Archimedes meets Einstein: a millennial geometric bridge

Self Cite

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.

“…Actually, there exists an even simpler construction without need of any auxiliary points 8 . In fact, the segments AB and MC are parallel, as can be proved by verifying that:…”

Section: Figurementioning

confidence: 99%

“…Another interesting contribution is [6], where the author put forward the pedagogical usefulness of Minkowski diagrams in momentum space to analyze the kinematics of relativistic collisions. Several aspects of the approach were later studied in more detail in [7,8].…”

Section: Introductionmentioning

confidence: 99%

Relativistic velocity addition from the geometry of momentum space

Paredes¹,

Prado²,

Mira³

2022

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“…The central idea of the present contribution is to put forward a formal analogy between the geometrical understanding of the bismar scale and that of elastic collisions. Other geometrical constructions for the illustration of elastic collisions in Newtonian mechanics have been presented in [12][13][14].…”

Section: Nonrelativistic Elastic Collisionsmentioning

confidence: 99%

The bismar scale and elastic collisions: a geometrical analogy

Prado,

Paredes,

Area

et al. 2024

Self Cite

Throughout history, scales have served as instrumental tools for quantifying the weight of objects, relying on a comparative assessment against a speciﬁed reference weight. Scales featuring uneven arms, such as the bismar scale, have proven particularly adept at gauging masses within a speciﬁc range relative to a predetermined reference mass. On the other hand, the kinematics of elastic collisions hinge on the inertial masses of the colliding entities. By observing the aftermath of a collision between a known reference mass and an object of unknown mass, one can deduce the latter’s mass. In this contribution, we highlight a fascinating and clear analogy between these two methodologies. We do so by adapting a geometric approach, initially applicable to the bismar scale, to both non-relativistic and relativistic elastic collisions, encompassing phenomena such as Compton scattering.

“…The first step in this dimensionally scaffolding sequence corresponds to a (1, 1) version of electromagnetism where magnitudes like charge, current, electric field, divergence, null cones or waves, as well as some of their relations, will be introduced in a straightforward way taking advantage of simple spacetime diagrams in two dimensions. A similar approach has been already used to introduce concepts like the mass-energy equivalence or the Compton effect in a visual way [29].…”

Section: One-dimensional Electromagnetismmentioning

confidence: 99%

Dimensional scaffolding of electromagnetism using geometric algebra

Prado¹,

Mira²

2020

Self Cite

Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like divergence, displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green’s functions and, with this, it is possible to obtain retarded Liénard–Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating exclusively at the speed of light in vacuum can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.