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Old Sep 29th 2017, 02:24 PM   #1
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A space time interval conundrum

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?
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A space time interval conundrum-fitted-hyperbolae.png  
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Old Sep 30th 2017, 01:31 PM   #2
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ok, I think I see where I went wrong in a couple of ways:
  • A moving observer has a "sheared" set of axis as the spatial and time axis rotate towards each other
  • Therefore the hyperbola would also be sheared
  • In hindsight makes more sense to move the origin to one of the points A or B and then draw a straight line through them
  • This is the line of simultaneity, its now a simple manner to calculate the world line that would produce this line of simultaneity
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Last edited by kiwiheretic; Sep 30th 2017 at 01:31 PM. Reason: spelling
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Old Sep 30th 2017, 08:58 PM   #3
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What was your goal in doing all of this? I'm not sure what it is that you're driving at.
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Old Sep 30th 2017, 09:29 PM   #4
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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.
It was to trying to find these other frames.
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Old Sep 30th 2017, 09:37 PM   #5
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Originally Posted by kiwiheretic View Post
It was to trying to find these other frames.
So why not use the Lorentz transformation instead of the spacetime interval? Do you think that you can get that information from the spacetime interval?
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Old Sep 30th 2017, 10:21 PM   #6
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Originally Posted by Pmb View Post
So why not use the Lorentz transformation instead of the spacetime interval? Do you think that you can get that information from the spacetime interval?
I don't know why I thought it was a hyperbola fitting exercise other than taking Collier's work down a wrong path. I admit I went off on a tangent. I could have saved myself some work.
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Old Sep 30th 2017, 10:36 PM   #7
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Originally Posted by kiwiheretic View Post
I don't know why I thought it was a hyperbola fitting exercise other than taking Collier's work down a wrong path. I admit I went off on a tangent. I could have saved myself some work.
Its never a waste of time. Consider it a learning experience. I hope you don't think professional physicists never take the wrong path, do you?
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