11.  The mystery of gravitation – a symbolic calculation

All above calculations were written down on the background of Minkowsky metrics – without any justification. Then it is better to take some more general metrics, to repeat all calculations on the background of this metrics and to put forward the demand of stability of the system of four worlds in relation to this metrics.

The lagrangian of “free space” and electron’s lagrangian of selfaction (see above) must be calculated for a curvilinear metrics. The extremum of the sum of these lagrangians in relation to metrics will give us equations similar to Einstein equations.

The meaning of the lagrangian of “free space” is simple. In the picture described above, any electron radiates electromagnetic waves on the frequency  – and is supported by these waves. The flow of these waves is going through any point in space-time, and for any point we have a density of energy. This density depends on curvilinear metrics – that is the lagrangian of “free space” for this metrics.

The situation is similar to “induced gravitation” of Sakharov [4], but here we have not the vacuum of quantum field theory. We have the “prequantum vacuum” that generates all local physics. This vacuum depends on observable astrophysical parameters and has no divergences. The Sakharov’s term has different sign in different worlds and the sum of these terms will be zero.

It is the first step in possible generalization of the notions of metrics, coordinates, numbers. These generalizations will be discussed in the next chapters together with the more general forms of prephysics.

In this chapter I begin from the usual lagrangian of electromagnetic field  in a curvilinear metrics  with curvature G.

          ÿ                                              (40)

According to the above section, any particle of the universe is oscillating with frequency (39). Hence at any point of space we have

                                                                         (41)

here   – velocity of any electron

           – density of particles in the universe

           – Hubble expansion

          R – distance to the horizon

For v ® 1 we have a natural cutoff that will be calculated later on (in another chapter), but now we can assume that because of fluctuations of the system of N particles in the universe we do not have scales smaller than .

Then we have from (41)

Putting this expression in (40) and putting ÿ in the first approximation to G to be zero, we have

Comparing this expression with the usual Einstein lagrangian for “free space” (k – gravitational constant):

          ,

we have

                                                                                                    (42)

If we have gravitational waves on the frequency (39), we don’t have the second term in (40), because on the frequency (39) there is no dumping (or enhancing) for any oscillating at this frequency entity in the universe. Then we have only fluctuations of the second term in (40), that are  times smaller than its usual magnitude. Hence k is  times greater.

Here beside approving Einstein’s intuition we have got some new results.

1. Gravitational constant depends on cosmological model. But cosmological model is the solution of Einstein equation with the same gravitational constant. Then we have here a condition of self-concordance. Only one model, expanding with zero curvature – passes through this condition.

2. Gravitational constant is generated by the expansion of the universe. Then any moving mass will change the magnitude of this constant. We have here effects that are  times greater in comparison to effects usually studied in general relativity.

3. Gravitational constant turns out to be  times greater on the undumping frequency.

The aim of this section and the two above sections, was to give you a general view of the topic. Now you know what you can expect, and I can proceed to concrete calculations.

But I shall start from history.

12.  History. The crazy idea of Wheeler and Feynman

There is a famous long standing problem of two signs of time.

The problem is:     Our basic physical equations are symmetric in two signs of time. Then why we take as solutions of these equations waves and particles which move only in one direction of time?

The approach of Wheeler and Feynman [3] to this problem was “crazy”: Let us take solutions that are also symmetric in two signs of time. Instead of today’s electrodynamics where all waves are retarded, Wheeler and Feynman put forward a new rule: field  generated by a moving electron is a half-sum of retarded and advanced solutions

          .

The result turned out to be highly unexpected: though the procedure of calculation changed drastically, all output formulas remained intact, i.e. the same as in the usual electrodynamics. The advanced waves from different sources precisely cancel each other.

I can not agree with this approach, because I think the two-signs-of-time symmetry of our basic equations is an artifact, a result of the described above sticking together of different worlds.

Then I put forward something “more crazy”: absolute negation. Absolute in the sense that I change the sign of time not only for waves but also for particles. Absolute in the sense that I change the sign of not only mechanical time, but also the sign of statistical time. Absolute in the sense that worlds with different arrows of mechanical or statistical time have nothing in common at the beginning.

Wheeler and Feynman [3] introduced the notion of response from the universe. This response had the form of a complex integral over all the universe – over all individual stars and galaxies, all bangs and collapses. And the magnitude of this integral turns out in Wheeler and Feynman to be precisely 1! Then Wheeler and Feynman decided that there is a principle that ensures this 1, and we must follow the principle without turning to this integral.

But in my theory of interacting worlds this integral (i.e. response from the universe) is not 1! Quite the opposite: this integral contains in itself all our physics (besides classical electrodynamics) and many new effects. And with other integrals of this kind we can go out of time and matter.

