Length Contraction Confusion

Question

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?

Answer 1

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.

Answer 2

I'll make it very very simple.

Yes, you can reach Galactic Center within your 30 years (you're in spaceship).

For Earthlings, you'll reach the Galactic Center after 28000 years because nothing can travel faster than light in any frame, and your age will be, yes, more than 28000 years for them.

Now, the calculations (to get the $v$ of the Galactic Center with respect to you)...

Your local clock, ageing and tissue decaying processes are all working with respect to frame of spaceship. So, you don't need to think about time dilation. But, there's a limitation. Speed of the Galactic Center will always be less than 1 lightyear per year (light is fastest in any frame). So, Galactic Center can move 30 lightyears at most in your 30 years. So, the Galactic Center can only reach you if it's less than 30 lightyears away (which is contracted distance for Earthlings).

Do I need to say more? Put 28000 ly on $L_0$ and 28 ly on $L$ in the Length contraction formula and solve for $v$ (Pay attention.. You can choose a value of $L$ between 0 to 30 ly; lower the distance, you need higher speed and greater energy). Done.