Additivity, rapidity, relativity Jean-Marc L´evy-Leblonda Physique Th´eorique, Universit´eParis VII, France Jean-Pierre Provostb Physique Th´eorique, Universit´ede Nice, France (Received 22 May 1978 ; accepted 8 August 1979) A simple and deep standard mathematical theorem asserts the existence, for any one-parameter differentiable group, of an additive parameter, such as the angle for rotations and the rapidity parameter for Lorentz transformations. The importance of this theorem for the applications of group theory in physics is stressed, and an elementary proof is given. The theorem then is applied to the construction from first principles of possible relativity groups. INTRODUCTION of position, and “rapidity” ϕ, as the natural (ad- ditive) group parameter1. Emphasizing the dis- Special relativity usually is introduced tinction helps in eliminating difficulties. For ins- through the Lorentz transformation formulas tance, students often ask whether the existence ′ x vt x = − of a limit velocity c (for material particles) does √1 v2 not imply the existence of a “limit reference fra- − ′ t vx me”, which would contradict the relativity prin- t = − , √1 v2 ciple. They also wonder how the consideration − expressed in terms of the relative velocity v of of velocities greater than c (for “unmaterial” the two reference frames (our units are such that objects, such as shadows—or for tachyons) can c = 1). These formulas, it must be admitted, be consistent with Einsteinian relativity. These are not very elegant and often seem complica- pseudoparadoxes disappear when it is realized ted to the student. They imply new and strange that the intrinsic parameter of the group law, properties for the velocity : existence of a limit expressing the transformation from one reference velocity c, and the curious composition law : frame to another, is rapidity, which has no limit and may increase indefinitely, while, conversely, v12 =(v1 + v2)/(1 + v1v2). (.1) there is no rapidity, i.e., no change of reference 2 At a later stage, the student, in modern courses frame, associated with superluminal velocities . at least, is introduced to the rapidity parameter Beyond its pedagogical utility, rapidity in the ϕ defined by recent years has become a common and efficient tool in particle physics. v = tanh ϕ (.2) It seems worthwhile, for these reasons, to in- in terms of which the Lorentz formulas take the troduce rapidity as soon as possible in the tea- simple form ching of relativity, namely, at the start. There x′ = cosh ϕx sinh ϕt is no need to go through the expressions of Lo- − rentz transformations using velocity, and then to t′ = cosh ϕt sinh ϕx, ( − “discover” the elegant properties of rapidity as and the composition law becomes if they resulted from some happy and unpredic- table circumstance. Indeed, there exists a deep, ϕ12 = ϕ1 + ϕ2. if simple, mathematical theorem establishing in The additivity of rapidity exhibits at once the advance the necessity of such a parameter. The fundamental group property of Lorentz transfor- theorem asserts the existence of an additive pa- mations. rameter for any one-parameter group (differen- Rapidity is not merely a convenient parame- tiable and connected). Unfortunately, this theo- ter. Its introduction in the teaching of Einstei- rem, despite its importance and simplicity, is nian relativity stresses the need to replace the Galilean velocity by two separate concepts : “ve- locity” v, as expressing the time rate of change 1 usually introduced in connection with the gene- 3) existence of an inversex ¯ for each element x : ral study of Lie groups and their algebras, requi- x x¯ =x ¯ x = e. ring rather elaborate mathematics, unfit to the ∗ ∗ needs of elementary physics. It is the purpose of We now suppose that the composition law has the present paper to give a simple proof of the some smoothness property. More precisely, we theorem, to illustrate it by elementary examples, ask that the function fx of left multiplication by and to use it as a basis for the derivation of relati- x, vity theory. Besides this specific use, the present df considerations may also be considered as an in- fx(y) = x y, (I.1) ∗ troduction to some important ideas of the group possesses a first-order derivative, and we require theory which play an essential role in contempo- it to differ from zero at the neutral element rary theoretical physics. ′ The paper is organized as follows. In Sec. I, the f (e) =0. (I.2) x 6 theorem is stated and proved. Section II is devo- Writing the associativity