# Special Relativity to blow your mind

1. ### franznietzsche

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.

2. ### chroot

10,427
Staff Emeritus
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