Foundations of Physics, Vol. 19, No. 11, 1989

The Relativistic Velocity Composition Paradox and the Thomas Rotation

Abraham A. Ungar 1

Received February 10, 1989," revised June 15, 1989

The relativistie velocity composition paradox o f Moeanu and its resolution are presented. The paradox, which rests on the bizarre and counterintuitive non- commutativity of the relativistic velocity composition operation, when applied to noncollinear admissible velocities, led Mocanu to claim that there are "some difficulties within the framework o f relativistic electrodynamics. "' The paradox is resolved in this article by means of the Thomas rotation, shedding light on the role played by composite velocities in special relativity, as opposed to the role they play in Galilean relativity.

1. I N T R O D U C T I O N

Following Feynman, a "paradox is a situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what would happen. Of course, in physics there are never any real paradoxes because there is only one correct answer." In physics, thus, "a paradox is only a confusion in our understanding. ''(2)

Special theory of relativity (STR) is a rich source of paradoxes in physics, several of which were listed by Sastry. (3) Attention to a paradox which arises from the noncommutativity of the relativistic composition of noncollinear velocities, not listed by Sastry, was recently drawn by Mocanu. (1) Having no known resolution, Mocanu regards the paradox as a source of several "difficulties within the framework of relativistic electrodynamics, ''(1) prompting him to elaborate "a new electrodynamics of moving bodies at relativistic velocities, different from that of Einstein and

1 Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105.

1385

0015-9018/89/I100-1385506.00/0 © 1989 Plenum Publishing Corporation

1386 Ungar

Minkowski," in which the difficulties are eliminated, resulting in a "Hertzian alternative to the special theory of relativity. ''(4)

The aim of this article is to present the Mocanu paradox and its resolution. The physical phenomenon which saves STR from the difficulties presented by the Mocanu paradox turns out to be the Thomas rotation, (5) which is in itself a peculiarity of STR, (6) so far badly neglected. ~7-22) The Thomas rotation was accidentally discovered by Thomas in 1926 as a means to reconcile a conflict in the spinning electron of the Goudsmit- Uhlenbeck model that gave twice the observed precession effect. (23) As recollected by Uhlenbeck, "it seemed unbelievable that a relativistic effect (that is, the Thomas rotation) could give a factor of 2 (called the Thomas 1/2) (24) instead of something of order v/c. ''~6) Some of the difficulties in STR that led Mocanu to propose his Hertzian theory of relativity (HTR) stem from the presence of the Thomas rotation. Thus, in order to eliminate the difficulties caused by the presence of the Thomas rotation in STR, Mocanu presented an alternative, HTR, in which there is a privileged reference frame and the Lorentz transformation group is degraded into a Hertz transformation semigroup. (4) It is therefore hoped that the present resolution of the Mocanu paradox will draw wide attention to the central role that the Thomas rotation plays in STR.

2. THE MOCANU PARADOX

Let u and v be two admissible velocities, that is, u, v ~ ~3 c where R3c is the set of all 3-vectors in the Euclidean 3-space ~3 with magnitude smaller than the speed of light c in empty space,

= { v LvI < c }

The velocity composition u * v of u and v is given by the equation ~25)

u + v 1 7 . u x ( u x v ) u * v l + u . v , c2-~/ c27~+1 l + u . v / c 2 , u , v ~ 3 c (1)

where 7, is the Lorentz factor,

1 1

7" - , / 1 - ( u l c ) 2 = x / 1 - (ulc) (2)

associated with the velocity u whose magnitude is u, u = lu], and where - and x signify the usual scalar and vector product between two vectors. The composite velocity u . v is asymmetric in u and v. Its magnitude is,

Relativistic Velocity Composition Paradox 1387

however, symmetric in u and v, the square of which is given by the equation

( u+v )2_1( )2 (u,v)Z=\a+u.v/c2j j -l+u.v/e2 / (3)

When the velocities u a n d v are collinear, u x v = 0, E q . ( 1 ) reduces to Einstein's addition theorem for (parallel) velocities, (26)

u + v u * v = 1 +: u -v/"c 2' u [1 v, u, v E ~ (4 )

which is both commutative and associative. As a peculiarity of STR, however, the general velocity composition law (1) is neither commutative nor associative. When c -+ c~ the set N~ deforms into N3 and the relativistic velocity composition law u , v (u, v e N3) continuously deforms into the Galilean velocity composition law u + v (u, v eff~ 3) which is both com- mutative and associative.

