Abstract
We suspect that, given our revision of the concepts of length, time and velocity, we will have to re-examine the magnitude of momentum and its conservation.
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- 1.
In fact, the quantity \( m_{0} c^{2} \) is the modulus of the four-vector of energy-momentum, which has as components the quantities \( E \), \( p_{x} c \), \( p_{y} c \) and \( p_{z} c \). The norm of this four-vector is, according to the definition, equal to \( m_{0}^{2} c^{4} = E^{2} - p_{x}^{2} c^{2} - p_{y}^{2} c^{2} - p_{z}^{2} c^{2} = E^{2} - p^{2} c^{2} \) and is invariant. The difference between the magnitude of this and the magnitude of a vector in a four-dimensional Euclidean space is in the negative signs which appear. For more on four-vectors, see Chap. 8.
References
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Christodoulides, C. (2016). Relativistic Dynamics. In: The Special Theory of Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-25274-2_6
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