Relativistic velocity addition from the geometry of momentum space

Abstract

With the goal of developing didactic tools, we consider the geometrization of the addition of velocities in special relativity by using Minkowski diagrams in momentum space. For the case of collinear velocities, we describe two ruler-and-compass constructions that provide simple graphical solutions working with the mass-shell hyperbola in a 1+1-dimensional energy-momentum plane. In the spirit of dimensional scaffolding, we use those results to build a generalization in 1+2 dimensions for the case of non-collinear velocities, providing in particular a graphical illustration of how the velocity transverse to a boost changes while the momentum remains fixed. We supplement the discussion with a number of interactive applets that implement the diagrammatic constructions and constitute a visual tool that should be useful for students to improve their understanding of the subtleties of special relativity.

Relativistic Velocity Addition on a Space-Time Diagram

“…Moreover, we see in Figure 5 that the line segment AB is 80% of the line segment AC. This is consistent with the first equation in Equation (7).…”

“…We present the relativistic addition of velocities on the Minkowski's space-time diagram. It is shown that we draw some world lines, measure the lengths of them In order to obtain the addition of velocities, we also used the momentum space [7], the velocity space [9], a geometric circle [10] and a euclidian space-time [11].…”

The bismar scale and elastic collisions: a geometrical analogy

“…Finally, we showcase the generalization of this construction to non-collinear momentum transfer scenarios, exemplified by Compton scattering (figure 6). It has been shown already [12][13][14][15][16][17][18][19][20][21][22][23][24] that geometric diagrams are helpful to relate relativistic problems to their non-relativistic counterparts, as well as to provide exact solutions. In particular [13], presents the concept of a spacetime lever, where the fulcrum embodies the reference frame in which the measurements are made simultaneously.…”

“…Improving education has been one of the goals of developing geometrical and illustrative approaches to special relativity, see e.g. [17][18][19][20], including geometrical approaches to elastic collisions [13,21,22]. In this context, and building on the idea of using scales together with spacetime diagrams to delve into relativistic mass and energy [23], it is natural to wonder whether the analogy between figures 2 and 3 can be generalized to the relativistic case.…”