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Abstract.

Theory of quantum gravity is still a mystery, as well as the derivation of fine structure constant and masses of elementary particles. (Fine structure constant is calculated from force between two elementary charges.) Possibility that elementary particles are black holes is offering itself intuitively, but nobody know how to prove this mathematically. I propose possibility that the fine structure constant can be calculated from Hawking radiation: A force acts between two black holes. Force acts because of Hawking photons which directly hit another black hole and because of the photons which deflect on this black hole. The first force is independent on gravitational constant and on masses of both black holes and it is inversely proportional with the square of mutual distance. The first force is only 52 times smaller than the force between two elementary charges. The second force is not inversely proportional with square of mutual distance, therefore it is not similar with force between two elementary charges. But it is possible that deeper use of quantum gravity (still unknown) could give that this force is similar with the force between two elementary charges. If further calculation will prove that total force is very close to force between two elementary charges, this is almost proof, that Hawking radiation is key to the fine structure constant. Hypotheses of quantum gravity can rarely be verified with numbers and hypothesis above is a such one. Other channels to explanation of fine structure constant are also article of Mark Hadley [1] and quantum electrodynamics (QED). But QED does not calculate value of fine structure constant.

PACS numbers: 04.70. Dy

COMMENT

This article is published in www.philica.com as number 49 [1].

1. INTRODUCTION

Let us imagine two black holes in thermical equilibrium, which are alone in the space. But, they are not yet in equilibrium. They radiate thermical radiation so they mutually influence each other. This influence of radiation can be divided in two parts:

1. Radiation which directly hits another black hole.

2. Radiation, which indirectly influences another black hole.

Calculation with direct radiation shows qualitative and quantitative similarity with elementary charge. Calculation with indirect radiation still ever shows qualitative similarity, but the problem is that this force is divergent, what is anti-intuitive.

2. RADIATION WHICH DIRECTLY INFLUENCE ANOTHER BLACK HOLE

We have two black holes with Schwarzschild radius r_s, with mutual distance R and mass of each one equals M. Temperature of the both black holes is assigned by T. Schwarzschild radius r_s of the black holes equals:

[tex]r_s = \frac {2GM}{c^2}[/tex] (2.1)

where G is gravitational constant.

A black hole radiates as black body [3], so thermical current j, radiated from this black hole, equals:

[tex]j = \sigma T^4\left(\frac{r_s}{R}\right)^2 [/tex] (2.2)

sigma is Stefan-Boltzmann’s constant [4], which can be expressed as:

[tex] \sigma = \frac{2 \pi^5k^4}{15c^2 h^3} [/tex] (2.3)

where h is Planck’s constant, c is speed of light and k is Boltzmann’s constant.

Temperature of a black hole is calculated as:

[tex]T = \frac{hc^3}{16\pi^2GMk} [/tex] (2.4)

These formulae together give that thermical current j on a distance R from the black hole, equals:

[tex]j = \frac{hc^2}{2^{11} 15\pi^3r_s^2 R^2 } [/tex] (2.5)

Let us respect that momentum of radiation, absorbed by the another black hole, becomes momentum of this black hole, so this black hole is a little pushed away by radiation. Black hole absorbs all radiation which comes closer to it than radius 1.5 r_s. Relativistic calculation [3] shows that all radiation, which approaches more than to radius 1.5 r_s, in flat space (not-curved from the black hole) would approach to radius (27/4)^(1/2) r_s (See Appendix A). So, absorption cross-section S_A of this black hole equals:

[tex]S_A = \frac{27\pi r_s}{4} [/tex] (2.6)

Force with which radiation current acts on the another black hole, equals:

[tex]F = \frac{S_Aj}{c} [/tex] (2.7)

Final recalculation gives that F is dependent of some constants and proportional to R^2, but independent of M, T or r_s. The elementary charge behaves on a same way and it is equal for all confirmed charged particles. (Quarks are not confirmed as isolated.) So let us calculate dimensionless constant alpha_H, which is defined similarly as the fine structure constant alpha [2]:

[tex]\alpha_H=\frac{2 \pi F R^2}{hc^2} = \frac{9}{2^{12}5\pi}=\frac{\alpha}{52.168} [/tex] (2.8)

alpha_H does not deviate much from alpha which equals 1/137.03599911(46). Number (46) means uncertainty of measurement on the last two digits. As already written, equation (2.8) does not respect force of radiation which is not absorbed by the black hole. This force will be described in the next section. But, because we know that the above calculation is valid also for big black holes, (which can be much different as elementary particle), this disagreement is not large.