property as ted to some examples. Section III uses the theo- rem to present a derivation of relativity theory, fx∗y(z)= fx[fy(z)] for one space dimension, from first principles, al- and differentiating it with respect to z at the ternative to a recent proposal. The new deriva- point z = e, we get, taking into account f (e)= tion allows for a straightforward generalization y y : to three-dimensional space, which is presented ′ ′ ′ in Sec. IV. fx∗y(e)= fx(y)fy(e). (I.3) The above requirement (I.2) therefore implies I ADDITIVE PARAMETER THEOREM that f ′ (y) = 0 for all y S, so that we have x 6 ∈ f (y1) = f (y2) and x y1 = x y2 whenever We consider a “one-parameter group”, i.e., a x 6 x ∗ 6 ∗ y1 = y2. For this reason we call a parametriza- group G, the elements of which are labeled by 6 real numbers. In more rigorous terms, the group tion with the above smoothness properties (I.1)- G is supposed to be in one-to-one correspon- (I.2) an “essential” parametrization. dence with some connected subset S of R, which Of course the labeling of the group is arbi- means that the set of parameters S is just one trary. If we consider the Lorentz group, for ins- single piece of the real line, finite or not. For tance, we can equally well characterize a given Lorentz transformation by its velocity v, or its instance, S =] , + [, if G is the translation 2 −1/2 group with the−∞ length∞ of the translation as a pa- rapidity ϕ or its “momentum” v(1 v ) = natural− rameter, and S =] c, +c[ if G is the Lorentz sinh ϕ. Whether there is some parame- group with the velocity− as a parameter. We de- trization for the abstract group G, precisely is note by g(x), x S, the generic element of G. the question we consider here. A general change ∈ of parametrization may be defined by some dif- The group structure requires the existence of a −1 composition law : the product g(x)g(y) of two ferentiable function λ such that λ exists as elements of G is an element of G labeled by a well. The group elements are now labeled by parameter which is a function of x and y and ξ = λ(x), η = λ(y), etc., and the composition will be denoted by x y. law is given by ∗ −1 −1 ξoη = λ[λ (ξ) λ (η)]. g(x)g(y)= g(x y) ∗ ∗ The additive parameter theorem, which we now We may as well identify each element of G with prove, mainly asserts that it is always possible its parameter x, considering the group as the set to choose a suitable λ such that the o law simply S endowed with the composition law . This law ∗ is the ordinary addition of real numbers ; that is, of course satisfies the group axioms, that is a function λ such that 1) associativity : λ(x y)= λ(x)+ λ(y). (I.4) (x y) z = x (y z), ∗ ∗ ∗ ∗ ∗ Differentiating (I.4) with respect to y, we obtain 2) existence of a neutral element (identity) e : the following equation for λ : e x = x e = x, λ′(x y).f ′ (y)= λ′(y), ∗ ∗ ∗ x American J. of Physics, Vol. 47, No. 12, December 1979 2 2 and remark that the identity (I.3) provides us at group. Since, setwise, Ê has the same cardina- once with a solution for this equation, namely : lity as Ê, we may always replace the two real pa- ′ ′ −1 rameters of the group by a single one using some λ (x)= f (e) . x bijection (such as the old trick of combining two The function λ defined by real numbers in decimal form by writing a single x number the odd and even digits of which are ′ −1 λ(x)= [ft (e)] dt, (I.5) those of the two numbers). The group now is a Ze “one-parameter” group (without any physical si- therefore obeys the additive property (I.4). The gnificance, of course). However, since no one-to- 2 Ê lower bound e in the integral is dictated, of one mapping Ê can be a measurable one, course, by the requirement that λ(e) = 0, an the conditions of↔ the theorem are not fulfilled immediate consequence of the addition law. and there is no overall additive parameter—and Is this parametrization unique ? Let us denote fortunately so, since the group is not abelian ! by ξ, η, etc. the additive parameters. Suppose that there exists another additive parametriza- II SOME EXAMPLES ′ ′ tion ξ , η , etc., given in terms of the first one by × ′ A. Consider the multiplicative group Ê of some function µ : ξ = µ(ξ), it must obey (positive) real numbers. We have here µ(ξ + η)= µ(ξ)+ µ(η). ′ fx(y)= xy, e =1, ft (e)= t. It results that µ is a linear function The additive parameter, the existence of which µ(ξ)= kξ, is guaranteed by the theorem, is explicitly given by formula (I.5): i.e., the additive parametrization is unique up x to a multiplicative constante.