Here is a problem I have been pondering for the last week or so:

In Peter Collier's book "A Most Incomprehensible Thing Notes Towards a Very Gentle Introduction to the Mathematics of Relativity" in section 3.3.5 he makes the following claim about the space time interval $\displaystyle \Delta S^2 =c^2 \Delta t^2 - \Delta x^2 $:

- Time-like interval - where $\displaystyle \Delta S^2 > 0$ , describes events within A's
lightcone. These events are causally related to A, and there will be some
inertial frame where A and C occur at the same place but at different
times. - Space-like interval - where $\displaystyle \Delta S^2 < 0$ , describes events outside A's
lightcone. These events are not causally related to A, and there will be
some inertial frame where A and C occur at the same time but at different
places. - Light-like interval - where $\displaystyle \Delta S^2 = 0$, describes events on A's lightcone.
These events are causally related to A, but they can only be linked to A by
a light signal. |

So I decided to play with this idea and proceeded to find a way to curve fit a set of hyperbolae to two points. I chose A(0,2) and B(2,6) where the units are (ct, x) and light travels at a 45 degree angle to both axis. So if A = (0,2) and B = (2,6) then we have $\displaystyle \Delta S^2 =2^2 - 4^2 = -12 \textrm{ which is} < 0 $ so we have a space-like curve. Therefore the two points (shown as red dots) should be on the same time hyperbola but they appear on opposing hyperbola?

The process I used to fit the curves was to:

- rotate points A,B 45 degrees clockwise around the origin
- Curve fit the rotated points to (x-x0)(y-y0) = 1 (basically find a displaced hyperbola).
- Rotate the resultant hyperbolae anticlockwise by 45 degrees around the origin.

I did this to avoid dealing with quadratic terms of $\displaystyle (ct-ct_0)^2 - (x-x_0)^2 = \Delta S^2$

As you can see from the diagram the curve fitting seems to have worked but the result wasn't what I expected. The points ended up on opposing curves. Can anyone see why?