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force...”

** **

**Here: “Gravitation as a
pressure force…”, part 4 = Sections 8 and 9 (Sect. 9 = conclusion), and
References**

** **

**8. Consistency with
observations and the question of the preferred-frame effects**

This tentative ether theory
is now rather complete, and it is self-consistent. The obvious question is:
does this theory agree with experiment? There is a vast amount of experimental
and observational data as regards gravitational physics [40], so that a very
detailed analysis should be performed, of course. For instance, there are
numerous tests of the "*weak
equivalence principle *", but the latter is none other, after all, than
the statement that gravitation is a universal force. Due to the basic equation
of motion in the theory, i.e. "Newton's second law" (19), this is obviously
true in this theory. There are also tests of the more specific statement that:
"in a local freely falling frame, the laws of non-gravitational physics
are the same as in SR", which is *Einstein's
equivalence principle* (EEP), and which we also call the equivalence
principle in the standard form, because a different equivalence principle is
postulated in the present theory (Sect. 3). It should be clear that EEP is *not* true in this theory, since EEP
implies Eq. (24) for continuum dynamics, as opposed to Eq. (37). Let us recall,
however, that these two equations are equivalent for the case of a constant
gravitational field, and note that the best-known theoretical frame for testing
EEP, known as the *TH**e**µ* formalism, is
restricted to static gravitational fields [40], and so does not allow to
analyse experiments that should decide between EEP and the proposed equivalence
principle. Furthermore, as well as for any theory based on EEP and Eq. (24),
our Equation (37) and the whole theory are in full agreement with the
assumption of a *universal coupling*,
since the same equation applies to any kind of matter and/or non-gravitational
field. For these two reasons, it would be probably quite difficult to find
laboratory experiments accurate enough to distinguish between the usual form of
the equivalence principle and the alternative form which is postulated in the
present theory.

Hence, we are inclined to believe that the main challenge
for this preferred-frame theory is to recover the "classical tests"
of GR, i.e. the effects of gravitation on light rays and the general
relativistic corrections to Newtonian celestial mechanics. The latter consist
essentially in the prediction, by GR, of Mercury's very small residual advance
in perihelion but, in our opinion, one should pose the question in a more
general way: does the theory produce a celestial mechanics which is more
accurate than Newton's theory?

In order to investigate the effects of gravitation on light
rays and the corrections made by this non-linear theory to Newtonian celestial mechanics, it is necessary, as well
as in GR, to develop an iterative approximation scheme, i.e., to develop a *post-Newtonian (pN) approximation scheme*.
The pN approximation *scheme* is the
method of asymptotic expansion of the dependent variables and the equations in
powers of a small parameter *e*, which is
defined by *U*_{max} */c*^{ 2} º *e*^{ 2}, with *U*_{max}
the maximum value of the Newtonian potential in the considered gravitating
system (assumed isolated, and the gravitational field being assumed weak and
slowly varying; *cf.* Fock [18],
Chandrasekhar [14], Weinberg [38], Misner *et
al. *[26], Will [40]). (The term *pN
approximation* alone usually makes reference to the approximation
immediately following the first, "Newtonian" approximation; this
second approximation is largely sufficient in the solar system.) Actually, the
small parameter will be taken simply as *e**'* = 1/*c*
as in Refs. 14 and 18. Note that, choosing the units such that *U*_{max} = 1, we get indeed
*e* = *e**'*. More generally,
constraining the units merely so that *U*_{max
}»
1,
we may take *e**'* = 1/*c* as the small parameter. Then all relevant quantities such as *U*, *v*,
etc., are O(1). Choosing the time coordinate as *x*^{ 0} = *T* (instead of *cT *), the assumption
of a slowly varying gravitational field is then automatically satisfied.
Moreover, only one term among two successive ones appears in the relevant
expansions. Whereas the usual explanation makes appeal to the behaviour under
time reversal [26, 40], we note that, in the proposed ether theory, all
non-Newtonian effects come from the "ether compressibility", *K *º1/*c*^{ 2}. So *K*
itself (or *U*_{max }*K*, if the units are not constrained so
that *U*_{max }» 1) could be
considered as the small parameter, whence the appearance of only one among two
successive terms in any expansion with respect to 1*/c* = Ö*K *- the leading
("Newtonian") term giving the parity. Finally, in a preferred-frame
theory, the absolute velocity *V* of
the mass-center of the system with respect to the ether frame should not exceed
the order *e **c*, as is the case for the typical orbital velocity *v* in the mass-center frame. We note
that, if *V* is approximately 300 km/s
for the solar system (as one finds if one assumes that the cosmic microwave
background is "at rest" with respect to the preferred frame [40]),
then one has indeed *V/c* £ *e* in the solar system [6], because there *e*^{ 2} º^{ }*U*_{max} */c*^{ 2 }» 10^{-}^{5} [26].

