Smooth loops and Thomas precession

The goal of the paper is to exhibit the theory of smooth loops, which the second author has developed since 1972, by employing it to describe the Thomas precession. The Thomas precession is a relativistic effect, which until recently rested in undeserved obscurity for too long. Presently, it plays an important role in nonassociative algebra. It is therefore appropriate to place the paper under review in the context of increasing interest in the abstract Thomas precession, called Thomas gyration. par Einstein's velocity addition is non-associative and, hence, cannot be a group operation. Luckily, however, it is accompanied by the Thomas precession which, "coincidentally" repairs the breakdown of (i) its associativity and (ii) its commutativity [A. A. Ungar, Amer. J. Phys. {bf 59} (1991), no.~9, 824--834; [msn] MR1126776 (92g:83003) [/msn]]. The Thomas precession is a relativistic rotation which is sensitive to the non-Euclidean nature of spacetime and, hence, has attracted NASA's interest to measuring it in the motion of gyroscopes of unprecedented accuracy in Earth orbit (NASA-Stanford GP-B Program). par Following its abstraction by the reviewer [Appl. Math. Lett. {bf 1} (1988), no.~4, 403--405; [msn] MR0975817 (90c:83003) [/msn]], the Thomas precession has attracted loop theorists as well. The reviewer's 1988 discovery [Found. Phys. Lett. {bf 1} (1988), no.~1, 57--89 962332 ] that the Thomas precession gives rise to a grouplike structure underlying the Einstein addition has become a classic among loop theorists. par A loop is a groupoid, that is, a non-empty set with a binary operation (G,oplus), with an identity element, in which each of the two equations aoplus x=b and xoplus a=b possesses a unique solution for its unknown x. Einstein's velocity addition, accompanied by Thomas' precession, became the first known concrete example of a non-associative loop operation which is both gyrocommutative and gyroassociative [A. A. Ungar, Results Math. {bf 17} (1990), no.~1-2, 149--168; [msn] MR1039282 (91d:83006) [/msn]; Found. Phys. {bf 27} (1997), no.~6, 881--951; [msn] MR1477047 (98k:83002) [/msn]]. Modeled on Einstein's addition with its Thomas precession, this loop became known as a gyrogroup. Like groups, gyrogroups are classified into gyrocommutative and non-gyrocommutative gyrogroups. Gyrocommutative gyrogroups are also known in the literature as K-loops and as Bruck loops. The second author has shown in previous publications [L. V. Sabinin, L. L. Sabinina and L. V. Sbitneva, Aequationes Math. {bf 56} (1998), no.~1-2, 11--17; [msn] MR1628291 (99i:83004) [/msn]; L. V. Sabinin, Uspekhi Mat. Nauk {bf 50} (1995), no.~5(305), 251--252 1365062 ] that a gyrocommutative gyrogroup is a left Bol loop with left Bruck identity. It is not clear why the original term "gyroassociativity", which was coined in [C. I. Mocanu, Found. Phys. Lett. {bf 5} (1992), no.~5, 443--456; [msn] MR1182667 (93g:83003) [/msn]] in order to emphasize the role that the Thomas precession-gyration plays in "repairing" the breakdown of associativity, became "quasi-associativity" in the paper under review. par Following the increased interest of non-group gyrocommutative gyrogroups in loop theory, (i) H. Karzel described the Thomas precession in terms of K-loops [{it Raum-Zeit-Welt und hyperbolische Geometrie}, Tech. Univ. München, Munich, 1994; [msn] MR1323631 (96b:51029) [/msn]], K-loop being a term coined in [A. A. Ungar, Results Math. {bf 16} (1989), no.~1-2, 168--179; [msn] MR1020224 (90j:83010) [/msn]] and named after Karzel in recognition of his pioneering work on algebraic structures with no available concrete example until the 1988 discovery of the gyrocommutative gyrogroup; and similarly (ii) Nesterov and Sabinin describe, in the paper under review, the Thomas precession in terms of Sabinin's theory of smooth loops. par The fact that Sabinin's smooth-loops theory (i) is well suited for the study of the Thomas precession and (ii) contains results that the reviewer discovered independently, but later, in the totally different context of relativity physics, demonstrates the vitality and usefulness of Sabinin's smooth-loops theory