In this note we show how to fold two speeds, if their global vectors obtained by transforming the Lorentz factor. The method allows to obtain the classical relativistic formula for such a composition [1] with the only help of vector algebra, without inertial and moving reference systems, clearly, and without the use of differentiation.

The global velocity vector (GVV) is a vector that defines the speed of the first point in the multidimensional space relative to its orthogonal basis \((\mathbf{j_0, j_1,\ldots ,j_n})\): \[\mathbf{V} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = v/c \qquad (1.1)\] where: \(\gamma = 1 / \sqrt{1 - \beta^2}\), the Lorentz factor. In figure (1a) this vector is depicted in blue.

Remember that display of infinite-dimensional vector on a two dimensional plane is very limited and we can show the strength of three-dimensional space.

Fig.1. GVV for the first point (a-b), two GVV for two points (c) and two GVV relative to a fixed coordinate system (d).

To simplify further presentation, let's turn GVV to the two-dimensional space (convolution property). This will not affect the more General proof, which can be viewed here. The vector will then become: \[\mathbf{V} = \mathbf{j_0} \frac{c}{\gamma} + \mathbf{j_1} v \qquad (1.2)\] we display in figure (1b). Note that the angle \(\alpha\) between \(\mathbf{V}\) and the axis \(\mathbf{j_0}\) remains the same, and the sine of that angle is now as follows: \[\sin(\alpha) = \beta \qquad (1.3)\] Here, we consider that the length of the GVV is always equal to the speed of light: \(|\mathbf{V}| = c\). Also, we then need the cosine of that angle: \[\cos(\alpha) = \sqrt{1 - \beta^2} \qquad (1.4)\] Plus in front of the root we chose based on the picture.

Now, place on the axis \(\mathbf{j_0}\) is another global vector \(\mathbf{V1}\), which will determine the movement of the second point. Figure (1c) we have painted it orange. It's clear that \(\alpha\) determines the angle between the two GVV. To obtain the final result now we need to introduce another orthogonal basis \((\mathbf{i_0, i_1,\ldots ,i_n})\) with respect to which both points will have their speed: the first \(v_1\), the second \(v_2\). This arrangement is shown in figure (1d). Then GVV these points will be: \[\mathbf{V} = \frac{c}{\gamma_1} \sum \limits_{n=0}^{\infty} \mathbf{i_n} \beta_1^n, \quad \beta_1 = v_1/c \qquad (1.5)\] \[\mathbf{V2} = \frac{c}{\gamma_1} \sum \limits_{n=0}^{\infty} \mathbf{i_n} \beta_2^n, \quad \beta_2 = v_2/c \qquad (1.6)\] it follows that we can find the cosine of the angle between these two vectors, but in these vectors by the formulas (1.5) and (1.6): \[\cos(\alpha) = {\mathbf{V}\cdot \mathbf{V2} \over |\mathbf{V}| |\mathbf{V2}|} = {1 \over \gamma_1 \gamma_2 (1 - \beta_1 \beta_2) } \qquad (1.7)\] Remains equal to (1.4) and (1.7), and to withdraw from the \(\beta\): \[\beta = {\beta_1 - \beta_2 \over 1 - \beta_1 \beta_2} \qquad (1.8)\] the Sign in front of this equality we have selected based on the figure (1c). If you move the speed of light from the left side to the right, we get the classical form of the relativistic sum of two velocities [1]: \[v = {v_1 - v_2 \over 1 - \beta_1 \beta_2} \qquad (1.9)\] empirically this result was first discovered Fizeau in 1851, the year [2].

Referring again to the formula (1.2) in which we presented a version of the convolution GVV and put there the result obtained earlier: \[\mathbf{V} = \mathbf{j_0} \frac{c}{\gamma_1 \gamma_2 (1 - \beta_1 \beta_2)} + \mathbf{j_1} {v_1 - v_2 \over 1 - \beta_1 \beta_2} \qquad (1.10)\] It is important in itself and for the global vector length, which we will discuss in the next section.