### Velocities in Special Relativity are not Vectors

von Janusz D. Laski

**Velocities in Special Relativity are not Vectors
Janusz D. Laski**

Beitrag zur 18. Internationalen Konferenz der Natural Philosophy Alliance

Juli 2011 – College Park, Maryland, USA

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In vector calculus the addition of two vectors means no more than the addition of their components. According to Special Relativity Theory (SRT) the addition of two velocities signifies more than that. Apart from adding the components we have to divide their sum by a coefficient 1 + (**v**/c)(**u**/c), which presence indicates that the velocity is not a vector in SRT. If the velocities in SRT were vectors they should be added up as vectors. The ratio of velocities **v**/c should not be considered as the hyperbolic tangent of an angel. The coefficient mentioned above appeared as the result of the baseless introduction of hyperbolic functions to the SRT formulae.

On that basis the formulae although fully consistent are evidently wrong. We have shown that they can easily be reduced to the correct formulae of the Galilean Transform. Also we have shown that in the case of a 3D space there are three different coefficients and three corresponding different times for one moving object. Therefore there is a choice to be made. On the one hand by dismissing the hyperbolic functions from SRT we annihilate SRT and on the other, by accepting them we reject (commonly accepted) the rules of vector calculus and obtain in 3D case, three different times instead of one.

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**4. Conclusion
**Ratio of Lorentz distance to Lorentz time gives formulae for velocity addition, which do not obey the rules of vector calculus.

According to Saa [3] the rules of four-vectors addition are the same as that for vectors in 3D space. Hence the introduction of the four-vectors does not solve the problem. The conclusions are as follows

1. The silent assumption that the ratio of any velocity to velocity of light equals the hyperbolic tangent of an argument (rapidity) has been made against the rules used in science.

2. Consequently Einstein’s formula for velocity addition does not agree with the correct formula for the addition of vectors 3. Minkowski formula and the Lorentz transform formulae are fully consistent with that erroneous Einstein formula hence all of them should be considered equally wrong.

4. Finally we propose to return to the Galilean transform formulae, which do agree with the rules of vector calculus.

- 30. Juni 2012
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