Observational evidence for general time dilation and stationary universe

Introduction

To show the relativistic mechanism that causes photons to experience dynamical friction we split the spacetime curvature into the curvature of space and the time dilation. The curvature of space alone can't cause a frequency shift of free falling photons since it does not affect their frequency. So the culprit is the time dilation that directly affects the frequency of the photons.

Furthermore, in a stationary situation, it has to be the time running slower at the source of light, similarly to the ordinary gravitational redshift but unlike in an ordinary gravitational redshift the dynamical friction causes a redshift isotropically. Such a redshift is obviously not possible in the Newtonian gravitation with its conservative gravitational field. So the redshift of photons in a stationary space must be a purely relativistic effect, impossible in Newtonian gravitation.

The Newtonian gravitation however, despite being not able to explain physics of dynamical friction of photons can be used as a tool to calculate it. It is so since math of the Newtonian gravitation is an exact model of the time dilation in the real world and the physical reason for the hypothesized effect is just the time dilation as we've just seen.

The additional hypothesis of the Newtonian model, that the Newtonian gravitational field is conservative, can be safely dropped since the principle of conservation of energy contradicts this hypothesis: This principle requires that photons exhibit at least some dynamical friction as interacting gravitationally with surroundings and in the Newtonian model they don't which is the same as recalling that Newtonian model doesn't apply to photons.

So knowing that the Newtonian model is defective not only because it ignores totally the space curvature, that luckily does not influence the frequency of photons, but also because it ignores the conservation of energy in case of photons, we can ignore the Newtonian hypothesis of a conservative field as applied to photons and then deal with the conservation of energy in the calculations directly, making sure that energy is always conserved. For this we just have to set a model with negligibly curved space in which photons with their speed of light don't show up.

Such a model is possible since we may apply our calculations not to photons but to a negligibly curved space with matter of negligible velocity comparing to the speed of light and investigate the imprint left in this space after the photons already left it. This way we get a theoretical basis for calculating dynamical friction of photons in a rigorous but in an easy to follow and understand Newtonian way.

Since it is so easy to understand we can see how we drift away from Newtonian magic. We are doing it by considering space after photons are gone and assume that they never be back to absorb somehow the lost energy. That's where the process becomes irreversible and the energy goes out of photons and never comes back. It is left in the space in a form of kinetic energy of the matter of the universe. This is the moment when the Newtonian field loses its conservative features. But we have to say that it is proper physics since there isn't even any way of getting the energy back into photons. The process of dynamical friction of photons in relativistic physics is irreversible.

It might be quite a bold statement to make so to be sure that we are right we have to verify our results observationally. To see if we have any luck and the nature still confirms Einsteinian predictions as these.

The dynamical friction of photons (redshift of photons in stationary space) has been always considered negligible by astrophysicists and cosmologists and to the best knowledge of this author it is not included in any contemporary cosmological hypothesis (for obvious reasons in none of these with symmetric metric tensor). 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 books of opponents of contemporary cosmological theories ^[15]. It is calculated below then.

Derivation of value of dynamical friction of photons

After the light has radiated out from its source, the space inside the sphere of photons is not homogeneous any more. The state of gravitational equilibrium of each dust particle inside the sphere has been upset by the the missing mass m(r) of matter that is now carried away in photons.

The Newtonian gravitational field created by the missing mass that has been carried away in photons, assuming that the movement of dust particles caused by the influence of photons is negligible, which means that field g(r) once created by the passage of the sphere of photons doesn't change with time, which assures the lack of feedback effects of matter in the space on photons in the sphere, is

g(r) = G m(r) r / r3

(1)

or in terms of gravitational potential Q(r)

g(r) = - dQ(r) / dr

(2)

The dust particles between sphere at r and a parallel sphere at r + dr have mass

dM(r) = 4π r2 ρ dr

(3)

and so its gravitational energy is

dE(r) = - Q(r) dM(r)

(4)

Dropping a useless from now on vectoral representation of r and keeping only its magnitude r and replacing Q with its differential we have the energy differential as

d2E(r) = - dQ(r) dM(r)

