Fünfdimensionale Physik

Relativistic addition of velocities in a five-dimensional spacetime

Dipl.-Phys. Roland Sprenger, Germany                                                                                                                         29.03.2024                                                                                          






Abstract


Another method of adding relativistic velocities is shown. It uses a fifth dimension of spacetime rotating the coordinate system of the Minkowski diagram into it and thus is an indication of the existence of a fifth dimension. As proof of correctness of the rotation method it is derived from the addition theorem of velocities. Photographs of a hardware model and diagrams of a computer-generated model illustrate how to find the resulting velocity by the rotation into the five-dimensional spacetime. Alongside the paradox  "c + u´ = c"  is resolved.

Content:
1. Introduction
2. The rotation angle
3. Proof of the procedure
4. Examples
5. Resolution of the paradox  "c + u´ = c"
6. Summary

1. Introduction


In Minkowski diagrams relativistic addition of velocities according to the addition theorem of velocities can be illustrated [1]. For example    v = 0.6 c   and     u´= 0.65 c results in     u = 0.899 c     and in the according Minkowski diagram (fig.1)      u = Δx / Δt = (15.5 Ls) / (17.2 s) = 0.901 c   can be read, which means the same numerical value within the accuracy of drawings.
                                                
                                               

Fig. 1: Minkowski diagram of a uniformly moved rocket in space at a speed of v = 0.6 c, which shoots protons at the coordinate origin in the direction of the movement at a speed of  u´= 0.65 c . In the referece system S of the observer the protons move on the red worldline.

The oblique angled reference system S´ with the coordinate axes x´ and t´ is the rest system of the rocket. One can imagine it being originated by turning the rectangular reference system S around the angle bisector by an angle φ into the fifth dimension (out of the drawing level), with the orthogonal projections of the axes x and t into the drawing level lying on the axes x´ and t´.

Hereinafter is shown that after the turning the projection of the worldline of the body with the speed u´ (a proton) in the rectangular coordinate system S´ of the body with the speed v (the rocket) into the drawing level coincides with its (red) worldline in System S. This means that velocities can be added relativistic assuming the existence of a fifth dimension and using it that way.
 


 

2. The rotation angle


The angle φ depends on the velocity v or on v/c in this way:

                                                                        cos φ = tan ⁡[ (π/4) - arctan (v/c) ]          for   0 ≤ v ≤ c                                         (1)

Proof: 
                                          

Fig. 2: Downside view (left) and side view (right) of the rotation of the angle δ out of the drawing level; ε is the orthogonal projection of the turned angle δ into the drawing level.

According to figure 2                                           b = a ∙ cos φ

as well as                                           tan δ = a


and                                                   tan ε = b     .


So                                                   tan ε = a ∙ cos φ


and                                                           tan ε = tan δ ∙ cos φ     .                                                                   (2)

In the case of turning reference system S to reference system S´ by the angle φ respectively the t-axis over the t´-axis applies according to fig. 3 for the turned angle δ = 45°  and tan δ = 1 . The angle  45°- α   then takes the place of angle ε . (δ and ε now are not the angles labeled as δ and ε in fig. 3 .)
 
From (2) then follows                                           tan [ (π/4) - α] = cos ⁡φ

According to fig. 1 applies  (see (6) )                          tan ⁡α = v / c
 
and thus                                                   cos⁡ φ = tan [ (π/4) - arctan⁡ (v/c) ]             s.a.

 
respectively                                           φ = arccos [ (tan ⁡(45°- α) ]                                                               (3)
for  0 ≤ α ≤ 45°   (graphs see attachment fig. 12)
For   - 45° ≤  α ≤ 0°    applies                                           φ = arccos [ (tan ⁡(45° + α) ]                     Proof s. [2]
The rotary axis in this case is the angle bisector in the 2. and 4. quadrant.

From the addition theorem of the tangent                   tan⁡ (x - y) = (tan x - tan y) / (1 + tan x ∙ tan y)              follows from (1)

                                                  cos⁡ φ = (1 - v/c) / (1 + v/c) ,
 
so                                                   cos⁡ φ = (c - v) / (c + v)                 for       0 ≤ v ≤ c .                   (4)

                                                                                          cos⁡ φ = (c + v) / (c - v)                  for     - c < v ≤ 0          see [2]

Now a second example is used to show that and how two relativistic velocities can be added by turning the rectangular coordinate system around the angle bisector w by the angle φ. For this the worldline of the second body (the proton, green in fig. 3) is drawn into the rectangular coordinate system. By the rotation by φ then the worldline of the second body in system S (red) results as the orthogonal projection of the turned (green) worldline of the second body.
 
