### Book review

*Beyond Einstein's Velocity Addition Law*. By Abraham A. Ungar. Fundamental Theories of Physics 117. Kluwer Academic, Dordrecht, The Netherlands, 2001, xlii + 413 pp., $138.00 (hardcover).

*Foundations of Physics* 32:327-330 (2002).

### Scott Walter

By most standards, the basic four-dimensional spacetime formalism
proposed by Hermann Minkowski in 1908 for use in special relativity
has performed passably well. Spacetime diagrams in particular are
ubiquitous to teaching and research alike. However, from the inception
of the spacetime approach, and afterwards at an interval of roughly
twenty years, rival formalisms have arisen with the intent either to
complement the dominant technique or to supplant it entirely. The most
recent entry in the lists is Abraham Ungar's gyrovector formalism,
which after its discovery in 1988 underwent significant development by
its author, culminating in the work under review. As with most new
formal methods, it has little to recommend its study in the way of new
physical results or insights, but may be compared with the spacetime
approach in terms of the elegance of respective proofs of selected
theorems. One might also compare Ungar's method to those of his immediate
forerunners. On both counts, the gyroformalism proves to be worthy of
physicists' attention.
When H. Minkowski began pondering the structure of the Lorentz group
in 1907, one of the first things he noticed was that geometrical
relations between velocity vectors measured in inertial frames of
reference are not Euclidean (as in classical kinematics), but
hyperbolic. The spacetime formalism he went on to develop for the
physics of relativity, however, did not exploit this insight, rather,
its reliance on an imaginary temporal coordinate tended to obscure the
non-Euclidean nature of the Minkowski metric. Nonetheless, his
system, developed by A. Sommerfeld and others into a full-blown
vectorial analysis, and doted with a visually intuitive model in the
form of a spacetime diagram, rose rapidly to dominate theoretical work
in relativity. Mathematicians like V. Varicak and E. Borel then saw
that by employing a *real* temporal coordinate, they could exploit
hyperbolic trigonometry, and went on inaugurate a new, non-Euclidean
style of relativity. This alternative style was largely neglected by
contemporary physicists, who also ignored Borel's subsequent discovery
of a kinematic phenomenon later known as Thomas precession. Even so,
the non-Euclidean style found steady employment in relativity
textbooks, where it was used to present velocity composition.
Over the years, there have been a handful of attempts to promote the
non-Euclidean style for use in problem solving in relativity and
electrodynamics, the failure of which to attract any substantial
following, compounded by the absence of any positive results must give
pause to anyone considering a similar undertaking. Until recently, no
one was in a position to offer an improvement on the tools available
since 1912. In his new book, Ungar furnishes the crucial missing
element from the panoply of the non-Euclidean style: an elegant
nonassociative algebraic formalism that fully exploits the
structure of Einstein's law of velocity composition. The formalism
relies on what the author calls the "missing link" between
Einstein's velocity addition formula and ordinary vector addition:
Thomas precession, or the angular difference in relative velocity that
results when one changes the order of frame velocities in the velocity
addition formula, the magnitude of this relative velocity being
invariant with respect to frame order.
Ungar lays out for the reader a sort of vector algebra in hyperbolic
space, based on the notion of a gyrovector. A gyrovector space differs
in general from a vector space in virtue of inclusion of Thomas
precession, and exclusion of the vector distributive law. As a result,
when expressed in terms of gyrovectors, Einstein's (non-commutative)
velocity addition law becomes "gyrocommutative", in that the two
possible expressions for relative velocity for two inertial frames in
non-parallel motion are related by a Thomas precession. One advantage
of this approach is that hyperbolic geometry segues into Euclidean
geometry, with notions such as group, vector, and line passing over to
their hyperbolic gyro-counterparts (gyrogroup, etc.). Ungar's book is
devoted to the presentation of the theory of gyrogroups and gyrovector
spaces derived from the formal wedding of velocity addition to Thomas
precession.
One might suppose that there is a price to pay in mathematical
regularity when replacing ordinary vector addition with Einstein's
addition, but Ungar shows that the latter supports gyrocommutative and
gyroassociative binary operations, in full analogy to the former.
Likewise, some gyrocommutative and gyroassociative binary operations
support scalar multiplication, giving rise to gyrovector spaces, which