(5)

After substituting values from (1), (2), and (3) to (5) we have finally the gravitational energy differential as

d2E = 4π G ρ m(r) dr2

(6)

Because of conservation of energy the energy gained by the particles in space has to be equal to the energy lost by the photons. The energy of photons gets transferred into gravitational energy of the particles of the space inside the sphere while the photons are experiencing dynamical friction. The Newtonian formalism lets us to do the calculations and to obtain the right results since we observe all the rules of getting right results within Newtonian approximation (negligible curvature of space and negligible velocities of particles in space). We remember that the real mechanism is just the time dilation at a source of photons so later we need to translate the results, whatever they are, into the time dilation at a source of photons to determine the amount of general time dilation.

The energy lost by photons is

dE = - c2 (dm(r) / dr) dr

(7)

From (6) and (7), after differentiating twice with respect to r to get rid of integrals, and replacing all constants by "cosmological constant of Einstein's universe"

Λ = 4π G ρ / c2

(8)

that shows up here (the reason is explained in the last paragraph of this section) we get

d2m(r) / dr2 = Λ m(r)

(9)

Solving (9), selecting from solutions the one that corresponds to conditions of the model, replacing inertial mass of photons m(r) by their coordinate frequency ν(r) to which m(r) is proportional by Planck's relation, and replacing Λ by 1/R² where R is known as "Einstein's radius of universe" (radius of curvature of space) to get a simpler form of the solution, we get

ν(r) = ν(0) exp(- r / R)

(10)

In Newtonian terms (of the above derivation) the effect expressed by equation (10) works as if photons moved against gravitational field of magnitude

ao = c2 / R

(11)

which becomes the value of the dynamical friction of photons and therefore a low limit of a dynamical friction of any other object.

In the world of the Newtonian model the effect simulates perfectly the mythical "tired light effect". But in the real world equation (10) represents the time dilation

dτ/dt = exp(- r / R)

(12)

where τ is the proper time at point in space and t is the coordinate time at the observer.

Differentiating (12) with respect to r we may notice that a stationary, homogeneous space has a property that the sum of the curvature of space (1/R) and of the change in the rate of proper time along distance vanishes at r = 0

(13)

Now we generalize this effect proposing a hypothesis of general time dilation that says that coupling of the rate of time dilation per unit of distance to space curvature is always such that their (tensoral) sum vanishes.

Since the effect is isotropic and is valid for any observer it is called the general time dilation to distinguish it from the ordinary gravitational time dilation that is a vector proportional to g and so it has to vanish in an isotropic space. The general time dilation, because of its tensoral character reflecting the curvature of space, vanishes only in a flat space.

Now it becomes obvious why the effect hasn't show up in Newtonian gravitation and had to be pulled out of it by the principle of conservation of energy. The effect depends on the curvature of space only and in the Newtonian gravitation the space is flat. However if the hypothetical principle of general time dilation is true then the curvature of space is coupled to the rate of the time dilation and the time dilation is coupled to the principle of conservation of energy. This way the calculations need only the time dilation, obtained through the conservation of energy, to produce the right results with the curvature of space forced to show up in the right amount in (8).

Some astrophysical implications and numerical predictions

Since, as it follows from (13), the amount of time dilation per unit of distance in the vicinity of an observer is 1/R then in terms of the apparent recession of the light sources Hubble's parameter of apparent expansion of space in the vicinity of any observer in the universe is

Ho = c / R

(16)

Hubble's parameter H_o = 70 km/s/Mpc (observed presently with standard deviation of about 3 km/s/Mpc) makes the radius of curvature of space R = (4.3 ± 0.2) Gpc, makes through the value of R and Λ in (8) the density of the universe ρ = (6 ± 0.5)x10^-27 kg/m³, and the amount of dynamical friction of photons equal to the low limit of any dynamical friction in the universe a_o = (7 ± 0.3)x10^-10 m/s². This number is about one standard deviation off the "anomalous" acceleration of Pioneer 10 and Pioneer 11 space probes (8.7 ± 1.3)x10^-10 m/s².

