Introduction

Because of the conservation of energy the photons have to experience some dynamical friction resulting in their redshift. If they didn't then in principle the matter could be carried away in photons and then returned in photons without any loss of energy. It would have caused a temporary change of gravitational field that could be used to create energy the same way as it is done in a tidal power plant but in this case the energy would be created from nothing. It wouldn't be possible in the world in which energy is conserved. It says that if energy is conserved the photons can't move around in a homogeneous, stationary space without any redshift and obviously there has to be a gravitational reason for it.

The reason can't be the mythical tired light effect (a redshift without a corresponding time dilation) since it is not possible in relativistic gravity where the whole physics is just the curvature of space (that is obviously causing no redshift) and the time dilation. So it has to be the time running slower at the source of light, the same as it is in the common gravitational redshift. However, unlike in the common gravitational redshift, where the redshift changes to the blueshift when the photon moves back by the same path, the effect of dynamical friction has to produce a redshift in any direction. Such a redshift is not possible in the Newtonian model of gravity with its conservative gravitational field so the effect must be a relativistic one and so the dynamical friction experienced by photons is, and is called here, relativistic dynamical friction. The physical reason for it (time running slower away from the observer and in any direction) is called here the general time dilation.

The implications of the existence of general time dilation are that the light signals in presence of matter are not time reversible and so the spacetime metric must contain time-space cross terms in any coordinate system. Therefore metric tensor of spacetime, to avoid a possibility of diagonalization, has to be non symmetric as proposed by Einstein ^[6] and degenerate as proposed here to remove some additional independencies of its terms arising from the presence of the antisymmetric part, and to couple the time dilation to the space curvature to keep the principle of conservation of energy valid ^[7]. One of the consequences of the above is that it implies a non Riemannian geometry of spacetime so all the theories based on Riemannian geometry must necessarily contradict the principle of conservation of energy and therefore they don't apply to the real world.

The general time dilation has been always considered negligible and to the best knowledge of this author it is not included in any contemporary cosmological theory (for obvious reasons in none of those based on Riemannian geometry). Possibly for this reason it has been never determined exactly or at least its derivation is not placed neither in textbooks on cosmology ^{[8] [9] [10] [11] [12] [13] [14]} nor in the books of the opponents of the contemporary cosmological theories ^[15]. To fill up this void it is calculated below and also there is proposed a (non Riemannian) spacetime metric consistent with this effect.

The tool that has been applied to do the calculations is the Newtonian model. It is possible to apply the Newtonian model rigorously here since what is calculated is only the gravitational energy gained by the arbitrarily slow matter in an arbitrarily flat space after the photons had passed by. The trick is to investigate the imprint that the photons have left on the slow matter and to conclude from this imprint what must have happened to the photons in the real world. This way the photons don't get directly into the calculations and so the calculations comply with both rules of correct application of Newtonian math: negligible velocities and negligible curvature of space. The "tired light effect", while not being a real effect, can be simulated (in a form of dynamical friction), similarly as other fictitious Newtonian entities (as e.g. gravitational attractive force or gravitational energy) are simulated without being related physically to the real world. Despite the lack of physical relation, their math, in a kind of magical way (in fact by analogy between the Newtonian math and the time dilation), produces arbitrarily accurate results whenever the space is arbitrarily flat. So whenever the physics assures that the Newtonian math is accurate, the calculations are done using locally the Newtonian magic, as it's done, even for cosmological applications, by other authors ^[16].

Derivation of the results

After the light has radiated out from its source, the space between the source and the plane of photons is not isotropic any more. The state of gravitational equilibrium of each dust particle between the plane at r = 0 and the sheet of light at r has been upset by the surface energy density -s(r)/c² on the side of the source (the missing mass that has been converted into light and transferred in photons to the other side of the particle), and the surface mass density +s(r)/c² on the side of the sheet of light.

Physically, what has changed between the source and the sheet of photons is that now there is a gradient of time rate that has changed the probabilities of finding dust particles at various points between the source and the sheet of photons. The calculations that would tell us about what happens based on those probabilities might be a nightmare. Fortunately we can avoid the horror of calculating probabilities by noticing that the rate of time in space is represented exactly by the Newtonian gravitational potential and so the Newtonian model can tell us exactly what happens. So in the magical world of Newtonian model each dust particle is pushed in the positive direction of the r axis as if the light dragged the particles behind itself. Each particle acquires Newtonian gravitational energy.

The acceleration at each dust particle expressed in terms of Newtonian potential Q(r), assuming that movement of dust particles (galaxies) caused by the influence of of photons is negligible or that Q(r) does not change with time), by elementary Newtonian calculations is

(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 r dr, where S is an arbitrary area of sheet of light and r 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

(3)

Taking second derivative with respect to r one gets from equations (1), (2), and (3)

(4)

This value of L makes it of course equal to the cosmological constant of Einstein's universe of mass density r. It is an interesting result since the derivation has been done using only the Newtonian magic.

Solving equation (4), selecting from the solutions the one that corresponds to conditions of the model, replacing mass of photons m(r) by their coordinate frequency n(r) to which m(r) is proportional, and replacing L by 1/R² where R is known as "Einstein's radius of the universe" (radius of curvature of space) to get a simpler form of the solution, we get

(8a)

Some numerical results and astrophysical implications

Equation (6) tells that the universe should look as if its apparent expansion were accelerating since the observed redshift from equation (6)

Another conclusion might be that the light radiating from a virialized cloud of dust (and possibly any other) must have redshift always greater than the gravitational redshift that is proper for that cloud and might be greater by many orders of magnitude. It is because the gravitational redshift of the light originating in the center of a cloud of dust and reaching its surface at distance r from the center is only

In a virialized cloud of dust the kinetic energy of the particles of dust acts as the isotropy of space in the model used for calculations, by effectively removing for the purpose of calculations the gravitational field and making the system stable, allowing this way the simple calculations leading to (9). As it is seen from comparing (9) and (13) the general time dilation in a virialized cloud of dust causes always greater redshift than gravitational redshift and how much greater depends only on the radius of the cloud (r) and its density (function of R).

Acknowledgments

References

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