Question

Adam is born on a space ship that is traveling at the speed 90% of the speed of light and headed towards Earth. Sarah is born on Earth.

Suppose from Sarah's reference frame, Adam is 10 light years away.
She'd calculate that in Adam's reference frame, he'd believe Sarah is about 3.162 light years away.

Suppose Adam is 3.162 light years away in his reference frame. He believes she is the one moving. He would do the same calculation to find that Sarah is about 1 light year away in her reference frame.

I'm a bit confused about this. Can someone clear this up?

Answer 1

What I mean is, is this a correct interpretation of relativity or am I doing it wrong? I know that in relativity people do not have to agree on simultaneity of events, but does that apply here?

Answer 2

I guess my issue is how would Adam know that it would take 10 years in Sarah's reference frame? He believe she is moving, not the other way around.

Answer 3

I don't understand it, sorry. But I hope someone will help you soon, wio!

Where they were born is irrelevant.

But I see what you mean.

Did they calculate wrong? Are they assuming an incorrect theory?

Each experience the same relative velocity of \(.90c\) where \(c\) is the speed of light.
For anyone else who might have need of it, I calculated the Lorentz factor to be:
\(\qquad\qquad\quad\)\(\gamma_u=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}\\\quad=\dfrac{1}{\sqrt{1-\dfrac{(.90c)^2}{c^2}}}\\\quad=\dfrac{1}{\sqrt{1-\dfrac{(.90)^2}{1}}}\\\quad=\dfrac{1}{\sqrt{1-.81}}\\\quad=\dfrac{1}{\sqrt{.19}}\approx2.29~...\approx2.3\approx\gamma_u\).

I forget a lot of this stuff - my class went through it so fast... But I found this on Wikipedia! I think Lorentz transformation is the topic, right? Not length contraction? I don't know!
\(x'=\gamma_u(x-v\ t)\)
http://en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_the_x-direction

And length contraction is \(L_0=L\ \gamma_u\).