Equation (12) tells that the universe should look as if its apparent expansion were accelerating since the observed redshift according to (12)

Z(r) = exp(r / R) - 1 = r / R + (r / R)2 / 2 + ...

(17)

makes Hubble's parameter as function of time, with the sense of increasing t toward past,

H(t) = Ho + Ho2 t / 2 + ...

(18)

In a uniformly expanding universe it would be seen approximately (neglecting the correction for relativistic effects of speed of expansion) as a hyperbola with the vertical asymptote at the moment of big bang at time t = 1/H_o

Hu(t) = Ho + Ho2 t + ...

(19)

The derivative of a difference between H and H_u with respect to time is the apparent acceleration of apparent expansion

dH/dt = Ho2 / 2 + ...

(20)

which for the above value of H_o is equal 2.5x10^-36 s^-2. Comparing to the 2006 Supernova Cosmology Project results ^[17], this value is 11% greater than the observational results while standard deviation of H_o² alone has been 8% in 2006.

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

Zg(r) = (r / R)2 / 6

(21)

while the redshift caused by the general time dilation is expressed by (17). It is so since the conditions in a virialized cloud of dust are identical to the conditions used as a model for the derivation of (12) from which (17) follows.

In a virialized cloud of dust the kinetic energy of the particles of dust acts as isotropy of space in the model used for the 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 (17). As it is seen from comparing (17) and (21) 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 and its density. Therefore the hypothesis of general time dilation may be used also as a justification for the existence of near quasars since then their redshift would depend mainly on the amount of dust around them and not that much on their distances from us.

We also look at the Cosmic Microwave Background Radiation (CMBR).

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 we can calculate the average size of the pieces of non luminous matter of the universe. This is because the chance of a photon hitting an obstacle on it's way, and transferring to it its energy, which then becomes thermal energy, is approximately proportional to the area of the obstacle (P₁ ~ D²) and to the number of obstacles along the photon's way (inversely proportional to the cube of the distance between obstacles P₂ ~ 1 / L³). Since for a fixed mass density of the whole space (already determined from Hubble's parameter) the distance between obstacles is proportional to their linear size (L ~ D => P₂ ~ 1 / D³), the chance of the photon hitting an obstacle becomes inversely proportional to its linear size: P = P₁P₂ ~ D² / D³ = 1 / D. So, knowing the temperature of the redshifted starlight, presumably re-emitted as a thermal radiation from the non luminous matter, 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.

Assuming that the spectral distribution of energy radiated by a star may be presented, with an accuracy to the absorption lines of its atmosphere, by equation

(22)

where c₁ and c₂ are constants and T_s is the temperature of the star's surface (with the peak value at ν = 2.82 T_s / c₂), according to ν(r) = ν(0) exp(- r / R) and (22) the distribution at distance r from the source is

(23)

therefore for any observer, the spectral distribution of radiation from all the stars is

(24)

where c₃ is a constant and p(r) is probability of light passing distance r without hitting any obstacle on its way, which is

(25)

where A is the average area of an obstacle and L is the average distance between obstacles, assuming that rA << L³. Combining (24) and (25) and making substitution z = c₂ ν/T_s, x = z exp(r / R), and a = AR / L³ one gets the spectral density of the radiation from all luminous sources as

(26)

where c₄ is a constant. It is visible from (26) 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 (26) is at z = 1.55 a^1/2 and therefore the temperature of a black body having the peak value of its distribution of radiated energy at the same frequency is

(27)

The average distance between the obstacles L may be determined from the relation ρ L³ = ρ_o D³ where ρ is as before the density of the universe, ρ_o 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 ρ can be determined from H = c / R = sqrt(4 π G ρ). After all the substitutions the average diameter of the obstacle is

(28)

Assuming value of Hubble's constant H = 2x10^-18 1/s, average temperature of stars T_s = 10⁴ K, temperature of thermal equilibrium of the universe 2.7 K, and the density of the matter of obstacles ρ_o = 10³ kg/m³ (H₂O), the average diameter of the obstacle is of order of 2 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.

Conclusions

Acknowledgments

References