How Do Lorentz Transformations Affect Perceptions of Time and Distance?

[space-time]

You are probably familiar with the following two Lorentz transformations:

x' = (x - vt) / sqrt(1 - v²/c²) and
t' = [t - (vx/c²)] / sqrt(1 - v²/c²)

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 - 0.97²) = 211.842693

t' = [50 - 0.97(100)] / sqrt(1 - 0.97²) = -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:

ds² = -c²dt² + dx² (I'm in 1D so we don't need dy and dz).

Well we also know that ds² 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:

ds² = 7500

Then when I plugged in my x' and t' values into this formula instead of normal x and t, I got:
ds² = 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.

 

[Dale]

Please use LaTeX in your posts, they are difficult to read otherwise.

Well I am having some issues interpreting what each variable refers to.

$t$ and $x$ are the time and space coordinates of one inertial frame. $t'$ and $x'$ are the time and space coordinates of the other inertial frame. $v$ is their relative velocity and $c$ is the speed of light.

 

[space-time]

Please use LaTeX in your posts, they are difficult to read otherwise. $t$ and $x$ are the time and space coordinates of one inertial frame. $t'$ and $x'$ are the time and space coordinates of the other inertial frame. $v$ is their relative velocity and $c$ is the speed of light.

Could you tell me what that would mean in terms of my example scenario? I know that an inertial frame is a frame of reference in which Newton's laws hold, but that doesn't really increase my understanding of the transformations.

 

[Nugatory]

Could you tell me what that would mean in terms of my example scenario? I know that an inertial frame is a frame of reference in which Newton's laws hold, but that doesn't really increase my understanding of the transformations.

Get hold of a sheet of graph paper. Call the vertical axis time and use the horizontal axis for space (only one dimension, the x dimension, in your example).

The event "sonic started running" is the point x=0,t=0. The event "sonic crossed the finish line" is the point x=100,t=100/.97. After 50 seconds have passed on tail's watch, he is at the point x=0,t=50 and sonic is at the point x=48.5,t=50. These are examples of x and t values using Tails's frame; you plug these into the Lorentz transforms to find the x' and t' values that Sonic perceives.

 

[space-time]

Get hold of a sheet of graph paper. Call the vertical axis time and use the horizontal axis for space (only one dimension, the x dimension, in your example).

The event "sonic started running" is the point x=0,t=0. The event "sonic crossed the finish line" is the point x=100,t=100/.97. After 50 seconds have passed on tail's watch, he is at the point x=0,t=50 and sonic is at the point x=48.5,t=50. These are examples of x and t values using Tails's frame; you plug these into the Lorentz transforms to find the x' and t' values that Sonic perceives.

Thanks Nugatory. I think I've just about got it now. I just want to do one last check to see if I understand, so please tell me if what I am about to say is correct:

I plugged in one of the example coordinates from Tails' frame of reference (the stationary frame) into the Lorentz transformations. Specifically, I plugged in the coordinates x = 0 t = 100/0.97 (These are the coordinates that Tails sees himself as having when Sonic crosses the finish line). When I plugged these into the Lorentz transformations I got:

x' = -411.3450349 and t' = 424.0670463
If I truly understand what you are saying correctly, then this means at the event "Sonic crosses the finish line", Sonic sees Tails as having coordinates:
(-411.3450349 , 424.0670463) which means that Sonic sees Tails as being 411.3450349 light seconds behind him and he believes that Tails has experienced a wait time of 424.0670463 seconds.

Is this correct?

 

[Nugatory]

x' = -411.3450349 and t' = 424.0670463
If I truly understand what you are saying correctly, then this means at the event "Sonic crosses the finish line", Sonic sees Tails as having coordinates:
(-411.3450349 , 424.0670463)

That looks right, but I haven't calculated it exactly. (By the way, If you're going to work through more of these problems, you might try choosing $v=\frac{3}{5}c=.6c$. Then the quantity $1/\sqrt{1-v^2}$ comes out to be exactly $\frac{5}{4}=1.25$, a nice round number that means you can just about do the arithmetic in your head).

which means that Sonic sees Tails as being 411.3450349 light seconds behind him and he believes that Tails has experienced a wait time of 424.0670463 seconds.

Not quite. It means that when Sonic's wristwatch reads 424.0670463 seconds Tails will be 411.3450349 light-seconds behind him... and if you think about it for a moment, this conclusion is obvious because as far as Sonic is concerned he is at rest while Tails is moving away from him at .97c - of course Tails is 411+ light-seconds away if he's been moving away at .97c for 424+ seconds.

It also means that according to Sonic, at the same time that Sonic's wristwatch reads 424+ Tails's wristwatch reads 100/.97. This is time dilation at work.

Note also that according to Tails, the events "Tails's wristwatch read 100/.97" and "Sonic passed the finish line" both happened at the same time, t=100/.97. However, according to Sonic these two events were not simultaneous, as he passed the finish line long before (transform x=100,t=100/.97 into Sonic's coordinates to see exactly when, and as a bonus you'll find the length-contracted distance that Sonic covered) his wristwatch read 424+ seconds. This is the relativity of simultaneity at work, and if you work through the numbers you'll find that it all comes out consistently.