sci.physics.research

On 2007-08-01, John M. Dlugosz <11lrhap02@ > wrote:

> My draft in progress can be seen at < /files/
> PhysFAQ-edit/Relativity/SR/ >.
>
> My question to the group concerns Figure 4. I followed the
> instructions in the original text, seen in the paragraph before the
> figure, and the two lines intersect! Based on the following
> paragraph, that should not happen, and the ship that starts out on the
> right should remain on the right, but draw ever closer in the lab-
> frame.
>
> What is going on here? Clearly the "K" in this formula is not at all
> related to the "k" used earlier, since the directions say "K>1" and
> the earlier "k" is the x-distance between ships in the lab-frame at
> launch. The units of space would make ">1" nonsense (one what?).
>
> In the Figure, the vertical height is 1 unit (- through )
> and the function iterates over t in that range. K=2. The original
> curve (and the left curve here) is the same function with K=1. Now,
> on the left curve, as with all the previous figures, note that the
> hyperbola appears just a few pixels right of the t axis. The function
> as written would give an x intercept of 1, so the illustration would
> have to be as wide as it is tall before you even start to see the
> curve. The two ships differ in their starting position, so the curves
> are shifted left by a different constant x for each ship. In Figure
> 4, the right curve is also shifted left to the same starting position
> as the second ship is always drawn at, though the natural position of
> the stated function is even farther right (intersects the x axis at
> x=K).

As I understand the text, it's prescribing two curves,
one x = sqrt(1+t^2) and the second x = sqrt(K^2+t^2). At t=0, the first
has x = 1, while the second has x = K. These are the initial positions
of the two rocket ships, so their initial lab frame separation is K-1.
If you draw these two curves, you'll see that they do not intersect, but
share a common asymptote, x = t. That is the picture that I believe
the text is describing.

If you want to shift the two curves, then you do so by shifting both of
them by the same amount, which won't change their relative asymptotic
properties. As you've noticed, if you shift them both by different
amounts, you can make them intersect.

The shape of each curve determines the precise acceleration that the
corresponding space ship experiences. The two curves, which I explicitly
gave above, have their accelerations and initial positions precisely
tuned to get the same asymptote. If change the shape of one of the
curves, or if you shift one curve with respect to the other, this
asymptotic property is broken.

Hope this helps. I'm happy to see that the FAQ is getting some new
illustrations. Good work!

Igor