Let Z, Z', and 22" be three inertial frames. The velocity of frame ,!,,, relative to frame Z' is v while the velocity of frame Z' relative to frame Z is u, as depicted in Fig. 1. The velocity of frame Z" relative to frame 22 is

Y

y,,

y ~

X

Fig. 1. Frame Z " moves with velocity v relative to frame Z ' while frame L" moves with velocity u relative to frame S. Time and one space dimension are suppressed for clarity.

1388 Ungar

the composite velocity u . v given by Eq. (1). Hence, by the velocity reciprocity principle, the velocity of frame 22 relative to frame 22" is

- ( u , v ) (5)

The velocity reciprocity principle in STR, used in the derivation of (5), was postulated by Einstein. He postulated that the relation between the coor- dinates of the two systems is linear, (27) so that the inverse transformation is also linear and the complete nonpreference of the one or the other system, then, demands that the inverse transformation shall be identical with the original one, except for a change of v to - v " . (z6)

Let us now derive the velocity of frame Z relative to frame Z" in another way. The velocity of frame Z relative to frame 22' is - u while the velocity of frame Z' relative to frame Z" is - v. Hence, the velocity of frame 2;" relative to frame 22" is

( - v ) • ( - u ) = - (v • u) (6)

By (5), the velocity of Z" relative to Z is u , v while, by (6), the velocity of Z" relative to Z is v • u. However, if u and v are noncollinear, then

u * v ~ v * u (7)

thus encountering the Mocanu paradox: Which one is the "correct" velocity of X" relative to Z? Is it u * v or v • u? The velocity of 22" relative to Z is an observable effect that appears contradictory. At first sight, the asymmetry in velocity composition does ho t , appear to be inherent in observable relationships between inertial frames.

In his words, the Mocanu paradox rests on the impossibility of deriving, in STR, "a unique and coherent transformation law of non- collinear velocities. ''(t) Following a presentation of the Thomas rotation and some of its properties in the next section, the Mocanu paradox will be resolved in Sec. 4.

3. T H E T H O M A S R O T A T I O N

Let Z, Z', and Z" be three inertial frames. The velocity of frame Z" relative to frame Z' is v while the velocity of frame Z' relative to frame 22 is u, as shown in Figs. 1 and 2a. The axes of both frames X and Z" have been constructed parallel to those of X' as seen by observers accompanying the moving system 22'. Nevertheless, an observer in X sees the axes of 22"

Relativistic Velocity Composition Paradox 1389

rotated relative to his own axes by a Thomas rotation angle, e, shown in Fig. 2b. This Thomas rotation is said to be generated by the velocities u and v, and is denoted by tom[u; v].

Figure 2 shows a Thomas rotation, t o m [ u ; v ] , generated by two velocities, u and v. It is a rotation about an axis parallel to the non- vanishing vector u x v through an angle E. If the rotation angle from u to v is denoted by 0 (Fig. 2), then the Thomas rotation angle e generated by u and v is given by the equations (283°)

(k + cos 0) 2 - sin 2 0 COS ~ =

(k + cos 0) 2 + sin 2 0 (8a)

- 2 ( k + cos O) sin 0 sin e = (k + cos 0) 2 + sin 2 0

where

k 2 - 7 " + 1 7 ~ + 1 k > l (8b) y , , - 1 ~ - 1'

It can be shown that lel < 101 and that e and 0 have opposite signs in the interval [ -~ r , rc]. (28) As c ~ o% we regain the Galilean relativity where velocity compositions, u + v, involve no Thomas rotations:

lira e = 0 C --~ o 9

Y Y

y .

X' Z'

~ . _ ,, ......... ,,,,,,, ,,,,

(a)

,,,,, X "

X / X

, , , , i ,,, J X

(b) Fig. 2. (a) Frame 22" moves with velocity v relative to frame 22' while flame .X' moves with velocity u relative to frame 22. As observed by an observer on Z' , the axes of frame Z'" are parallel to those of frame Z", and the axes of frame _r' are parallel to those of frame Z'. (b) By the Mocanu paradox, the velocity of Z'" relative to X is represented by two distinct expres- sions, u • v and v • u, which are only seemingly inconsistent with each other (see Sec. 4 for the resolution of the paradox); and the axes of Z" are rotated relative to S by the Thomas rotation angle e of Eqs. (8). Time and one space dimension are suppressed for clarity.

