Entries by robphy

Addition of Velocities (Velocity Composition) in Special Relativity

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The “Addition of Velocities” formula (more correctly, the “Composition of Velocities” formula) in Special Relativity [tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c} }{1 + \frac{v_{AB}}{c} \frac{v_{BC}}{c}}[/tex] is a non-intuitive result that arises from a “hyperbolic-tangent of a sum”-identity in Minkowski spacetime geometry, with its use of hyperbolic trigonometry. However, I claim it is difficult to obtain this by looking at…

Relativity on Rotated Graph Paper (a graphical motivation)

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(based on https://www.physicsforums.com/threads/teaching-sr-without-simultaneity.1011051/post-6588952 and https://physics.stackexchange.com/a/689291/148184 ) In my earlier Insight Spacetime Diagrams of Light Clocks, I stated without proof that the areas of all light-clock diamonds are equal. In my article, “Relativity on Rotated Graph Paper”, Am. J. Phys. 84, 344 (2016) I provided algebraic proof. In the penultimate draft, I had a non-algebraic motivating…

The Electric Field Seen by an Observer: A Relativistic Calculation with Tensors

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This Insight was inspired by the discussion in “electric field seen by an observer in motion“, which tries to understand the relation between two expressions: the definition of the electric field as seen by an observer (expressed as an observer-dependent 4-vector, as decomposed from the Maxwell field tensor ##E_{a}=F_{ab}v^b##, as found in Wald’s General Relativity…

Learn Relativity Using the Bondi K-calculus

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Although Special Relativity was formulated by Einstein (1905), and given a spacetime interpretation by Minkowski (1908) [which helped make special relativity more accessible and acceptable], it could be argued that these approaches are still too abstract and too mathematical for most students. In the early 1960s, Hermann Bondi advocated a presentation of special relativity (called…

Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity

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Traditional presentations of special relativity place emphasis on “velocity”, which of course has an important physical interpretation… carried over from Galilean physics. However, in many situations, working only with velocities can be tedious in calculations and may obscure the underlying geometry. First, a story (inspired by my post in the recent Relativistic Relative Velocity thread)….

Learn About Relativity on Rotated Graph Paper

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This Insight is a follow-up to my earlier tutorial Insight (Spacetime Diagrams of Light Clocks). I gave it a different name because I am placing more emphasis on Relativity and invariance, and I want to discuss more advanced methods involving “causal diamonds” (rather than just the “clock diamonds” in the earlier Insight).  I will assume…

Learn About Spacetime Diagrams of Light Clocks

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We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper that has been rotated by 45 degrees. Many quantitative results can be read off a spacetime diagram simply by counting boxes, with very little algebra—I like to describe this as a method for doing “visual calculations” in relativity. The details are…