as in the standard spacetime diagram. There the axes are measured by clock and meter-stick, more familiar benchmarks, and the basis of spacetime theory. So the delineation of rapidity as hyperbolic parameter of beam-space is a reference to the seventeenth century origin of our precious transcendental functions, and a supplement to spacetime diagramming. Lorentz boost The rapidity arises in the linear representation of a Lorentz boost as a vector-matrix product . The matrix is of the type with and satisfying , so that 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. This action may be depicted in a spacetime diagram. In matrix exponential notation, can be expressed as , where is the negative of the anti-diagonal unit matrix It is not hard to prove that . This establishes the useful additive property of rapidity: if , and are frames of reference, then where denotes the rapidity of a frame of reference relative to a frame of reference . 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 , so the rapidity is implicitly used as a hyperbolic angle in the Lorentz transformation expressions using and β. We relate rapidities to the velocity-addition formula by recognizing and so 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. The product of and appears frequently, and is from the above arguments Exponential and logarithmic relations From the above expressions we have and thus or explicitly The Doppler-shift factor associated with rapidity is . In experimental particle physics The energy and scalar momentum of a particle of non-zero (rest) mass are given by: With the definition of and thus with the energy and scalar momentum can be written as: So, rapidity can be calculated from measured energy and momentum by