1390 Ungar

When [ul, jv[ ~ c, the Thomas rotation angle e generated by u, v e ~3 c approaches the negative rotation angle 0 from u to v:

lim e = - 0 lul,lv[ ~c

An orthogonal matrix representation, relative to the frame 22 of Fig. 2, of the Thomas rotation is given by the equation (28)

f I + sin tom[u; v] = ~I,

O(u, v) O2(u, v) lul Ivl sin0 + ( 1 - c ° s Q luxvl -------T' u × v # O

U X V = 0

where I is the 3 x 3 identity matrix and ~ is the skew symmetric matrix

(9)

0 --(03 (2) 0 ) ~(U, V) ~- 0) 3 0 -- 1 (10a)

--602 0)1

representing the linear transformation of cross product with to, that is, f2r = to x r for a 3-vector r. The entries 0)k, 1 <~ k <~ 3, of the matrix 12 are the components of the vector product to = u x v measured in the frame 2; of Fig. 2,

O,)=((DI,fD2,0)3)=IIXV=(U2V3--U31)2, U3Vl--UlV3, UlV2--U2Vl) (lOb)

A graphical presentation of cos ~ and sin e as functions of 0 is presented in Ref. 28 for several values of k.

The resultant velocities u , v and v , u have equal magnitudes [Eq. (3)]. Hence, one of them can be transformed into the other by a rotation in the plane of u and v. This rotation turns out to be the Thomas rotation generated by the velocity parameters u and v,

u* v = t o m [ u ; v ] ( v * u) (11)

indicating that the Thomas rotation inverse to tom[u; v] is tom[v; u]. The Thomas rotation, hence, gives rise to a weak commutative law [Eq. (11)] .for the velocity composition operator ,. The Thomas rotation, moreover, gives rise to a right and a left weak associative law, thus establishing a weakly associative group structure for the set ~ of the relativistically admissible velocities, with the group operation given by the relativistic velocity composition: (28 30) For all u, v, w ~ ~3 c we have

Relativistic Velocity Composition Paradox 1391

(i) (ii)

(iiia) (iiib)

(iv) (v)

u * v e ~ u *v = tomEu; v](v • u)

u , (v • w) = (u * v) * tomEu; v] w (u * v) • w = u * (v * tomEv; u] w)

0 * U = U * 0 = U

( - u ) , u = u , ( - u ) = 0

Closure

Weak commutative law

Right weak associative law

Left weak associative law

Existence of identity

Existence of inverse

In general, a group is associative by definition. The term weakly associative group for the pair ([~, , ) is, however, justified in view of the properties listed in (i)-(v) enabling one, for instance, to solve the equations a • x = b and x • a = b for the unknown x(a, b, x e [~3) (29-31)

4. RESOLVING THE MOCANU PARADOX

Having the Thomas rotation in hand, we are now in a position to resolve the Mocanu paradox. Let (t, x, y, z)' (the exponent t indicates transposition) and (t', x', y', z ') ~ be the time-space coordinates of an event, measured respectively in the inertial frame S and in the inertial frame 2,". If the velocity of Z" relative to X is v, then the two time-space coordinates of the event are connected by a rotation-free Lorentz transformation B(v),

=B(v) y,

Z ~

(12)

Rotation-free Lorentz transformations are called, in the jargon, boosts. The boost B(v) of Eq. (12) is parametrized by a velocity parameter v and, relative representation o2~

7~vl

B(v) = T.V2

~7~Vs

to a frame in which v = ( v l , v2, v3), it has the matrix

C-2yvVl C- 27vV2

2 2 1 + c-2 ? f ; 1 v~ c-2 ?v+lT---"~ vl v2

c_2 1 + c 2 'iLv 7~+ 1 G + 1

C_ 2 ~d2v i, i y 3 C 2 )12v Y2Y3 9%+1 y ~ + l

c 2~vv3 \

c 2 ~2 VlV3

7~+I

c-2 71 v2v3 ?v+ 1

l + c -2 71 ?v + 1 vj]~ ,

(13)

