To the References of the paper (= end
of this page)

__6. THE MODIFIED NEWTON LAW IMPLIES GEODESIC MOTION OF
LIGHT-LIKE PARTICLES__

A basic prediction of GR,
also found with the ETG (Eq. (19)), is that clocks are slowed down in a
gravitation field. One empirical confirmation is the observed gravitational red
shift of electromagnetic spectra. To deduce the red shift from the slowing down
of clocks, the frequency

n_{0} of any monochromatic wave is considered constant
throughout its light path, when expressed in terms of the time coordinate t -
in the case of a stationary field where the space-time metric (g_{ab}) does not depend on t. This constancy of n_{0} in turn is derived from the wave
equation f=0 with f=exp(iy), y=w(x^{1},x^{2},x^{3})t + f(x^{1},x^{2},x^{3}) and f from Eq. (24) :
Laue proved this for a static field [17] but his arguments extend to a
stationary one, in the way outlined in the treatise [18]. Laue [17] also
deduced from the same way of reasoning that light rays follow the null geodesic
lines of **g** . These arguments also apply in the
ETG, since they depend only on the equation
f=0, in the
limit of geometrical optics. It is, however, interesting to show that in the
ETG the constancy of n_{0} as well as the geodesic propagation can be deduced
also from the modified Newton law (20) with the mass of the energy E in the
place of the inertial mass m(u) :

in which n is the frequency as measured with the local
(momentarily coinciding) clock, n=dn/dt** _{x }**, as imposed by the local nature of quantum phenomena
(h is Planck's constant and n is the number of oscillations). The constancy of n

E p_{e }= E_{0 }p_{e0 }= Const. along the light path. (31)

We first prove that for a static field (p_{e}
time-independent), (31) follows from (30) and the fact that, as measured with
physical clocks and rods (bound to *E*), the modulus of the velocity of the photon is always
equal to c :

(the usual notations for scalar products and squares, **u.v** and **u**^{2}, refer to the curved space metric **g**). Combining
(30)_{1} and (32)_{2} we get :

but according to Eq. (5) where the grad operator refers to the metric **g** and with p_{e}=r_{e}c^{2}, we have

**g.**d**x** = - (grad p_{e }**. **d**x**)/ r_{e }= - c^{2} dp_{e}/p_{e }.

Therefore, (33) may be rewritten as

- E c^{2} dp_{e}/p_{e} = c^{2} dE , (35)

which means that d(Ep_{e})=0 along the path, i.e. Eq. (31) is
obtained.

Now we prove that Eqs.
(30) and (31) imply that the photon moves along a null geodesic line of **g **(*during this
proof, we omit the index *e* in *p* and *r* for simplicity*). Let **x**_{0} be any fixed point in M (i.e. **x**_{0} is at rest in the ether), outside the gravitation
field (i.e. far enough). We introduce the universal time coordinate x^{0}=ct with t=t_{x}_{0}. Whether the field p depends on t or not, we may
adopt, at any time t, a space coordinate system (x^{i}) on M
such that p=Const. is equivalent to x^{1}=Const.,
and such that the Euclidean metric **g**^{0} is diagonal in the natural basis (**e**_{i}) :

(g^{0}_{ij}) = diag (_{}).

Then, assumption (A) and Eq. (16) with p=r_{ }c^{2} mean that **g** also is diagonal, and that:

(g_{ij}) = diag _{}, (36)

with p^{¥}^{ }= p(**x**_{0}). In the case where such an "isopotential"
coordinate system is bound to the frame *E*, the
space-time metric (21) writes in the local coordinates (x^{a}) (a=0,3) of the space-time **R**´M:

(g_{ab}) =
diag _{}, (37)

with b_{i} = - a_{i} for 1 £ i £ 3. This case includes the static case (¶p/¶t =0) and that of spherical symmetry around a fixed
point in ether. The Christoffel symbols G and G^{ }’ of the metrics **g** and **g **,
respectively, have been calculated in the previous work [8b]. The first-kind
symbols verify :

G
'_{ijk} = - G_{ijk} (i,j,k = 1,2,3); (38)

G '_{aaa} = b_{a,a}/2, G ' _{abg} = 0 if a¹b¹g¹a,
(39)