provide the setting for various models of hyperbolic geometry, just as
vector spaces form the setting for the common model of Euclidean
geometry. In particular, Einstein gyrovector spaces provide the
setting for the Beltrami ball model of hyperbolic geometry, while
Möbius gyrovector spaces provide the setting for the Poincaré ball
model of hyperbolic geometry.
The book begins with a potted history of Thomas precession and its
abstract counterpart, Thomas "gyration" (thus explaining the
plethora of "gyro" prefixes). This chapter will be of most interest
to physicists, as it presents the basic method of gyrovectors and
their motivation in a nutshell. Subsequent chapters are more formal,
laying out in definition-lemma-theorem style the theory of gyrogroups
and gyrovector spaces, with reference to the Beltrami and Poincaré
models of hyperbolic geometry, including a nice discussion of common
and hyperbolic parallel transport, studied with the methods of
nonassociative algebra rather than those of differential geometry. In
the book's final chapters, one finds a parametrization of the abstract
Lorentz boost by gyrovectors, and two further parametrizations of
nonlinear, pseudo-Lorentz transformations.
A trip "Beyond the Einstein Addition Law" will require a grasp of the
basics of group theory, as well as access to a computer algebra
program. For the perplexed, the author points to a wealth of references
on both elementary and advanced concepts. Further assistance is
provided by over eighty figures, most of which employ either Beltrami
or Poincaré discs. The figures do yeoman service in aiding the reader
to follow the geometric arguments, the assimilation of which may be
tested by working out the exercises that close each chapter.
Unfortunately, the text suffers from inadequate editing, with
unnecessary repetition between chapters. A self-congratulatory tone,
and an irrelevant recapitulation of an ongoing priority dispute are
also to be regretted. Perhaps in relation to the latter, the author
refers again and again to his first publication on K-loops, when he
could have saved space and time by simply including the paper in an
appendix.
Minor flaws in presentation aside, what can be said of the
epistemic value of Ungar's formalism? There are three areas
susceptible to application of the gyrogroup method: physics,
non-Euclidean geometry and abstract algebra. All three areas appear
ready to benefit from the method, which simplifies certain
calculations and exploits Euclidean space intuitions. But will these
advantages bring about new insights, and new results? For abstract
algebra and non-Euclidean geometry, the author summarizes several
recent results, for example, a new model of hyperbolic geometry (the
"Ungar model"), and the discovery of cohyperbolic geometry (in which
the angle sum of a triangle is
$\pi $, but triangle medians are not
concurrent).
For physics, however, one still wonders if a compact, Euclidean vector
notation accompanied by Beltrami or Poincaré models will ever
outperform tensor calculus aligned with spacetime diagrams. To argue
the case for gyrogroup methods, Ungar revives the age-old question of
the reality of the Lorentz-FitzGerald contraction (LFC). He applies
his gyrogroup-theoretic method to the problem of the apparent form of
a sphere in uniform motion, and finds that it is measured as a
flattened ellipsoid by an observer at rest. According to Ungar, this
result vindicates Einstein's stance on the appearance of a sphere in
uniform motion against the well-known argument of R. Penrose (1959),
which holds that such flattening is not photographically observable.
Penrose, however, considers light propagating with finite velocity
from the sphere to an ideal photographic plate at rest, where Ungar
assumes infinite signal velocity, a difference which clearly moots the
comparison. For the case considered by the author, a more fruitful
analogy might be drawn with the three-dimensional, mental "snapshots"
described by J. Synge in his 1955 textbook on special relativity.
Given the form of an object in motion as judged by a comoving inertial
observer, Synge's method determines the object form as judged by an
observer at rest, via iterative Lorentz transformation of surface
points, where Ungar's algebra recovers the entire form in one fell
swoop.
Although apodictic proof of the gyrogroup method's creative power of
discovery in physics is still forthcoming, physicists will appreciate
the new geometric view of Einstein's velocity addition law and Thomas
precession. Announced as a likely companion text to both undergraduate
and graduate courses in physics, non-Euclidean geometry and abstract
algebra, the book stands to be of most profit to advanced students and
researchers. It will be a useful addition to all research libraries.

Scott WALTER

LPHS-Archives Poincaré (UMR 7117)

Université Nancy 2

23 bd Albert 1er

54015 Nancy Cedex

France