In fig.3 a rocket at the speed of   v = 0.2 c   fires a proton with the speed   u´ = 0.3 c   in the rest system of the rocket. The t´-axis is the worldline of the rocket in system S, the green line is the worldline of the proton in S´ before the turning and 
the red line is the worldline of the proton in S respectively the projection of its worldline in S´ after the turning. So the resulting velocity can be found turning the green worldline by the angle φ .

                                                       
                                                 
Fig. 3: Minkowski-diagram for the relativistic addition of the velocities  v = 0.2 c  and   u´ = 0.3 c ; The addition theorem of velocities results in u = 0.4717 c , the drawing in       u = 5.1 Ls / 10.8 s = 0.472 c . (The rotation angle is φ = 48.19°, measurable in fig.7. With  tan γ = u/c = 0.4717  follows    γ = 25,25° , see below) .


3. Proof of the procedure

 
As proof that the turning of S´ into the fifth dimension exactly and in each case results in the same relativistic sum of velocities as the Minkowski diagram and as the addition theorem of velocities now is shown that the angle ε is the orthogonal projection of angle δ when δ was turned by the angle φ around the angle bisector w.
 
The from the Lorentz transformations derived addition theorem of velocities  u = (u´+ v) / (1 + u´ v / c2)   is assumed as true. In the special cases with opposite orientations of the movements  v = c  and  u´ = - c  as well as  v = - c  and  u´ = c  however follows  u = 0 / 0  . Therefore this formula is valid only for all v with   | v | < c   and all u´ with   | u´ | ≤ c   . The restriction on | v | < c  makes sense physically, for a body emitting a quantum with velocity c or  - c  cannot have lightspeed itself; and quanta at lightspeed do not emit quanta at lightspeed. In the theoretical special cases   v = c  and u´ = c   as well as   v = - c  and u´ = - c however the addition theorem of velocities still applies. 
 

As c ≠ 0 applies            u = (u´ + v) / (1 + u´v / c2) = [ (u´ + v) ∙ c2] / [ (1 + u´v/c2) ∙ c2] = [ (u´ + v) ∙ c2] / (c2 + u´ v) .

                                                                             u ∙ (c2 + u´ v) = (u´ + v) ∙ c2        as (c2 + u´ v) ≠ 0   as per requirement  | v | < c  

                                                                             (u c2 + u u´v) ∙ 2 = (u´c2 + v c2) ∙ 2

                                                          u c2 + u u´v + u c2 + u u´v =  u´c2 + v c2 +u´c2 + v c2

On both sides is added    +c3 - u v c - u u´c + u´v c    . 

                        c3 + u c2 - v c2 - u v c - u´c2 - u u´c + u´v c + u u´v = c3 + u´c2 + v c2 + u´v c - u c2 - u v c -u u´c - u u´v

                         c ∙ (c2 + u c - v c - u v) - u´ ∙ (c2 + u c - v c - u v) = c ∙ (c2 + v c + u´c + u´v) - u ∙ (c2 + v c + u´c + u´v)

                                                        (c - u´) ∙ (c2 + u c - v c - u v) = (c - u) ∙ (c2 + v c + u´c + u´v)

                                                      (c - u´) [ c (c + u) - v (c + u) ] = (c - u) [ c (c + v) +u´ (c + v) ]

                                                                   (c - u´) (c - v) (c + u) = (c - u) (c + u´) (c + v)

Now on both sides is divided by  [ (c + u´) (c + v) (c + u) ] . For this must apply  u´, v, u ≠ - c . v ≠ - c  is required above. Necessary at this point is the further restriction u´ ≠ - c  . From   v ≠ - c   and   u´ ≠ - c   follows   u ≠ - c  too.

