I think that the time actually slows down as you approach the speed of light and since we keep it co

Assuming the speed of light is constant travel trailers used for BoxGuy travel trailers used relative to himself, the time between Pulse and Reflect is equal to the time between Reflect and Return. This is because the distance the light travels relative to him is d in both cases.
With the same assumptions for PlatGirl, the time between Pulse and Reflect is less than the time between Reflect and Return. This is because the mirror will travel 2 * d on the away trip (because travel trailers used when light has traveled 2 * d , the mirror travel trailers used will be d farther to the left, so both the mirror and pulse will be in the same location), but only 2/3 * d on the return trip (using similar travel trailers used logic).
Assuming travel trailers used that the light pulse is in the same location for all observers at any given moment, Pulse has to occur simultaneously for both BoxGuy and PlatGirl, Reflect has to occur simultaneously for BoxGuy and PlatGirl, and Return has to occur simultaneously for BoxGuy and PlatGirl.
Finally, if we try to figure out the relative passage of time for BoxGuy and PlatGirl with the above, we get that time travels faster for PlatGirl than for BoxGuy during Pulse-Reflect. This is because light travels farther for her ( 2*d ) than him ( d ) during that time. With similar logic, we get that time travels travel trailers used slower for PlatGirl than for BoxGuy during Reflect-Return.
The last conclusions do not make sense, since the coming or going of a beam of light should travel trailers used not affect the relative time-lapse for two observers. For example, if this were the case what would happen if another travel trailers used pulse was emitted the moment the first pulse is reflected? Time cannot move faster AND slower for both of them.
As mentioned by other users, d will be shorter for PlatGirl than for BoxBoy according to SR. However, the duration of Pulse-Reflect is still shorter than Reflect-Return for PlatGirl, and the durations are equal for BoxBoy.
In response to my question on Mark's answer, we can use the Lorentz Transform to calculate PlatGirl's space-time coordinate travel trailers used for BoxGuy's Reflect observed event, which happens at (d,d/c) in his frame of reference:
The thing is that the constantcy of the speed of light is both (a) a consequence of the symmetries travel trailers used of Maxwell s equations and (b) an experimentaly measured fact ( since 1887 and many times since then). travel trailers used In anycase in your list of possibilities you left out the distance between points may be different travel trailers used for different travel trailers used observers
"Simultaneous" travel trailers used refers to two different events that occur at the same time in some particular reference frame, but you're applying it to the same event in two different frames. So it doesn't make sense to say "Pulse has to occur simultaneously for both BoxGuy and PlatGirl." That's a single event - it can't be simultaneous all by itself, even when observed by two different people.
You could, if you want, set the origins of the coordinate systems they are using so PlatGirl and BoxGuy assign the same time coordinate to Pulse. If you do, they will not assign the same time coordinate to Reflect. The time between the events travel trailers used Pulse and Reflect is different in different frames.
Additionally, PlatGirl and BoxGuy will not agree on the length of the boxcar. Your calculation assumes travel trailers used they both measure the length to be $d$, but actually PlatGirl will observe the boxcar to be Lorentz-contracted.
In reply to your edit, yes the durations from Pulse to Reflect and Reflect to Return are the same for BoxGuy and different for PlatGirl. That is just a fact. That's how it is. Notice, though, that the spatial separations are also different. travel trailers used For BoxGuy, these events the same distance apart. For PlatGirl, they are different distances apart. What's the same between frames is the interval $\Delta x^2 - \Delta t^2$.
Ok, so we set the space-time-origin (0,0) to Pulse at the Light for both BoxGuy and PlatGirl. The space-origin moves with the Light for BoxGuy, but is fixed relative to the platform for PlatGirl. At (d,d/c) travel trailers used and (0, 2d/c) for BoxGuy, Reflect and Return happen respectively. What is the formula for translating these space-time coordinates directly back to PlatGirl s coordinate system?
Nevermind, I ve updated my question notes with the Lorenz travel trailers used Transform. travel trailers used From this, it appears time observed at a point in space is warped according to the direction of relative travel. For example, at the time of the Pulse, BoxGuy s mirror is at $(d,0)$, which maps to PlatGirl s $(\frac{2\sqrt{3}}{3}d, \frac{\sqrt{3}}{3}d/c)$. Thus, PlatGirl observes a current version of BoxGuy (for them both), but BoxGuy s current mirror is a future travel trailers used version of PlatGirl s current mirror. By symmetry, if the mirror and origin were reversed, PlatGirl s mirror would be the future one. Is this correct?
