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It is shown that the principle of conservation of energy
implies that gik tensor of spacetime must have an
antisymmetric component (postulated in 1950 by Einstein, for a
different reason).
The resulting metric redshift simulates expansion
of the universe with Hubble's constant H0 = c/R,
where c is speed of light and R is Einstein's radius of the
universe.
When translated from Einsteinian gravity to Newtonian the
effect simulates a Newtonian drag, acting on any moving object
in the universe, equal c2/R, which, when applied to
photons, simulates "tired light effect".
If the entire observed expansion of the universe is apparent
and Hubble's constant is ~70 km/s/Mpc,
then some of the testable results are:
(i) that the mass density of the universe is
~6x10-27 kg/m3,
(ii) that there should be a lower limit on dynamic friction
acting on any moving object in the universe of
~7x10-10 m/s2,
(iii) that the apparent expansion of the universe should look
as if it were accelerating with acceleration
(dH/dt)at t=0 = ~2.5x10-36 s-2,
(iv) that the cosmic background radiation is thermal radiation
of the non luminous matter of the universe heated by the
redshifted starlight,
(v) that the average size of chunks of non luminous matter is
~1 m if their specific density is
~103 kg/m3.
The necessity of existence of non-symmetric component of gik tensor (component that was postulated by Einstein [Einstein, 1950]) may be demonstrated with an example of a photon traveling through a non empty stationary (non-expanding) universe:
The energy of the photon influences the spacetime geometry so the photon is in principle detectable by some gravitational detectors in the universe. Certain amount of information (and therefore energy) has to be delivered to those detectors. The total energy of the universe is conserved so the photon's energy at certain distance from its source has to be smaller than it had been at the source by the amount that shows up in all possible detectors of the universe that detect this passing-by photon. If it were not so then it would be possible in principle to build a tidal power plant powered by the energy from the passing-by photons without energy of those photons ever diminishing. This power plant would create an infinite amount of energy from nothing, which would contradict the principle of conservation of energy and therefore it is impossible.
The necessary conclusion is that frequency of a photon (that is proportional to its energy by Plank's relation E = hf), as observed by the observers located along the photon's path, has to be diminishing along the photon's path. Since it is not possible for the energy of a photon to drop while the photon is on its way because there are no "gravitational forces" acting on the photon (for the lack of "gravitational forces" acting at the distance in the real world) the light coming from distant sources in a stationary universe has to be redshifted roughly proportionally to the distance of the source of the light to keep energy conseved.
Thus, since this effect, which might have been already observed in the universe as Hubble's redshift (a.k.a. cosmological redshift), can't be explained by the loss of energy of the photon (a.k.a. tired light effect). The geometry of the spacetime itself has to be responsible for this effect. It means that this redshift is a metric redshift and that therefore, in homogeneous space, the time rate (defined as derivative of proper time with respect to coordinate time) at the source of the photon has to be smaller than the time rate at any observer. It means that for any stationary observer in a non-empty stationary homogeneous universe the time has to run slower in any observed location of that universe roughly proportionally to the distance to that location for energy to be conserved.
For the above reason the effect might be called general time dilation. It has to be of course a purely relativistic effect that, unlike the common gravitational redshift, has no counterpart in the Newtonian gravity since Newtonian gravitational field is a conservative field. Such behavior of time may look at first glance logically impossible but it does not lead to any logical contradiction as it can be immediately verified by the existence of an obvious physical model that would behave the same way (a universe with accelerating time rate). This behavior of time is simply another relativistic paradox illustrating the non Newtonian nature of the real world and that the time is not absolute.
Because of this effect the light signals in the real world (unlike in simplified cosmological models described by symmetric metric tensors) are not time reversible and therefore the metric of the real spacetime must have non vanishing time-space cross terms. Because of possibility of diagonalization of any symmetric tensor, gik tensor has to have an antisymmetric component to resist diagonalization and assure this way the presence of cross terms in the metric.
The amount by which time slows down with distance from the observer might be determined from the amount of Newtonian potential energy that matter in space would gain from passing by photons since the Newtonian potential in Newtonian gravity corresponds directly to the time dilation in the real world. The calculations might be done by calculating only the change of potential energy of matter in space without considering any apparent attractive gravitational forces (that couldn't be legally applied to particles as fast as photons). This way the calculation becomes simple and free of any suspicion of an unwarranted approximation since it can be done at conditions of arbitrarily small space curvatures and arbitrarily low velocities of matter at which Newtonian solutions are arbitrarily accurate. See Appendix A for the derivation of the amount of the redshift and the projected metric of the spacetime.
The amount of the postulated metric redshift per unit of distance turns out to be 1/R where R is a constant that is equal to the value of Einstein's radius of universe and in terms of the apparent recession of the light source Hubble's constant of the apparent expansion of a stationary universe, which is due to the postulated redshift, is therefore in first approximation equal to c/R, where c is speed of light. Let's see whether this redshift may be identified with the observed Hubble's redshift and therefore used for the calculation of R from the observed value of Hubble's constant.