3. THERMAL RADIATION WHICH INDIRECTLY INFLUENCE ON ANOTHER BLACK HOLE

Let us assume that R is much larger than r_s, therefore the another black hole is reached by an almost parallel radiation. We expect that a momentum of the black hole is influenced only by a radiation, which approaches very close to the black hole; the force of other, distant, radiation is negligible. But, the next calculation shows that precisely this is a problem. Let us discuss energy current j, which passing black hole by a distance r, which is much larger than distance r_s. Including [5], angle of deflection delta of this radiation equals:

[tex]\delta = \frac{2r_s}{r} + ..[/tex] (3.1)

Parts with larger power of (r_s/r) follow, but they are negligible at big r’s.

Change of an energy current delta j in the direction of radiation is calculated as:

[tex]\Delta j = j (sin(\delta))^2 = 4j(r_s/r)^2 [/tex] (3.2)

We assumed that angle is so small that:

[tex] sin(x)=x [/tex] (3.3)

Force, with which this indirect thermical radiation acts on the black hole, equals:

[tex]F = \frac{2 \pi}{c} \int_{a}^{\infty} \Delta j\, r\, dr = \frac{hc}{2^8 15 \pi^2 R^2} \int_{a/r_s}^{\infty} (r_s/r)\;d(r/r_s) [/tex] (3.4)

Due to simplification, only forces from one big radius a to infinity are included. It is evident that this force is divergent - (ln(r/r_s)) and this is a problem. If the value of the above integral would be finite, force would be proportional only to R^(-2) (independent of r_s), what is in accordance with behaving of forces of two elementary charges (or differently written, alpha_H would be independent of R).

Otherwise, it can be respected that current j is not infinitely wide and it is not oriented only in one direction, but in this case, calculated value of alpha_H is very dependent from R. There is a lot of possible corrections used within general relativity, but they do not give hope that alpha_H would be negligibly dependent from R. (The best one is ray-ray gravitational influence.) These calculations assume that the black holes are together infinitely long time, which is not real. For instance, if delta j calculated in (3.1), in not calculated in infinite interval, but in finite interval, integral (3.4) becomes very convergent. I expect finite interval in the case, if we treats photons and not classical electromagnetic radiation. Probably, we should pass into regime of quantum gravity to get convergent integral (3.4). So we can get value alpha_H, which is almost independent from R, and I hope that this value equals alpha.

Cramer’s interpretation of quantum mechanics [6] gives that the photon checks firstly where it will fly to and then it takes off. (So Cramer makes real and virtual photons more similar.) So, if we have two black holes, alone in the space, photons fly only in the direction of the another black hole. Therefore, there are not photons, which fly in other directions, and they indirectly do not give momentum to the black hole. If there are particles in the space, in that case thermal current fly to them and so it influences also indirectly on black’s hole momentum. But, I suppose that elementary particles (and therefore black holes), builds up space and so quantum gravity is maybe different from general relativity. So I speculate that divergence, obtained above, does not exist in quantum gravity.

4. CONCLUSION

It is shown that repulsion of two elementary charges can be a consequence of the Hawking radiation. The calculation indicates that this is possible, but at some phase of calculation it becomes divergent, so anti-intuitive, and for that reason uncompleted. I hope that bigger consideration of quantum influences eliminates this divergence. Quantum influences should be respected, because the average wavelength of radiation is approximately of the same largeness as r_s. The above derivation also hints that elementary particles are black holes. It is not known how this is possible, but no one proved that this is not possible. Some theories are in article [7]. In any case, this article is a now a hint for quantum gravity theory.

