The hypothesis of generalized time dilation

W. Jim Jastrzebski

File 3263-u.htm: June 2007, first issue: February 1985

It is shown by an example of dynamical friction of photons in Einstein's universe that for any observer at the center of polar coordinates at rest in the space of this universe, the time dilation per unit of distance along radial coordinate (-d2τ/dtdr) equals the curvature of space (1/R). It is hypothesized that this fact is an action of a principle of Einstienian physics, called here a principle of generalized time dilation, that says that the sum of the rate of time dilation per unit of distance and the space curvature vanish identically. The theory, eccept predicting near quasars, predicts also three other observations that are given below with their theoretical and observed values for the radius of curvature of space R = (4.3 ± 0.2) Gpc that has been adjusted to the observed value of Hubble parameter H0 = c / R = (70 ± 3) km/s/Mpc: (i) density of space ρ = c2 / (4πG R2) [10-27kg/m3] = 6.0 ± 0.5, observed 5.5 ± 4.5, (ii) accelerating expansion of space (dH/dt) / H02 = 0.50, observed 0.45 ± ?, (iii) 'anomalous' acceleration of space probes a0 = c2 / R [10-10 m/s2] = 7.0 ± 0.3, observed 8.7 ± 1.3 The hypothesized metric of Einstein's universe (because of its isotopy shown only for t,r coordinates) is ds2 = exp(-2r/R)dt2 + 2sinh(2r/R)dtdr - exp(2r/R)dr2.

Introduction

The purpose of this paper is to point to a fact that there must exist a mechanism responsible for the Hubble redshift in a stationary Einstein's universe [1] since the photons in it should experience at least some dynamical friction [2] [3] [4] [5] [6] because of the conservation of energy.

To show the Einsteinian 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 homogeneous space causes no redshift since it can't change energy of free falling photons. So the culptrit must be the time dilation as there is no other physics beyond curvature of space and the time dilation in Einstein's universe.

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, not existing in the 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 hypothsized effect is just the time dilation.

The additional hypothesis of the Newtonian model, that the Newtonian gravitational field is conservative, can be safely ignored while doing calculations for photons since the principle of conservation of energy contradicts this hypothesis: The principle requires that photons exhibit at least some dynamical friction as interacting gravitationally with surroundings and in the Newtonian model they don't.

So knowing that the Newonian model is defective not only because it ignores totally the space curvature 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 as if all the requirements of the Newtonian formalism were fulfilled. For this we just have to set a model with negligibly curved space in which photons 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.

Remark on the conservation of energy in gravitation: Since the principle of conservation of energy is needed in calculations that use a concept of gravitational energy, the gravitational energy has to be found and proved that together with kinetic energy it makes an invariant quantity. It may be done in the following way: As it is well known, the total energy of any object is E(x) = m(x)c2sqrt(g00(x)), where m(x) is inertial mass of the object and g00(x) is the time-time term of metric, which makes C2(x) = c2sqrt(g00(x)) a generalized speed of light squared. During a free fall along x the object gets into space where the time runs at different rate and where there is a different amount of space. In general it changes inertial mass of the object m(x) and speed of light C at location of the object as perceived by a non local observer. It turns out that object's internal energy that corresponds to this new speed of light plus the object's kinetic energy in this frame is conserved as we would expect (as is readily demonstrated through simple algebraic transformations [7] resulting in dE(x)/dx = 0). What we might not expect though is that internal energy of the object E0(x) = m0C2(x) is also its gravitational energy since derivative of this energy with respect to distance dE0(x)/dx turns out to be equal to the Newtonian gravitational force in direction -dx. It proves that the sum of gravitational and kinetic energy is conserved in a free fall and we can use this fact in the derivation of results.

The dynamical friction of photons (redshit 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

Let in a stationary, closed, homogeneous, nearly flat space filled with dust of mass density ρ a small ball of matter of negligible size is radiated out isotropically creating a sphere of photons of negligible thickness, radius r, and inertial mass m(r) at this radius. Also let the photons not collide with the particles of dust.

