4 ms·
This isn't wrong, but also somewhat misses the point. Four-velocity is change of spatio-temporal position (Δx,Δt) in an inertial frame per unit of time Δτ meas
by cygx 6y ago
This isn't wrong, but also somewhat misses the point.
Four-velocity is change of spatio-temporal position (Δx,Δt) in an inertial frame per unit of time Δτ measured by the object in motion. For a concrete example, think a race track with distance markers accompanied by (synchronized) stationary clocks, and a vehicle carrying its own clock. Three-velocity will be given by Δx/Δt (the distance along the track divided by the time on the stationary clock), whereas the spatial component of four-velocity will be given by Δx/Δτ (the distance along the track divided by the time on the vehicle's clock). The temporal component of four-velocity will be c·Δt/Δτ, ie proportional to the clock ratio.
Turns out no matter the speed of the vehicle, it will always hold that
(c·Δt/Δτ)² - (Δx/Δτ)² = c²
A priori, we could certainly imagine this relation not to hold!
However, once we've baked this relation into the geometry of spacetime, we can of course take the more abstract perspective described above and think about reparametrization-invariant dynamics, with choice of eigentime as parameter an insignificant way to fix an arbitrary gauge.
- jules 6y agoI see. Good point, it depends on how you define four velocity and how exactly you interpret "moving through time with the speed of light". I personally think that "we move through time at the speed of light" is a very confusing way to explain that physical fact about how clocks tick. I'd explain the same fact as follows: If you have a clock moving through spacetime along some path p(r), put tickmarks at regular intervals [1] of its arc length ∫|p'(r)|dr. Those tickmarks indicate when the clock ticks. [1] e.g. choosing the length of an interval by matching it to a one second tick of a stationary reference clock.