the solar-system scale!). Newton's gravity is characterized by Poisson's equation for the gravity acceleration**g**:

with*pi* the usual number of trigonometry, *G* Newton's
constant and *rho* the mass density of matter. (div is
the divergence operator.) Together with Eq. (1) for **g**,
the requirement that Poisson's equation (3) is
recovered in the incompressible case, i.e.,
*rho_e* = Const., leads immediately to an equation for the field *p*_{e}
:

*Delta* (*p*_{e})
= 4 *pi G rho* *rho_e,
*(4)

On the other hand, we know that Newton's gravitation propagates instantaneously. Therefore, it should correspond to the limiting case of an incompressible fluid. The latter should be an excellent approximation in many cases, because Newton's gravity is already very accurate in certain situations (in particular at the solar-system scale!). Newton's gravity is characterized by Poisson's equation for the gravity acceleration

div(**g**) = `-`4 *pi G rho*
(3)

with

where *Delta* = div grad is the Laplace operator.
It is natural to assume that this equation remains valid for the
"real" case with a compressibility, in the particular situation where
the effects of propagation can be neglected -- i.e., in the static
situation (e.g.
one unique massive body, possibly with "test particles" that do not
affect
the attraction field). Thus, in that case, *rho_e*
in Eq. (4) becomes a function of the local ether pressure, *rho_e* =
*rho_e *(*p _{e}*). However,
in the general situation for the compressible case, there should be
pressure waves, thus

* g
*) *Provisional conclusion and acknowledgment*

I have tried to expand on the heuristic concept of the theory
that I am investigating. It seems to me that this concept makes
gravitation
understandable. I could go further, but the development of the theory
naturally
leads to the equations playing a more and more important role.
Equations
and their *derivable* consequences are the most important thing
in
a theory, from the point of view of *physical science* (as
opposed
to "intuitive physics"). From this point of view, the heuristics
developed
in points *a*) to *e*) is just a way to the
beginning
of the scalar theory, in particular to the expression of **g**.
(There
is indeed another, phenomenological, way to the expression of **g**
:
see [A16],
§ 3.) The interested reader might have a look on the guide below.
Although most papers are rather technical, several ones contain an
important amount of non-technical comments, especially [A18],
[A28],
[B13],
and [B17].

Many persons expressed encouragements to this theory and/or made
interesting remarks about it. They are too numerous to be quoted here.
However, I wish
to express my particular gratitude to the late Eugen Soos and to
Pierre
Guélin, both of whom gave me a lot of their time, and to my wife
Marie-Alix Arminjon, who is enduring since quite a lot of years my
devotion to this curious activity.

5. A
short guide to the new scalar theory of gravitation

a) *To get an idea
*b)

c)

g)

In short, this is a preferred-frame bimetric theory, in which the
scalar gravitational field both determines the gravity acceleration
vector and influences the metric: there is a gravitational dilation of
physically measured distances, as compared with distances evaluated
with the Euclidean space metric; and there is a gravitational
contraction of measured time intervals, as compared with the intervals
of the "absolute time" (the preferred time coordinate of the theory).
This gives a "curved space-time", as in GR, but here with a simple
origin, and together with a flat space-time (depending on which of the
two metrics is chosen). The dynamical equations are based on a
consistent formulation of Newton’s second law in a curved space-time.

Ref. [B25] is a short, readable summary of the construction of the theory, in the context of some modification whose necessity appeared rather recently [see point c) iii) below]; a more extended summary of that modified theory is Ref. [B26]. A quick technical presentation of the main equations of the initial theory and its first test in celestial mechanics (spring 2002) can be found in Ref. [B21]. The first part of Ref. [A28] gives a rather detailed summary of the motivation for the theory and the construction of it (Sect. 2; a more incisive introduction is Ref. [B17]); the status (at summer 2001) of the observational confrontation is then reviewed (Sect. 3 of [A28]); afterwards the application of the theory to cosmology is discussed. In those three papers, reference is made to the earlier work.

b)
*The development of the theory; the dynamics in the theory*

The bulk of the theory was presented in Ref. [A18],
including the interpretation of gravity as a pressure force and the
derivation of the field equation, and together with some applications -
that to the spherically symmetrical case includes the gravitational
collapse of a dust sphere. (Ref. [A8]
presented an extension of Newtonian mechanics to fluid inertial frames
and described in more detail the beginning of the construction of the
theory ; Ref. [A9]
described in detail a further step of this construction and the first
test of the theory.) In Ref. [A15],
the
question of the equations of motion in the theory (geodesics or
Newton's
2nd law?) is reexamined in the context of obtaining a conservation law
for
energy, and a simpler form of the gravitational equations is got,
moreover
the r.h.s. of the equation for the scalar gravitational field is
precised.
(Due to the review process, Ref. [A18] was published after [A15].)