**i**) *Expansion of the metric and the field
equation in the preferred frame*

The leading expansion is that of the scalar
field, *b* or *f *= *b*^{ 2} :

*b* = 1-*U/c*^{ 2} + *S/c*^{
4 }+ ... , *f* = 1-2*U/c*^{ 2}+(*U*^{ 2} + 2*S *)*/c*^{ 4} + ... = 1-2*U/c*^{ 2 }+ *A/ c*^{
4} + ... (64)

The space metric deduced from the Euclidean
metric **g**^{0} by assumption (A) (Sect. 3) is then obtained
[6] as

*g*_{ij}* *= *g*^{0}* _{ij}* + (2

(with

*g*^{0}* _{ij}* =

in Cartesian coordinates),
and the space-time metric is

*g*_{00} = *c*^{ 2}*f* = *c*^{ 2}(1
- 2*U/c*^{ 2} + *A/ c*^{ 4} + ...), *g*_{0i } = 0, *g** _{ij
}* = -

The mass-energy density

*r* = [(*T*^{ 00})_{E} ]* /c*^{ 2}

may be written in the form

*r* = *r*^{
0}_{ } + *w*^{ 1}*/c*^{ 2}
+ ..., (67)

where *r*^{
0}_{ }is the conserved mass density which is found at the first approximation
(expanding, for a perfect fluid, the energy equation (34), one finds that *r*^{
0}_{ }obeys the usual continuity equation and that mass is conserved at the
pN approximation also). The pN expansion of the field equation (13) follows
easily from Eqs. (64) and (67) :

D_{0}*U* = - 4p*G* *r*^{
0}, (68)

D_{0}*S* = 4p*G* *w*^{ 1} - D_{0}*U*^{ 2}*/ *2 - *¶*^{ 2}*U/**¶** t*^{ 2}, or D_{0}*A* = 8p*G* *w*^{ 1} - 2*¶*^{ 2}*U/**¶** t*^{ 2}. (69)

**ii**) *Post-Newtonian equations of motion for a
test particle in the preferred frame*

Using the energy equation (28), one first
rewrites Newton's second law for a free test particle, Eq. (19) with **F**_{0} = 0, as [6]

_{} (*x*^{ 0} = *T *), (70)

where the G '^{ m}* _{nr}* symbols are the
Christoffel symbols of the space-time metric and the G

_{}_{}. (71)

An important point is that Eq. (71), derived
from Newton's second law, is nevertheless *undistinguishable
from the pN expansion of the equation for null space-time geodesics*. For a
mass point, the expanded equation is

_{}

_{}. (72)

Note that, in the pN equation of motion for a
photon, Eq. (71), the "Newtonian" gravity acceleration **g**^{0} º grad_{0} *U* (with components *U _{, i }* in the
Cartesian coordinates utilized) intervenes at the same order in

**iii**) *Transition to a moving frame. Application to
the effects of a weak gravitational field on light rays*

In order that the pN motion may be considered
as a perturbation of the problem in classical celestial mechanics, one has to
work in the mass-center frame, as in classical mechanics. Let **V**(*T
*) be the current absolute velocity of the mass-center. We define the
mass-center frame E** _{V}** as the frame
that undergoes a pure translation, with velocity

