The second property determines the length of the global acceleration vector \(\mathbf{R}_t' = \mathbf{A}\), which is obtained by differentiating the global velocity vector with respect to time: \[|\mathbf{A}| = {\beta'(t) \over 1 - \beta(t)^2}, \quad \beta(t) = {v(t) \over c} \tag{3.1}\] The next property is very unusual and follows from the previous one.

We will prove that the integral of the global acceleration modulus is equal to “speed” or “hypervelocity” [1]. To do this, we integrate over time formula (3.1): \[ \int \limits_0^t |\mathbf{A}|\, \Bbb{d}t = \int \limits_0^t {\beta'(t) \Bbb{d}t \over 1 - \beta(t)^2} = \int \limits_0^t {\Bbb{d}\beta(t) \over 1 - \beta(t)^2} \tag{3.2}\] Such an integral is tabular and is taken as follows [2]: \[ \int \limits_0^t {\Bbb{d}\beta(t) \over 1 - \beta(t)^2} = \frac12 \ln{1 + \beta(t) \over 1 - \beta(t)} \tag{3.3}\] Recall that to obtain the global velocity or acceleration vector in SI units, we need to multiply \(\mathbf{R}\) or \(\mathbf{A}\) by \(c\). After that, we can obtain the Lorentz "rapidity" [1]: \[ \frac{c}{2} \ln{1 + \beta(t) \over 1 - \beta(t)} = \theta \tag{3.4}\] Here: \(\theta\) -- rapidity.

Speed is used in physics when it is necessary, for example, to go from low speeds to high ones -- close to the speed of light. Then, instead of the usual speed, its relativistic analogue -- speed is substituted. But in the theory of a single space, there is no problem of transition, since at any speed its global vector has the same form. Applying this theory, we can obtain the necessary relativistic formulas by fairly simple methods, and here imaginary coordinates or angles are not used!

As an example demonstrating the convenience of using speed, we will give the addition of two (or more) speeds. In this case, the velocities are simply added up: \[ \theta = \theta_1 + \theta_2 \tag{3.5}\] Here: \(\theta\) -- the velocities of the total velocity, \(\theta_1, \theta_2\) -- the velocities of velocity 1 and 2, respectively.

From here we can obtain the relativistic formula for the addition of velocities in the following way: \[ \frac{c}{2} \ln{1 + \beta \over 1 - \beta} = \frac{c}{2} \ln \left[ {1 + \beta_1 \over 1 - \beta_1} {1 + \beta_2 \over 1 - \beta_2} \right] \tag{3.6}\] From which we can deduce: \[ \beta = {\beta_1 + \beta_2 \over 1 + \beta_1 \beta_2} \tag{3.7}\] Here: \(\beta\) -- the total relative velocity, and \(\beta_1, \beta_2\) -- relative velocities 1 and 2, respectively. Note that the same result can be obtained using geometric constructions.

The 3rd property also assumes that the sum of the absolute values of the global acceleration vectors will be equal to the absolute value of the total acceleration vector: \[ \int |\mathbf{A}|\, \Bbb{d}t = \sum \limits^n \int |\mathbf{A_n}|\, \Bbb{d}t \tag{3.8}\] Here: \(n\) -- global acceleration vector number. This formula clearly demonstrates a significant difference between the theory of a single space and the classical theory of relativity.