Addition of relativistic velocities in arbitrary directions in four-dimensional space

Dipl.-Phys. Roland Alfred Sprenger November 28, 2025

A vectorial construction method for adding relativistic velocities in arbitrary directions with the aid of a fifth dimension is presented and justified.

Sections:

1. Introduction

2. The special case of equal directions

3. Directions perpendicular to each other

4. Arbitrary acute angles between u' and v

5. Addition of the four-dimensional vectors u'* and v*

6. Obtuse angles between u' and v

7. Summary and outlook

Attachment

1. Introduction

As usual, for simplicity, the x-axis of the observer system S is placed in the direction of motion of a first body, e.g., a rocket with velocity v (bold type indicates vector, normal type indicates magnitude of vector). In its rest system S', when the two reference systems coincide, a second small body, e.g., a proton, is fired from the rocket at velocity u´ in any direction. The x-axes of both inertial systems are parallel. The observer in S measures the velocity u of the proton with the components u_x, u_y, u_z. The following applies:

u_x = (u´_x + v) / [1 + (u´_x ∙ v) / c² ] (1)

u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ] with γ´ = √ (1 - v²/ c² ) (2)
u_z= (u´_z ∙ γ´) / [1 +(u´_x ∙ v) / c² ] . (3)

2. The special case of equal directions

If the velocities of the first and second bodies are in the same direction, the following applies

u_y= u_z= 0 and u_x= (u´_x+ v) / [1 + (u´_x ∙ v) / c² ] , with u´_x= u´ , i.e., the addition theorem of velocities.

The numerator contains the classical arithmetic sum of the velocity magnitudes; the denominator shortens them relativistically. In the scripts “Relativistic addition of velocities in R⁵” [1] and “Rest length and time in five-dimensional spacetime” [2], the introduction of a fourth spatial auxiliary dimension proves to be advantageous. Therefore, it is assumed here as well as there that the magnitude of the classically added velocity vector is preserved in that four-dimensional space (see Fig. 1), but that only its corresponding three components occur in three-dimensional space, i.e., the orthogonal projection of the four-dimensional velocity vector into three-dimensional space.

Then the relativistically shortened vector u is the cosine component of the sum vector u´+ v, which is rotated by an angle ψ into this fourth spatial dimension and is then called u*. This means that

u = (u´ + v) cos ⁡ψ .

With (1) follows that cos⁡ ψ = 1 /[1 +(u´_x ∙ v) / c² ) . (4)

If ψ is calculated using (4) and an arc with radius u´+ v is drawn around the origin, the perpendicular from its intersection with the free leg of ψ to the x-axis determines the velocity u (see Fig. 1).

Fig. 1: Geometric relativistic “addition” (better: combination) of parallel velocity vectors u´ and v using a fourth dimension in the w-direction; corresponding numerical example in the attachment

3. Perpendicular directions

The first body (the rocket) moves again in the direction of the positive x-axis at velocity v. The second body (the proton) is fired at the coincidence of the systems at the origin perpendicular to the direction of motion of the first body at velocity u´. For simplicity, the y´-axis of the rest frame S´ of the first body is set by the vector u´. Then u´_x= 0 , u´_y= u´ , u´_z= 0 .

From Eq. 1, with u´_x= 0, it follows that: u_x= v .

Eq. 2: u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ] is simplified with u´_x= 0 to u_y= u´_y ∙ γ´

and from Eq. 4 it follows that ψ = 0° .

The velocity component u_y is therefore the velocity u´_y = u´ compressed by the factor γ´. It can be obtained by geometric construction by drawing a quarter circle in the y-w-plane from the point (0 | u´_y | 0) (see Fig. 2), then plotting an angle ω with

cos ω = γ´ (5)

from the origin and the y-axis in the y-w-plane (or x-y-plane) and then dropping the perpendicular to the y-axis from the intersection of the quarter circle with the free leg of ω. This is because in the resulting right-angled triangle (Fig. 2), the adjacent side has the length u´_y∙ cos⁡ ω = u_y .

Fig. 2: Geometric relativistic combination of mutually perpendicular velocity vectors u´ and v using a fourth dimension in the w-direction; corresponding numerical example in the attachment

The vector u' is rotated by the angle ω into the fourth dimension and then added as vector u'* to vector v to form vector u*. The perpendicular projection of u* onto the x-y plane is the desired velocity u. Due to the right angle between u_y and v , the following applies u = √ (v²+ u_y² ) .