The link
between Newton's second law and Einstein's geodesic motion has then
been
found in a general context [A16]. In
Ref. [A20],
the equation for continuum dynamics (replacing the equation of GR) is
derived, and applied to show the occurrence of matter creation or
destruction by a reversible exchange with the gravitational field.
References [A16
, A20]
are valid for the modified version of the theory [point c) iii)
below]
as well, because the dynamical equations are unchanged.

The whole construction of the theory was reviewed in Ref. [B13],
together with its observational status as of 1998.

However, there was in fact a bifurcation in the construction of
the theory, and it happens that the initially-chosen branch was the
wrong one, see point c) iii) below -- but the
main features of the theory are preserved.

c) *Asymptotic
post-Newtonian approximation (PNA) and equations of motion*

i) The PNA is initiated for test particles in Ref. [A19] and
applied there to light rays, with the consequence that the predictions of the scalar theory for
light rays are the same as those of general relativity. (This
remains true [A35] for
the modified version of the scalar theory.) A systematic study of that
approximation is presented for extended bodies in Ref. [A23],
which derives the expanded local equations and the expanded boundary
conditions. It is also shown there that the standard
(Fock-Chandrasekhar) PNA does not pertain to the usual method of
asymptotic expansions. However, in the particular case of a test
particle in a Schwarzschild field, the asymptotic and standard PNA's
are equivalent [A29]. In
Refs. [A25]
and [A26],
the local equations are integrated inside the volume of the bodies, to
get global translational equations of motion for the mass centers.
However, it turns out to be necessary to introduce a definite
asymptotic framework for the small parameter *eta* that
quantifies a good separation between the gravitating bodies [A32]. As
a result, the final, tractable equations of motion depend explicitly on
structure parameters.

ii) The same has been found for general relativity in the harmonic
gauge, by applying the same method: first, using the asymptotic PNA,
one expands the exact local equations for a perfect-fluid system; then
one integrates the local PN equations in the bodies; and finally one
expands these integrated equations w.r.t. the separation parameter eta. Apart from corrections
that cancel when there is exact spherical symmetry at Newtonian order,
there is in the final equations of
motion one additional term, as compared with the Lorentz-Droste
(Einstein-Infeld-Hoffmann) acceleration [A36 , B27].
This term depends on the spin of that
body whose the acceleration is calculated (self-acceleration term) and
on its internal structure (universality-violating term), and does not
seem negligible for the giant planets.

iii) Moreover, for the scalar theory, it has been found [A33] that the equation of motion for a test particle does not coincide with the point-particle limit of the equation of motion for extended bodies. This violation of the weak equivalence principle is due to the anisotropy of the spatial metric and, therefore, might also be true for GR, depending on the gauge [A33] (see a summary in [B23]), but this would be more difficult to check (due to the greater complexity of GR). For the scalar theory, this problem is not difficult to solve: one modifies the spatial metric to get it isotropic, which is nearly as natural as the initial assumption of a unidirectional space contraction [O3 , B25]. This leaves the dynamical equations of the theory unchanged (apart from the different metric), but, to keep the energy conservation law, it is necessary to modify the equation for the scalar gravitational field: now its right-hand side is just the (flat) wave or d'Alembert operator. This does solve the problem with the weak equivalence principle [A35]. However, the final, tractable equations of motion for the mass centers of a system of extended bodies, according to the modified version, are yet to be published.

To check the theory, detailed calculations of solar system ephemerides are being done, and compared with ephemerides of the Jet Propulsion Laboratory. To this end, an adjustment program has been built, that loops on the numerical integration of the equations of motion for the mass centers, in order to optimize the parameters [A31]. This code has been tested by investigating in which measure one may reproduce (over one century) the predictions of the DE403 ephemeris, by using purely Newtonian equations of motion [A31]. It has also been applied to adjust over 60 centuries a less simplified model, in which the PN corrections in the Schwarzschild field of the Sun are also considered [A30].When one implements in this code the equations of motion obtained with the asymptotic PNA of the scalar theory, one finds [O1] a small, but significant deviation from the JPL ephemeris, that is based on the standard PNA of GR. This difference comes in the first place from using the asymptotic PNA instead of the standard PNA. The difference is larger if the spins of the planets are accounted for [O2, § 4.6.2]. Ref. [O1] contains also a summary of the "asymptotic" PNA used for getting the equations of motion. The "asymptotic" PNA is compared with the standard PNA and it is conjectured that, if a general asymptotic PNA can be built also for general relativity (which seems difficult), its equations and numerical results should also differ from the standard PNA. This has now been verified specifically for the case of GR in the harmonic gauge (which is the gauge used in relativistic celestial mechanics), see point c) ii) above.