G '_{aab} = - G '_{baa} = g_{aa,b}/2 = b_{a,b}/2 if a¹b;

- G '_{0aa} = G '_{aa}_{0} = G '_{a}_{0}_{a } = b_{a}_{,0}/2, G '_{a}_{0}_{b} = G '_{ab}_{0} = G ' _{0}_{ab} = 0 if (a¹0 and b¹0 and a¹b), (40)

G
'_{00}_{a}_{ }= G '_{0}_{a}_{0}_{ }= - G '_{a}_{00}_{ }= b_{0,}_{a}_{ }/2 = p p_{,}_{a }/(p^{¥})^{2}_{ }. (41)

The second-kind Christoffel’s are defined by

Since b_{i} = - a_{i} for 1 £ i £ 3, we thus have from (38):

_{}

The geodesic equation for a=0, writes in view of (40) and (42) :

and since we expect a null geodesic, this equation serves to determine
the time dependence of the parameter l: y=dx^{0}/dl. It comes from
(41) and (42) that :

which is nil, unless if i=1. We get from (44) and (45) :

in which dp/dx^{0} is due
only to the spatial variation of p along the trajectory. Eq. (46) gives

dy/y + 2 dp/p = 0 , y = dt/dl
= 1/p^{2} ,
(47)

in which dt/dl has been substituted for dx^{0}/dl , since l may be replaced by a proportional parameter. By (43),
the left-hand side of the geodesic equation writes for i=1,3 :

Now by (31) and (19), we write the neo-Newtonian law (30) as:

_{} , (49)

whence by (47)_{2} and (18) :

_{} , (50)

thus the two first terms in the last member of (48) represent g^{i}/[p^{2 }(p^{¥})^{2}]. From
(5) and (36) :

_{} (51)

Hence, by (40), the geodesic equation G^{i}=0 is also verified for i=2,3, and the last is

_{}. (52)

But from (37), (41), (42) and (47), we get:

_{} . (53)

Hence the geodesic equation for photons (or other particles moving at
the velocity of light) is proved independently of any field equation, from
Newton's second law in the frame *E*. Like for ordinary particles, we assume here, by
induction from this result obtained for a static field, that the geodesic
motion holds true in the general case where a Newton law has not been defined.

__7. SPHERICAL SYMMETRY : STATIC SOLUTION AND HYDROSTATIC EQUILIBRIUM __

A spherical gravitation
field has to be defined as the case where p_{e} or r_{e} depend only on the radial distance r (as measured
with the Euclidean metric **g**^{0} making the macro-ether a rigid body M) and the time t
(as measured with one clock at a given point **x**_{0}ÎM, i.e. bound to *E*). Then one "isopotential" coordinate
system, in which the metrics **g** and **g** are
given by (36) and (37), is the spherical coordinate system (r,q,f). Thus the
field equation (23) writes, in view of (19) and (25) :

_{} , (54)

in which one may moreover assume that p_{e}^{¥} does not depend on t, i.e. a static situation at
infinity (or far enough from the massive body). Equation (54) implies that the density
of mass-energy r also depends on t and r only. The *static* spherical solution (r_{e} and *hence* r independent of t) follows easily
from (54) and was given for a general distribution r(r) with finite
integral M=M(r=¥)<¥ in the previous work [8b] (no confusion
can occur between the mass M and the 3D-manifold M consisting of the
macro-ether):

_{} , (55)

where M(r) = 4p_{}; the integration constant C is
non-positive, and negative only in

the unphysical case where an (1/r^{2})-repulsion occurs for r®0 and at the same time the field **g** is equivalent to GM^{a}/r^{2} with M^{a}<M at
large r. In the physical case C=0, the space-time line

element is thus, by Eq. (37) (with _{} ):

_{} (56)

and moreover the field **g**
given by Eq. (5) (where the grad operator refers to the curved metric **g**), i.e. the
gravity acceleration field in the sense of the modified Newton law (20) or
(30), is the exact Newtonian field

**g** = - G M(r) **e**_{r}/r^{2 } (57)^{}

(this is proved in [8b]). We emphasize that all of this is true for a
general function r(r); in particular, if r(r)=0 for r>R, it is valid for r£R as well as for r³R. In that case, the metric (56) for r³R is the exact Schwarzschild exterior metric (U_{N}(r)=GM/r). However, the metric for r<R is *not* that predicted by GR. E.g. in the
case of a uniform density, r(r)ºr_{0} for r<R, the term U_{N}(r) in (56) is given, according to Eq. (55)_{2}, by