APPENDIX A: DEFLECTION OF LIGHT-RAY

Deflection of light by black hole is described by the next equation:

[tex]\theta = \int_0^y \{(1-\rho^2) [1-(1+\frac{\rho^2}{1+\rho})\frac{r_s}{r_0}]\} ^{-1/2} d\rho[/tex] (A.1)

This is written in polar coordinates with centre in a black hole. rho=r_0/r, r_0 is radius, at which current j is the closest to the black hole. If we put r_s = 0, only part 1-rho^2 stays inside of brackets. This part so describes straight line. Line is almost straight also at very small rho’s. In this case the second part equals to 1-r_s/r_0. If we prolong the line at these small value to y = 1, the closest distance r_1 between the prolonged line and the black hole is

[tex]r_1 = r_0 (1-r_s/r_0)^{-1/2}[/tex] (A.2)

This can be derived so, that we put

[tex]sin(\theta\; -\;\delta/2) = r/r_1[/tex] (A.3)

delta/2 equals theta – pi/2 when y = 1 and delta is total angle of deflection. Calculation at big r’s and use of equations (A.1) and (A.3) gives (A.2).

References:

[1] Diemer, T. & Hadley, M. J. (1999). Charge and the topology of spacetime. Classical and Quantum Gravity, 16, No 11, 3567-3577. arXiv: gr-qc/9905069

[7] Kokosar J 2006 The Variable Gravitational Constant G, General Relativity Theory, Elementary Particles, Quantum Mechanics, Time’s Arrow and Consciousness PHILICA.COM Article number 49 (http://www.philica.com/display_article.php?article_id=49)

[3] Hawking S W 1975 Particle creation by black holes, Commun.Math.Phys. 43 199-220 (http://www.slac.stanford.edu/spires/find/hep/www?key=165336 )

[4] Yao W M et al. 2006 1. Physical constants J. Phys. G 33 1 (http://pdg.lbl.gov/2006/reviews/consrpp.pdf)

[5] Einstein A 1916 The Foundation of the General Theory of Relativity Annalen der Physik 49 769-822 (http://www.Alberteinstein.info/gallery/gtext3.html ) Derivation of deflection of light can be found also in (http://www.mathpages.com/rr/s6-03/6-03.htm)

[6] Cramer J G 1988 An Overview of the Transactional Interpretation International Journal of Theoretical Physics 27 227-236 (http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html )

[7] Kokosar J 2006 The Variable Gravitational Constant G, General Relativity Theory, Elementary Particles, Quantum Mechanics, Time’s Arrow and Consciousness PHILICA.COM Article number 17 (http://www.philica.com/display_article.php?article_id=17)

Theory of quantum gravity is still a mystery, as well as the derivation of fine structure constant and masses of elementary particles. (Fine structure constant is calculated from force between two elementary charges.) Possibility that elementary particles are black holes is offering itself intuitively, but nobody know how to prove this mathematically. I propose possibility that the fine structure constant can be calculated from Hawking radiation: A force acts between two black holes. Force acts because of Hawking photons which directly hit another black hole and because of the photons which deflect on this black hole. The first force is independent on gravitational constant and on masses of both black holes and it is inversely proportional with the square of mutual distance. The first force is only 52 times smaller than the force between two elementary charges. The second force is not inversely proportional with square of mutual distance, therefore it is not similar with force between two elementary charges. But it is possible that deeper use of quantum gravity (still unknown) could give that this force is similar with the force between two elementary charges. If further calculation will prove that total force is very close to force between two elementary charges, this is almost proof, that Hawking radiation is key to the fine structure constant. Hypotheses of quantum gravity can rarely be verified with numbers and hypothesis above is a such one. Other channels to explanation of fine structure constant are also article of Mark Hadley [1] and quantum electrodynamics (QED). But QED does not calculate value of fine structure constant.

PACS numbers: 04.70. Dy

COMMENT

This article is published in www.philica.com as number 49 [1].

1. INTRODUCTION

Let us imagine two black holes in thermical equilibrium, which are alone in the space. But, they are not yet in equilibrium. They radiate thermical radiation so they mutually influence each other. This influence of radiation can be divided in two parts:

1. Radiation which directly hits another black hole.

2. Radiation, which indirectly influences another black hole.

Calculation with direct radiation shows qualitative and quantitative similarity with elementary charge. Calculation with indirect radiation still ever shows qualitative similarity, but the problem is that this force is divergent, what is anti-intuitive.

2. RADIATION WHICH DIRECTLY INFLUENCE ANOTHER BLACK HOLE

We have two black holes with Schwarzschild radius r_s, with mutual distance R and mass of each one equals M. Temperature of the both black holes is assigned by T. Schwarzschild radius r_s of the black holes equals:

[tex]r_s = \frac {2GM}{c^2}[/tex] (2.1)

where G is gravitational constant.