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 (1a)

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

dM(r) = 4π r2 ρ dr (1b)
and so its gravitational energy is
dE(r) = - Q(r) dM(r) (1c)

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) (1d)

After substituting values from (1), (1a), and (1b) to (1d) we have finally the gravitational energy differential as

d2E = 4π G ρ m(r) dr2 (2)

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 generalized time dilation.

The energy lost by photons is

dE = - c2 (dm(r) / dr) dr (3)

From (2) and (3), 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 (3a)
that shows up here (but not as accidentally as it might seem and a possible reason is explained in conclusions) we get
d2m(r) / dr2 = Λ m(r) (4)

Solving (4), 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) (5)

In Newtonian terms (of the above derivation) the effect expressed by equation (5) works as if photons moved against gravitational field of magnitude

a0 = c2 / R (5a)
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 (5) represents the time dilation

dτ/dt = exp(- r / R) (6)
where τ is the proper time at point in space and t is the coordinate time at the observer.

Differentiating (6) 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

d^2 tau/dtdr + 1/R = 0 (6a)
Now we generalize this effect proposing a hypohtesis of generalized 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.

Conclusions

Since the effect is isotropic and is valid for any observer it is called the generalized 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 a space with a central symmetry, in particular in an isotropic space. The generalized 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 gravtitaiotn the space is flat.

However if the hypohtetical principle of generalized 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 out to show up in (3a).

Some astrophysical implications and numerical predictions

The geometrical implications of existence of the effect are that the spacetime metric must contain time-space cross terms as proposed already by Einstein [16] (which makes this metric non symmetric) and be degenerate as it is proposed here to remove additional independencies of its terms arising from the presence of the antisymmetric part and to couple the curvature of space to the time dilation required by the conservation of energy. The metric tensor of isotropic (t, r) spacetime that produces the effect is
gik = [ e-2r/R    -e-2r/R ]
[ e2r/R    -e2r/R ]
(7)
which is degenerate and so a non Riemannian, yet it produces quite a decent metric, approaching Minkowski for r << R
ds2 = exp(-2r/R)dt2 + 2sinh(2r/R)dtdr - exp(2r/R)dr2 (8)
with a spatial part of assumed curvature 1/R to be consistent with the spatial part of Einstein's universe for which the above derivation has been carried out.

Since, as it follows from (6a), 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

H0 = c / R (8a)

Hubble's parameter H0 = 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 (3a) 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 a0 = (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 (6) tells that the universe should look as if its apparent expansion were accelerating since the observed redshift according to (6)

Z(r) = exp(r / R) - 1 = r / R + (r / R)2 / 2 + ... (9)
makes Hubble's parameter as function of time, with the sense of increasing t toward past,
H(t) = H0 + H02 t / 2 + ... (10)
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/H0
Hu(t) = H0 + H02 t + ... (11)
The derivative of a difference between H and Hu with respect to time is the apparent acceleration of apparent expansion
dH/dt = H02 / 2 + ... (12)
which for the above value of H0 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 H02 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 thah the gravitational redshift that is poper 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 (13)
while the redshift caused by the generalized time dilation is expressed by (9). 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 (6) from which (9) 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 (9). As it is seen from comparing (9) and (13) the generalized 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 hypohtesis of generalized 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.

Acknowledgments

The author expresses his gratitude to Dr. Helmut A. Abt, Dr. Chris E. Adamson, Prof. John Baez, Dr. Tadeusz Balaban, Prof. A. Gigli Berzolari, Dr. Philip Campbell, Dr. Michal Chodorowski, 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. Marek Kalinowski, Dr. Don A. Lautman, Dr. Alan P. Lightman, Dr. David McAnally, Dr. M A H MacCallum, Prof. Richard Michalski, Prof. Joe M. Namyslowski, 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 have spent discussing with the author the subject of this paper and related issues. Most thanks goes to Dr. Michal Chodorowski of the Astronomical Center of Polish Accademy of Sience, without whose friendly critique many of the ideas expressed in this paper might have never developed to a legible state and Prof. Joe M. Namyslowski without whose help it could have never been published in any scientific journal.

References

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