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.

1.
L. H.
Thomas
, “
Motion of the spinning electron
,”
Nature (London)
117
,
514
(
1926
);
L. H.
Thomas
, “
The Kinematics of an electron with an axis
,”
Philos. Mag.
3
,
1
23
(
1927
).
2.
E. P.
Wigner
, “
On unitary representations of the inhomogeneous Lorentz group
,”
Ann. Math.
40
,
149
204
(
1939
).
3.
G. P.
Fisher
, “
The Thomas precession
,”
Am. J. Phys.
40
,
1772
1781
(
1972
). This article derives the Thomas–Wigner rotation and Thomas precession in several different ways and gives an excellent summary of previous treatments.
4.
J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998), 3rd ed., pp. 548–553, 563–564, 571.
5.
R. D. Sard, Relativistic Mechanics (Benjamin, New York, 1970), Chap. 5.
6.
H. Arzeliès, Relativistic Kinematics (Pergamon, New York, 1966), pp. 173–180, 198, 201–203.
7.
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison–Wesley, New York, 2002), 3rd ed., pp. 282–285.
8.
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,
R. A.
Muller
, “
Thomas precession: Where is the torque?
,”
Am. J. Phys.
60
,
313
317
(
1992
) and
H.
Kroemer
, “
The Thomas precession factor in spin-orbit interaction
,”
Am. J. Phys.
72
,
51
52
(
2004
).
9.
S. Gasiorowicz, Quantum Physics (Wiley, New York, 1996), 2nd ed., p. 282.
10.
R. L. Liboff, Introductory Quantum Mechanics (Addison–Wesley, San Francisco, 2003), 4th ed., p. 586.
11.
R. Shankar, Principles of Quantum Physics (Plenum, New York, 1981), p. 477.
12.
J. J. Sakurai and S. F. Tuan, Modern Quantum Mechanics (Addison–Wesley–Longman, New York, 1994), p. 305.
13.
One exception, which gives an elementary and clear example of the Thomas–Wigner rotation, is the recent paper by
J. P.
Costella
,
B. H. J.
McKellar
, and
A. A.
Rawlinson
, “
The Thomas rotation
,”
Am. J. Phys.
69
,
837
847
(
2001
).
14.
A. A.
Ungar
, “
Thomas precession and its associated grouplike structure
,”
Am. J. Phys.
59
,
824
834
(
1991
).
15.
A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovectors (Kluwer, Dordrecht, 2001).
16.
A.
Ben-Menahem
, “
Wigner’s rotation revisited
,”
Am. J. Phys.
53
,
62
66
(
1985
).
17.
H.
Urbantke
, “
Physical holonomy, Thomas precession, and Clifford algebra
,”
Am. J. Phys.
58
,
747
750
(
1990
).
18.
G. H.
Goedecke
, “
Geometry of the Thomas precession
,”
Am. J. Phys.
46
,
1055
1056
(
1978
).
19.
J. D.
Hamilton
, “
Relativistic precession
,”
Am. J. Phys.
64
,
1197
11201
(
1996
).
20.
E. G. P.
Rowe
, “
Rest frames for a point particle in special relativity
,”
Am. J. Phys.
64
,
1184
1196
(
1996
).
21.
L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, New York, 1975), p. 36. The first edition of this book was published by Addison–Wesley in 1951.
22.
W. Pauli, Theory of Relativity (Dover, New York, 1981), pp. 73–74.
23.
B. A. Rosenfeld, History of Non-Euclidean Geometry (Springer-Verlag, New York, 1987), pp. 270–273.
24.
P. K.
Aravind
, “
The Wigner angle as an anholonomy in rapidity space
,”
Am. J. Phys.
65
,
634
636
(
1997
).
25.
C.
Criado
and
N.
Alamo
, “
A link between the bounds on relativistic velocities and areas of hyperbolic triangles
,”
Am. J. Phys.
69
,
306
310
(
2000
).
26.
W. Rindler, Relativity: Special, General and Cosmological (Oxford U.P., Oxford, 2001), pp. 43–44, 46.
27.
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.
30.
L.
Parker
and
G. M.
Schmieg
, “
Special relativity and diagonal transformations
,”
Am. J. Phys.
38
,
218
222
(
1970
).
31.
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=±1 and g=±r. 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.
41.
G.
Muñoz
, “
Spin-orbit interaction and the Thomas precession: A Comment on the Lab Point of View
,”
Am. J. Phys.
69
,
554
556
(
2001
).
42.
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 −A 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).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.