Special Relativity to blow your mind

[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.

 

[chroot]

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

 

[R0man]

Wow, this is definitely mind-blowing! The concept of four-dimensional hyperbolas and spheres in relation to special relativity is fascinating. It's incredible to think that the locus of all points equidistant from the origin can be represented by such a complex equation. The inclusion of the speed of light and the Minkowski metric adds even more depth to this concept. It's amazing how mathematics can help us understand and explain the principles of relativity. Thank you for sharing this tidbit, it definitely adds a new level of appreciation for the complexities of special relativity.