For relativistic uses the Lorentz factor (aka gamma)
\gamma = { c \over {\sqrt{c^2-v^2}}} is useful.
Stationary it is unity.
0.9 c is about 2.3,
0.99999999 c, is about 7000.
Asymptotic to infinity as v \rightarrow c
It is more useful because it directly feeds into useful derived physics, and conveys the impact of the velocity in a manner lots of 9s doesn’t.
Time dilation, length contraction, relativistic momentum, and mass are all just a matter of multiplying or dividing by gamma.
Not sure how directly useful that might be. However, in astronomy one can define a redshift factor of an observed object: for example, if an object emits some wavelength of light that is subsequently observed at twice that wavelength, the redshift factor is z=1. There have been galaxies observed with confirmed redshifts greater than 10.
Well it beats typing long strings of nines and then, if we needed to do the actual maths, needing to carefully count the number of nines.
The only flaw with this convention is it’s not made it into common parlance. A thread titled “A car moving at 11 nines c…” would of course begin with several responses of basically What chu talkin bout, Willis?
I would find { c -10^{-11}} more intuitive.
Presumably c(1-10^{-11}).
Too bad they screwed up the notation. They say the formula is nines = -log_{10}{(1-x)}, but they also say that 99.95% is 3.5 nines, or 3N5/N3.5. But in fact 99.95% should be 3.3 nines due to the logarithm. They mention this later as an alternative representation, but now how does one know which version is being used?
Yeah, I screwed that up, thanks.
More intuitive or useful for what purpose, is what I am asking. For example, suppose I tell you that a proton is coming out of this accelerator with an energy of 5 TeV. That seems a lot more intuitive than saying it is accelerated to— I’ll probably screw this up— 99.9996% of the speed of light.
0.999999982c for a 5 TeV proton.
But your general point is correct. There is not really any practical value to using speed as a measure for such highly relativistic objects. When done for pedagogy, (1-\frac{v}{c}) would be the go-to way to simplify the notation. The “number of nines” approach is used in engineering applications (e.g., I might request “five nines” nitrogen, meaning a nitrogen supply that is 99.999% pure), but I would never use that for talking about a relativistic object’s speed.
For particles, energy and/or momentum is the most salient thing, and the \gamma factor also has lots of practical value.
Something less well known but pretty neat is “rapidity”, which I’ll label as w. Rapidity is related to speed v, but it removes the annoyances that speed (and some other things) have when switching between reference frames. Of particular note, rapidities add linearly between frames, and light travels at w=\infty.
So, while velocity requires a clumsy formula for dealing with things like “a fast object emits another fast object; what do we see?”, rapidities add in the way you “want” them to. If someone on a train shoots a relativistic bullet with rapidity w_b down the train car, then someone sitting at the train station watching the train go by them at rapidity w_t will see the bullet travelling at rapidity w_t \pm w_b (with the sign being dependent on which way the gun was pointing relative to the train’s direction).
Rapidity is given by w=\tanh^{-1}\left(\frac{v}{c}\right), where \tanh^{-1} or equivalently \rm{arctanh} is the inverse hyperbolic tangent. The presence of this specific function hints at the inherently hyperbolic nature of the geometry of spacetime.
Rapidity doesn’t see much action in the wild – as noted, energies and/or momenta are usually the most relevant measures – but it does show up in applications in a few specific corners.
Especially since something going at 0.999991c is very different from something going at 0.999998c, but they’re both “five nines”.
And almost all of the relevant calculations use gamma, anyway, so you might as well just make that the way you talk about things.