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

Roland Alfred Sprenger November 28, 2025

A vectorial construction method for adding relativistic velocities in arbitrary directions with the aid of an auxiliary dimension w 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. Construction with front view, side view, and top view

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/ 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´ and u_x = u ,

i.e., the addition theorem of velocities u = (u´+ v) / [1 + (u´∙ v) / c² ] .

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 .

From the addition theorem of velocities follows that cos⁡ ψ = 1 / [1 +(u´∙ 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

Instead of calculating ψ numerically, ψ can also be constructed purely geometrically [3]: First, construct the product p = (u´/c) (v/c) from Eq. 4 using the intercept theorem (see Fig. 2) by drawing a line parallel to the line segment between points (1|0) and (0|u´) through point (v|0). For the distances applies (without the unit of measurement c), p : u´ = v : 1 and thus p = u´ v . This constructs the distance 1+p , i.e., the denominator of Eq. 4 . Then construct a right-angled triangle with the adjacent side 1 and the hypotenuse 1+p . The angle at the foot of the adjacent side is ψ, because cos⁡ψ = 1 / (1 + p).

Fig. 2: Geometric construction of angle ψ from Fig. 1 in top view

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. 3), 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. 3), the adjacent side has the length u´_y∙ cos⁡ ω = u_y .

Fig. 3: 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 ) .

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 cos²ω = 1 - (v/c)²

⇒ cos²ω + (v/c)² = 1

⇒ sin⁡ ω = v / c (6)

This can also be used to construct the angle ω, see Fig. 4. 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, because in the resulting right-angled triangle with the unit of measurement c for the distances, applies sin ω = v : c.

Fig. 4: Construction of angle ω from Fig. 2 in side view (with the numbers from Example 2, see Attachment).

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. 5), just using
u_x= (u´_x+ v) ∙ cos ⁡ψ = u´_x∙ cos ⁡ψ + 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. 5). 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. 5: 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

The x and y component vectors of u´ are rotated into the fourth dimension (green) as described above and then added vectorially to the vector u´* (light green). The vector v is also rotated into the fourth dimension (light blue) and added to u´* to form u* (light red). The vector u* is then projected perpendicularly onto the plane spanned by the vector v* and the free leg of the angle ω. The projected vector is then projected orthogonally onto the x-y plane once again. The three-dimensional vector u is therefore twice the projection of the four-dimensional vector u*.

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 .

Using Eqs. 4 and 5, equations 1 to 3 are
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. Construction with front view, side view, and top view

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., as in technical drawings the vector labels indicate the respective projections of the vectors as well), limited to the necessary vector representations

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. 7. Because then

cos⁡ ψ´ = (u´_x+ v) / u_x

and with Eq. 1 cos⁡ ψ´ = 1 + (u´_x ∙ v) / c² < 1 (10)

because u´_x < 0 . The geometric construction of u is shown in Fig. 7.

I.e. now applies u_x = (u´_x + v) / cos⁡ ψ´ .

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. 7: 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. The rotations replace the scale changes in the Minkowski diagrams. When directions are identical or perpendicular to each other, the combined velocity in the observer's system is obtained by simple projection. This explains why all velocities there are limited by the speed of light. This may be an indication that this auxiliary dimension has physical significance also. If a lot of evidence [1], [2], [5], [6] from different areas of physics, such as cosmology [4], 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] Chatgpt 5.2 Thinking

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

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

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

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

[6] 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. 3:

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. 5:

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 4 for Fig. 7:

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

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. Construction with front view, side view, and top view

6. Obtuse angles between u´ and v

7. Summary and outlook

References

Attachment