Empirical observation exercises
The sub-pages below link to some example exercises in this context.
For the measurement of very small time differences and/or observations at extreme speeds, a variety of simulators capable of generating empirical data may come in handy. Simulators also have the advantage that we can change parameters like the speed of light at will, something that is much more difficult to do in everyday life.
This page is about empirical observation exercises which put (e.g. introductory physics) students into the shoes of scientific investigators, and which give them a visceral feeling for:
(i) limits to the assumption of time as universal in Minkowski's space-time version of Pythagoras' theorem, like (cδτ)2 = (cδt)2 - (δx)2 in (1+1)D, where c becomes literally the number of meters in a second,
Note: Is there a more descriptive name for "action time-derivative" energy functions like L, other than "Lagrangian", in celebration of Joseph-Lewis Lagrange (1736-1813)?
Aside: More generally, the relativistic Lagrangian L ≡ δS/δt = -mc2/γ, which at low speeds is roughly kinetic K minus potential U energy so that differential aging becomes γ ≡ δt/δτ ≈ mc2/(U-K) ≈ 1+ΔL/mc2, where rest energy mc2 is included in potential energy U. Here S is the scalar quantity known as action, with units (like Planck's constant) of energy × time, which is useful for predicting motion.
(iii) the everyday effects of differential aging using γ ≡ δt/δτ ≈ 1+ΔL/mc2 at low speeds where L is a kinetic or potential energy value, and Δ refers to the difference in that value for our traveler relative to its value in the reference, bookkeeper, or "map" frame where x and t are measured, and that is used to define simultaneity.
(ii) the usefulness of frame-invariant proper time τ, synchrony-free proper velocity w ≡ δx/δτ = p/m, and frame-invariant (also cell-phone detectable) proper-acceleration/force α ≡ Fo/m, and/or