_{}, (58)

whereas Schwarzschild's interior solution is :

_{}, (59)

1/R'^{ 2} = 8 p G r_{0 }/(3c^{2})

with A>0 and B=1/2 (see e.g. [19]). The main difference between the
interior solutions according to GR and the ETG is the following one : according
to the ETG, the ratio

d*l */ d*l*^{0 }= Ög_{r r }= Ö(- g_{r r }) >1

between the radial distances as evaluated with the physical metric **g** and the
Euclidean one **g**^{0} (which is equal to **g** at infinity) continues to increase as r goes from R
to 0, whereas it decreases towards 1 for GR. *The result of the ETG is in agreement with what is found from the EP*,
that the metric effects of gravity are given (for weak fields) by the Newtonian
potential (Sect. 4), since U_{N}(r) in
Eqs. (55),(56) and (58) turns out to be nothing else than the Newtonian
potential - however, no restriction to the field strength is imposed here.
Another difference with GR is that, independently of the state equation, *the radius* R *of the body cannot be smaller than the Schwarzschild radius* r_{S} = 2 GM/c^{2}, for otherwise Eq. (55) (with C=0, and U_{N}(r)=GM/r for r³R) would lead to a negative square p_{e}^{2}. The number R is the radius of the body as evaluated
with the Euclidean metric **g**^{0}, hence the physical radius is greater. On the other
hand, the mass-energy density r is evaluated with the physical metric **g** , hence M= òrÖg^{0} dr dq df = òr dV^{0} is smaller than the mass of the whole energy creating
the field, M' = òr dV = òrÖg dr dq df , since

Ög^{0}/Ög = dV^{0} /dV = Ö(1-2 U_{N}(r)/c^{2}) < 1.

The same occurs in GR, where M is interpreted as the sum of M' and the
negative energy of the gravitational field [4]. A discussion of the
gravitational energy in the present theory will be given elsewhere.

Since pressure does
contribute to gravity in the ETG as in GR, one may expect that, as in GR, the
static equilibrium is impossible if the integral M is too large : first, SR at
the local scale and the definition of the energy-momentum tensor **T** hold true in the ETG; second, if one
admits the geodesic formulation of motion in the general case (as is done in
this paper), then the "conservation equation" (CE), div_{g}**T**=0, can be used in the ETG (the geodesic motion of
free particles is equivalent to the CE for **T**
with "dust" matter; the essential difference in the ETG is that
geodesic motion and the CE are not a consequence of the field equation for
gravity). In that case, the static equilibrium writes exactly as in GR (see
e.g. Weinberg [4]). For the case of spherical symmetry, it gives using the
static solution (56)-(57) :

_{}

which is not identical to the Oppenheimer-Volkoff equation. However, it
may be shown that it gives the same qualitative result, i.e. the existence of a
superior limit for the mass M.

__8. SPHERICAL SYMMETRY : GRAVITATIONAL COLLAPSE AND GRAVITATION WAVES __

From above, we take it
for granted that a very massive object will necessarily undergo a dynamic
transformation, either an explosion or, in the case where the (ordinary)
pressure p in the body is unable to balance the gravitation force, an *implosion*. Since pressure increases
gravity also in the ETG, we expect that, as in GR, some qualitative features of
the implosion are rather resistant to the change of the unknown state equation
and therefore can be found from the analysis of the oversimplified situation of
dust matter. Thus, the pressure p is *neglected*.
The particles of the collapsing body are hence free and they have a geodesic
motion. Furthermore, we assume here that the body, made of freely moving dust
particles, is *comoving with the
macro-ether* (see the end of Sect. 5 : the assumption of a constitutive
ether leads naturally to this hypothese for very high densities such that
elementary particles are more or less contiguous to each other). We thus write
in any coordinates bound to the frame *E* :

_{}

which is equivalent to

_{}

For the case of spherical symmetry around the center O of the massive
body which is assumed to be comoving with ether, the spherical coordinates x^{1}=r, x^{2}=q, x^{3}=f of the
Euclidean space M (the body "macro-ether"), with origin O, are indeed
bound to the frame *E* and moreover are such that p_{e} does not depend on x^{2} and x^{3}. Thus the space-time metric is
given by (37) and we get by (41) and (42) :