However,
at the pN approximation, a photon follows a null geodesic of the physical
space-time metric **g**, and, by the
Lorentz transformation, the components of this space-time tensor transform thus
like a (twice covariant) tensor, of course: this gives an alternative way to
get the pN equations of motion for a photon in the frame E** _{V}**. Neglecting O(1

frame E, it is indeed
constant in the frame E** _{V }**which moves with the
spherical massive body (e.g. the Sun) that creates the relevant field, so that
the geodesic equation is really that deduced from Schwarzschild's metric. We
conclude that,

**iv**) *Remarks on the adjustment of astrodynamical
constants and the preferred-frame effects*

In contrast to the pN acceleration (71) for a
photon, which is invariant by a Lorentz transformation of the flat metric
(provided the velocity *V* of the moving frame is compatible with the
pN approximation, i.e. such that *V/c*
= O(*e *)), the pN
acceleration for a mass point, Eq. (72), is *not* invariant. For a mass point, the pN
acceleration in the moving frame is, in space vector form,

*d _{ }*

where (*d _{
}*

However,
one may consider that things are less simple than this. The main point is that,
in classical celestial mechanics, the astrodynamical constants such as the
Newtonian masses *M*^{ }_{i }^{N}* _{
}*of the celestial bodies (or rather the products

_{}. (74)

In reality, no couple of celestial bodies is
exactly isolated, and perturbation theory allows to modify the astrodynamical
constants by successive corrections, always remaining in the frame of Newtonian
theory (NG). With modern computers, one may calculate hundreds of Newtonian
parameters by a global fitting based on a numerical approximation scheme of NG
and using hundreds of input data from various observation devices (*cf.* Müller *et al.* [28]).

We
know that the correct theory of gravitation is not NG. Let us assume for a
moment that it is instead some non-linear theory (T), the equations of which
reduce to NG in the first approximation. The pN approximation of theory (T)
will introduce, in addition to the first-approximation masses *M*^{ }_{i }^{0 }(and higher-order multipole moments of
the mass density *r*^{ 0} of the first
approximation, etc.), some pN parameters such as the pN corrections to the
active masses, say *M*^{ }_{i }^{1}. Obviously,
the fitting of observational data must now be carried out *within the pN approximation of theory* *(T ). *There is simply no reason that the first-approximation masses
*M*^{ }_{i }^{0},* _{ }*which are obtained (together with
the corrections

To
investigate the consequence of this state of things, let us call *D _{j}*

*D ' _{j }*

differing from the correct pN predictions* D _{j }*

The
foregoing argument applies *a priori*
also to *general relativity* (although
there is no preferred-frame effect in GR, there are of course non-Newtonian,
"relativistic" effects). As a relevant example, the standard pN
approximation of GR, using the harmonic gauge condition [38], introduces,
besides the "Newtonian" (first-approximation) potential *U* (denoted -*f* by Weinberg [38]), still another scalar
potential obeying a Poisson equation. Namely, it introduces the pN potential
denoted *y* by Weinberg
[38], which plays exactly the same role as *A*/2
in the present theory (cf. Eqs. (66) and (69) hereabove). Weinberg writes
explicitly that "we can take account of *y* by simply replacing *f* everywhere by *y* + *f** * " [or *y* */c*^{ 2
}+ *f*, since Weinberg sets *c* = 1], consistently
with our statement that Newtonian astrodynamics *in praxi* does not coincide with the first approximation of the
non-linear theory, here GR.

Coming
back to the proposed theory, the foregoing argument is not a proof, of course,
that the theory does correctly explain all minute discrepancies between
classical celestial mechanics and observations in the solar system. This
argument merely suggests that the present theory should not be *a priori* rejected on the basis that it
predicts preferred-frame effects in celestial mechanics - the more so as this theory
correctly explains the gravitational effects on light rays, which are the most
striking and the best established predictions of GR.

**9. Conclusion**

A rather complete theory of gravitation may
be built from an extremely simple heuristics, already imagined by Euler, and
that sees gravity as Archimedes' thrust in a perfectly fluid ether. The theory
thus obtained is very simple also, as it is a scalar bimetric theory in which
the metric effects of gravitation are essentially the same as the metric
effects of uniform motion, and with the field equation being a modification of
d'Alembert's equation. Perhaps the most important finding is that Newton's
second law may be extended to a general space-time curved by gravitation, in a
way that is both consistent and seemingly compelling. This extension is not
restricted to point particles: it applies also to any kind of continuous
medium, in fact it is Newton's second law for the electromagnetic *field continuum* that gives the Maxwell
equations of the theory.

