Expert: Philip Stahl - 9/29/2010

Question
In a previous question about time dilation (http://en.allexperts.com/q/Astronomy-1360/2010/8/Theory-relativity-Time-Dialatio...) you compute that astronauts traveling 4.2 light years at 0.95c would only observer 1.37 years passing.  Given they traveled the full 4.2 light years, would that mean that they observed their own speed as nearly 3.1c?

Or would it mean they observed the distance they traveled as only 1.3 light years?

Thank you,
Rick Seiden  

Answer
Hello,

Of course, the latter is true - since travel at superlight speed is simply not on. The key point is not to mix up the two distinct observers: the one on Earth (observing the trip, for whom the distance is still 4.2 Ly) and the ones actually making the trip - who record a distance of ~ 1.3 Ly.

Again, this is fully analogous to the example of muons entering the Earth's atmosphere, and the key again is to keep the frames of reference separate.

Consider then, the case of a muon formed high in the atmosphere and travelling at 0.99c for a distance of 4.6 km (4,600 m)  before it decays: e.g.

muon -> (e-) + neutrino + (anti-neutrino)

How long does the muon survive as measured in *its own rest frame*, and how far does the muon travel as measured *in its own frame*? If time dilation applies, we expect that the time in its own rest frame will be significantly longer than for a stationary observer (e.g. on Earth's surface) observing it. We also expect its distance traveled will be much shorter!

Given a proper time, t': and t = t'/ y

where y = [1 - v^2/c^2]^1/2 then t' = ty

So the proper time t' = t, where

t = x/c = (4.6 x 10^3 m/s)/ 2.98 x 10^8 m/s) = 1.55 x 10^-5 s

which is the muon lifetime relative to an observer on Earth.

and so: t' = ( 1.55 x 10^-5 s) (o.141) = 2.18 x 10^-6 s or ~ 2.2 us (micro-seconds)

which is the commonly observed lifetime for muons in their frame of reference.

The distance traveled is: L = x/y = (4.6 x 10^3 m) (0.141) = 649 m

Using only L, one might suppose the muons would never reach Earth's surface since the path length is "too short". But it is precisely *time dilation* that accounts for the fact a large number DO reach the Earth and are detected.

The same principle used above is applicable to the astronauts case.