4 ms·
Intuition for relativistic velocity works a lot better if you imagine changing units so that velocities between (0,c) become velocities in (0, infinity). Then t
by ajkjk 8y ago
Intuition for relativistic velocity works a lot better if you imagine changing units so that velocities between (0,c) become velocities in (0, infinity). Then two infinities added is just infinity, and adding two numbers which are close to infinity is a number even closer to infinity.
This amounts to measuring 'rapidity' w (https://en.wikipedia.org/wiki/Rapidity https://en.wikipedia.org/wiki/Rapidity) instead of velocity. Rapidity captures what velocity 'actually is': a hyperbolic angle between the spatial coordinates and the time axis, given by w = arctanh(v/c). For low velocities, w = v/c, so you can write v = wc and just treat it like a regular velocity and it basically adds as you'd expect. But as v/c -> 1, w -> infinity.
(Unfortunately, non-trivial rapidities along different spatial directions don't exactly add, and it's complicated. Boosting in x and y can result in a rotation in the xy plane also. Or something like that; I haven't looked at the math in a while.)
I sometimes wonder if it would make more sense to define cw = v as the 'correct' extension of velocity to high energies, and just say "oh, meters per second -- that was wrong. That's not what velocity is". Thinking of the speed of light as 'infinite' seems very appealing.
- jakeinspace 8y agoThe small angle approximation strikes again!
- deleted 8y ago[deleted]
- noobermin 8y agoThe problem of course is when position is actually important in a problem, you need to integrate velocities instead of rapidity. The reason velocity addition trips people up is because adding velocities inherently means "seeing velocity as measured by another frame," which then requires Lorentz transform. For many problems, you confine yourself to a single inertial frame, you can still do physics comfortably and you never have to worry about adding velocities.
- tzs 8y agoSpeaking of special relativity, people learning basic special relativity might find this simple special relativity calculator I made useful [1]. It's fairly primitive. The original intent was that it would be a base to play around with CSS on, and to maybe rewrite using various front end frameworks to learn them, and add features like named variables, showing Minkowski diagrams, cloning rows, and things like that. I never got any time for any of that, so it never got past the very simple stage, but is fine for doing things like figuring out the ladder paradox and things of that ilk when learning about special relativity. [1] https://github.com/tzs/Physics-special-relativity-calculator https://github.com/tzs/Physics-special-relativity-calculator