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Is there a spacetime geometric formulation of elastic collisions of particles?

Terry
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1 Answers1

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Yes, here are some references.
There are likely many more.

Eugene J. Saletan (1997) "Minkowski diagrams in momentum space" American Journal of Physics 65, 799 (1997); https://doi.org/10.1119/1.18651

  • Minkowski diagrams in configuration space, with points representing events are often used in undergraduate courses on special relativity. Similar diagrams in momentum space are seldom shown and the object of this note is to demonstrate their pedagogical usefulness in discussing particle in teractions. In conf‌iguration space each point has coordinates $(t,x)$; in momentum space the coordinates are $(E,p)$. Two examples should be suff‌icient to show how such diagrams can be used.

Nandor Bokor (2011). Analysing collisions using Minkowski diagrams in momentum space May 2011 European Journal of Physics 32(3):773-782 DOI: http://doi.org/10.1088/0143-0807/32/3/013

  • Momentum space and Minkowski diagrams are powerful tools for interpreting and analysing relativistic collisions in one or two spatial dimensions. All relevant quantities that characterize a collision, including the mass, velocity, momentum and energy of the interacting particles, both before and after collision, can be directly seen from a single Minkowski diagram. Such diagrams can also be useful for analysing the differences between Newtonian and relativistic mechanics. As an interesting example, a simple derivation of the Compton wavelength shift formula, based on the geometrical properties of such momentum space diagrams, is also presented.

Ogura, A. (2018) Diagrammatic Approach for Investigating Two Dimensional Elastic Collisions in Momentum Space II: Special Relativity. World Journal of Mechanics, 8, 353-361. doi: http://doi.org/10.4236/wjm.2018.89026.

  • The diagrammatic approach to the collision problems in Newtonian mechanics is useful. We show in this article that the same technique can be applied to the case of the special relativity. The two circles play an important role in Newtonian mechanics, while in the special relativity, we need one circle and one ellipse. The circle shows the collision in the center-of-mass system. And the ellipse shows the collision in the laboratory system. These two figures give all information on two dimensional elastic collisions in the special relativity.

Ogura, A. (2019) Elastic Collisions in Minkowski Momentum Space with Lorentz Transformations. World Journal of Mechanics, 9, 267-284. doi: http://doi.org/10.4236/wjm.2019.912018.

  • We reexamined the elastic collision problems in the special relativity for both one and two dimensions from a different point of view. In order to obtain the final states in the laboratory system of the collision problems, almost all textbooks in the special relativity calculated the simultaneous equations. In contrast to this, we make a detour through the center-of-mass system. The two frames of references are connected by the Lorentz transformation with the velocity of the center-of-mass. This route for obtaining the final states is easy for students to understand the collision problems. For one dimensional case, we also give an example for illustrating the states of the particles in the Minkowski momentum space, which shows the whole story of the collision.

Here are my recent posts on collisions (not-necessarily elastic).
Momentum diagram for two colliding Particles

Here's another:

Ananda Dasgupta (2007) "Relativistic kinetics from the Bondi K-calculus" European Journal of Physics, v28 n5 p817-831 ; http://doi.org/10.1088/0143-0807/28/5/005

  • The Bondi K-calculus is a delightful method that has been used to provide rich insights into relativistic kinematics. In this paper, we will try to show how several important results of relativistic kinetics can be derived simply by using this approach. In addition, we will also indicate how the K-calculus can be used to simplify certain calculations in this topic.
robphy
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