Assuming Hubble's constant ~70 km/s/Mpc the result is R = ~4 Gpc, which implies the density of a stationary universe ~6x10-27 kg/m3 (see Appendix A for the relation between Einstein's cosmological constant LAMBDA, R, and the density of a stationary universe). If this predicted value happens to be the real density of the universe then it seems that it is safe to identify our metric redshift with the Hubble's redshift.
If this metric redshift is the observed Hubble's redshift then the spacetime metric may now remain stationary (no real expansion of space needed) to be consistent with observations. It seems, based on the obtained result for the density of the universe, that it may be safely assumed that the whole observed expansion of the universe is an illusion (which is the justification for the title of this paper).
Another conclusion from the above is the existence of an apparent drag acting on any moving object in the universe and simulating the postulated by many authors tired light effect. The amount of the drag is c2/R, which, for the above assumed value of Hubble's constant, is ~7x10-10 m/s2. It is fairly close to the drag experienced by Pioneer 10 and Pioneer 11 space probes. This observation was not known (at least to the author) at the time of writing the original version of this paper in 1985 but it seems to confirm (possibly only accidentally) its main idea so it has been included in this version as a second piece of evidence in favor of the theory under discussion (the first and only up to this point piece of evidence being the value of Hubble's constant roughly agreeing with the estimated mass density of the universe).
Another observation that was not known at the time of writing the original
version of this paper in 1985 and which can be used either as the third piece
of evidence in favor of the theory under discussion or as means to falsify it,
depending on the results of the actual observations, is that the universe
should look as if its expansion were accelerating.
Since the theory predicts (at least for the distances small enough for the
curvature of space to be safely neglected) the observed redshift as
expressed by eq. (6):
Z(r) = exp(r/R) - 1 = r/R + (r/R)2/2 + (higher order components),
and so it predicts that Hubble's constant should be seen as function of time
(while looking backwards in time, and ignoring the higher order components) as
H(t)predicted = H0 + H02t/2,
while in a uniformy expanding universe it would be seen as
H(t)uniform = H0 + H02t,
then the apparent acceleration of the (apparent) expansion of the universe
is predicted to be:
(dH/dt)at t=0 = H02/2,
which for the assumed above value of Hubble's constant it is
~2.5x10-36 s-2.
If a precise observation of this acceleraton proves that the observed acceleration differs from this predicted value then the theory under discussion has to be abandoned. It has no adjustable parameters that could save it since it is derived from the first principles only.
In case the theory can't be falsified by observation of the acceleration since it predicts the acceleration correctly (within the observational error), then to foresee other possible applications of the above a connection with the cosmic background radiation has been investigated. This radiation cannot be just the redshifted starlight since then it could not have the black body spectrum that it has. It seems therefore that it has to be the radiation from non-luminous matter that is in thermal equilibrium with the redshifted starlight. If it is so then it presents an opportunity to calculate the average size of the pieces of non luminous matter of the universe. It is because the chance of a photon hitting an obstacle on it's way (and transferring to the obstacle its energy that then becomes thermal energy) is approximately proportional to the area of the obstacle (square of its linear dimension) and to the number of obstacles along the photon's way (inversely proportional to the cube of the distance between obstacles). Since for a fixed mass density of the whole space (already determined from Hubble's constant) the distance between obstacles is proportional to their linear size, the chance of the photon hitting an obstacle becomes inversely proportional to its linear size.
So, knowing the temperature of the redshifted starlight, presumably re-emitted as a thermal radiation from the non luminous matter (~2.7 oK), and assuming specific density of the matter that the non luminous matter is made of, one may determine the average size of the pieces of non luminous matter of the universe. It comes out ~1 m for the specific density of that matter is ~103 kg/m3 (see Appendix B for details), which seems a size large enough to absorb photons redshifted to the millimeter wavelength range.
The author expresses his gratitude to Dr. Helmut A. Abt, Dr. Chris E. Adamson, Dr. John Baez, Dr. Tadeusz Balaban, Prof. A. Gigli Berzolari, Dr. Philip Campbell, Dr. Tom Cohoe, Dr. Marijke van Gans, Prof. Roy J. Glauber, Dr. Mike Guillen, Dr. Alan Guth, Dr. Martin J. Hardcastle, Dr. Franz Heymann, Dr. Chris Hillman, Dr. Don A. Lautman, Dr. Alan P. Lightman, Dr. David McAnally, Dr. M A H MacCallum, Prof. Richard Michalski, Dr. Bjarne G. Nilsen, Dr. Bohdan Paczynski, Dr. Janina Pisera, Dr. Ramon Prasad, Dr. Frank E. Reed, Dr. S. Refsdal, Dr. Clay Spence, Dr. Andre Szechter, Dr. Michael S. Turner, Dr. Slava Turyshev, Dr. Clifford M. Will, Prof. Ned Wright, and anonymous referees from various scientific journals, for the time they had spent discussing with the author the subject of this paper and related issues.
