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We reproduce hereafter parts of a chapter relating to "Stars of Uniform Density" taken from the the book of Steven Weinberg "GRAVITATION AND COSMOLOGY" (Steven WEINBERG, 1972).
<< General relativity finds an interesting application to one other class of stable stars, those consisting of incompressible fluids... These stars are of interest, not because they actually exist, but because they are simple enough to allow an exact solution of Einstein's equations, and because THEY SET AN UPPER LIMIT TO THE GRAVITATIONAL RED SHIFT OF SPECTRAL LINES FROM THE SURFACE OF ANY STARS.
The red shift is given by the relation
1 + z = (1 - 2MG/R)^-1/2 (1)
and the structure of the Einstein' equations imposes the absolute upper limit 4/9 to MG/R, which means that z of the light signals leaving those stars is always less than 2.
However, THERE IS NO THEOREM THAT LIMITS THE RED SHIFTS OF LIGHT SIGNALS FROM THE INTERIOR OF STATIC SPHERICALLY SYMMETRIC BODIES. For instance, a light signal FROM THE CENTER of a transparent uniform star would have a red shift given by the relation
1 + z = 2 / (3(1 - (2MG/R))^1/2 - 1 (2)
z becoming infinite as MG/R approaches the maximum value 4/9.
In those relations, M and R are respectively the mass and the radius of the star, and G is the gravitational constant divided by c^2. >>
We replace the mass M of the star by (4 Pi R^3 dy) / 3 were Pi is the constant of the same name and dy the density of the star, and we replace G by G/c^2.
MG/Rc^2 becomes then 4 Pi G R^2 dy / 3 c^2.
For any density dy, we may replace 4 Pi G dy / 3 by a constant K^2, which gives
2MG/Rc^2 = 2 K^2 R^2 / c^2
For a stable star of radius R whose density is the same as the mean density of the universe (the value 2.10^-29 g / cm^3 is often retained),
K = 2,36.10^-18 / sec and 1/K = 4,23.10^17 sec or 1,34.10^10 years which is analogous to the Hubble time.
Let us remember that for General Relativity, the radius of such a star is limited by the relation
MG/Rc^2 < 4/9 or K^2 R^2 / c^2 < 4/9, i.e. KR/c < 2/3,
which means that the maximum radius of such a star is 2 c / 3 K, or about 9 billions light-years.
Let us also observe that relations (1) and (2) become respectively
1 + z = (1 - 2 R^2 / R0^2)^-1/2 (3) and
1 + z = 2 / (3(1 - 2R^2/R0^2)^1/2 - 1) (4)
for those stars of density 2.10^-29 g / cm^3 and R0 = c/K.
With the help of equation (4), we may express R in terms of z and R0:
R/R0 = ((2z(2z + 3))^1/2) / 3(1 + z) (5)
We retain from those remarks that for General Relativity, a light signal coming from the center of a "stable", i. e. not contracting nor expanding, star whose radius is less than c/K undergoes a "gravitational braking" and a corresponding loss of energy which express itself by a red shift given by relation (4). We observe also that the constant K is similar to the Hubble constant.
We shall investigate hereafter if relation (4) is compatible with the cosmological data.
We admit that the universe is euclidian, homogeneous, isotropic and spherical, and that its radius Ru is much larger than the radius R0 = c/H of the "observable" universe (c is the speed of light and H the Hubble constant). In fact, we consider that Ru is as big as we want it to be (a simple definition of infinity).
We isolate from this universe a sphere of radius R. The theorem of GAUSS tells us that a body situated at the center of the sphere and having a velocity V moves away from the center till a distance R according to relation V = KR (6). At the distance R from the center, its velocity cancels out.
Let us remember here that K^2 = 4 G Pi dy / 3, and that K has the same caracteristics as the Hubble constant.
At any distance r < R from the center, the velocity of the body is given by the relation
v = V cosKt (7),
and the distance r is attained after the time t according to relation
r = R sin Kt = V sin Kt / K (8).