1392 Ungar

In general, two successive boosts are not equivalent to a boost; the combination of two successive boosts is equivalent to a net boost preceded, or followed, by a Thomas rotation(28):

B(u) B(v) = B(u * v) Tom[u; v] (t4a)

and

B(u) B(v) =Tom[u; v] B(v * u) (14b)

Here Tom[u; v] is a 4 x 4 matrix representing space rotation of time-space coordinates by a Thomas rotation, and is given by the equation

Tom[u; v] = (10 0 ) (15) tom[u; v]

in terms of its effects on time-space coordinates. The left-hand sides of Eqs. (14) correspond to the two successive

boosts in Fig. 2a while the right-hand sides of Eqs. (14) correspond to the resultant boost preceded, or followed, by the Thomas rotation in F!g. 2b. According to Eq. (14a) the velocity of Z" relative to Z, in Fig. 2, is

u * v

(that is, the composition of the velocity u preceded by the velocity v) and the Thomas rotation Tom[u; v] must be performed, in the transition from Z" to Z, before applying the boost B(u • v) of the velocity u ,v. According to Eq. (14b), on the other hand, the velocity"of Z" relative to Z, in Fig. 2, is

V * U

(that is, the composition of the velocity u followed by the velocity v) and the Thomas rotation Tom[u; v] must be performed, in the transition from Z" to X, after applying the boost B(v • u) of the velocity v • u.

According to Eqs. (14), both composite velocities u , v and v , u describe correctly, but partially, the boost composition B(u)B(v). Both u • v and v • u are associated with the Thomas rotation tom[u; v] of the composite boost B(u)B(v). However,

(i) if the composite velocity u • v is selected to parametrize the com- posite boost B(u)B(v), then the Thomas rotation Tom[u;v], associated with the composite boost B(u)B(v), must be applied before the application of the resulting boost B(u • v) of u * v; and

Relativistic Velocity Composition Paradox 1393

(ii) if the composite velocity v • u is selected to parametrize the com- posite boost B(u)B(v), then the Thomas rotation T o m [ u ; v ] , associated with the composite boost B(u)B(v), must be applied after the application of the resulting boost B(v • u) of v • u.

The resolution of the Mocanu paradox is now clear. Unlike Galilean composite velocities u + v (u,v~N3), relativistic composite velocities u * v (u, v ~ N3) embody space rotations, tom[u; v]. When we say that the (composite) velocity of frame Z'" relative to frame Z (Fig. 2) is u , v (respectively, v * u), we mean that in the transition from the time-space coordinates of an event measured in 2"" into the time-space coordinates of the event measured in £" we have to apply a boost with velocity parameter u , v (resp., v , u) preceded (resp., followed) by the Thomas rotation tom[u; v]. Noticing that composite velocities in STR embody space rota- tions, we see that the two distinct relativistic composite velocities u • v and v * u of the velocities u and v are both correct. The two distinct composite velocities of u and v, u * v and v , u, embody a velocity operation and a space rotation in two different orders. We may interject here that since velocities and rotations are intimately connected in STR via the Thomas rotation, Lorentz transformations in 1 + 3 dimensions cannot be parametrized by velocities alone. They are, rather, parametrized by velocities and orientations in such a way that composite Lorentz transfor- mations correspong to composite velocities and composite orientations, which are interrelated by Thomas rotationsJ 28)

Paradoxes are useful in clarifying confusions. The Mocanu paradox illustrates the role composite velocities play in STR. While composite velocities in Galilean relativity completely determine relationships between inertial frames, their relativistic counterparts do not! In addition to com- posite relativistic velocities between inertial frames, one must also take into account their associated Thomas rotations: A composite velocity, u , v, preceded by its associated Thomas rotation, tom[u; v], is equivalent to the reversely ordered composite velocity, v , u, followed by the Thomas rota- tion tom[u; v]. The two distinct composite velocities u , v and v , u are mutually compatible. The asymmetry in the relativistic velocity composi- tion law is thus attributed to the presence of the Thomas rotation by means of which the Mocanu paradox is resolved.

