i'm having a bit of trouble wrapping my head around special relativity, so i'd like to explain what I think is going on, to see whether or not I have understood.
The question i'm thinking about is this;
Can a person, in principle, travel from Earth to the galactic centre
(which is about 28000 ly distant) in a normal lifetime? Explain using
both time-dilation and length-contraction arguments. What constant
velocity would be needed to make the trip in 30 years?
I assume that I have a very fast space ship, capable of travelling at $0.9c$. The person on board the space ship, I am considering to be in the rest frame. I imagine this 28,000 light year distance to be like a ruler in space. Relative to the spaceship, the ruler is moving at $0.9c$ and so length contraction will be observed.
$$l = l_{0} \sqrt{1 – \frac{v^{2}}{c^{2}}} $$
Which works out to be $\approx 12,205$ ly
Now, is this the actual distance the spacecraft must cover to get to the galactic center?
i.e relative to the spacecraft, it will take $\frac{l}{0.9}$ years to get there?
Best Answer
Yes, for any pair of objects moving inertially and at rest relative to each other, a distance $l_0$ apart in their rest frame, the distance between them in the frame of an inertial observer who sees them moving at speed v (along the same axis that joins them) will be $l = l_0 \sqrt{1 - v^2/c^2}$. So when this observer passes one object, the other object is moving at $v$ towards him starting from a distance $l$ away, so as you'd expect the time the observer measures before passing the second object is just $l/v$. Note that in the objects' own frame, the time for the observer to move from one object to the other is $l_0 /v$, but in this frame the observer's clock is slowed by a factor of $\sqrt{1 - v^2/c^2}$, so in this frame you can use this to predict that the observer's clock ticks forward by $(l_0 / v) * \sqrt{1 - v^2/c^2} = l/v$--same predicted time elapsed on the clock, but using a different argument.
For a rocket that is continuously accelerating, it would be possible to reach very distant locations within a human lifetime--see the table of examples on the relativistic rocket page.