Although
the general extension of Newton's second law is unique, its exact expression
still depends on which assumption is stated for the gravity acceleration. It is
at this point that the ether heuristics makes its demarcation from the logic of
space-time which leads to Einstein's geodesic assumption and Einstein's
equivalence principle. The ether heuristics leads indeed to postulate that the
gravity acceleration depends only on the local state of the imagined fluid,
whereas geodesic motion can be true in a general, time-dependent metric, only
if the gravity acceleration depends also on the velocity of the particle - and this, in such a way
that the velocity-dependent part of the gravitation force does work [5]. Hence,
the here-assumed gravity acceleration is in general incompatible with geodesic
motion and, in connection with this, it leads to an equation for continuum
dynamics, Eq. (37), which differs from the equation that goes with geodesic
assumption and Einstein's equivalence principle, Eq. (24). It also enforces
that the theory is a preferred-frame theory, which is independently enforced by
the scalar character of the theory. It is interesting to note that the
non-Newtonian effects, as well as the preferred-frame character, come once a
non-zero compressibility is attributed to the ether, just like in Dmitriev's
theory [16].

The
alternative continuum dynamics implies that, in a variable gravitational field
at a high pressure, matter is produced or destroyed, by a reversible exchange
with the gravitational field, Eq. (44). The tenuous amounts seem compatible
with the experimental evidence on "mass conservation". An
experimental confirmation or refutation of this new form of energy exchange
would be extremely interesting. In the same way, the alternative continuum
dynamics leads to Maxwell equations that contain the possibility of electric
charge production/destruction in an electromagnetic field *subjected to* a variable gravitational field. However, the amounts
cannot be estimated without having recourse to a numerical work, which is yet
to be done. This is obviously a dangerous point for the theory, although it is
also an interesting one.

As
to the classical tests, it has been shown that no preferred-frame effect exists
for light rays at the first post-Newtonian approximation, and that in fact the
gravitational effects on light rays are correctly predicted. It has also been
shown that preferred-frame effects do exist in celestial mechanics, and it has
been argued that this does not kill the theory. Furthermore, such effects might
play an interesting role at larger scales, e.g. they might contribute to
explain rotation curves in galaxies. We mention also that, contrary to general relativity,
the theory predicts a *bounce *instead
of a *singularity *for the
gravitational collapse of a dust sphere [3]; and that, like general relativity,
it leads to a "quadrupole formula" that rules the energy *loss*, thus the correct sign, for a
gravitating system with rapidly varying field (this is less shortly outlined in
Ref. 3).

Finally,
it has not been attempted to attack the impressive task of linking the present
theory with quantum mechanics and quantum field theory. Only some preliminary
remarks may be done: first, the abandon of the relativity principle (in the
presence of a gravitational field) and its replacement by a preferred frame
(presumably the average rest frame of the universe/ether) change drastically
the problematic. In particular, it makes sense, in this framework, to assign a
non-punctual (albeit fuzzy) extension to particles. Further, the quantum
non-separability, and the unity of matter and fields, are strongly compatible
with the heuristic interpretation of particles as organized flows in the fluid
ether. Another remark is that quantum theory is based on the
Hamiltonian/Lagrangian formalism, and that the present theory is not amenable
to this formalism, except for a constant gravitational field. But Newton's
second law is a more general tool than these variational principles, since it
makes sense also in non-conservative situations. However, the existence, in the
present theory, of a local conserved energy (made of the gravitational energy,
plus the energy of matter and non-gravitational fields), should play an
important role.

**Acknowledgement**

** **

I am grateful to Prof. E. SOÓS (Institute of
Mathematics of the Romanian Academy, Bucharest) and to Prof. P. GUÉLIN (Laboratoire
"Sols, Solides, Structures", Grenoble) for helpful discussions.

(NB: when putting this paper on the web, the status of the author’s papers has been updated; this is the only modification to the original text)

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