
#1
Dec1603, 06:53 PM

P: 1,782

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_1x_1)^2+(y_2x_2)^2+(y_3x_3)^2c^2(y_4x_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. 



#2
Dec1603, 07:06 PM

Emeritus
Sci Advisor
PF Gold
P: 10,424

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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