Then the approach of Wheeler and Feynman does not give us new physics, but they give some elements of new language that I use in derivation of new physics.

In a particular case of the assumption (27) my results are the same as the results of Wheeler and Feynman. The assumption (26) is erroneous but it is convenient as a first step because of simplicity of calculations.

Then I start from rewriting some calculations of Wheeler and Feynman.

13.  Response from the universe according to Wheeler and Feynman

In the usual electrodynamics radiation propagates out and practically does not return to the electron. But with loops in time, the radiation returns (as advanced) and forms the response from the universe.

That response can be calculated in 5 steps:

1. Let us have an electron moving according to the Newton’s law

                                                                                                                 (43)

on the background of the universe in the form of unlimited flat space filled with electrons with density .

According to (7) we would have at a point  at the distance  the radiated retarded field of intensity (for a small )

                                                                                 (44)

Here is the unit vector along .

2. An electron at a point  of the universe has the acceleration:

3. This acceleration generates an advanced radiated field that at the place of the initial electron has the form (I am using (44))

4. We must add this field to the external field in (43):

5. The sum over all particles of the universe (– density of particles) is

Now I must take into account the multiple scattering. The index of refraction depends on frequency  and because of that I must rewrite the above calculations for oscillations at a definite .

We have the same five steps:

1. For an electron oscillating with frequency  we would have in vacuum at a distance r the retarded radiated field:

Its phase shift on the background of the matter of the universe is as usually

          ,

Here the index of refraction

                                                                    (45)

therefore we have at the point

2. Then we calculate the acceleration of the electron at the point

3. and the advanced field E radiated by this electron at the point  (we are using (45)), which is

Here we can take field in vacuum because all the scattering processes have been taken into account.

We must take into account also another possibility: a wave can be radiated as advanced (along the past cone) and return as retarded.

4. After summing up over all particles of the universe and integration over angles we obtain:

                   (46)

5. Integrating over the universe (actually with a small dissipation – see [1]) we have:

                         (47)

That is the usual expression for radiation dumping.

You could ask: “What if we have moving medium, mirrors, strong absorption and so on?” But all that can be taken into account (see [3]) – the output will be the same. The situation is different in my version of the time-symmetrization – we would have here many new effects.

14.  Response from the expanding universe according to the unity of the worlds

The calculations of the preceding section can be repeated on the background of the cosmological model of Section 2. We must take into account the change of scales along the ray.

The response (46) is proportional to the factors:

(from  to ), (from  to ) and .

Here  does not depend on scale.

 at  is changed from  at  by the factors  (here T is the cosmic time at the moment of radiation).

 at y is changed from  at  by the factors  (we must divide the final time by the beginning time).

Hence

does not depend on scale.

Hence (47) does not change in expanding universe according to Wheeler –Feynman.

In my notation this result can be rewritten as (in Wheeler – Feynman all signals are causal)

                                                    (48)

does not depend on scale.

According to my principles (see (23)) sign of time and sign of causality can be changed only together, then the right factor in (46) is

For anticausal waves we have in (3) negative mass and because of that we have different sign of time in

Then instead of (48) we have

                                       (49)

If this consideration seems unusual to you, you can repeat the derivation in the system of reference with constant scale, but for particles moving with the Hubble velocities. You would have the factor  from the Lorentz force from retarded wave and you would have the factor  for advanced anticausal field from (9). I will not write down this calculation because it is cumbersome.

Now we must repeat the derivation of (46) taking into account all four solutions (7‑10). We have four kinds of loops, depicted on Fig. 8.

Fig. 8.   Four kinds of loops

            1. forward cone, causal

            2. past cone, causal

            3. forward cone, anticausal

            4. past cone, anticausal

The contribution of any loop is the result of multiplication of four factors:

          {phase shift} × result of absorption ×

These factors are (see (46), is the coefficient of dumping):

For loops of the first kind:            

For loops of the second kind:       

Here  changes sign because t (or r) changes sign.

For loops of the third kind:           

Here m changes sign because in anticausal world any spectrum of mass is limited from above (not from below as in our world).

For loops of the fourth kind:         

Summing up these four contributions, we have the following expression for mass:

                            (50)

Here we have a divergence at v = 1, but in complete theory we shall find a natural cutoff here. As a provisional consideration you can take into account that here physics is generated by all N particles of the universe. Then no number can be smaller than  (because of fluctuations). Then the limit of integration can be put equal to

                                                                                                   (51)

and (50) can be rewritten as (for small )

                                    (52)

It seems that there are many more important factors near the beginning of the universe. But at the undumping frequency  (see below), where the integral (51) is used, these factors turn out to be unimportant.

The cutoff on  will be calculated in one of the next chapters. It will be shown that for small distances (or high temperatures) the interaction-of-the-worlds approach predicts drastic change of dynamics.