nLab
rapidity

Contents

Idea

For vectors in Minkowski spacetime p,1\mathbb{R}^{p,1} (regarded as the inner product space ( p+1,η)(\mathbb{R}^{p+1}, \eta) for η\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 p,1\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,|x) p,1x = (x^0, \vert x) \in \mathbb{R}^{p,1} be a vector with Minkowski norm-square

|x| η 2(x 0) 2|x| 20. -{\vert x\vert}_\eta^2 \coloneqq (x^0)^2 - {\vert \vec x\vert}^2 \;\geq 0\; \,.

and

x 00. x^0 \geq 0 \,.

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

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

and

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

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

In contrast, the velocity of xx is the product of the speed of light cc with the hyperbolic tangent of the rapidity:

v |x|t =c|x|x 0 =csinh(z)cosh(z) =ctanh(z) \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}

References

Last revised on November 24, 2017 at 16:48:19. See the history of this page for a list of all contributions to it.