Einstein, A., 1950, "On the Generalized Theory of Gravitation", Scientific American, Vol. 82, No. 4.
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| (1) |
where G is Newtonian gravitational constant, and other terms as described before. The gravitational energy acquired by an element of space containing particles of mass dm is Q(r) dm. Putting dm = S rho dr, where S is an arbitrary area of sheet of light and rho is mass density of the universe, one gets the energy of dust particles between plane at r = 0 and the parallel plane at r over area S as
| (2) |
Because of the principle of conservation of energy this energy has to be equal energy lost by the light
![E(r) = [sigma(0) - sigma(r)] S](pic/eq03.jpg){kind=small}
| (3) |
Taking second derivative with respect to r one gets from equations (2) and (3)
| (4) |
and from equations (1) and (4)
| (5) |
This value of LAMBDA makes it of course equal to the cosmological constant of Einstein's universe of mass density rho. It is an interesting result since the derivation has been done using only Newtonian equations. Replacing surface energy of sheet of light by its coordinate frequency nu(r) to which sigma(r) is proportional by Plank's formula (E = h nu), one obtains
| (5a) |
Solving equation (5a) one obtains frequency of light as function of distance r as
| (6) |
In Newtonian terms the effect expressed by equation (6) works as if light moved against gravitational field c2/R where R = LAMBDA-1/2, or so called "Einstein's radius of universe". The redshift caused by the effect is of course Z = exp(r/R) -1. For r << R one may drop higher order terms in Z and it simplifies the formula for the redshift to Z = r/R. The effect therefore implies that a stationary universe should look as if it were expanding with Hubble's constant of that apparent expansion
| (7) |
The metric of spacetime that would produce the effect
expressed by equation (6) could be determined by placing the
redshift into the time-time term replacing
nu(r)/ nu(0) by d tau(r)/dt, which is the
rate of proper time at distance r from the observer), and
requiring zero light interval. Such a metric in
t, r spacetime could be:
gtt = exp(-2r/R),
gtr = -exp(-2r/R),
grt = exp(2r/R),
grr = -exp(2r/R).
Obviously such a matrix has identically vanishing determinant which would show that time and space are not independent from each other (that e.g. the shape of space determins the shape of time as we might have expected anyway), and that the metric is therefore non Riemannian:
| (8) |
with assumed spatial part dr2 of curvature 1/R.
It's worth noting that, as follows from equ. (8), the isotropic spacetime might have a property that at any stationary observer the sum of curvature of space (1/R) and of the change in rate of proper time along distance vanishes:
| (9) |
![I_o_(nu) = c_1 nu^3 / [exp(c_2 nu / T_s) - 1]](pic/eq10.jpg){kind=small}
| (10) |
where c1 and c2 are constants and Ts is the temperature of the star's surface (with the peak value at nu = 2.82 Ts/c2), according to equations (6) and (10) the distribution at distance r from the source is
![I(nu, r) = (c_1[nu exp(r/R)]^3)/(exp[c_2 nu exp(r/R) / Ts] - 1)](pic/eq11.jpg)
| (11) |
therefore for any observer, the spectral distribution of radiation from all the stars is
![I(nu)=c_3 Integ 0..oo ((p(r)[nu exp(r/R)]^3)/(exp[c_2 nu exp(r/R)/T_s]-1))dr](pic/eq12.jpg)
| (12) |
where c3 is a constant and p(r) is probability of light passing distance r without hitting any obstacle on its way, which is
| (13) |
where A is the average area of an obstacle and L is the average distance between obstacles, assuming that r A << L3. Combining equations (12) and (13) and making substitution z = c2 nu/Ts, x = z exp(r/R), and a = A R/L3 one gets the spectral density of the radiation from all luminous sources as
| (14) |
where c4 is a constant. It is visible from the equation (14) that this distribution is not a black body distribution, and therefore the background radiation is not just the redshifted starlight. Therefore the background radiation must be a radiation from the non-luminous matter of the universe, matter that is in thermal equilibrium with the redshifted starlight. For a << 1 the peak value of this distribution represented by equation (14) is at z = 1.55 a1/2 and therefore the temperature of a black body having the peak value of its distribution of radiated energy at the same frequency is
| (15) |
The average distance between the obstacles L may be determined from the relation rho L3 = rhoo D3 where rho is as before the density of the universe, rhoo is the density of the obstacle, and D is the diameter of the obstacle (assuming that the obstacles are roughly spherical objects for which approximately A = D2, and that almost the whole matter of the universe is composed of such obstacles). R and rho can be determined from equation (7). After all the substitutions the average diameter of the obstacle is
| (16) |
Assuming value of Hubble's constant H = ~10-18 s-1, Average temperature of stars Ts = ~104 oK, temperature of thermal equilibrium of the universe ~2.7 oK, and the density of the matter of obstacles rhoo = ~103 kg/m3, the average diameter of the obstacle is of order of 1 m. It is large enough size to make the non luminous matter of the universe responsible for the absorption of light in the millimeter wavelength range.