A black hole radiates as black body [3], so thermical current j, radiated from this black hole, equals:

[tex]j = \sigma T^4\left(\frac{r_s}{R}\right)^2 [/tex] (2.2)

sigma is Stefan-Boltzmann’s constant [4], which can be expressed as:

[tex] \sigma = \frac{2 \pi^5k^4}{15c^2 h^3} [/tex] (2.3)

where h is Planck’s constant, c is speed of light and k is Boltzmann’s constant.

Temperature of a black hole is calculated as:

[tex]T = \frac{hc^3}{16\pi^2GMk} [/tex] (2.4)

These formulae together give that thermical current j on a distance R from the black hole, equals:

[tex]j = \frac{hc^2}{2^{11} 15\pi^3r_s^2 R^2 } [/tex] (2.5)

Let us respect that momentum of radiation, absorbed by the another black hole, becomes momentum of this black hole, so this black hole is a little pushed away by radiation. Black hole absorbs all radiation which comes closer to it than radius 1.5 r_s. Relativistic calculation [3] shows that all radiation, which approaches more than to radius 1.5 r_s, in flat space (not-curved from the black hole) would approach to radius (27/4)^(1/2) r_s (See Appendix A). So, absorption cross-section S_A of this black hole equals:

[tex]S_A = \frac{27\pi r_s}{4} [/tex] (2.6)

Force with which radiation current acts on the another black hole, equals:

[tex]F = \frac{S_Aj}{c} [/tex] (2.7)

Final recalculation gives that F is dependent of some constants and proportional to R^2, but independent of M, T or r_s. The elementary charge behaves on a same way and it is equal for all confirmed charged particles. (Quarks are not confirmed as isolated.) So let us calculate dimensionless constant alpha_H, which is defined similarly as the fine structure constant alpha [2]:

[tex]\alpha_H=\frac{2 \pi F R^2}{hc^2} = \frac{9}{2^{12}5\pi}=\frac{\alpha}{52.168} [/tex] (2.8)

alpha_H does not deviate much from alpha which equals 1/137.03599911(46). Number (46) means uncertainty of measurement on the last two digits. As already written, equation (2.8) does not respect force of radiation which is not absorbed by the black hole. This force will be described in the next section. But, because we know that the above calculation is valid also for big black holes, (which can be much different as elementary particle), this disagreement is not large.

3. THERMAL RADIATION WHICH INDIRECTLY INFLUENCE ON ANOTHER BLACK HOLE

Let us assume that R is much larger than r_s, therefore the another black hole is reached by an almost parallel radiation. We expect that a momentum of the black hole is influenced only by a radiation, which approaches very close to the black hole; the force of other, distant, radiation is negligible. But, the next calculation shows that precisely this is a problem. Let us discuss energy current j, which passing black hole by a distance r, which is much larger than distance r_s. Including [5], angle of deflection delta of this radiation equals:

[tex]\delta = \frac{2r_s}{r} + ..[/tex] (3.1)

Parts with larger power of (r_s/r) follow, but they are negligible at big r’s.

Change of an energy current delta j in the direction of radiation is calculated as:

[tex]\Delta j = j (sin(\delta))^2 = 4j(r_s/r)^2 [/tex] (3.2)

We assumed that angle is so small that:

[tex] sin(x)=x [/tex] (3.3)

Force, with which this indirect thermical radiation acts on the black hole, equals:

[tex]F = \frac{2 \pi}{c} \int_{a}^{\infty} \Delta j\, r\, dr = \frac{hc}{2^8 15 \pi^2 R^2} \int_{a/r_s}^{\infty} (r_s/r)\;d(r/r_s) [/tex] (3.4)

Due to simplification, only forces from one big radius a to infinity are included. It is evident that this force is divergent - (ln(r/r_s)) and this is a problem. If the value of the above integral would be finite, force would be proportional only to R^(-2) (independent of r_s), what is in accordance with behaving of forces of two elementary charges (or differently written, alpha_H would be independent of R).

