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On a nonstandard Lorentz invariant model of the photon
Acta Astronautica 69 (2011) 373–374
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
On a nonstandard Lorentz invariant model of the photon B.V. Kuksenko Heroes Panfilovsev street, 29, 2, 244, Moscow, Russian Federation
a r t i c l e i n f o
abstract
Article history: Received 15 November 2010 Received in revised form 6 April 2011 Accepted 6 May 2011 Available online 26 May 2011
In the recent paper, a nonstandard model of a photon, as a ‘‘flat figure compressed up to nil thickness’’ has been postulated. In the framework of Newtonian gravity, a deflection of such a photon in the gravitational field of the Sun corresponds to the observable data. In the present paper, a formal (mathematical) foundation of the above model is presented. By this, one concludes that the nonstandard model of a photon is Lorentz invariant. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Lorentz invariance Gravitation
1. At first, consider a point mass M in its own reference frame, which is placed into the origin of the standard spherical coordinate system ðr, W, jÞ. Let us turn to the Newtonian gravitational law describing the force F of interaction between M and a test mass m, distanced from M by r, as F¼
GMm , r2
ð1Þ 11
where G ¼ 6:6725 10 m3 =kg s2 is the gravitational constant. Keeping in mind (1), we use the usual derivation presented in many textbooks, where a gravitating mass determines force lines that determine directions and intensity of momentum fluxes in the gravitational field of this mass. Thus, calculating the full momentum flux, we integrate (1) over the solid angle on the sphere of radius r Z 2p Z p GMm 2 r dO ¼ 4pGMm, ð2Þ r2 0 0 where dO ¼ sinW dW dj. The flux (2) just enters into the mass M and, due to the third Newtonian law, leaves M. Dividing (2) by m and by the full solid angle 4p, one obtains a density of the momentum flux r ¼ GM. 2. Now we assume that the point mass M acquires a ~ . The spherical coordinate system constant velocity V connected with the moving mass M is as above ðr, W, jÞ, whereas the spherical coordinate system connected with 0 ~ the observer is denoted as ðr 0 , W , jÞ. A direction of V E-mail address: [email protected] 0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.05.010
coincides with the positive direction of the main axis, thus W ¼ 0. Then, by the cylindrical symmetry, all the expressions under consideration have no dependency on the azimuth angle j. In correspondence with special relativity, the picture of the force lines has to be compressed. Thus, one has to 0 have r ¼ rðW Þ instead of r ¼ GM ¼ const because the observer sees a compression of distances in the direction of moving. For a cluster of fluxes of momentum leaving the mass this means an affine compression. A coefficient qffiffiffiffiffiffiffiffiffiffiffiffi 2 of the compression is the Lorentzian radical 1b , ~ j=c ¼ V=c. where b ¼ jV 3. As an example of using the special relativity formulae, we consider an intercept of a straight line. In the system of the moving mass M, one edge of the intercept is placed into the origin, the other edge is placed into the point ðr, W, jÞ; whereas in the system of the observer, one edge of the same intercept is placed into the origin, too, 0 the other edge is placed into the point ðr 0 , W , jÞ. The rules of special relativity lead to the expression for the density of the ‘‘compressed’’ momentum flux: 2
rðW0 Þ ¼
GMð1b Þ 2
2
ð1b sin W Þ3=2 0
:
ð3Þ
Ignoring a bit different notations, this formulae is identical to the expression for the special relativity deformations of the electric charge field in electrodynamics presented in textbook [2].
