Relativistic Pythagorean Theorem
Sep. 23rd, 2005 11:06 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jgraftonIn physics today, while doing linear transformations with Lorentz transformations, we learned that the Pythagorean Theorem seems to not hold up - the hypotenuse is shorter than its legs.
Could math seriously be broken? Of course not - you just need to add the relativistic correctional factor. A little bit of scribbling in my notes reveals the following:
And so the world is still complete. Obviously this does not corrupt the original definition, since for relatively low v the correctional factor can be ignored.
(Yes, I know this is horribly broken, and probably not symmetric and probably doesn't make sense. But I can have fun anyway, right?)
Could math seriously be broken? Of course not - you just need to add the relativistic correctional factor. A little bit of scribbling in my notes reveals the following:
And so the world is still complete. Obviously this does not corrupt the original definition, since for relatively low v the correctional factor can be ignored.
(Yes, I know this is horribly broken, and probably not symmetric and probably doesn't make sense. But I can have fun anyway, right?)
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(no subject)
Date: 2005-09-23 04:12 pm (UTC)Still reading Wikipedia about Lorentz transformations in hopes of fully understanding this though...
You should post this to
(no subject)
Date: 2005-09-23 04:16 pm (UTC)(no subject)
Date: 2005-09-23 05:13 pm (UTC)My existance has been validated.
(no subject)
Date: 2005-09-23 05:17 pm (UTC)$(ds)^2 = \sum_{i=1}^n (dx_i)^2$ whereas an n dimensional Minkowski metric, using $t = x_1$ to denote the timelike axis, has $(ds)^2 = -(dt)^2 + \sum_{i=2}^n (dx_i)^2$.
The upshot of this is that the pythagorean theorem holds in all space-like dimensions of a Minkowski manifold, but that the shortest distance between two time-separated points is no longer a straight line (as it would be if the hypotenuse were the longest leg). One instead finds that for (t_1, x) and (t_2,x) that the straight line between them is the longest path between them.
Hope that helps.
(no subject)
Date: 2005-09-23 05:38 pm (UTC)(no subject)
Date: 2005-09-23 05:42 pm (UTC)(no subject)
Date: 2005-09-23 09:01 pm (UTC)...
...
(no subject)
Date: 2005-09-23 09:03 pm (UTC)That was me, by the way. Stupid work computers.