4. Arbitrary acute angles between u´ and v

Now, with conditions otherwise the same as in Section 3, let the angle between u´ and v be arbitrary in the range from 0° to 90°. The component u_x is constructed with u´_x and v in the same way as shown in Section 2 (see fig. 3). The following applies:

u_x= (u´_x+ v) ∙ cos ⁡ψ .

Eq. 2: u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ] states that the component vector u´_y is compressed twice in succession, namely by the factor 1 / [1 +(u´_x ∙ v) / c² ] = cos ψ and by the factor γ´ = √ (1 - v²/ c² ) = cos ω .

To do this, draw a quarter circle with radius u´_y around the origin in the y-w-plane, because in Fig. 1 the angle ψ is already in the y-w-plane. The angles ω and (ψ + ω) are then plotted at the origin from the y-axis (see Fig. 3). From the intersection of the free leg of (ψ + ω) with the quarter circle, the perpendicular is dropped onto the free leg of the angle ω. In the resulting right-angled triangle with angle ψ, the hypotenuse has the length u´_y and the adjacent side has the length u´_y∙ cos ψ . A second perpendicular is now dropped from the foot of the perpendicular onto the y-axis. In the resulting right-angled triangle with angle ω, the hypotenuse has the length u´_y∙ cos ψ and the adjacent side has the desired length u´_y∙ cos ψ ∙ cos ω . This is the component u_y we are looking for, because

u´_y∙ cos ψ ∙ cos ω = u´_y∙ {1 / [1 + (u´_x ∙ v) / c² ] } ∙ γ´ = u_y .

Fig. 3: Geometric relativistic combination of the velocity vectors u' and v, between which there is an arbitrary angle, using a fourth dimension in the w direction; corresponding numerical example in the attachment

In three-dimensional space, the velocity u is calculated, because the y-axis was aligned with u', as u = √ (u_x²+ u_y² ) .

Here, because of (3), u_z= 0 .

The three-dimensional vector u is the projection of the four-dimensional vector u* with the w-component

u_w= (u´_x+ v) ∙ sin ⁡ψ (see Fig. 3). (6)

Using Eqs. 4 and 5, equations 1 to 3 are accordingly

u_x= (u´_x+ v) ∙ cos ⁡ψ (7)

u_y= u´_y∙ cos ⁡ω ∙ cos ⁡ψ (8)

u_z=u´_z∙ cos ⁡ω ∙ cos ⁡ψ (9)

5. Addition of the four-dimensional vectors u´* and v*

The procedure carried out in sections 2 and 4 for the sum u´_x+ v can also be carried out for the individual summands, and then the four-dimensional vectors u´* and v* can be added graphically in x-y-w-space to form the vector u*. The vector u is obtained as a projection in the w-direction onto the x-y-plane, and the velocity u as its magnitude. With equations 6 to 9, the following applies

The example in Section 4 is shown in this way in Fig. 4 und Fig. 5 (as a numerical example in the attachment).

Fig. 4: Vector addition of the four-dimensional velocity vectors u´* and v* with any angle between them (between 0° and 90°) to the same numerical example as in Fig. 3 (see attachment) with the calculated vectors

Fig. 5: Vector addition of the four-dimensional velocity vectors u´* and v* of Fig. 4 by geometric construction; Since the z-components were set to zero, the z-dimension can be omitted.

To simplify the drawing of the quarter circles and to avoid having to calculate the position of their intersections with the free legs, the construction in Fig. 5 can also be carried out in front view, side view, and top view, see Fig. 6.

Fig. 6: Geometric construction of the combined velocity u with the fourth dimension in the w-direction and resolved into front view, side view, and top view (i.e., the vector labels only indicate the respective projections of the vectors), limited to the necessary vector representations; for the complete representation, see Attachment Fig. 9

Instead of using γ´ = √ (1 - v²/ c² ), i.e. cos ⁡ω = √ (1 - v²/ c² ), the angle ω can be calculated more easily using

sin ⁡ω = v / c,

because it follows that cos² ω = 1 - (v / c)²

⇒ cos² ω + (v / c)² = 1

⇒ sin⁡ ω = v / c (10)

This allows us to construct the angle ω, see Fig. 7. To do this, draw a semicircle with radius 0.5 c around the point (0 | 0.5 | 0) c and then an arc with radius v/c around the point (0 | 1 | 0) c . The free leg of ω runs through their intersection point.