374
B.V. Kuksenko / Acta Astronautica 69 (2011) 373–374
4. To reconstruct the model presented in [1], one needs 0 to carry out the passage to the limit b-1 at W ¼ p=2. By this, we just try to understand the structure of the gravitational field of the object, when its force lines are compressed into the plane with nil thickness. Then, however, the formula (3) acquires the evident singularity. Below, we suggest a technique that allows the information be conserved under such a passage to the limit. The main requirement for implementation of such a technique is as follows. One has to conserve a non-infinitesimal thickness of the density of the momentum flux during all the procedure steps, up to its finish. The passage to the limit with the nil thickness is carried out as the final step. The limit process has to be finalized at the equator at W ¼ W0 ¼ p=2. By this, we choose a small square, specifically at the equator, and calculate the momentum flux through it with all the values of b. More attention has to be paid to the neighborhood of b ¼ 1. For such a derivation, we introduce a small parameter d 4 0 : d ¼ 1b. A small square is pre0 sented by inequalities p=2e=2 r W r p=2 þ e=2 and B=2 r j r þ B=2, where e and B are the other small parameters with evident interpretation. The parameter B is introduced into the expressions linearly and it is not connected with the effect under consideration; also, B is independent from the other parameter e. Thus, we have introduced B only to present the full picture, and without loss of generality, we will equalize B ¼ 1. 5. Now, let us calculate the momentum flux through the above constructed small square: Z B=2 Z p=2 þ e=2 2 GMð1b Þ 0 0 I¼ sinW dW dj 2 2 0 3=2 B=2 p=2e=2 ð1b sin W Þ Z p=2 þ e=2 0 0 sinW dW 2 ¼ GM Bð1b Þ 2 2 0 3=2 p=2e=2 ð1b sin W Þ GM2sinðe=2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Iðe, bÞ, ð4Þ 2 1b cos2 ðe=2Þ in the final expression we have set B ¼ 1. Now, let us find the limits. It is obvious that for b o1 one has lime-0 Iðe, bÞ ¼ 0. However, for b ¼ 1 the formula (4) yields Iðe,1Þ ¼ 2GM:
ð5Þ
This is the momentum flux through an angle of one radian j (not solid angle). The full flux is 2p times greater and is equal to 4pGM, which is equal to the full flux (2) (when m¼1) through the solid angle for the rest mass M. 6. Formula (5), in fact, presents the density of the flux for the maximally compressed object. One can see that (5) is two times greater than the density of the flux that corresponds to (2). This determines the main result: considering the momentum flux in the orthogonal direction one finds that for the maximally compressed object the flux is equal to the flux of a not moving (spherically symmetrical) object with the double mass. Due to this property of the nonstandard model of the photon, we have obtained [1] the observable deflection of light in the gravitational field of the Sun. It is important to note that the limit process presented above has a formal (mathematical) sense only. It should not be interpreted as the passage to the limit into the ‘‘reference frame of the photon’’ that is principally impossible in the framework of special relativity. The mass M in (5) cannot be interpreted as the rest mass, only as a mass related to the photon energy. Also, now we do not pretend that the suggested photon model could be used in any other experiments except of the deflection of the light. 7. At last, the suggested model allows to conclude that the non-standard model of the photon is Lorentz invariant. Indeed, for two observers, moving with a constant relative velocity, a trajectory of the photon is rotated by the angle corresponding to the rule of special relativity. For each of the observers the maximally compressed plane (modeling the photon) is orthogonal to the trajectory. Thus, the plane is rotated by the same angle that corresponds to the special relativity. This just means that the model is Lorentz invariant. References [1] B.V. Kuksenko, On the photon path deviation in the Sun gravity field, Acta Astronautica 65 (2009) 1437–1439. [2] E.M. Purcell, Electricity and Magnetism, Berkeley Physics Course, vol. 2, Mcgraw-Hill Book Company, New York, 1963.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
On a nonstandard Lorentz invariant model of the photon B.V. Kuksenko Heroes Panfilovsev street, 29, 2, 244, Moscow, Russian Federation
a r t i c l e i n f o
abstract
Article history: Received 15 November 2010 Received in revised form 6 April 2011 Accepted 6 May 2011 Available online 26 May 2011
In the recent paper, a nonstandard model of a photon, as a ‘‘flat figure compressed up to nil thickness’’ has been postulated. In the framework of Newtonian gravity, a deflection of such a photon in the gravitational field of the Sun corresponds to the observable data. In the present paper, a formal (mathematical) foundation of the above model is presented. By this, one concludes that the nonstandard model of a photon is Lorentz invariant. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Lorentz invariance Gravitation
1. At first, consider a point mass M in its own reference frame, which is placed into the origin of the standard spherical coordinate system ðr, W, jÞ. Let us turn to the Newtonian gravitational law describing the force F of interaction between M and a test mass m, distanced from M by r, as F¼
GMm , r2
ð1Þ 11
where G ¼ 6:6725 10 m3 =kg s2 is the gravitational constant. Keeping in mind (1), we use the usual derivation presented in many textbooks, where a gravitating mass determines force lines that determine directions and intensity of momentum fluxes in the gravitational field of this mass. Thus, calculating the full momentum flux, we integrate (1) over the solid angle on the sphere of radius r Z 2p Z p GMm 2 r dO ¼ 4pGMm, ð2Þ r2 0 0 where dO ¼ sinW dW dj. The flux (2) just enters into the mass M and, due to the third Newtonian law, leaves M. Dividing (2) by m and by the full solid angle 4p, one obtains a density of the momentum flux r ¼ GM. 2. Now we assume that the point mass M acquires a ~ . The spherical coordinate system constant velocity V connected with the moving mass M is as above ðr, W, jÞ, whereas the spherical coordinate system connected with 0 ~ the observer is denoted as ðr 0 , W , jÞ. A direction of V E-mail address: [email protected] 0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.05.010
coincides with the positive direction of the main axis, thus W ¼ 0. Then, by the cylindrical symmetry, all the expressions under consideration have no dependency on the azimuth angle j. In correspondence with special relativity, the picture of the force lines has to be compressed. Thus, one has to 0 have r ¼ rðW Þ instead of r ¼ GM ¼ const because the observer sees a compression of distances in the direction of moving. For a cluster of fluxes of momentum leaving the mass this means an affine compression. A coefficient qffiffiffiffiffiffiffiffiffiffiffiffi 2 of the compression is the Lorentzian radical 1b , ~ j=c ¼ V=c. where b ¼ jV 3. As an example of using the special relativity formulae, we consider an intercept of a straight line. In the system of the moving mass M, one edge of the intercept is placed into the origin, the other edge is placed into the point ðr, W, jÞ; whereas in the system of the observer, one edge of the same intercept is placed into the origin, too, 0 the other edge is placed into the point ðr 0 , W , jÞ. The rules of special relativity lead to the expression for the density of the ‘‘compressed’’ momentum flux: 2
rðW0 Þ ¼
GMð1b Þ 2
2
ð1b sin W Þ3=2 0
:
ð3Þ
Ignoring a bit different notations, this formulae is identical to the expression for the special relativity deformations of the electric charge field in electrodynamics presented in textbook [2].
