Elastic collisions between particles: the relativistic point of view
Synopsis
In this work, we compose the theoretical model of the elastic collision between two particles for the special case where the particles are merged into one point -their center of mass- and then after their interaction are emitted with new constant velocities (2). During this process, each particle does not lose its identity; the (rest) mass of each particle remains invariant. This simple situation of emerging and emitting particles is a necessary restriction for the composition of the relativistic model: it implies that in the center-of-mass-frame there is a time-moment that the velocities of the particles are zero and the spatial parts of the four-forces by which the particles interact, are of the Newtonian action-reaction form. Hence, one can imply the four-momentum conservation during the particles' interaction, in any inertial-Cartesian reference frame.
Key-Concepts
Minkowski space - World lines of moving particles - Proper-time - World-time - Four-velocity - Four-momentum - Minkowski force - Inertial reference systems in Cartesian coordinates - The Lorentz transformations - The equations of motion in a Minkowski space - System of interacting particles - Center-of-Mass inertial system of reference - Four-momentum of a system - The four-momentum conservation for a system of particles
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References
Mechanics: Goldstein, Poole & Safko, Addison Wesley, 3d edition.
Electrodynamics and Classical Theory of Fields and Particles: A.O. Barut, Dover edition 1980.
Introduction to Classical Mechanics with Problems and Solutions: David Morin Harvard University, Cambridge University Press 2008.
Einstein's General Theory of Relativity: Oyvind Gron - Sigbjorn Hervik, Springer ed.
In this work, we compose the theoretical model of the elastic collision between two particles for the special case where the particles are merged into one point -their center of mass- and then after their interaction are emitted with new constant velocities (2). During this process, each particle does not lose its identity; the (rest) mass of each particle remains invariant. This simple situation of emerging and emitting particles is a necessary restriction for the composition of the relativistic model: it implies that in the center-of-mass-frame there is a time-moment that the velocities of the particles are zero and the spatial parts of the four-forces by which the particles interact, are of the Newtonian action-reaction form. Hence, one can imply the four-momentum conservation during the particles' interaction, in any inertial-Cartesian reference frame.