Contents

1. Quantum theory of gravitation
2. Quantization of gravitational field
   1. 1. Avoid stereotyping
   2. 2. The cause of quantization of the electromagnetic field is the presence of the elementary electric charge
   3. 3. A new approach to quantum gravity
   4. 4. Is there a minimum elementary mass?
   5. 5. Is there the elementary mass in the relativity theory?
   6. 6. Is the velocity discrete?
   7. 7. Are there elementary energy, momentum, and frequency?
   8. 8. Is the translational energy of an electron, atom, or molecule quantized?
   9. 9. Quantization of gravitational field
   10. 10. Revision of the graviton concept
       1. 10.1. Gravitons lead to speed of light catastrophe
       2. 10.2. Equation for speed of light does not contain gravitational constant G
       3. 10.3. Equivalence principle disproves graviton concept
       4. 10.4. Graviton cannot leave event horizon while gravitational field can
   11. 11. Aristotle’s principle of sufficient reason and Occam’s razor
   12. 12. The criterion of falsifiability
   13. 13. Conclusions
   14. References
3. Quantum gravitational collapse and quantum solution of singularities problem in general relativity
   1. 1. Introduction
   2. 2. Does general relativity predict singularities in black holes?
   3. 3. First quantum solution of gravitational collapse problem: Pauli principle
   4. 4. Second quantum solution of gravitational collapse problem: Heisenberg’s uncertainty principle
   5. 5. Third solution: QED
   6. 6. Derivation of the maximal density in the quantum theory of gravitation
   7. 7. Gravitational field of the quantum black hole and maximal curvature of spacetime in quantum gravity
   8. 8. The inevitability of a big bang as a result of a big crunch
   9. 9. Conclusions and consequences
   10. References

Quantization of gravitational field

Abstract:

I derive equations of quantum theory of gravitation containing all 3 fundamental constants — G, c and h.

I introduce and critically analize a new approach to quantization of energy-momentum tensor T_ik which fully correlates with experiments.

I derive quantum solution of singularities’ problem in general relativity, maximal density of quantum matter and maximal curvature of spacetime. I show that quantum gravitational collapse is not unlimited compression; for star of any big mass it stops at the stage of maximal density of quantum matter ~10²⁷ kg/m³ with maximal spacetime curvation ~10² m^-2 without formation of singularity inside black hole.

The model predicts inevitable big bang after big crunch.

1. Avoid stereotyping

Newton’s theory of gravitation has only one fundamental constant — gravitational constant G. Relativistic theory of gravitation (general relativity) has 2 constants — gravitational constant G and speed of light c. Quantum theory of gravitation by definition must have all 3 fundamental constants — G, c and Planck constant h. Exactly such theory is derived here.

I derive description of gravitational interaction at quantum level, quantum solution of singularities’ problem in general relativity, maximal density of quantum matter and maximal curvature of spacetime. The model predicts inevitable big bang after big crunch.

And, in spite of absence of experimental observations, just using analogy, stereotyping with photon, a graviton hypothesis was put forward. Graviton is postulated as a hypothetical quantum of gravitational field, which is postulated as Bose-condensate of virtual gravitons.

However, during more than a century since derivation of general relativity, all attempts to describe gravitational field as Bose-condensate of virtual gravitons failed and had fundamental difficulties. And all such theories are non-renormalizable and don’t solve the problem [1—21], e.g. a recent review [16]. And it’ particularly noteworthy that the only renormalizable variant [17] uses fake particles. Therefore the contemporary approach is experimentally inapproppriate. And graviton was not yet found experimentally [22—28].

Therefore we should think in the opposite direction.

And many physicists admit necessity of fundamentally new approaches to derivation of quantum theory of gravitation:

In order to make the greatest scientific discoveries, great scientists of the past always were induced to go against the generally accepted opinion, against dogmas and challenging the crowd stereotypes, e.g.:

* Copernicus, Bruno and Galileo proved heliocentrism while everyone believed in geocentrism. And they were right.

* Columbus sailed West in search of India with the belief that the Earth was spherical, when the whole world considered the Earth flat. And Columbus was right, too!