374
B.V. Kuksenko / Acta Astronautica 69 (2011) 373–374
4. To reconstruct the model presented in [1], one needs 0 to carry out the passage to the limit b-1 at W ¼ p=2. By this, we just try to understand the structure of the gravitational field of the object, when its force lines are compressed into the plane with nil thickness. Then, however, the formula (3) acquires the evident singularity. Below, we suggest a technique that allows the information be conserved under such a passage to the limit. The main requirement for implementation of such a technique is as follows. One has to conserve a non-infinitesimal thickness of the density of the momentum flux during all the procedure steps, up to its finish. The passage to the limit with the nil thickness is carried out as the final step. The limit process has to be finalized at the equator at W ¼ W0 ¼ p=2. By this, we choose a small square, specifically at the equator, and calculate the momentum flux through it with all the values of b. More attention has to be paid to the neighborhood of b ¼ 1. For such a derivation, we introduce a small parameter d 4 0 : d ¼ 1b. A small square is pre0 sented by inequalities p=2e=2 r W r p=2 þ e=2 and B=2 r j r þ B=2, where e and B are the other small parameters with evident interpretation. The parameter B is introduced into the expressions linearly and it is not connected with the effect under consideration; also, B is independent from the other parameter e. Thus, we have introduced B only to present the full picture, and without loss of generality, we will equalize B ¼ 1. 5. Now, let us calculate the momentum flux through the above constructed small square: Z B=2 Z p=2 þ e=2 2 GMð1b Þ 0 0 I¼ sinW dW dj 2 2 0 3=2 B=2 p=2e=2 ð1b sin W Þ Z p=2 þ e=2 0 0 sinW dW 2 ¼ GM Bð1b Þ 2 2 0 3=2 p=2e=2 ð1b sin W Þ GM2sinðe=2Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Iðe, bÞ, ð4Þ 2 1b cos2 ðe=2Þ in the final expression we have set B ¼ 1. Now, let us find the limits. It is obvious that for b o1 one has lime-0 Iðe, bÞ ¼ 0. However, for b ¼ 1 the formula (4) yields Iðe,1Þ ¼ 2GM:
ð5Þ
This is the momentum flux through an angle of one radian j (not solid angle). The full flux is 2p times greater and is equal to 4pGM, which is equal to the full flux (2) (when m¼1) through the solid angle for the rest mass M. 6. Formula (5), in fact, presents the density of the flux for the maximally compressed object. One can see that (5) is two times greater than the density of the flux that corresponds to (2). This determines the main result: considering the momentum flux in the orthogonal direction one finds that for the maximally compressed object the flux is equal to the flux of a not moving (spherically symmetrical) object with the double mass. Due to this property of the nonstandard model of the photon, we have obtained [1] the observable deflection of light in the gravitational field of the Sun. It is important to note that the limit process presented above has a formal (mathematical) sense only. It should not be interpreted as the passage to the limit into the ‘‘reference frame of the photon’’ that is principally impossible in the framework of special relativity. The mass M in (5) cannot be interpreted as the rest mass, only as a mass related to the photon energy. Also, now we do not pretend that the suggested photon model could be used in any other experiments except of the deflection of the light. 7. At last, the suggested model allows to conclude that the non-standard model of the photon is Lorentz invariant. Indeed, for two observers, moving with a constant relative velocity, a trajectory of the photon is rotated by the angle corresponding to the rule of special relativity. For each of the observers the maximally compressed plane (modeling the photon) is orthogonal to the trajectory. Thus, the plane is rotated by the same angle that corresponds to the special relativity. This just means that the model is Lorentz invariant. References [1] B.V. Kuksenko, On the photon path deviation in the Sun gravity field, Acta Astronautica 65 (2009) 1437–1439. [2] E.M. Purcell, Electricity and Magnetism, Berkeley Physics Course, vol. 2, Mcgraw-Hill Book Company, New York, 1963.