The resolution of the Mocanu paradox by means of the Thomas rota- tion recalls to mind the group structure of relativistic velocities mentioned in Sec. 3 and elaborated in Ref. 30. Galilean velocities form a group, (N3, + ), under Galilean velocity composition. However, their relativistic counterparts do not form a group unless associated Thomas rotations are properly incorporated: With the appropriate incorporation of the Thomas

825/19/11-8

1394 Ungar

ro t a t i on , re la t iv is t ic veloci t ies do form a (weakly assoc ia t ive) g roup , (~c,3 tom, ,),(33) In te res t ing ly , the weak ly associa t ive g r o u p s t ruc tu re of (R3c, , )

is n o t a n i so la ted m a t h e m a t i c a l s t ruc tu re in special re la t iv i ty ; it was f o u n d

by Karze l a n d s tud ied by K e r b y a n d by Wefe lshe id in a to ta l ly different context.(34)

The M o c a n u p a r a d o x a n d its r e so lu t i on expose the cen t ra l role tha t

the T h o m a s r o t a t i o n p lays in STR, It is h o p e d tha t fo l lowing this exposi-

t ion no ser ious b o o k on S T R will fail in the fu ture to m e n t i o n the T h o m a s

r o t a t i o n a n d its resu l t ing weak ly associa t ive g r o u p of re lat ivis t icat ly admiss ib l e velocities.

R E F E R E N C E S A N D N O T E S

1. Constantin I. Mocanu, "Some difficulties within the framework of relativistic electro- dynamics," Arch, Elektroteeh. 69, 97-110 (1986).

2. R. P. Feynman, R. B, Leighton, and M. Sands, 7"he Feynman Lectures on Physics (Addison-Wesley, Reading, Massachusetts 1964), Vol, II, Sec. 17-4.

3. G. P. Sastry, "Is length contraction really paradoxical?" Am. J. Phys. 55, 943-945 (1987). 4. Constantin I. Mocanu, Electrodynamics of Moving Bodies at Relativistic Velocities (Publ.

House of Roum. Acad., Bucharest, 1985); "Hertzian alternative to special theory of relativity. I. Qualitative analysis of Maxwell's equations for motionless media," Hadronic J. 10, 61-74 (1987), and references therein.

5. L. H. Thomas, "The motion of the spinning electron," Nature (London) 11"/, 514 (1926); "The kinematics of an electron with an axis," Philos. Mag. 3, 1-22 (1927); see also Ref. 28.

6. George E. Uhlenbeck, "Fifty years of spin: Personal Reminiscences," Phys. Today 29, 43M8 (June, 1976).

7. Presently, most books on STR make no mention of the Thomas rotation (or precession). Several outstanding exceptions are: E.F. Taylor and J, A. Wheeler, Spacetime Physics, H.M. Foley and M.A. Ruderman, eds. (Freeman, San Francisco, 1966); M.C. M¢ller, The Theory of Relativity (Clarendon Press, Oxford, 1952); J.D. Jackson, Classical Eleetrodynamics (Wiley, New York, 1975); and H.P. Robertson and T.W. Noonan, Relativity and Cosmology (Saunders, Philadelphia, 1968). Several articles on the Thomas precession (rotation) are listed in Refs. 8-22.

8. Lanfranco Belloni and Cesare Reina, "Sommerfeld's way to the Thomas precession," Eur. J. Phys. 7, 55-61 (1986).

9. Ari Ben-Menahem, "Wigner's rotation revisited," Am. J. Phys. 53, 62~66 (1985). In this article, as well as in several others, the Thomas rotation is referred to as the Wigner rota- tion. The use of the term "Wigner rotation" for the description of the Thomas rotation apparently came into the English literature from a text by S. Gasiorowicz, Elementary Particle Physics (Wiley, New York, t967), p, 74, who copied the term from the German literature. An objection to the use of this term for the description of the Thomas rotation is based on the claim that the "correct" Wigner rotation is the Thomas rotation measured in the frame in which the accelerated particle is at rest; see Ref. 25 of Ref. 28.

