# Special Relativity to blow your mind

by franznietzsche
Tags: blow, mind, relativity, special
 P: 1,769 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: $$d^2=(y_1-x_1)^2+(y_2-x_2)^2+(y_3-x_3)^2-c^2(y_4-x_4)^2$$ where $$c$$ 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 $$-c^2$$ 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.
 PF Patron Sci Advisor Emeritus P: 10,400 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 $$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu$$ where $\eta$ is the metric of Minkowski (flat) spacetime. - Warren

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