We develop a relativistic velocity space called *rapidity space* from the single assumption of Lorentz invariance, and use it to visualize and calculate effects resulting from the successive application of non-collinear Lorentz boosts. In particular, we show how rapidity space provides a geometric approach to Wigner rotation and Thomas precession in the same way that space–time provides a geometrical approach to kinematic effects in special relativity.

## REFERENCES

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L. H.

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Some authors derive an approximation that is valid to second order in β, which is all that is needed to calculate the relativistic correction to the spin–orbit term in hydrogen. See, for example,

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One exception, which gives an elementary and clear example of the Thomas–Wigner rotation, is the recent paper by

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In Sec. VIII C we discuss the more general case of the three-dimensional relativistic velocity space obtained from four-dimensional space–time.

28.

Reference 26, pp. 52–53.

29.

We thank one of the referees for pointing this out to us.

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In this case, by “conformal,” we mean that angle measurements using a natural metric induced from Minkowski space (in a manner similar to the one developed in this section) coincide with those using the metric arising from viewing the surface as embedded in Euclidean 3-space.

32.

Because our model involves only two coordinates, $x1$ and $x2,$ by “conformal” we mean that angle measurements coincide with those using the Euclidean metric in 2-space.

33.

There are other choices of $A,$ $B$, and $C$ that will solve the three equations. First, if $A=C=0,$ then $B=\xb11$ and $g=\xb1r.$ These solutions, however, make the denominator on the right-hand side of Eq. (44) vanish. Second, if $B=0,$ then $A$ can be arbitrary as long as $C=1/(4A).$ In this case, however, if we use any value of $A$ other than $A=1/2,$ we end up with the same model of a disk, but one whose radius is not equal to one. In this case everything is the same, just scaled appropriately.

34.

Although the coordinates in a two- or three-dimensional velocity space are the components of the velocity, the coordinates in rapidity space are not components of the rapidity; rather, they are proportional to the coordinates of the velocity, with a proportionality factor that varies from point to point. The space is called rapidity space simply to emphasize that the distance from the origin to any point in it is the rapidity of that point.

35.

This calculation implicitly assumes that the distance $s$ is to be evaluated along a straight line. By definition, the distance between two points is the minimum of all the path integrals connecting them, which means that distances are evaluated along geodesics (unless otherwise stated). Thus, we are assuming that any geodesic that includes the origin is a straight line. We prove this assumption in Sec. VI.

36.

See, for example, Ref. 16, pp. 62–63. We give an algebraic proof of this result in Sec. VIII D.

37.

M. L. Boas,

*Mathematical Methods for the Physical Sciences*(Wiley, New York, 1983), p. 433.38.

This fact is easily proved by noting that any four-sided Euclidean figure can be constructed from two triangles.

39.

See, for example, Ref. 4, p. 552.

40.

One exception is given in Ref. 41 in which the result is derived in the lab frame.

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G.

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D. J. Griffiths,

*Introduction to Quantum Mechanics*(Prentice–Hall, New York, 1994), pp. 240–241.43.

H. Haken and H. C. Wolf,

*The Physics of Atoms and Quanta*(Springer-Verlag, New York, 2000), p. 193.44.

There are actually two such matrices associated with any Möbius transformation, because the matrices $A$ and $\u2212A$ correspond to the same transformation.

45.

John G. Ratcliffe,

*Foundations of Hyperbolic Manifolds*(Springer-Verlag, New York, 1994), p. 129.46.

This is the exercise from Ref. 21 discussed in Sec. I.

47.

Note that because of the non-commutativity of quaternions, the formula for this Möbius transformation must be expressed with the “denominator” on the right.

48.

For a more algebraic approach to the spinor map, see G. L. Naber,

*The Geometry of Minkowski Spacetime*(Springer-Verlag, New York, 1992), or M. Carmeli and S. Malin,*An Introduction to the Theory of Spinors*(World Scientific, Singapore, 2000).
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