Lorentz Time Transformation Problem

[Vorde]

Homework Statement

1) An alien spacecraft flies past the Earth with a speed of 0.866 c., according to an earthbound observer. One year later that observer witnesses a newly discovered planet start to pass between the Earth and its star (an event known as a transit) 20 ly away. When and where would an observer aboard the ship say the transit started?

x=20 ly
t=1 y
v= .866 c

Homework Equations

t'= (t-(vx/c))/(root(1-(v/c)^2)

(Sorry, the math notation wouldn't work, anyways it's the Lorentz transformation for time).

The Attempt at a Solution

Plug in: (20-(.866*1/1))/(root(1-(.866/1)^2)

Solve.

t'= 38.268 years


The problem is that this seems to say that an object speeding towards an event would see the event after the observer farther away (38.268 years rather than 20 years), which even in the context of special relativity doesn't make sense to me?

Can anyone help me out here?

 

[anathema]

As an observer on the spacecraft, you would see the transit starting at the same time as the earthbound observer, but the transit would appear to be happening 38.268 years away from your perspective. This is due to time dilation and length contraction effects caused by the high speed of the spacecraft. Essentially, time would appear to be passing slower for you and distances would appear to be shorter, compared to the earthbound observer. So while the transit would appear to be happening 20 light years away from the earthbound observer, it would appear to be happening 38.268 light years away from you on the spacecraft. This is a result of the relativity of simultaneity, which states that events that are simultaneous for one observer may not be simultaneous for another observer in a different frame of reference.