This picture assumes that Boxguy is standing travel trailers used next to the lamp, and that the flash leaves the lamp just as it passes PlatGirl. (If, for example, BoxGuy were standing next to the mirror, the picture would look a little different.)
The blue near-vertical line is BoxGuy's worldline (and the lamp's). Each of the other blue lines is "the world at a particular instant" according to Boxguy. He measures distances along any one of these lines.
Both Platgirl and BoxGuy will agree that the gold lines traverse $x$ units of space in $x$ units of time (I am taking the speed of light to be 1.) That's because travel trailers used the light rays are at "45 degree angles" to the axes in both PlatGirls's and Boxguy's opinion. (Don't forget that Boxguy views the two thick vertical lines as perpendicular in spacetime, and note that the gold line bisects the angle between them.)
travel trailers used By staring long enough at this picture, you ought to be able to describe exactly what's happening from both BoxGuy's and Platgirls' points of view, and to see how they're two different ways of describing the same thing (i.e. two different ways of assigning coordinates to points on the same gold line).
[It helps to remember that points along a vertical line all occupy travel trailers used the same location in space according to Platgirl, and that points along a line parallel to his worldline all occupy travel trailers used the same location in space according to BoxGuy. I didn't draw these gridlines for fear of making the diagram look too intimidating, but it might help to add them.]
It's not just that the concept of absolute velocity is impossible to know its that the concept itself doesn't have any meaning (in relativistic theory). travel trailers used Saying "I am approaching the speed of light" is an ambiguous and often misleading statement. The correct formulation (relativistically speaking) would be "there is an observer in a certain frame of reference who is currently measuring my velocity as seen from his/her frame of reference travel trailers used to be approaching the speed of light".
Saying the speed of light is absolute is really a short cut for the following statement: travel trailers used "No matter in which frame of reference travel trailers used an observer sits every time he or she measures the speed of light he or she appears to get the same value". This is an experimental statement which is postulated as a physical law because nobody has credibly demonstrated an experiment contradicting this.
Once again, travel trailers used saying "I have no way of knowing whether I am at rest or moving" is equivalent to saying "I have no way of knowing whether I am taller or shorter". Taller or shorter than what? At rest or moving with respect to what?
Please correct me if I am wrong. But I think that the speed of light, measured distance and time from a frame of reference are concepts defined relatively to each other. travel trailers used In the sense that we fix the speed of light and define distance and time relatively to it.
In particular I am not really travel trailers used convinced that the measured speed of light is the same in every reference frame. It is the same only if you measure it with a mirror so that the beam once travels away from the observer and once towards him.
I think that the time actually slows down as you approach the speed of light and since we keep it constant the concepts of simultaneity and distance are redefined though a Lorentz travel trailers used transformation according travel trailers used to which the events ahead the moving observer look closer in space-time and the ones behind him look farther.
This whole assumption of constant speed of light is a consequence of the ambiguity between rest and motion. travel trailers used In fact an observer within an inertial frame of reference has no way to know his velocity thus he assumes that he is at rest and uses the speed of light to measure distances and simultaneity around him. This results in a geometry described by Lorentz transformations. travel trailers used If he knew his velocity he could have a better knowledge of simultaneity since you can move away or towards light sources.
An observer can easily measure a higher or lower speed of light with a 'simple' setup. A spaceship, a sensor and a light screen. When he crosses travel trailers used the sensor check point the light screen turns on. If he travels through the screen and doesn't know the distance between the sensor and the screen he will use the speed of light to judge the screen closer to him. The opposite is valid if he travels away from the screen.
Another possibility is that the observer measures the distance between check point and screen travel trailers used in a frame of reference where there is no relative velocity. Then when he passes the check point he measures the time needed to detect the light screen and knowing the actual distance he would have measured a faster (or slower) travel trailers used speed of light which he can use to measure his own speed and correct the Lorentz induced distortion of space time.
Ok I have done some research, here are two articles that can clarify where the confusion arises. Is seems that Lorentz transformations were based on an anisotropic one-way speed of light meaning travel trailers used that you would measure a different one-way speed of light depending if you move toward or away from a source. Still the two-way speed of light is the same as in special relativity and this is actually the only way to currently measure it accurately (it is hard to get accurate synchronization and also to approach the speed of light).
Einstein took Lorentz equations to build his special relativity the