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x' = (x - vt) / sqrt(1 - v

^{2}/c

^{2}) and

t' = [t - (vx/c

^{2})] / sqrt(1 - v

^{2}/c

^{2})

Well I am having some issues interpreting what each variable refers to. You see, here is how I've been thinking of it:

If you have one stationary observer, one moving observer and some destination point, then x is the distance that the stationary observer measures between himself and the destination point. x' is the distance that the moving observer sees between himself and the destination point. t is the amount of time that the moving observer has been moving as measured by the stationary observer. t' is questionably the amount of time that the moving observer thinks that he has been moving (I was fuzzy about this one). Finally, v is the velocity relative to the stationary observer that the moving observer is moving at.

That is how I thought of it, at least until I did some example calculations. I created the following example scenario for myself (Note that c = 1 light second per second in the units I used) :

Imagine that both Sonic and Tails start at a starting line. Now Tails decides that he himself does not want to run, but he just wants Sonic to run (so Tails stays stationary). Now Tails (who is stationary) observes that the finish line is 100 light seconds away (in the x - direction, I am only working in 1D here for simplicity sake). This means that x = 100. Meanwhile, Sonic starts running towards the finish line with a constant velocity of 97% of the speed of light (which is just v = 0.97 since c = 1). Now Tails, who has a stop watch, measures that thus far in his run, Sonic has been running for 50 seconds, so t = 50. Now that we have all of this info, we can do some Lorentz transformations.

Under these circumstances:

x' = [100 - 0.97(50)] / sqrt(1 - 0.97

^{2}) = 211.842693

t' = [50 - 0.97(100)] / sqrt(1 - 0.97

^{2}) = -193.3321664

Now the above results are what confused me. For starters, the x' that I derived is greater than the distance from the starting line to the finish line that the stationary observer observed (and this is supposed to be the distance that Sonic sees remaining between himself and the finish line after he has already run for so long if my interpretation of the variables is correct). In other words, after running for what Tails measures to be 50 seconds at 0.97c, Sonic would see himself as having just over 211 light seconds to go before he gets to the finish line. How could running at 97% of the speed of light towards the finish line make Sonic see himself as being a greater distance away from the finish line than Tails (who is at the starting line and sees himself as being 100 light seconds away from the finish line)? This just doesn't make sense to me.

The fact that t' turned out negative is just really baffling to me. By the logic of my original interpretation, this would mean that Sonic either thought he went back in time to just over 193 seconds ago, or he believes that he actually passed the finish line over 193 seconds ago. I don't think the latter would be possible however, since light itself would take 100 seconds to reach the finish line, and Tails measured Sonic (who was moving slower than light) to have been running for only 50 seconds. Of course I don't think the former is possible either.

This leads me to believe that my interpretations of the meanings of the variables may not be accurate. The only possible explanation to rectify my situation is this:

We are familiar with the space-time interval:

ds

^{2}= -c

^{2}dt

^{2}+ dx

^{2}(I'm in 1D so we don't need dy and dz).

Well we also know that ds

^{2}is supposed to be invariant regardless of frame of reference.

Well, when I plugged in my x and t values (for Tails' frame of reference, the stationary observer), I got:

ds

^{2}= 7500

Then when I plugged in my x' and t' values into this formula instead of normal x and t, I got:

ds

^{2}= 7500.000017 (which is a very negligible difference).

Noticing that the invariance notion seems to hold here, I am hypothesizing that this all means that the proper distance from the starting line to the finish line is sqrt(7500) light seconds.

Is my hypothesis correct or wrong? Is my interpretation of the variables correct or wrong? If either or both are wrong, could any of you please explain to me where the error in my interpretation was and what each of the variables in these Lorentz transformations really refers to? It would also greatly help if you used or modified my example scenario when explaining it to me. Thank you.