JimiJams said:
I'm not sure what relativity of simultaneity is, how would that have an effect on how the observer sees the photon moving away at c? This might be too large a can of worms to open here, but if it pertains or if you have links I'd be interested.
The best thing you can do to grasp what's going on with Special Relativity (where gravity is ignored) is to learn what the Lorentz Transformation is and one of the easiest ways to do that is to draw spacetime diagrams. I use a simplified version of the LT where the value of c is 1 so that it drops out of the equations and I like to use units of feet and nanoseconds and define c to be 1 foot per nanosecond. I also like to limit all motion and activity to the x-axis so that we only have two equations (plus the calculation of gamma, γ) to deal with. I also use speed as a fraction of c and we call that beta, β, which is equal to v/c. As is customary, I use the primed variables for the coordinates of the transformed frame and the unprimed variables for the coordinates of the original frame. And we always use inertial frames (non-accelerating). So here are the three equations:
γ = 1/√(1-β
2)
x' = γ(x-βt)
t' = γ(t-βx)
There are certain values of β that make γ come out as a rational number and one of them is 0.6 where γ=1.25. So that's what I'm going to use in the following example.
You mentioned earlier a light box so that's what I'll use for my example. Consider a box with two mirrors on opposite sides spaced 3 feet apart. We start two photons (or flashes of light) going back and forth between the mirrors such that they cross paths in the middle of the box. Then we draw a spacetime diagram to depict this scenario:
Hopefully this diagram makes sense to you. The dots represent 1-nanosecond increments of time along the two worldlines for the ends of the box where the mirrors are. Each dot is a separate "event" with its own x and t coordinates. (It also has y and z coordinates but we set them equal to zero and they remain zero even after transformation.) Note that since the light travels at 1 foot per nanosecond, it follows the thin lines drawn at 45-degree angles.
Also note that an observer at the blue end of the box can only see what is going at x=0. So he can see that a flash of light reflects off his mirror every 3 nanoseconds. And since he knows that there are two flashes of light, he knows that it takes 3 nanoseconds for each flash to make the round trip. And since he defines the speed of light to be 1 foot per nanosecond in his rest frame, he declares that the two flashes of light hit the opposite mirrors at the same time, in other words simultaneously, although he cannot see this happening. That's what we show in the diagram. Every 3 nanoseconds, there is a flash of light reflecting off of each mirror.
Now we transform from the blue observer's rest frame to one that is moving at -0.6c. This will make it look like the light box is moving to the right at 0.6c:
Now, all of a sudden, we see Time Dilation, Length Contraction and Relativity of Simultaneity. But first, let's make sure you know how the LT works. Pick any dot. Let's start with the blue one at the top. It's coordinates are x=0 and t=18 in the original frame. So we plug those numbers into the LT equations, one at a time:
x' = γ(x-βt) = 1.25(0-0.6*18) = 1.25(0.6*18) = 13.5
t' = γ(t-βx) = 1.25(18-0.6*0) = 1.25(18) = 22.5
As you can see on the transformed diagram, the top blue dot has coordinates matching those values. You can continue with all the other dots. Or you can cheat and just do it for the top two and bottom two dots and then just fill in the remaining dots by proportionally spacing them.
OK, now that we're clear you know how to do the LT, let's look at the coordinates of some of the events shown as dots on the diagram in the two diagrams. First, we note that the Coordinate Time increment for the dots is spaced farther apart in the second diagram. That is what we mean by Time Dilation, a stretching out of the Proper Time (depicted by the dots) for a moving clock or observer. Note that the stretching out factor is equal to γ, in this case 1.25.
Second, we note that the distance between the two worldlines is closer together. Look at the spacing between the blue and red lines at t=5 and you'll see that it is about 2.4 feet. This is what we mean by Length Contraction and it is equal to the Proper Length (from its rest frame) divided by γ, in this case, 3/1.25 which equals 2.4 feet.
Finally, we note that the two events of the flashes of light reflection off both mirrors at the same time is no longer true. There is a difference of over 2 nanoseconds between what used to be simultaneous events. This, of course, is demonstrating the Relativity of Simultaneity, which doesn't have any special factor like LC and TD.
Here is another very important observation to make and that is that each observer in a scenario, sees, measures, and observes everything in exactly the same way in each reference frame. The blue observer continues to see a flash every 3 nanoseconds according to the Proper Time on his clock or watch. So to answer your question, Relativity of Simultaneity has no effect on how the observer sees the photon moving away because he cannot see the propagation of light. LC, TD and RoS are all coordinate effects and not directly observable by any observer unless he takes extra effort to actually collect a lot of data, apply Einstein's second postulate and draw his own spacetime diagram.
Now you have had a crash course on the most important aspect of Special Relativity, the Lorentz Transformation.
Does this all make perfect sense to you? Can you see how it relates to all the previous answers you have been given? Any more questions?