                                                             [ (c - u´) / (c + u´) ] ∙ [ (c - v) / (c + v) ] = [ (c - u) / (c + u) ]                                (5)

From  1 Ls = c ∙ 1 s  follows  v = s / t = (a Ls) / (b s) = (a / b) (c s / s) = (a / b) c     (s. fig. 1)  .   It follows                  v / c = a / b

and from tan⁡ α = a / b         follows

tan α = v / c               and according fig. 3                  tan γ =u / c        ,      tan ⁡(45° - δ) = u´ / c .                      (6)

For the first fraction in (5) applies with (6)

                                     [ (c - u´) / (c + u´) ] = [ (c - u´) (1 / c) ] / [ (c + u´) (1 / c) ] = [1 - (u´ / c)  ] / [1 + (u´ / c) ]

                         = [1 - tan⁡ (45° - δ) ] / [1 + tan⁡ (45° + δ) ] = [1 - (1 - tan ⁡δ) / (1 + tan ⁡δ) ] / [1 + (1 - tan ⁡δ) / (1 + tan ⁡δ) ]

                          = { [1 + tan⁡ δ - (1 - tan⁡ δ) ]  / [1 + tan ⁡δ ] } / { [1 + tan⁡ δ + 1 - tan⁡ δ] / [1 + tan ⁡δ] } = (2 tan ⁡δ) / 2 = tan ⁡δ   .

For the second fraction in (5) applies (4)            (c - v) / (c + v) = cos ⁡φ .


For the third fraction in (5) applies    (c - u) / (c + u) = [ (c-u)  1/c ] / [ (c + u) 1/c ] = (1 - u / c) / (1 + u / c)

and with (6)                        = (1 - tan ⁡γ) / (1 + tan ⁡γ) = tan ⁡(45° - γ) .
From this follows for (5)                                tan ⁡δ ∙ cos ⁡φ = tan ⁡(45° - γ) .                                                (7)

According fig. 3 applies 45°- γ = ε  .  So the angles δ and ε of fig. 3 fulfil equation 2         tan⁡ δ ∙ cos ⁡φ = tan ⁡ε
for the rotation of an angle δ around one of its legs which here is the angle bisector by an angle φ and with the orthogonal projection ε of angle δ on the initial surface. With this is shown that the relativistic addition of velocities can be done by the described procedure of the rotation of the rectangular reference system S´ into a fifth dimension as well for all  | v | < c   and all u´ with  - c < u´ ≤ c  .



4. Examples

 
This result shall be illustrated on the basis of the second example with    v = 0.2 c       and      u´ = 0.3 c. 
If a straight line of origin is given by the unit vector (n1 ; n2 ; n3 ) then any point in R3 is turned by the rotation matrix  

                                                    
by the angle α around that straight line [3]. For  α := φ = 48.19°  and the unit vector of the angle bisector between the coordinate axes x and t   (1/√2 ; 1/√2 ; 0)   follows the rotation matrix     
                                                        .
The point (3|10|0) of the worldline of the proton in reference system S´ before the turning is mapped by this matrix to the point (4.167|8.833|3.689) which is positioned exactly over the worldline of the proton in reference system S. For with     tanγ = 4.167/8.833 follows      γ = 25.26°   , which is the same angle as calculated at fig. 3 (the difference of 0.01° is a rounding error) i.e. the turned green worldline actually is positioned above the red worldline.
This means, that instead of the addition theorem of velocities or a Minkowski diagram a rotation matrix can be used, for with  tan γ = u/c  follows  u = 0.4718 c  (the difference of 0.0001 is a rounding error).

Moreover the angel γ and from that the velocity u can be calculated with analytical geometry using the rotating by the angle φ too [5] (see the lines d and e in fig. 3).

The 5d-model is confirmed at a hardware-model, see the photographs in figures 5, 6, 8, 10 .  Cameras do not produce parallel projections but central projections. That is why there are slight distortions and the right point of taking the pictures had to be chosen from a distance as large as possible. So the pictures had to be enlarged which caused fuzziness. 

                                               
Fig. 5: Right-angled geo-triangle rotated by the angle φ  at  v = 0.2 c and   u´ = 0.3 c . The dark worldline of the proton on the geo-triangle transitions differentiably into its red worldline in coordinate system S (the gap is caused by the flattened edge of the geo-triangle). 
                                                 
Fig. 6: The same object as in fig. 5 in side views which are orthogonal to each other; because of the central projection the angle φ is a bit bigger than 48,19° (50,5°). The parallel projection in fig. 7 shows an angle φ = 48.2° .

