Rapidity

Lorentz boost

The transformations relating reference frames are associated with Hendrik Lorentz. To make a moving frame with rapidity w into the rest frame with perpendicular axes of time and space, one applies a hyperbolic rotation of parameter −w. Since cosh(−w) = cosh w and sinh(−w) = −sinh w, the following matrix representation of the hyperbolic rotation will bring the moving frame into perpendicularity (though all frames keep hyperbolic orthogonality since that relation is invariant under hyperbolic rotation).

A Lorentz boost is a vector-matrix product $${\displaystyle \left(\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\end{array}\right)=\left(\begin{array}{cc}\cosh w& -\sinh w\\ -\sinh w& \cosh w\end{array}\right)\left(\begin{array}{c}ct\\ x\end{array}\right)=\mathbf{\Lambda }(w)\left(\begin{array}{c}ct\\ x\end{array}\right).}$$

The matrix Λ(w) is of the type ${\displaystyle \left(\begin{array}{cc}p& q\\ q& p\end{array}\right)}$ with p and q satisfying p² – q² = 1, so that (p, q) lies on the unit hyperbola. Such matrices form the indefinite orthogonal group O(1,1) with one-dimensional Lie algebra spanned by the anti-diagonal unit matrix, showing that the rapidity is the coordinate on this Lie algebra. In matrix exponential notation, Λ(w) can be expressed as ${\displaystyle \mathbf{\Lambda }(w)=e^{\mathbf{Z}w}}$, where Z is the negative of the anti-diagonal unit matrix ${\displaystyle \mathbf{Z}=\left(\begin{array}{cc}0& -1\\ -1& 0\end{array}\right).}$ Since Z² is the identity, Z is a hyperbolic unit.

Velocity addition

A key property of the matrix exponential is ${\displaystyle e^{\mathbf{X}(s+t)}=e^{\mathbf{X}s}{e}^{\mathbf{X}t}}$ from which immediately follows that $${\displaystyle \mathbf{\Lambda }(w_1+w_2)=\mathbf{\Lambda }(w_1)\mathbf{\Lambda }(w_2).}$$ This establishes the useful additive property of rapidity: if A, B and C are frames of reference, then $${\displaystyle w_{\text{AC}}=w_{\text{AB}}+w_{\text{BC}},}$$ where w_PQ denotes the rapidity of a frame of reference Q relative to a frame of reference P. The simplicity of this formula contrasts with the complexity of the corresponding velocity-addition formula.

As we can see from the Lorentz transformation above, the Lorentz factor identifies with cosh w $${\displaystyle \gamma =\frac{1}{\sqrt{1-v^2/c^2}}=\frac{1}{\sqrt{1-{\beta }^2}}\equiv \cosh w,}$$ so the rapidity w is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using γ and β. We relate rapidities to the velocity-addition formula $${\displaystyle u=\frac{u_1+u_2}{1+\frac{u_1{u}_2}{c^2}}}$$ by recognizing $${\displaystyle {\beta }_i=\frac{u_i}{c}=\tanh w_i}$$ and so $${\displaystyle \tanh w=\frac{\tanh w_1+\tanh w_2}{1+\tanh w_1\tanh w_2}=\tanh (w_1+w_2)}$$

Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to proper time (time as measured by the object undergoing acceleration itself). Therefore, the rapidity of an object in a given frame can be viewed simply as the velocity of that object as would be calculated non-relativistically by an inertial guidance system on board the object itself if it accelerated from rest in that frame to its given speed.

Hyperbolic functions

The product of β and γ appears frequently in the equations of special relativity. As a result, some authors define an explicit parameter ${\displaystyle \alpha }$ for this expression, which is, from above: $${\displaystyle \alpha =\beta \gamma =\tanh w\cosh w=\sinh w}$$

This relationship uses hyperbolic functions of the rapidity ${\displaystyle w}$ to relate these parameters of special relativity, as Minkowski had observed:

$${\displaystyle \alpha =\sinh w}$$ $${\displaystyle \beta =\tanh w}$$ $${\displaystyle \gamma =\cosh w}$$

Exponential and logarithmic relations

From the above expressions we have $${\displaystyle e^w=\cosh (w)+\sinh (w)=\gamma (1+\beta )=\frac{1+\beta }{\sqrt{1-{\beta }^2}}=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}},}$$ and thus $${\displaystyle e^{-w}=\cosh (w)-\sinh (w)=\gamma (1-\beta )=\frac{1-\beta }{\sqrt{1-{\beta }^2}}=\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}.}$$ or explicitly $${\displaystyle w=\ln \left[\gamma (1+\beta )\right]=-\ln \left[\gamma (1-\beta )\right].}$$

Doppler effect

The Doppler-shift factor, for the longitudinal case with source and receiver moving directly towards or away from each other, that is associated with rapidity w is ${\displaystyle k=e^w}$.

In experimental particle physics

The energy E and scalar momentum |p| of a particle of non-zero (rest) mass m are given by: $${\displaystyle \begin{array}{cc}E& =\gamma m{c}^2\\ \left|\mathbf{p}\right|& =\gamma mv.\end{array}}$$ With the definition of w $${\displaystyle w=\mathrm{artanh}\frac{v}{c}=\mathrm{artanh}\beta ,}$$ and thus with $${\displaystyle \cosh w=\cosh \left(\mathrm{artanh}\frac{v}{c}\right)=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\gamma }$$ $${\displaystyle \sinh w=\sinh \left(\mathrm{artanh}\frac{v}{c}\right)=\frac{\frac{v}{c}}{\sqrt{1-\frac{v^2}{c^2}}}=\beta \gamma =\alpha ,}$$ the energy and scalar momentum can be written as: $${\displaystyle \begin{array}{cc}E& =m{c}^2\cosh w\\ \left|\mathbf{p}\right|& =mc\sinh w.\end{array}}$$

So, rapidity can be calculated from measured energy and momentum by $${\displaystyle w=\mathrm{artanh}\frac{|\mathbf{p}|c}{E}=\frac{1}{2}\ln \frac{E+|\mathbf{p}|c}{E-|\mathbf{p}|c}=\ln \frac{E+|\mathbf{p}|c}{m{c}^2}.}$$

However, experimental particle physicists often use a modified definition of rapidity relative to a beam axis $${\displaystyle y=\frac{1}{2}\ln \frac{E+p_{z}c}{E-p_{z}c},}$$ where p_z is the component of momentum along the beam axis. This is the rapidity of the boost along the beam axis which takes an observer from the lab frame to a frame in which the particle moves only perpendicular to the beam. Related to this is the concept of pseudorapidity.

Rapidity relative to a beam axis can also be expressed as $${\displaystyle y=\ln \frac{E+p_{z}c}{\sqrt{m^2{c}^4+p_T^2{c}^2}}.}$$