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.

Fig.1. Two global velocity vector and their orthogonal bases.

So, suppose we have two global velocity vector (Fig. 1) \(\mathbf{V1}\) and \(\mathbf{V2}\), and we know how they are determined relative to their common orthogonal basis \((\mathbf{j_0, j_1,\ldots ,j_n})\). This basis we will further reference. It should also be noted that in the figure, we can only display three coordinates of their infinite number. The first vector is deployed based on the speed \(v_1\) from a real space, which is taken relative to the reference basis: \[\mathbf{V1} = \frac{c}{\gamma_1} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta_1^n, \quad \beta_1 = v_1/c \qquad (1.1)\] the Second vector is deployed based on the speed \(v\), which is taken as the reference basis, but it represents the relativistic sum of speeds \(v_1\) and \(v_2\), which we will find in this work: \[\mathbf{V2} = \frac{c}{\gamma} \sum \limits_{n=0}^{\infty} \mathbf{j_n} \beta^n, \quad \beta = v/c \qquad (1.2)\] where: \(\gamma_1,\, \gamma\) — Lorentz factors. Note: because we believe that the space of the first and second vectors have the same characteristics, the signs (+ or -) before the coefficients must also be identical, and hence we can choose any, because then they will be multiplied in pairs. Here we have selected all the pros, but the result is the same in any other combinations.

As we know from vector algebra, the cosine of the angle between such vectors will be like this: \[\cos(\alpha) = {\mathbf{V1}\cdot \mathbf{V2} \over |\mathbf{V1}| |\mathbf{V2}|} \qquad (1.3)\] the Modules of these vectors is always equal to \(c\): \[|\mathbf{V1}| = |\mathbf{V2}| = c \qquad (1.4)\] Scalar multiply these two vectors we obtain the desired cosine of the angle: \[\cos(\alpha) = {1 \over \gamma_1 \gamma (1 - \beta_1 \beta)} \qquad (1.5)\] Also, we can create another orthogonal basis, which we will post on the basis of the first vector \((\mathbf{i_0, i_1,\ldots ,i_n})\), where \[\mathbf{i_0} = {\mathbf{V1} \over |\mathbf{V1}|}, \quad |\mathbf{i_0}| = 1 \qquad (1.6)\] Then, using the transformation of the Lorentz factor is already in the new basis we get that the second vector on the first, can be written as: \[\mathbf{V2} = \frac{c}{\gamma_2} \sum \limits_{n=0}^{\infty} \mathbf{i_n} \beta_2^n, \quad \beta_2 = v_2/c \qquad (1.7)\] where: \(v_2\) is the velocity in real space, in the coordinate system of the first vector. Thus it turns out that the speed \(v_1\) can be regarded as the speed of the first point relative to the reference basis, and a speed \(v_2\) can be regarded as the speed of the second point, but regarding the first.

Now we can find the same cosine of the angle, but between the \(\mathbf{i_0}\) and \(\mathbf{V2}\): \[\cos(\alpha) = {\mathbf{i_0}\cdot \mathbf{V2} \over |1| |\mathbf{V2}|} = \frac{1}{\gamma_2} \qquad (1.8)\] Given, what is \(\gamma_2 = {1 / \sqrt{1 - \beta_2^2}}\), and that the cosine of the angle in the formula (1.5) coincides with (1.8), is the formula to equal \[\gamma_1 \gamma (1 - \beta_1 \beta) = \gamma_2 \qquad (1.9)\] and deduce \(\beta_2\): \[\beta_2 = \pm {\beta - \beta_1 \over 1 - \beta_1\beta_2} \qquad (1.10)\] from our figure (1) we must choose the plus sign, therefore the final formula will be: \[\beta_2 = {\beta - \beta_1 \over 1 - \beta\beta_1} \qquad (1.11)\] the formula itself shows the relativistic result, but we can convert it to classic view, in which basis \((\mathbf{j_0, j_1,\ldots ,j_n})\) be fixed, a basis \((\mathbf{i_0, i_1,\ldots ,i_n})\) will move relative to him with velocity \(v_1\). Regarding the latter point will move with velocity \(v_2\). You want to find \(v\), representing the sum of the velocity \(v_1\) and \(v_2\) with respect to the first basis. For this it is enough to \(\beta\) from the formula (1.11): \[\beta = {\beta_1 + \beta_2 \over 1 + \beta_1\beta_2}\, \quad v = {v_1 + v_2 \over 1 + v_1 v_2 / c^2} \qquad (1.12)\] the Formula (1.12) is the classical form of the relativistic sum of two velocities [1]. Empirically this result was first discovered Fizeau in 1851, the year [2].