_{}, _{} ,

_{}, _{}.
(63)

This implies that Eq. (62)_{2} is
equivalent to

p_{e, i} = 0
(i = 1,...,3) , (64)

which means that the field p_{e} has a uniform value p_{e}^{0} in the
massive body, occupying a domain W in the space M. This result demands that we are able
to give a sense to our basic assumption (A) in a case where the direction of the
field **g** , asked for by Eq. (16), is
undetermined. We restrict the discussion of this degenerate situation to the
case where the field p_{e} is
furthermore uniform at infinity. Then, either the uniform value p_{e}^{0} is p_{e}^{¥}, in which case there is no space contraction in the
considered domain, which must extend to infinity. Or p_{e}^{0} < p_{e}^{¥}, in which case the domain W, where p_{e}º p_{e}^{0} , has to be bounded. Thus, the direction of
contraction is determined at the boundary ¶W, by continuity with the exterior of W where grad p_{e} is not nil, and the contraction ratio b=p_{e}/p_{e}^{¥} has the same value at each point of W and ¶W. This may be
sufficient to determine a unique direction of contraction at any point of W It is at least
true in the present case of spherical symmetry, where the direction must
obviously be the radial one. Now let us examine whether the condition (62)_{1 }gives a further requirement which should be satisfied
by dust matter comoving with ether, in addition to (64). With Eqs. (63) and
(64), we calculate

_{}

and

_{}

Hence (62)_{1} is
automatically verified if p_{e} depends
only on t within the body.

Let the free fall begin at
time t=0 : the initial distribution p_{e0} must be independent of r. This is not compatible with a static
situation at t=0 since, due to Eq. (55)_{1} (with C=0), it would mean that the Newtonian potential U_{N}(r) does not depend on r inside the body - that is,
for r<R - and this in turn implies that r_{0}(r) is nil for all r. *The collapse in free fall cannot start from
a static situation*. The space-time metric **g** is given by Eq. (37) with _{} in particular g_{0i }=0 (this
is general in the frame *E*, Sect. 4). For
dust matter, the mixed components of the energy-momentum tensor in the comoving
frame *E* are therefore

and T^{lm} = _{}^{ }/ b^{2} with b=p_{e}(t)/p_{e}^{¥} for the contravariant ones; it follows that only the
component a=0 of the CE for **T** is not
trivially verified, and gives

_{}

We thus get :

_{}

Inserting (69) into the field equation (54) in which p_{e} depends only on t, we obtain :

_{} , (70)

or after substitution of the proper time t, which flows uniformly in the body
(dt/dt = p_{e}/p_{e}^{¥}):

_{} . (71)

Note that the constancy of *a* and *b*, i.e. the fact that the initial mass-energy density r_{0 }actually does
not depend on r, is a consequence of the field equation with a uniform field p_{e}(t) inside the body, but in turn, by (74), implies
that r remains uniform : *the collapse in
free fall can occur only with a uniform material density* r. We thus have
already two important differences with GR.

The resolution of Eq. (71) is
standard : one introduces the slope z=dp_{e}/dt = z(p_{e}) as an auxiliary function,
considered as one of pe, and one
writes

_{}

whence with (71) (where *b*=1 by
changing the time unit) :

_{}

which gives t as function of p_{e} :

_{}

One starts with a positive value pe0 at t = t =0. The initial slope z_{0} determines the integration constant C, which is positive (Eq. (73a)). In
that case, if one restricts the discussion to curves which cannot go accross
the (p_{e}=0)-axis, the solution pe to Eq. (71) is a periodic function of t , which
cancels at regular intervals dt = 2 F(p_{e max}) with p_{e max} =
(3C/2)^{1/3} (Eq. (73a)); at those times t where p_{e} cancels, the slope z undergoes a discontinuity,
passing from -ÖC to ÖC (Eq. (73b)). Whatever the sign of
z_{0} = (dp_{e}/dt)(t=0), the macroscopic ether pressure p_{e} inside the body cancels after a finite proper time. Actually, the sign
of z_{0} is related to the time evolution of
the radius R' of the body, as evaluated with the measuring rods of the freely
falling "observer" (a robust one) : assumption (A) means that R' is
expressed as function of the constant radius R (as evaluated with the Euclidean
metric bound with ether, as long as this remains comoving with matter) as R' =
(p_{e}^{¥}^{ }/p_{e}(t)) R , hence

