Relative time dilation at relativistic speeds

Question

As I understand it (feel free to correct me if I'm wrong here), the core tenet of Einstein's theory of special relativity is the idea that if you were on a sealed train moving at any constant velocity from $0$ to $c$, there would be no way to tell from within the train what velocity that is, without interacting with the world outside the train. In other words, velocity is relative, and experience is the same to an observer within any given inertial reference frame regardless of the choice of frame.
With that being said, let's use our time dilation formula for a thought experiment:
$t = frac{t_0}{sqrt{1-frac{v^2}{c^2}}}$
Usain Bolt was clocked at the 2009 World Championships in Berlin running at just over 44.7 km/h (27.7 mph).
I'll be ambitious and say I can run at half that speed, so 22.3 km/h (13.9 mph).
In my eternal penchant for losing bets, I challenge Usain Bolt to a 100-meter dash on a train moving at 45 km/h shy of $c$, so 99.9999958% of the speed of light. We line up at the starting line, a scantily-clad antiquated metaphor waves a checkered flag, and we're off!
I reach full (and constant) speed, look to my left, and mentally count off a second in my reference frame. Because I'm running at 22.3 km/h from my perspective, and the train is moving at 45 km/h short of $c$, my speed to an observer outside the train is 99.9999979% of $c$. Plug that into the time dilation formula and I experience a second of this outside observer's time in about 1 hour 21 minutes. Bolt is running at 44.7 km/h, so his speed to the outside observer is 99.9999999722% of $c$. Plug that into the time dilation formula and the outside observer's second takes Bolt a full 11 hours 46 minutes to experience. At that rate, Bolt takes 8.7 times longer to complete each second than I do.
In my reference frame, at an 8.7 times speed difference, Bolt should be looking thoroughly slo-mo, and this race should be starting to feel just a bit better.
Doesn't that mean I could figure out by looking at Bolt that the train is moving at nearly the speed of light? That seems to be at odds with Einstein's frame invariance.

Wolphram jonny · Answer

You will not see anything strange when you look at Bolt. The relative speed between the two of you is small, so time dilation will be undetectable. Your confusion arises because you assume that you perceive in Bolt the same time dilation that an observer at rest with earth sees. The guys on Earth see Bolt's clock running slower than yours, but they see Bolt of course running faster than you do (he will always be in front of you).
In addition there is an error in the way you used the speeds to compute time dilation, but that is described in Francis's answer

Charles Francis · Answer

You are not using the correct relativistic formula for your speed and Bolt's speed in the reference frame of the observer.
$$  v_{mathrm{net}}  =  frac {( u  +  v )} {1 +uv}$$
where  $v_{mathrm{net}}$, $u$ and $v$ are expressed as fractions of the speed of light. If you use the correct formula, you will find that the train, yourself and Bolt, are  moving respectively at 99.9999958304488%, 99.9999958304490% and 99.9999958304492%
of the speed of light in the reference frame of the outside observer.
Also, if you were to simply plug these into the time dilation formula used by the outside observer, you could still get the wrong answer because you are not allowing that time dilation results from a difference in the definitions of synchroneity for the three observers, not from the fact that any observer experiences time any differently from any other. IOW, you could not use a simple proportionality to the outside observer's clock, but would have to work out a correct relativistic formula (this is not commonly calculated, because it is not interesting).
Finally, your method is fundamentally wrong. You were calculating by asking the outside observer. Irrespective of the fact that you used the wrong formulae, that means you were attempting to interact with the world outside the train, in conflict with the fundamental postulate which you did actually get right.

Relative time dilation at relativistic speeds

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