Otherwise, it can be respected that current j is not infinitely wide and it is not oriented only in one direction, but in this case, calculated value of alpha_H is very dependent from R. There is a lot of possible corrections used within general relativity, but they do not give hope that alpha_H would be negligibly dependent from R. (The best one is ray-ray gravitational influence.) These calculations assume that the black holes are together infinitely long time, which is not real. For instance, if delta j calculated in (3.1), in not calculated in infinite interval, but in finite interval, integral (3.4) becomes very convergent. I expect finite interval in the case, if we treats photons and not classical electromagnetic radiation. Probably, we should pass into regime of quantum gravity to get convergent integral (3.4). So we can get value alpha_H, which is almost independent from R, and I hope that this value equals alpha.

Cramer’s interpretation of quantum mechanics [6] gives that the photon checks firstly where it will fly to and then it takes off. (So Cramer makes real and virtual photons more similar.) So, if we have two black holes, alone in the space, photons fly only in the direction of the another black hole. Therefore, there are not photons, which fly in other directions, and they indirectly do not give momentum to the black hole. If there are particles in the space, in that case thermal current fly to them and so it influences also indirectly on black’s hole momentum. But, I suppose that elementary particles (and therefore black holes), builds up space and so quantum gravity is maybe different from general relativity. So I speculate that divergence, obtained above, does not exist in quantum gravity.

4. CONCLUSION

It is shown that repulsion of two elementary charges can be a consequence of the Hawking radiation. The calculation indicates that this is possible, but at some phase of calculation it becomes divergent, so anti-intuitive, and for that reason uncompleted. I hope that bigger consideration of quantum influences eliminates this divergence. Quantum influences should be respected, because the average wavelength of radiation is approximately of the same largeness as r_s. The above derivation also hints that elementary particles are black holes. It is not known how this is possible, but no one proved that this is not possible. Some theories are in article [7]. In any case, this article is a now a hint for quantum gravity theory.

APPENDIX A: DEFLECTION OF LIGHT-RAY

Deflection of light by black hole is described by the next equation:

[tex]\theta = \int_0^y \{(1-\rho^2) [1-(1+\frac{\rho^2}{1+\rho})\frac{r_s}{r_0}]\} ^{-1/2} d\rho[/tex] (A.1)

This is written in polar coordinates with centre in a black hole. rho=r_0/r, r_0 is radius, at which current j is the closest to the black hole. If we put r_s = 0, only part 1-rho^2 stays inside of brackets. This part so describes straight line. Line is almost straight also at very small rho’s. In this case the second part equals to 1-r_s/r_0. If we prolong the line at these small value to y = 1, the closest distance r_1 between the prolonged line and the black hole is

[tex]r_1 = r_0 (1-r_s/r_0)^{-1/2}[/tex] (A.2)

This can be derived so, that we put

[tex]sin(\theta\; -\;\delta/2) = r/r_1[/tex] (A.3)

delta/2 equals theta – pi/2 when y = 1 and delta is total angle of deflection. Calculation at big r’s and use of equations (A.1) and (A.3) gives (A.2).

References:

[1] Diemer, T. & Hadley, M. J. (1999). Charge and the topology of spacetime. Classical and Quantum Gravity, 16, No 11, 3567-3577. arXiv: gr-qc/9905069

[7] Kokosar J 2006 The Variable Gravitational Constant G, General Relativity Theory, Elementary Particles, Quantum Mechanics, Time’s Arrow and Consciousness PHILICA.COM Article number 49 (http://www.philica.com/display_article.php?article_id=49)

[3] Hawking S W 1975 Particle creation by black holes, Commun.Math.Phys. 43 199-220 (http://www.slac.stanford.edu/spires/find/hep/www?key=165336 )

[4] Yao W M et al. 2006 1. Physical constants J. Phys. G 33 1 (http://pdg.lbl.gov/2006/reviews/consrpp.pdf)

[5] Einstein A 1916 The Foundation of the General Theory of Relativity Annalen der Physik 49 769-822 (http://www.Alberteinstein.info/gallery/gtext3.html ) Derivation of deflection of light can be found also in (http://www.mathpages.com/rr/s6-03/6-03.htm)

[6] Cramer J G 1988 An Overview of the Transactional Interpretation International Journal of Theoretical Physics 27 227-236 (http://mist.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html )

[7] Kokosar J 2006 The Variable Gravitational Constant G, General Relativity Theory, Elementary Particles, Quantum Mechanics, Time’s Arrow and Consciousness PHILICA.COM Article number 17 (http://www.philica.com/display_article.php?article_id=17)

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