_{}

Thus, if the local
"observer" finds an implosion at t=0, one must have z_{0 }> 0, but then the slope z cancels at the proper time

_{}

so that the sign of the time evolution of R' is changed : *the implosion becomes an explosion* !
moreover, as t continues to increase, p_{e} cancels
at t_{2 }= t_{1} + F(p_{e max}) and R'®¥ as t_{ }® t_{2} : the
nil value of p_{e} = r_{e }c^{2} means that the finite amount of ether contained in the domain r<R,
and which is comoving with the body, is diluted in a domain of infinite size -
as evaluated with physical rods. Once the value p_{e max} has been passed beyond, the density of matter decreases as p_{e }, since r=r_{0} p_{e}/p_{e0} (Eq. (69)). While r continues to decrease, say when it
becomes much smaller than its initial value r_{0}, the argument of Sect. 5, that all
ether would be involved in material particles, cannot be used any more to admit
that the ether is comoving with matter (but, on the other hand, since all is
uniform within the body, any reason to change this situation should have to
come from an interaction with the exterior).** **Thus, the calculations have only an indicative value after** **p_{e} and r have gone below the values they took at the start of the free fall. *During the raid in the domain of very high
densities, however, the comoving of ether and matter is nearly implied by the
assumption of the constitutive ether (see Sect. 5) and in turn it precludes
that the collapse in free fall lead to any singularity*.

The measuring rods of
sufficiently far observers are not affected by gravitation (p_{e }» p_{e}^{¥}), thus one could be tempted to say that, as long as
its density is high enough so that ether is comoving with matter, the body has
a constant radius for remote observers : if the macroscopic motions of ether
and matter coincide, a change in the physical size of a massive object can
occur only with a change in the gravitational contraction. But as soon as the
metric is influenced by gravity - and thus by such objects - it is difficult
for a remote observer to define the size of an object, e.g. because the light
paths become curved. By the way, can light rays escape the freely falling body
in finite time t, or does it become a "black hole" ? A first point is
that the time t (of far clocks) becomes *infinite*
during the proper time t_{2} (of freely falling clocks in the body or at its
surface) that it takes for the value p_{e} inside the body to cancel, since

_{}

However, to answer the question about light rays, we should be able to
solve the equation

_{}

which implies that we should know the field p_{e}(t, r). In GR, Birkhoff's theorem allows one to choose
space-time coordinates in which the exterior metric is static, and thus to find
ones in which it is the Schwarzschild metric. This means that, according to GR,
gravitation waves cannot exist outside the massive body, in the case of
spherical symmetry (see e.g. Weinberg [4]). *But
it is not so in the proposed theory*. The field equation (54) is a
hyperbolic quasi-linear differential equation in the variables (t, r), since it
writes

_{},

_{}. (79)

The characteristics are therefore the radial light rays :

_{}

(which shows in a demonstrative way that gravity propagates with the
velocity of light in the ETG, though it is already clear from the general equation
(23)). Thus, depending on the initial and boundary conditions, the solution to
(79) is generally made of both outgoing and ingoing waves. In the present case,
the initial condition f(t=0, r)=f_{0}(r) and (¶f/¶t)(t=0, r)=q_{0}(r) for all r ³ 0 determines uniquely the solution in the domain r £ f(t), where (r=f(t)) is the outgoing characteristic starting from the
origin (t=r=0) [20], and we have seen that the initial data cannot be static,
that is, q_{0}(r)¹0. Therefore, a non-static situation (with waves) is certainly present
for t>0, and since it resolves in a uniform (but time-dependent) situation
for r<R, we may confidently expect that outgoing *and ingoing* waves are present. Now we observe that the wave
velocity w=|dr/dt|=cf=cg_{00} *decreases*
along the path of the ingoing wave, because the wave moves towards the
direction where p_{e}
decreases and the clocks are slowed down; we have in general :

