nLab rapidity

Idea

For vectors in Minkowski spacetime $\mathbb{R}^{p,1}$ (regarded as the inner product space $(\mathbb{R}^{p+1}, \eta)$ for $\eta$ the Minkowski metric) rapidity is the analog in Lorentzian geometry of what in Euclidean geometry would be the angle to the time-axis. If vectors in $\mathbb{R}^{p,1}$ are thought of as tangent vectors to trajectories in Minkowski spacetime, their rapidity is a measure for the velocity of the trajectory, which is well-adapted to Lorentz transformations.

Definition

Let $x = (x^0, \vert x) \in \mathbb{R}^{p,1}$ be a vector with Minkowski norm-square

$$-{\vert x\vert}_\eta^2 \coloneqq (x^0)^2 - {\vert \vec x\vert}^2 \;\geq 0\; \,.$$

and

$$x^0 \geq 0 \,.$$

Them its rapidity is the unique $z \in \mathbb{R}$ such that

$$x^0 = \sqrt{-{\vert x\vert}^2_\eta} \, \cosh(z)$$

and

$${\vert \vec x\vert} = \sqrt{-{\vert x\vert}^2} \, \sinh(z)$$

where $\cosh, \sinh$ denote the hyperbolic cosine and hyperbolic sine, respectively.

In contrast, the velocity of $x$ is the product of the speed of light $c$ with the hyperbolic tangent of the rapidity:

$$\begin{aligned} v & \coloneqq \frac{ { \vert \vec x\vert} }{t} \\ & = c \frac{ {\vert \vec x\vert} }{x^0} \\ & = c \frac{ \sinh(z) }{ \cosh(z)} \\ & = c \tanh(z) \end{aligned}$$

Related concepts

• velocity, speed

• special relativity

• singular support of causal propagator

References

• Wikipedia, Rapidity