Maybe I missed something but at 13:58 that last equation does not appear to be correct based on the previous equation. Regrouping the previous equation by terms, it would be (sinh φ x) + (cosh φ c t) σt + (cosh φ x) σx - (sinh φ ct) σt σx + …; in the equation on the screen, the term with no sigma x nor sigma t somehow picked up a sigma t, and the term with sigma x AND sigma t somehow lost the sigma t?
That's a good question! It's because paravectors experience Lorentz contractions in their direction of motion whereas biparavectors experience Lorentz contractions perpendicular to their direction of motion! The dagger allows the Lorentz rotor to anticommute with orthogonal directions to motion and thereby cancel out, while for biparavectors the Clifford conjugate allows the Lorentz rotor to cancel out in directions of motion but stay in orthogonal directions!
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/tr.png)
Eccentric, This made me so happy! I liked and subscribed!
Maybe I missed something but at 13:58 that last equation does not appear to be correct based on the previous equation. Regrouping the previous equation by terms, it would be (sinh φ x) + (cosh φ c t) σt + (cosh φ x) σx - (sinh φ ct) σt σx + …; in the equation on the screen, the term with no sigma x nor sigma t somehow picked up a sigma t, and the term with sigma x AND sigma t somehow lost the sigma t?
Why does the position transform like ΛrΛ+ but forces transform ΛFΛ~? Shouldn't all of them be hyperbolic rotations?
That's a good question! It's because paravectors experience Lorentz contractions in their direction of motion whereas biparavectors experience Lorentz contractions perpendicular to their direction of motion!
The dagger allows the Lorentz rotor to anticommute with orthogonal directions to motion and thereby cancel out, while for biparavectors the Clifford conjugate allows the Lorentz rotor to cancel out in directions of motion but stay in orthogonal directions!