_{}, (81)

thus our statement is certainly true inside the body (r £ R) for t ³ t_{1} (Eq. (76)), since we have then ¶f/¶r =0 and ¶f/¶t <0, whence dw/dr>0 for the ingoing wave
(dr<0, e=-1); this strict inequality extends to a larger domain r < R+h(t);
the inequality is also ensured at sufficiently large r, where the wave starts
from a static situation. But a wave whose velocity decreases along its
direction of propagation may become a shock wave if its propagation length is
sufficient, as discussed by Whitham [20] (the other condition ensuring the
appearance of a shock wave is that the decrease in the wave speed should be
stronger than the damping of the wave; intuitively, we would say that here an
ingoing wave should be reinforced rather than damped, but this must be
checked). If we admit on the above basis that w decreases along ingoing
characteristics in the whole domain r £ f(t) (and that ether is still
comoving with matter), then we have necessarily enough time t for a shock wave
to develop before the "catastrophe" p_{e}=0, since it takes an infinite time t [Eq. (77)] [the catastrophe relies
precisely in the fact that a finite proper time corresponds to an infinite time
of remote clocks (and the reverse for the spatial size), a very strange
situation]. According to our theory based on a constitutive ether, a shock
wave, i.e. a discontinuity in the ether pressure, should destroy the material
particles made of stable local flows in ether (perhaps vortices). Thus, once a
shock wave would have reached the body, an enormous amount of energy would be
released, thereby ending the collapse (otherwise, the whole mass would be
destroyed) and at the same time reducing the mass of the body, i.e. the cause
of the collapse. Winterberg already suggested the possibility of a shock wave
in gravitational collapse, within his different theory [7] (see below). This
implosion-explosion cycle could repeat until the mass is small enough to allow
static equilibrium. These considerations might be of interest for the
interpretation of g bursts.

__9. CONCLUSION__

Starting from a tentative
interpretation of Newtonian gravity as a pressure force in an incompressible
perfect fluid or ether, one may first obtain a new theory of gravity within
Newtonian mechanics, simply by giving a compressibility to the ether; this
results in a finite velocity of propagation. This does not seem to have been
proposed earlier, and the reason may be that one : surprisingly, the
"ether pressure" p_{e} must
decrease towards the gravitational attraction whereas the average mass density
in the particles depends on p_{e} only and
decreases with p_{e}, and thus also decreases towards
the attraction - thus the particles would swell in the gravitation field, i.e.
there, where they are closer from one another! But the density in the particles
is not the density of matter (the number of particles per unit volume), hence
there is no absurdity and moreover one sees then at a glance that matter cannot
be indefinitely compressed (since the particles would then encroach up on one
another). When one accounts for the metric effects of gravity, the essential
point is the coexistence of two spatial metrics in the reference frame *E* of ether : an
Euclidean one **g**^{0}, which is bound to the ether (when the latter is
considered at a macroscopic scale), i.e. which makes it a rigid
"body"; and the physical one **g**, which is related to **g**^{0} by a contraction of measuring rods in the ratio p_{e}/p_{e}^{¥} (with p_{e}^{¥} the value of p_{e} in remote regions, free from gravitation), depending on time in the
non-static situation. The clocks bound with *E *are assumed to be slowed down in the same ratio, and
special relativity allows to pass to moving local frames. This assumption of
gravitational contraction (resp. dilation) of space (resp. time) standards
expresses the principle of equivalence between the metric effects of motion and
gravitation, which has a basis in the ETG. Independently of any field equation,
it allows, *in the static case*, to
deduce the geodesic characterization of motion of ordinary as well as
light-like particles from Newton's second law expressed with physical standards
affected by gravitation.

The field equation for p_{e} obtained within Newtonian mechanics in assuming an
ether compressibility keeps the same form in the modification integrating the
metric effects, but it becomes non-linear in p_{e}. In the spherical static situation, it leads to Schwarzschild's
space-time metric **g** outside
the body, but the interior metric is different and agrees with the EP.
Furthermore, the associated "acceleration" field **g** (Eqs. (5) and (20)) is the exact Newtonian one, inside and
outside. The same interior metric as here has been found (in the particular
case of uniform density) by Winterberg in his theory, also based on ether [7].
However, Winterberg's theory is very different, first in the concept of ether :
he assumes that the ether is made of quantum particles, obeys (nonrelativistic)
mechanics and is subjected to gravitation. Here it is continuous at any scale,
the macroscopic ether of the gravitation theory defines a global inertial frame
(when there is one), and gravity is caused by ether as its pressure force.
Winterberg's theory needs ether plus an independent absolute space which would
be a Newtonian inertial frame for the moving ether (Winterberg's ether has a
macroscopic motion even in the static case). Winterberg's field equation also
is very different, since it is one for a 4-vector potential (instead of scalar
here) and it is linear in this potential, very much like the equations for the
electromagnetic vector and scalar potentials.

