Rapidity (physics)

The rapidity is an alternative measure of speed , which in the special theory of relativity is used. Usually (small theta ) is used as a symbol . The term was first formulated in 1911 by Alfred Robb . ${\displaystyle \theta }$

The rapidity is defined as

${\displaystyle \theta =\operatorname {artanh} (v/c),}$

in which

• ${\displaystyle v}$ the speed and
• ${\displaystyle c}$the speed of light
• ${\displaystyle \operatorname {artanh} }$is the hyperbolic areatangent function.

Rapidity measures the speed in units of the speed of light that an accelerated body would have without relativistic effects. The rapidity is therefore unlimited ( range of values ), which is a more natural approach than the limitation of the actual speed, which can never exceed the speed of light (range of values ). ${\displaystyle \theta =(-\infty ,+\infty )}$ ${\displaystyle v=(-c,+c)}$

In addition, rapidity has the advantage that two rapidities can simply be added, while the relativistic addition theorem must be used for velocities .

For non-relativistic velocities, the rapidity approaches the value of : ${\displaystyle {\frac {v}{c}}}$

${\displaystyle v\ll c\Leftrightarrow v/c\ll 1\Rightarrow \theta \approx v/c.}$

In particle physics

Instead of the exact formula

${\displaystyle \theta ={\frac {1}{2}}\cdot \ln \left({\frac {1+\beta }{1-\beta }}\right)={\frac {1}{2}}\cdot \ln \left({\frac {E+cp}{E-cp}}\right)}$

With

• ${\displaystyle \beta =v/c}$
• energy ${\displaystyle E}$
• pulse ${\displaystyle p=|{\vec {p}}|}$

In experimental particle physics , a rapidity is often defined relative to the beam axis according to

${\displaystyle \theta _{\text{Beamline}}={\frac {1}{2}}\cdot \ln \left({\frac {E+cp_{L}}{E-cp_{L}}}\right)}$

is used, wherein the longitudinal pulse is the pulse component along the beam axis. ${\displaystyle p_{L}}$

The related amount of pseudorapidity is defined as

${\displaystyle \eta ={\frac {1}{2}}\cdot \ln \left({\frac {p+p_{L}}{p-p_{L}}}\right).}$

Individual evidence

1. ^ Theoretical Physics , page 327 by Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf, Springer-Verlag, 2014, accessed on February 19, 2016.
2. ^ Theoretical Physics , page 372 by Matthias Bartelmann, Björn Feuerbacher, Timm Krüger, Dieter Lüst, Anton Rebhan, Andreas Wipf, Springer-Verlag, 2014, accessed on February 19, 2016.