10. Shahar Ben-Menahem, "The Thomas precession and velocity space curvature," J. Math. Phys. 27, 1284-1286 (1986).

11. James T. Cushing, "Vector Lorentz transformations," Am. J. Phys. 35, 858-862 (1967).

Relativistic Velocity Composition Paradox 1395

12. Sidney Dancoff and D. R. Inglis, "On the Thomas precession of accelerated axes," Phys. Rev. 50, 784 (1936).

13. S. F. Farago, "Derivation of the spin-orbit interaction," Am. J. Phys. 35, 246-249 (1967). This article containes an incorrect statement about the Thomas rotation, see Ref. 17 of Ref. 28;

14. George P. Fisher, "The electric dipole moment of a moving magnetic dipole," Am. o r. Phys. 39, 1528-I533 (I971) and "The Thomas precession," 40, 1772-t781 (1972).

15. W. H. Furry "Lorentz transformation and the Thomas precession," Am. J. Phys. 23, 517-525 (1955).

16. Roger R. Haar and Lorenzo J. Curtis, "The Thomas precession gives g ~ - 1, not g j2," Am. J. Phys. 55, 1044-1045 (1987).

17. A. C. Hirshfeld and F. Metzger, "A simple formula for combining rotations and Lorentz boosts," Am. J. Phys. 54, 550-552 (1986).

18. K. R. MacKenzie "Thomas precession and the clock paradox," Am..I . Phys. 40, 1661-1663 (1972).

19. E. G. Peter Rowe "The Thomas precession," Eur. J. Phys. 5, 40-45 (1984). 20. Nikos Salingaros "The Lorentz group and the Thomas precession. II. Exact results for the

product of two boosts," J. Math. Phys. 27, 157-162 (1986); and "Erratum: The Lorentz group and the Thomas precession. II. Exact results for the product of two boosts," J. Math. Phys. 29, 1265 (1988).

21. David Shelupsky, "Derivation of the Thomas precession formula," Am. J. Phys. 35, 650-651 (1967).

22. N. W. P. Strandberg, "Special relativity completed: The source of some 2s in the magnitude of physical phenomena," Am. J. Phys. 54, 321-331 (1986).

23. L. H. Thomas, "Recollections of the discovery of the Thomas precessional frequency," AIP Conf. Proc. No. 95, High Energy Spin Physics, G.M. Bunce, ed. (Brookhaven National Lab, 1982), pp. 4-12.

24. See, for instance, V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Quantum Electrodynamics, trans, by J. B. Sykes and J. S. Bell (Pergamon Press, New York, 1982), p. 126.

25. See, for instance, J. T. Cushing. 01) 26. Albert Einstein, "Zur Elektrodynamik Bewegter K6rper (On the Electrodynamics of

Moving Bodies)," Ann. Phys. (Leipzig) 17, 891-921 (1905). For English translation, see H. M. Schwartz "Einstein's first paper on relativity" (covers the first of the two parts of Einstein's paper), Am. J. Phys. 45, 18-25 (1977); and H.A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity, trans, by W. Perrett and G.B. Jeffery, (Dover, New York) 1952), first published in 1923.

27. Einstein stated that "it is clear that the equations must be linear owing to the homogeneity properties which we attribute to space and time. ''~26) In his Ref. 19, Schwartz (26) justifiably comments that "This is a laconic statement in need of amplification."

28. A. A. Ungar, "Thomas rotation and the parametrization of the Lorentz transformation group," Found. Phys. Lett. 1, 57-89 (1988).

29. A. A. Ungar, "The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities," Appl. Math. Lett. 1, 403~405 (1988).

30. A. A. Ungar, "The relativistic noncommutative nonassociative group of velocities and the Thomas rotation," Results Math. 16, 168-179 (1989).

31. A set F with a binary operation • such that for all a, b e F each equation a * x = b and x * a = b has exactly one solution in F and there is a neutral element 0 s F with 0 * a = a * 0 = a is known in group theory as a loop; see, for instance, R. H. Bruck, A Survey of Binary Systems, 2nd edn. (Springer, New York, 1966).

1396 Ungar

32. M. C. Moiler, The Theory of Relativity (Clarendon Press, Oxford, 1952), p. 42. 33. A. A. Ungar, "Weakly associative groups," preprint. 34. H. Wefelscheid, Proc. Edinburgh Math. Soc. 23, 9 (1980); W. Kerby and H. Wefelscheid,

"The maximal sub near-field of a near-domain," J. Algebra 28, 319-325 (1974); H. W/ihling, Theorie der Fastk6rper (Thales, W. Germany, 1987), and references therein.