Relativistic velocity addition from the geometry of momentum space

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.

Relativistically invariant Bohmian trajectories of photons

Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here we derive expressions for the relativistic velocity and spacetime trajectories of photons in a Michelson-Sagnac-type interferometer. The trajectories satisfy quantum-mechanical continuity, the relativistic velocity addition rule. Our new velocity equation can be operationally defined in terms of weak measurements of momentum and energy. We finally propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.

Time Dilation and the Equivalence of Inertial Frames

We have compared the data of three clocks A, B and D moving in relative uniform motion with relative speed/velocity between A and B set at 0.6c, relative speed/velocity between A and D set at 0.8c and relative speed between B and D set at (5c/13) = 0.3846c as per the velocity addition formula (0.8-0.6)/(1-0.8*0.6). We have compared the time readings of the clocks when they meet at three events. Event 1 meeting of A and B, Event 2 meeting of A and D, Event 3 meeting of B and D.

Time Dilation and the Equivalence of Inertial Frames

We have compared the data of three clocks A, B and D moving in relative uniform motion with relative speed/velocity between A and B set at 0.6c, relative speed/velocity between A and D set at 0.8c and relative speed between B and D set at (5c/13) = 0.3846c as per the velocity addition formula (0.8-0.6)/(1-0.8*0.6). We have compared the time readings of the clocks when they meet at three events. Event 1 meeting of A and B, Event 2 meeting of A and D, Event 3 meeting of B and D.

The four basic errors of special theory of relativity and the solution offered by extended Galilean relativity

Is the special theory of relativity (STR) a “simple” or “tricky” theory? They who think that it is a simple theory say (i) that its postulates are simple, that Nature is such, (ii) that the mathematics of STR is perfect, and (iii) that experiments support it. I consider its two postulates to be very true, whereas the mathematics of the STR has a shortcoming, and, as for the experiments, the question must be posed: which theory do they support best? The problem for STR lies in the transition from its postulates to its basic equations, i.e., Lorentz transformation and the velocity addition formula. The passage from the principle of relativity and the constancy of the speed of light to the basic equations of the STR is affected by four fundamental errors—three physical and one mathematical. Continuous attempts to reconcile these latent mistakes have made STR increasingly tricky. As a result, it is in a similar situation to Ptolemy's geocentric model after “improvements” thereto by Tycho Brahe. However, the “Copernican solution” for relative motion—offered by extended Galilean relativity—is very simple and effective.

Special Relativity: Putting it All Together

We have introduced the ideas of special relativity in quick succession because none of those ideas can really be understood in isolation. This chapter works through examples in some detail so you can practice applying the ideas and solidifying your understanding.We start with an overview of how to use spacetime diagrams to solve problems in special relativity, then we walk through examples ofmeasuring the length of a moving object; the train‐in‐tunnel paradox; velocity addition; and how clock readings are arranged so that each observer measures the other’s clocks as ticking slowly.

Doppler Effect and Velocity Addition Law

We now pivot from relationships between frames to look at the effect of motion on communications between specific observers.This will help us look at the twin paradox in the next chapter, and will prove crucial to understanding the effects of gravity on time. Along the way, we develop an understanding of the Doppler effect; a key tool in many areas of modern science. We find that Doppler effects are reciprocal (Alice observes the same effect on Bob’s signals as Bob observes on Alice’s signals) and that Doppler effects compound over multiple frame changes. We then use the compounding of Doppler effects to deduce the algebraic formof the velocity addition law. We show that this Einstein velocity addition law reduces to the Galilean law at low speeds.