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Special Relativity to blow your mind

by franznietzsche
Tags: blow, mind, relativity, special
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franznietzsche
#1
Dec16-03, 06:53 PM
P: 1,783
Here's a few kickers for those of you who don't know a lot about the mathematics of relativity:

The locus of all points equidistant from the origin is a four dimensional hyberbola given by:
[tex]
d^2=(y_1-x_1)^2+(y_2-x_2)^2+(y_3-x_3)^2-c^2(y_4-x_4)^2
[/tex]
where [tex] c [/tex] is the speed of light. Also the cross section of this perpendicular to the time axis (x_4) is a sphere, the euclidean locus of equidistant points.

the reason the [tex] -c^2 [/tex] is in the equation is the Minkowski metric which also determines the lorentz transformation that makes inertia increase as velocity increases etc. Hope someone else finds this tidbit entertaining.
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chroot
#2
Dec16-03, 07:06 PM
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What you've posted is a sort of ugly form of the "line element," which defines the distance between two neighboring points in spacetime. In special relativity, the line element is most succinctly expressed as

[tex]ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu[/tex]

where [itex]\eta[/itex] is the metric of Minkowski (flat) spacetime.

- Warren


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