Inconsistent distance measurements with relativity I was trying to explain relativity to a friend when I accidentally created an example that I did not understand. Essentially, the idea is that a rocket is travelling at $0.5c$ and as it passes Earth, someone on Earth shines a light parallel to the rocket. 
I have run the calculations (to the best of my ability), and I have determined that after two seconds (from the rocket's perspective), the rocket will perceive the distance between Earth and the rocket as 1 light second and the distance between the rocket and the light flash as 2 light seconds. In addition, after two seconds (still from the rocket's perspective) the person on Earth will perceive the distance between the Earth and the rocket as 1.12 light seconds and the rocket and the light flash as another 1.12 light seconds. 
I know that relativity describes length contraction to explain the different measures of the distance between Earth and the rocket, but why would the distance between Earth and the light flash be measured so differently?
 A: The length contraction formula does not apply in this case. The length contraction formula relates a length that is constant and at rest in one frame to a length that is constant and moving in another frame. In this case the length is not constant, so you cannot use the length contraction formula. 
You need to use the full Lorentz transform instead. The Lorentz transform automatically takes care of length contraction, time dilation, and the relativity of simultaneity. It automatically simplifies to the length contraction or time dilation formulas when appropriate, and avoids them when they are not appropriate (as in this case). Therefore, I recommend new students always start with the Lorentz transform instead of the length contraction and time dilation formulas. 
A: First of all, I think you should learn the basic concepts for this kind of problems. There are three:


*

*objects

*events

*reference frames.
Instead, forget words like "perspective" and "perceive". Relativity  is about physical entities and measurements, not perceptions and subjective appearances.
As to the three kinds of concepts, let me put the list at work in your problem. I can identify three objects: the Earth, the rocket, the flash of light. A little harder to identify the relevant events because your description is not very clear. I could try the following:


*

*Rocket passes Earth and a flash is emitted from their common location. I shall call this event $E_0$. At the same time, rocket clock is set to 0.

*The rocket clock says 2 secs. Let me call $E_1$ this event. 
Here a problem rises, as you write the rocket will perceive the distance between Earth and the rocket as 1 light second and the distance between the rocket and the light flash as 2 light seconds. Here the third concept comes into play. You are thinking of two reference frames:


*

*The Earth frame (I shall call it $K_E$).

*The rocket frame $K_R$
The meaning of your statement should be that you are examining measurements done in $K_R$. It is understood, when dealing with time in one frame, that you have as many synchronized clocks as you need, in whichever place you like. Then what you mean is:
There is a clock which at time 2 secs (time of $K_R$) is at the same place of Earth. Which is its distance from the rocket? 


*

*Here we have a new event $E_2$: a clock at rest in frame $K_R$ is coincident with Earth and marks 2 secs.


The same we can do for another event:


*

*Event $E_3$: a clock at rest in frame $K_R$ is passed by the flash when it marks 2 secs.


Unfortunately we are far from finished... We have still to define two other events interpreting your words: In addition, after two seconds (still from the rocket's perspective) the person on Earth will perceive... 
But this I cannot understand. A length measurement can be defined in several ways, but the usual meaning in SR (the one involved in length contraction) is that you are given two objects (note: objects, not events), at rest in some frame $K$. You measure their distance in $K$, then repeat the measurement in another frame $K'$, where the same objects are moving (with the same speed). It is understood that the measurement is done between the space positions the objects occupy at the same time in $K'$. Length contraction means that the second measurement gives a result less than the first, by a factor $\gamma$.
So your wording looks contradictory: you write of two seconds from the rocket perspective, and this suggests you are still thinking of frame $K_R$. But then you say "the person on Earth": have you changed mind (that is, frame)? Are you thinking now of frame $K_E$? This is not in accordance with the SR definition of length.
I could now engage in trying to find a self-consistent interpretation of your thoughts, but I believe it would be a much more fruitful exercise for you to try that yourself. Keep in mind the three concepts I listed above (objects, events, frames) and you will find the answer.
