Question

The Andromeda galaxy is two million light-years from Earth, measured in the common rest frame of Earth and Andromeda. Suppose you took a fast spaceship to Andromeda, so it got you there in 35 years as measured on the ship. If you sent a radio message home as soon as you reached Andromeda, how long after you left Earth would it arrive, according to timekeepers on Earth?
My

Answer 1

Here, assume the Andromeda galaxy doesn't move an appreciable distance towards us in all this time.

Clearly the message takes 2 million years to travel to Earth from 2 million light-years away, but the real question is how long did it take you to get over there (as measured from Earth).

Call your speed as a fraction of the speed of light β. The time it takes you to travel the distance of 2 million light years as measured by Earth is therefore:

2000000 / β years

Again, you see it as 35 years, why? The lorentz factor:

2000000 / β / γ = 35
=> 57143 / β = γ
=> 57143 = βγ

γ actually depends on β, and they are linked by this equation:

γ = 1 / sqrt(1 - β^2)

From above -
57143 = β / sqrt(1 - β^2)
=> 57143sqrt(1 - β^2) = β
=> 3265306123 * (1 - β^2) = β^2
=> 3265306123 - 3265306123*β^2 = β^2
=> 3265306123 = 3265306124 * β^2

=> β^2 = 3265306123 / 3265306124

=> β^2 = 0.99999999969375
=> β = 0.9999999996875

99.99999996875% of the speed of light is pretty nippy. So it takes just over 2 million years for you to travel the distance as measured from Earth:

2000000 / 0.9999999996875 = 2000000.000625 years

Therefore, 2000000.000625 years to get there, plus another 2 million for the message to get back, the folks on Earth have been waiting 4000000.000625 years for your call.