# Rapidity in 4-vector Transformation

In Lorentz transformation we have a concept of rapidity as related to boost. Rapidity is defined as a hyperbolic angle α such that $$\tanh(α)=v/c .$$ This further defines a matrix for Lorentz transformation in term of sinh(α) and cosh(α). This is also said to represent LT under a rotation of the frame of reference by an imaginary angle. I could not understand how this idea of using hyperbolic angle struck to Einstein or for that matter whosoever proposed it.

I will show you the idea with a simple example: a 1+1 space-time.

In ordinary Euclidean 2-d space $$\mathbb{R}^2$$, the rotation is all operation on points $$(x^1,x^2)$$ which holds the quantity

$$(x^1)^2+(x^2)^2=C$$

invariant. So with the fact that $$\cos^2 \theta+\sin^2 \theta=1$$ one can simply write down the form:

$$x^1=\sqrt{C} \cos\theta \ ; \ x^2=\sqrt{C}\sin\theta$$

And the variation of $$\theta$$ satisfies our requirement automatically.

Then for a Minkowski 1+1 space, the invariance of light speed show that the "rotation", or the transform between frames should hold:

$$(x^1)^2-(x^2)^2=C$$

invariant. So with the fact that $$\cosh^2\theta-\sinh^2\theta=1$$ one can directly see how Lorentz boost connects to the hyperbolic functions.

• (x)That is so nice. I was thinking on line that just as the solution to In the second order differential equation y''(x)+a2 *y(x)=0. is y = Acos(ax)+B sin(ax), where A and B are constants. Similarly the solution to y''(x)−a2*y(x) = 0 (note the sign change in front of a2) is y = Acosh(ax) + B sinh(ax). (a2 stands for square of a) – Ashwani Kumar Apr 9 at 13:42