Parallel projections on the other hand are produced by a computer code written with GNU Octave [4] (code see attachment, diagrams fig. 7). 
                                             
                                               
Fig. 7: The example of fig. 3 and 5, upside left side the spatial representation with the axes x, t and x5, the blue worldline for v = 0.2 c, the dotted green worldline for u´ = 0.3 c, the red worldline of the resulting velocity u = 0.4717 c, the turned black t-axis and the turned green worldline. Upside right the vertical projection shows that the blue worldline is the exact projection of the t´-axis and the covered red worldline is the exact projection of the turned green worldline for they are still visible at the upper ends. Downside left some calculated data concerning example 2 and downside right a side view with the measurably exact angle 48.2°.


Running the GNU-Octave-Program und clicking the icon with the double arrow (rotate) in one of the figures 7 the figure can be rotated respectively the point of view can be changed. This way the vertical projection and the side view can be produced. The computer program “Video_turning_to_plan.mp4” (see attachment after fig. 9) shows the rotation from an oblique view to the plan view. 


                                               

Fig. 8:  Comparison of the central projection (left side) and the parallel projection (right side) for   v = 0.3 c   and   u´ = 0.8 c   (u = 0.8871 c). The red worldline is covered by the green worldline. Comparable angles are equal in the left and right picture; they are independent of the scales. See also attachment fig. 10.


The resulting velocity u can also be calculated by means of analytic geometry using the blue lines g, d and e in fig. 3 [5]. And u can as well be calculated from the angles φ, α and β´ [5].

On the website https://www.youtube.com/watch?v=pd8bspQomdw a video with the measurement of the resulting velocity u is shown at a three-dimensional model using the fifth dimension.

 

5. Resolution of the paradox   "c + u´ = c" 


In this fife-dimensional model of the world resolves the paradox   "c + u´ = c" . In the theoretical or approximate case of   v = c   the worldline of the first object (blue) is the angle bisector and the turning angle results in   φ = 90° , see fig. 9.

                                              
Fig. 9: Left side applies   0° ≤ φ < 90°   , right side applies  φ = 90° .

If   φ = 90°   the orthogonal projection of the green worldline of the second object coincides with the angle bisector w for any angle β´ (0 <  β´ ≤ 45°). So the resulting velocity in S always is c. Paradoxically the first and the second object move together in S although the first object emits the second one. But in the fifth dimension they move away from each other at the speed   u´ = (tan β´) ∙ c  .

 

6. Summary


The shown method of adding relativistic velocities with its derivation from the addition theorem of velocities supports the ingenious idea of Theodor Kaluza, who introduced a fifth dimension as the first one in 1921 [6], as well as the Kaluza-Klein-theory [7]. Furthermore a five-dimensional spacetime may give the possibility to explain some more physical phenomena, for example the orbitals of atoms being standing waves of the  three-dimensional space or the redshift of galaxy spectra beeing caused by a curvature of the threedimensional space within the fife-dimensional spacetime. 
 
 

References

[1] Metzler Physik, J.B. Metzlersche Verlagsbuchhandlung. Stuttgart. 1988. p. 343-349

[2] R. Sprenger. Moving emitters and absorbers of photons. DOI10.5281/zenodo.10974861

[3] https://de.wikipedia.org/wiki/Drehmatrix
[4] John W. Eaton, David Bateman, Søren Hauberg, Rik Wehbring (2024).
     GNU Octave version 9.1.0 manual: a high-level interactive language for numerical computations.
     URL https://www.gnu.org/software/octave/doc/v9.1.0/

[5] www.roland-sprenger.de. Relativistische Addition von Geschwindigkeiten im R5 . fig. 10 et seq.

[6] T. Kaluza: Zum Unitätsproblem der Physik. In: Sitzungsberichte Preußische Akademie der Wissenschaften. 1921. S. 966–972. archive.org.                                 

[7] Klein, O. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Physik 37, 895–906 (1926).
     https://doi.org/10.1007/BF01397481   

[8] R. Sprenger. Universe without expansion and big bang. www.roland-sprenger.de
     DOI 10.5281/zenodo.13336248
            
 
 

Attachment


GNU-Octave-Program for fig. 7:
                                                                 
First type this code into the editor window (as shown here), because it can be corrected only there. Store it with a name followed by .m ; then copy it to the command window and start it with that name but without .m

                                               
Fig. 10: Comparison of the central projection (left side) and the parallel projection (right side) for   v = 0.6 c   and   u´ = 0.6 c   (u = 0.8824 c).

GNU-Octave-program for the “Video_turning_to_plan.mp4”:                                                                              
                                                                                                First type in the command window ‘pkg load video’, copy the code to the command window and start it with ‘open Video_turnig_to_plan.mp4’.