Dipl.-Phys. Roland Sprenger                                                                                                               October 9, 2025

 

 

 

It is shown how, based on the special theory of relativity, in a five-dimensional spacetime   through geometric construction without oblique coordinate systems and scale changes rest lengths and rest times can be determined. The rest system of the relativistically moving body is the observer´s system rotated around the time axis into a fifth auxiliary dimension. Superluminal velocities can also occur there.

Sections:
1. Introduction
2. Determination of rest length and time

3. Justification of the construction method

4. Adjustment by a fifth dimension

5. Movement in R⁵

6. Observer in the reference system S´

7. Superluminal velocity at t´< l 

8. Summary and outlook

 

 

The script “Relativistic Addition of Velocities in R5” [1] takes up Theodor Kaluza´s idea of a fifth dimension from 1921 [2]. In [1], unlike Minkowsk diagrams, velocities are determined without specifying the changed lengths and times in fife-dimensional spacetime. This script shows construction methods for the latter and describes the motion of a body in R⁵.

Example 1: A rod lying parallel to the x-axis moves at v = 0.6 c in the positive x-direction and passes the origin with its tip at time t = t´ = 0 s. Its length is measured as l = 3 Ls. What is its length in its rest frame S´ and how long does it take in S´ for its end to pass the origin, i.e., for it to fly past?

 

In S, the following applies                             t = l / v = (3 Ls) / (0,6 c) = 5 s                 (see Fig. 1) .

At the coincidence of the reference systems S and S´, a flash of light emits a spherical wave at the origin, which is shown at time t = 5 s (Fig. 1). A light clock (light green) moves with the rod from x = 0 to x = 3. (In a light clock, a photon/light wave is constantly reflected back and forth at the mirrored ends.) In the rest frame of the rod, the shorter time t´ (blue) has elapsed for the observer as it passes by, and its length l´(red) is greater there.

Fig. 1: Construction method for the dilated time t´(blue) and the rod length l´(red) in the rest frame S´ of the rod; rod length l = 3 Ls in the observer´s frame S purple, time t = 5 s green and light clocks light green; As the scales for Ls and s are equal the line A-B represents the time t´ as well as the distance flown by the photon in the light clock in z-direction.
 
 

The rest time t´ is constructed by taking the distance of time t (here 5 s) in the compass and drawing an arc around the origin O. A line parallel to the t-axis is drawn through the endpoint A of the distance l (here (3|0)). It intersects the arc at point B. The length of the distance A-B corresponds to the desired time t´.
 

The rest length l´ is constructed by plotting the rod length l in S (here 3 Ls) on the t-axis and then drawing a parallel line to the x-axis from its endpoint C, see Fig. 1. Its intersection with the line O-B is point D. The line A-D is the rest length l´ we are looking for, i.e., the rod length in its rest frame, which the observer in S measures as contracted.
 

In this example, the known formulas yield t´= 4 s and l´ = 3.75 Ls.

 

With                                         γ´ = 1/γ = √ (1 - v²/c² ) 

the following applies to the times                                    t´ = γ´ ∙  t 

and for the lengths                                             l = γ´ ∙  l´ .

In Example 1, γ´ = 0,8 .

 

From                                            t´ = √ (1 - v²/c2 ) ∙ t ,                    (1)

follow                                     t´² = (1 - v²/c²) ∙ t² = t² - (v²  t²)/c² = t² - l ²/c² 

and                                                     t´² + (l /c)² = t² .                            (2)

 

This is the Pythagorean theorem for the right-angled triangle in Fig. 1, who´s right angle is at point A and who´s long cathetus is the time t´ we are looking for. Its short cathetus is only called l (i.e., not l /c), but because l is plotted in Ls, the values of l and l /c are equal: l = 3 Ls and l /c = 3 Ls / c = 3 ∙ c ∙ 1s / c = 3 s . Since the scales on the x-axis und the t-axis are the same, both distances are equal in length.



The physical reason for the existence of this right-angled triangle and for Eq. 1 is the constancy of the vacuum speed of light c for all inertial systems. This is because when the tip of the rod triggers a flash of light as it passes the origin, a spherical light wave propagates in the wave model, while in the particle model, photons fly in all directions, e.g., also into a light clock that is parallel to the z-axis at x = 0, see Fig. 2. If the light clock now moves to the right together with the tip of the rod at the same speed v, a photon in it travels the same distance on the relevant radius beam of the spherical wave as a photon in a light clock that has stopped at x = 0 in the t-direction, due to the constancy of the speed of light, see Fig. 1. However, the observer in S evaluates and measures as time t´ in his reference system S only the component of the photon´s motion parallel to his t-axis in the light clock moving with the rod. This results in the right-angled triangle O-A-B with the hypotenuse of length t, which satisfies Eq. 2.

The spherical sphere defines simultaneity – unlike in the Minkowski diagram, where parallels to the t-axis define simultaneity. Unlike there, however, the scales remain unchanged in this model.

 

According to Eq. 1, the following equation applies to lengths

 

 

 

5. Motion in R⁵

The reference system S´ is the system S rotated around the t-axis by an angle ψ, with 

                                                                                    cos ψ = l / l´ = γ´ = √ (1 - v²/c² ) .                                         (5)

When the rod and the observer swap reference frames, the rod moves to the left for the observer, who is now in reference frame S´. According to the principle of relativity, the observer also measures a contracted rod length l in a longer time t. The rest length and time are constructed according to the same rules as before (Fig 4).

                                            
Fig. 4: Now the observer is in the system S´ and the rod in S moves to the left. The construction of the rest time t (blue) and the rest length l (red) follows the same rules as in Figs. 2 and 3.

In the following second example results t´ < l, which is shown in Fig. 5 using the same construction method as before. A rod moves in positive x-direction at a speed v = 0.8 c. Its length is measured as l = 4.8.Ls. It therefore takes t = 6 s to pass the origin. With  γ´ = 0.6  , the following applies:    t´ = 3.6 s   ,    l´ = 8 Ls    and    s = 6.4 Ls .

                                                     

 

The geometric methods shown for constructing rest length and time do not require affine coordinate systems or scale changes, and are therefore simpler and more intuitive than Minkowski diagrams. They may also be a further indication [1] of the existence of a fifth dimension of spacetime. However, further evidence would need to be found for this, e.g., in the interpretation of the redshift of galactic light [3], the interpretation of atomic orbitals as standing waves in R⁵, or the explanation of the entanglement of quantum objects.


 

 

 

References