From the relations (7) and (8), we can express r in terms of v, or v in terms of r, with the help of relations
r = R (1 - v^2 /V^2)^1/2 (9) and v = V (1 - r^2/R^2)^1/2 (10)
Relation (10) can also take the form v = (V^2 - K^2 r^2)^1/2, or
v^2 = V^2 - K^2 r^2 (11)
It is interesting to note that K^2 r^2 corresponds to the square of a virtual velocity vr acquired at the center of the sphere by a body initially at rest at a distance r from the center and virtually falling towards it. On the other hand, K^2 r^2 gives the real energy loss of a body having left the center with a velocity V. Indeed, for a body with mass m,
m v^2 / 2 = m V^2 / 2 - m vr^2 / 2 (12)
We admit that the red shift is given by the non-relativistic relation z = (v/c) / (1 - v/c). We admit also that the relevant velocity here is the velocity vr of relation (12).
In fact, vr = Kr can be considered as a virtual velocity whose counterpart is the velocity Hr considered as real by the theory of the universe in expansion, thus
z = (vr/c) / (1 - vr/c) (13)
Let us remember that the radius of the observable universe is the radius R0 = c/K (or c/H if we liken K to the Hubble constant).
For the observable universe, vr = KR, c = KR0, and
vr/c = R/R0. It follows from (13) that
z = (R/R0) / (1 - R/R0) (14) and
R/R0 = z / (1 + z) (15)
By the way, with the relativistic Doppler shift,
(1 + z)^2 = ( 1 + v/c) / (1 - v/c), we would have obtained
(1 + z)^2 = (1 + r / R0) / (1 - r / R0) (16) and
r/R0 = ((1 + z)^2 -1) / ((1 + z)^2 + 1))
FINALLY, LET US REMEMBER THAT RELATION (4) FROM GENERAL RELATIVITY IMPOSES THAT FOR AN OBJECT OBSERVED AT A DISTANCE R >= 2 R0 / 3 FROM THE CENTER, Z BECOMES INFINITE.
IF WE REPLACE R/R0 BY 2/3 IN OUR RELATION (14), WE OBTAIN A MAXIMUM REDSHIFT OF 2. BEING EXCEEDED BY A GREAT NUMBER OF GALAXIES AND QUASARS, THAT VALUE MEANS THAT GENERAL RELATIVITY IS AT VARIANCE WITH THE OBSERVED FACTS, AND THUS CANNOT BE RIGHT.
Using classical mechanics, we have demonstrated that an electomagnetic signal coming from the center of a "stable homogeneous star" having the density of our universe and a radius R0 = c/K undergoes a red shift in its travel, and that the relation between the distance covered by the signal and its red shift is given by relation (15) which is different from that established for an expanding universe. The existence of that relation is beyond doubt. If we admit the Cosmological Principle according to which all positions in the universe are essentially equivalent, we have also to admit that every point in the universe can be considered as the center of a sphere of radius R0, and consequently that the wavelength of any light or electomagnetic signal coming from any point of the universe will increase in proportion of the travelled distance according to relation (15). That position is as justified as that of the proponents of the theory of an expanding universe who consider that any point of the universe coincide with the center of the expansion.
As for us, we started from the hypothese of a stable euclidian universe, using the rules of classical mechanics. It follows on the one hand, that the theory of an expanding universe is superfluous, and on the other hand, that the theory of General Relativity that leads to a wholly different relation is here NOT ONLY SUPERFLUOUS BUT ALSO ERRONEOUS.
Furthermore, the same way as an electromagnetic signal undergoes a red shift and thus a loss of energy owing to its interaction with the universe, any moving body having a rest mass loses a fraction of its kinetic energy in proportion of the distance it travels in the universe, in accordance to relation (12). This phenomena is clearly absolute.
Steven Weinberg, Gravitation and Cosmology,1972, John Wiley & Sons, Inc., pp 330-335
Les Adrets de l’Estérel, October 27, 1997.