No matter who you take — Magellan, Newton, Maxwell, Planck, and other geniuses — none of the great scientists was a dogmatist, did not obey the herd instinct. Because dogmatism clings to the old ideas and prevents the creation and even understanding of a new scientific discovery that has already been created. That is why dogmatists do not make discoveries, but even do not understand the pioneers.

Similarly, to realize a scientific discovery, a breakthrough in any field, we will have to do as all geniuses did — move away from the crowd and overcome our herd instinct — after all, we are not a herd of sheep to follow the crowd, but we are thinking people, Homo Sapiens. Therefore, we should be guided by reason, and not by animal herd instinct.

To form such an approach, we need critical thinking and a departure from the usual patterns and stereotypes in thinking:

So let’s use critical thinking and let’s move away from the herd instinct, template thinking and stereotyping.

2. The cause of quantization of the electromagnetic field is the presence of the elementary electric charge

In order to create the quantum theory of gravitation, I use a fundamentally new, innovative approach — the search for the elementary mass and the elementary energy-momentum tensor, similar to the elementary electric charge in quantum electrodynamics.

The physical cause of the quantization of electromagnetic field is that there is a minimal, elementary electric charge e, such that all other charges are multiples of it:

In nature there is no an electric charge that is not a multiple of the charge of the electron. All charges are multiples of it. The charge of any elementary particle is a multiple of the charge of an electron. The electric charge does not take either chaotic or infinitesimal values, but only strictly discrete values that are multiples of the electron charge.

Therefore, a proton and an electron, having equal elementary charges (modulus), can exchange one virtual photon, so they can make an elementary act of electromagnetic interaction. An elementary electric charge can emit an elementary electromagnetic field — its quantum is a virtual photon.

Therefore elementary electric charge is included in the equations of quantum electrodynamics.

3. A new approach to quantum gravity

Similarly, to derive the equations of quantum gravity, I propose to try a new approach — to search for and measure the elementary, minimum values of mass, energy, and momentum. And these elementary mass, energy, and momentum can be considered as the source of the assumed graviton.

Only after measuring the elementary values of mass, energy, and momentum, these values can be used in the equations of the quantum theory of gravitation.

I want to draw the reader’s attention to an important fact about quantization of energy and momentum.

In general relativity, the source of the gravitational field is the energy-momentum tensor, a measure of the energy and momentum density in a volume:

Therefore, to find the quantum of the gravitational field — the assumed graviton — we need to find the minimal, elementary values of energy and momentum.

Yes, energy and momentum are quantized:

And usually physicists believe that the fact of quantization of energy and momentum (and hence the energy-momentum tensor in general relativity) is an obvious argument in favor of the graviton and the quantum nature of the gravitational field. But any attempt to create a quantum theory of gravitation faces insurmountable difficulties — all these models are non-renormalizable.

So this is a dead end? What is the problem?

The fact is that the question of quantization of energy and momentum is actually much more complicated.

Planck’s constant, unlike electric charge, is neither an elementary energy nor an elementary momentum.

And the frequency \nu[1] in expression (3) can take a variety of values. Yes, electromagnetic waves are quantized, but a quantum of infrared light has less energy than UV, and a quantum of UV light has less energy than an X-ray quantum.

Therefore, the quantum of electromagnetic radiation (3) is not yet an elementary minimum energy, since in nature there are smaller values of radiation quanta of other wavelengths.

Therefore, the question of the existence of elementary energy depends not only on the Planck constant, but also on the question of the existence of an elementary frequency.

This is the question I want to focus the reader’s attention on. This is the question that must be solved in order to create a quantum theory of gravitation.

Therefore, my approach is the following: just as elementary electric charge is necessary to construct QED, so also elementary minimum values of mass, energy, momentum, and frequency are necessary to construct quantum theory of gravitation.

For a more complete understanding, let’s look at the graphs of the dependence of electric charge, energy, and momentum.

 Since this editing platform has not LaTeX converter and Greek alphabet, then I have to use LaTeX notation and graphics instead of Greek alphabet.