The test of the theory in celestial mechanics should be redone with
the modified version of the theory [see point c) iii)].
My guess is that it should be at least as good as with the initial
version. There is a lot of specialized parameter adjustment in
celestial mechanics [O2, Sect.
4.6]. Therefore, I now believe that the question of the link with
quantum theory may be a more urgent work for a theorist [point g) ii) below].

e) *Gravitational radiation and pulsar energy loss* [A34 , B24]

An asymptotic scheme of post-Minkowskian (PM) approximation is built
by
associating a conceptual family of systems with the given
weakly-gravitating
system. It is more general than the post-Newtonian scheme in that the
velocity
may be comparable with *c*. This allows to justify why the 0PM
approximation of
the energy rate may be equated to the rate of the Newtonian energy, as
is
usually done. At the 0PM approximation of this theory, an isolated
system loses
energy by quadrupole radiation, without any monopole or dipole term. It
seems
plausible that the observations on binary pulsars (the pulse data)
could be
nicely fitted with a timing model based on this theory.

f)
*Cosmology *[A28]

An analytical cosmological solution is obtained for a general form of the energy-momentum tensor. According to that theory, expansion is necessarily accelerated, both by the vacuum and even by matter. In one case, the theory predicts expansion, the density increasing without limit as time goes back to infinity. In the other case, the Universe follows a sequence of (non-identical) contraction-expansion cycles, each with finite maximum energy density; the current expansion phase will end by infinite dilution in some six billions of years.

g)
*Link with other parts of physics: electromagnetism, quantum
theory*

i) Electromagnetism:
Section 7 of the review paper [B13]
summarizes the derivation of gravitationally-modified Maxwell equations
and the proof of the consistency of these equations with photon
trajectories in the theory.

ii) Quantum
theory: In Ref. [B15],
Schrödinger's original wave mechanics is analyzed from the
viewpoint of the modern theory of linear wave equations and their
dispersion relations. This allows to extend the Klein-Gordon
relativistic wave equation to the case where a constant gravitational
field is present. It is argued that Schrödinger’s wave mechanics
can be extended
to the case with a variable gravitational field only if one accepts
that the wave equation is a preferred-frame one. (Recent work suggests
that a preferred-frame equation is indeed a possibility, but not the
only one [A39].) From
this viewpoint, generally-covariant extensions of the wave equations of
quantum mechanics seem rather formal. Finally, it is conjectured that
there is no need for a quantum
gravity.

But the formulation of quantum
mechanics in a gravitational field remains a problem, and I am
pursuing its study along this line of research; in particular, I am
studying the formulation of the Dirac theory in a curved space-time [A37, A38 , A39]. Thus, the Dirac
equation has been derived directly from wave mechanics, i.e., from the
classical Hamiltonian [A37].
This derivation applies either in the free case or with an
electromagnetic field, as also in a gravitational field, be it a static
one [A37]
or a general one [A39].
However, two distinct
gravitational Dirac equations may be thus derived, none of which does
coincide with the standard gravitational Dirac equation, that is due to
Weyl and to Fock [A39].
Moreover, the transformation behaviour of the Dirac equation has to be
changed: instead of leaving the Dirac matrices unchanged and the wave
function follow the (counterintuitive) spinor transformation, one may
transform the set of the Dirac matrices as a third-order tensor and
transform the Dirac wave function as a (spacetime) vector [A37, A39]. This makes the
Dirac equation covariant as well, and in fact for general coordinate
changes.

To the top of this page (Lorentz-Poincaré relativity and a scalar theory of gravitation)

To the **Open Directory
Project,** Directory Science: Physics

To the page **Faster
Than Light versus Minkowski and Aristotle space-time**
(B. Chaverondier)

(The consistent
possibility that Lorentz symmetry might not always apply, already in
the absence of gravitation)

Copyright Mayeul Arminjon, 25 September 2002. Last updated 30 May 2007.

** **