The basic assumption that
the material particles are local flows in the fluid ether [13] leads to admit
that, in the case of very high density, the macroscopic motions of ether and
matter coincide. The study of
gravitational collapse in free fall (CFF) with spherical symmetry, made under
this assumption, exhibits marked differences with GR. If nothing would stop the
free "fall", the implosion would necessarily be followed by an
explosion in such a way that, as seen from local observers, the space within
the body would become infinite. This is the exact contrary of the point
singularity of GR; perhaps our calculation is less justifiable when the density
ceases to be very high, but it does apply to preclude the occurrence of a
singularity with infinite density. Moreover, according to the proposed theory,
gravitation waves (moving at the velocity of light) may well exist outside a
body also in the case of spherical symmetry, and indeed in the situation of CFF
the occurrence of an ingoing shock wave seems quite plausible, which would
release energy and stop the free fall.

It is emphasized that the
present theory is non-covariant; this is logically consistent since it assumes
a privileged frame. But therefore, the fact that Schwarzschild's exterior
metric is found in the static case with spherical symmetry does not completely assure
that such observational data as the advance in the perihelion of planetary
orbits are satisfactorily predicted : it is necessary to build an equivalent of
the post-Newtonian (pN) approximation of GR and to check that it agrees with
observational data. Progress is currently being made in this direction and the
results obtained so far seem to remain compatible with this requirement,
although a further investigation is still needed. In particular, this research
in progress has already shown that no gravitation wave can be found in the
first pN correction. The pN approximation also shows that the theory with
Newtonian space and time (Sect. 2) is not a consistent approximation of the
present theory, since the effects of the compressibility cannot be separated
from the metric effects.

A recent result is that a
consistent Newton law can also be defined in the general (time-dependent)
situation, and leads to a true conservation equation for the energy; this
includes both the (negative) potential energy of matter in the gravitation
field and the positive energy of the gravitation field itself. However, the
Newton law is *not* compatible with
geodesic motion in time-dependent situations. The energy conservation equation,
together with the weak-field, linear approximation of the present theory, allow
to calculate the gravitational radiation flux in a way similar to that followed
in linearized GR. In particular, if the center of mass of the gravitating
system is at rest in the privileged frame or "ether", this radiation
flux corresponds to an energy *loss*
which depends only on the third time derivative of the quadrupole tensor and
involves a *G/c*^{5} factor
(this is much as the famous "quadrupole formulae" of GR, on which are
based the estimates of the secular decrease in the orbital period for a binary
system). For this question also, however, it remains to verify that the effect
of changing the reference frame is small for expectable absolute velocities.

Let us finally comment on
the Lense-Thirring effect: this is a prediction of GR, according to which the
rotation of a massive body should manifest itself by a "magnetic"
component of the gravitation field, thereby causing a specific precession of
the orbit of any test particle around the body. This effect comes from the fact
that in GR: (i) particles move along geodesics and (ii) non-zero *g*_{0i}
components of metric **g** are
predicted in the case of stationary rotation [4,18]. In the present theory, a
stationary rotation of a massive body, with respect to the privileged frame *E*, causes a
constant field of "ether pressure" *p _{e}* and, in that case, this theory also predicts geodesic
motion of test particles- but the

ACKNOWLEDGEMENTS

I am very grateful to Prof. P. Guélin for many helpful discussions and
comments on this work. It is also a pleasure to thank Prof. E. Soos for his
interesting remarks on this subject. The interest of the questions of a
referee, on the gravitational radiation and the Lense-Thirring effect, is also
acknowledged.

*Received June 26, 1995*

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To the beginning of “Scalar theory of gravity as a pressure force”