Fig. 7: Construction of angle ω using the numbers from Example 3.

6. Obtuse angles between u´ and v

If the angle α between the velocity vectors u´ and v is obtuse, i.e., 90° < α ≤ 180°, then the x-component u´_x of u´ is negative. Then, according to the previous definition (4), the angle ψ with cos ⁡ψ = 1 / [1 + (u´_x ∙ v) / c² ] is not defined because cos ψ > 1 .

Eq. 1 u_x= (u´_x+ v) / [1 + (u´_x ∙ v) / c² ) states that in this case, the numerator vector u´_x+ v is not compressed as before by the factor 1 / [1 + (u´_x ∙ v) / c² ], but rather stretched. This is possible in a right-angled triangle with angle ψ´, in which the distance u´_x+ v is the adjacent side and u_x = u_x* is the hypotenuse, see Fig. 8. Because then

cos⁡ ψ´ = (u´_x+ v) / u_x

and with Eq. 1 cos⁡ ψ´ = 1 + (u´_x ∙ v) / c² < 1 (11)

because u´_x < 0 . The geometric construction of u is shown in Fig. 8.

The construction of u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ] is performed as before with the angles ψ´ and ω. In the side view, as in the top view for the triangles with ψ´, the tangent section is used instead of the perpendicular. Because cos ω = γ´ < 1, the definition of the angle ω remains unchanged.

Fig. 8: Geometric construction of the combined velocity u with the fourth dimension in the w direction corresponding to Fig. 6, but with an obtuse angle α between u´ and v ; numerical example see Attachment

7. Summary and outlook

The combination of relativistic velocities with arbitrary directions can be clearly illustrated as a geometric construction in four-dimensional space in the form of vector addition. The relativistic shortening of velocity vectors can be explained more simply by their rotation into a fourth auxiliary spatial dimension, whereby only their corresponding components can appear in three-dimensional space. This explains why all velocities there are limited by the speed of light. This may be an indication that this auxiliary dimension also has physical significance. If a lot of evidence [1], [2], [4], [5] from different areas of physics, such as cosmology [3], is found, it may be possible to provide direct proof at some point, as was the case with the spherical shape of the Earth and the heliocentric world view.

References

[1] www.zenodo.org ; DOI 10.5281/zenodo.10966257

www.roland-sprenger.de: Relativistic addition of velocities in R⁵

[2] www.zenodo.org ; DOI 10.5281/zenodo.17266293
www.roland-sprenger.de: Rest length and dilated time in five-dimensional spacetime

[3] www.zenodo.org; DOI 10.5281/zenodo.13336221;

www.roland-sprenger.de: Universe without expansion and Big Bang

[4] T. Kaluza: Zum Unitätsproblem der Physik. In: Sitzungsberichte Preußische

Akademie der Wissenschaften, 1921, S. 966–972, archive.org.

[5] Klein, O. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Physik 37,

895–906 (1926). https://doi.org/10.1007/BF01397481.

EOS | Quantum Gravity in the First Half of the Twentieth Century | Oskar Klein (1926):

Attachment

Example 1 for Fig. 1:

v = 0.6 c ; u´ = 0.8 c ; u´ + v = 1.4 c

cos⁡ ψ = 1 / [1 + (u´ ∙ v) / c² ] = 0.6757 ; Ψ = 47.5°

(u´+ v) cos ψ = 1.4 c ∙ 0.6757 = 0.9460 c

Test: u = (u´+v) / [1 + (u´ ∙ v) / c² ] = 0.9459 c

Apart from a rounding error, the results match.

Example 2 for Fig. 2:

v = 0.5 c ; u´= u´_y = 0.7 c ; u´_x = u´_z= 0

u_x = 0.5 c

γ´ = √ (1 - 0.5² ) = 0.8660 ; cos ω = γ´ = 0.8660 ; ω = 30°

u_y = u´ ∙ cos ω = 0.7 c ∙ 0.8660 = 0.6062 c

u = √ (u_x²+ u_y² ) = √ (0.5²+ 0.6062² ) c = 0.7858 c

Test:

Eq. 2: u_y= (u´_y∙ γ´) / [1 +(u´_x ∙ v) / c² ] = (0.7 c ∙ 0.8660) / (1 + 0) = 0.6062 c

u = √ (u_x²+ u_y² ) = √ (0.5²+ 0.6062² ) c = 0.7858 c

Example 3 for Fig. 3:

v = 0.4 c ; u´ = 0.6 c ; u´_x = 0.3 c ⇒ u´_y = 0.5196 c

cos⁡ ψ = 1 / [1 + (u´_x∙ v) / c² ] = 1 / (1 + 0.3 ∙ 0.4) = 0.8929 ⇒ ψ = 26.77°

u_x= (u´_x+ v) ∙ cos ⁡ψ = (0.3 + 0.4) c ∙ 0.8929 = 0.625 c

sin⁡ ω = v / c = 0.4 ⇒ ω = 23.6° ⇒ cos ω = 0.9165

u_y= u´_y∙ cos ⁡ω ∙ cos ⁡ψ = 0.5196 ∙ 0.9165 ∙ 0.8929 = 0.4252

u_z= u´_z∙ cos ⁡ω ∙ cos ⁡ψ = 0 ∙ 0.9165 ∙ 0.8929 = 0

u_w= (u´_x+ v) ∙ sin ⁡ψ = (0.3 + 0.4) ∙ sin 26.77° = 0.3153

u* = √ (u_x²+ u_y²+ u_z²+ u_w² ) = √ (0.625²+ 0.4252²+ 0²+ 0.3153²) c = 0.8190 c

u = √ (u_x²+ u_y² ) = √ (0.625²+0.4252² ) c = 0.7559 c

u* = (0.625 | 0.4252 | 0 | 0.3153) c

Test:

Eq. 1: u_x= (u´_x+ v) / [1 + (u´_x∙ v) / c² ] = (0.3 c + 0.4 c) / (1 + 0.3 ∙ 0.4) = 0.625 c

γ´ = √ (1 - 0.4² ) = 0.9165

Eq. 2: u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ] = (0.5196 c ∙ 0.9165) / (1 + 0.3 ∙ 0.4) = 0.4252 c

u = √ (u_x²+ u_y² ) = √ (0.625²+0.4252² ) c = 0.7559 c

Example 3 for Fig. 4, 5 and 6 as vector addition:

v = 0.4 c ; u´ = 0.6 c ; α´ = 60°

ψ= 26.77 ; cos ψ = 0.8928 ; sin ψ = 0.4504

ω = 23.58° ; cos ω = 0.9165 ; sin ω = 0.4

Test see above.
u* = (0.625 | 0.4252 | 0 | 0.3152) c

Projection of u* onto the x-y plane: u = (0.6250 | 0.4252 | 0) c

Magnitude of u : u = √ (u_x²+ u_y²+ u_z²) = √ (0.6250²+0.4252²+ 0²) c = 0.7559 c

Geometric construction with front view, side view, and top view:

Fig.9 : Geometric construction of the combined velocity u with the fourth dimension in the w-direction and resolved into front view, side view, and top view (i.e., the vector labels only indicate the respective projections of the vectors). Complete representation of all vectors involved

Example 4 for Fig. 8:

v = 0.6 c ; u´= 0.4 c ; α´ = 120°

u´_x = u´ cos 120° = -0.2 c ; u´_y = u´ sin 120° = 0.3464 c

u´_x + v = 0.4 c ; cos ψ´ = 0.88 ; ψ´ = 28.36°

u_x = u_x* = (u´_x + v) / cos ψ´ = 0.4546 c

u´_y / cos ψ´ = 0.3936 c

cos ω = 0.8 ; ω = 36.87°

u_y = (u´_y / cos ψ´) cos ω = 0.3149 c

u = √ (u_x²+ u_y² ) = 0.5530 c

Test:

Eq. 1: u_x= (u´_x+ v) / [1 + (u´_x ∙ v) / c² ] = (-0.2 c + 0.6 c) / (1 - 0.2 ∙ 0.6) = 0.4545 c

γ´= √ (1 -0.6²) = 0.8

Eq. 2: u_y= (u´_y ∙ γ´) / [1 + (u´_x ∙ v) / c² ) = (0.3464 c ∙ 0.8) / (1 - 0.2 ∙ 0.6) = 0.3149 c

u = √ (u_x²+ u_y² ) = √ (0.4545²+0.3149² ) c = 0.5529 c

Except for a rounding error, the results for u match.

Addition of relativistic velocities in arbitrary directions in four-dimensional space

Contents

1. Introduction

3. Perpendicular directions

4. Arbitrary acute angles between u´ and v

5. Addition of the four-dimensional vectors u´* and v*

6. Obtuse angles between u´ and v

7. Summary and outlook

References

Attachment