Additivity, , 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 , of an additive parameter, such as the for rotations and the rapidity parameter for Lorentz transformations. The importance of this theorem for the applications of group theory in 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 , and “rapidity” ϕ, as the natural (ad- ditive) group parameter1. Emphasizing the dis- usually is introduced tinction helps in eliminating difficulties. For ins- through the formulas tance, students often ask whether the existence ′ x vt x = − of a limit 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 v of of greater than c (for “unmaterial” the two reference frames (our units are such that objects, such as shadows—or for ) 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 . 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 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 , unfit to the ∗ ∗ needs of elementary physics. It is the purpose of We now suppose that the composition law has the 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 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 . ′ 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 , 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 , 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 2 −1/2 group with the−∞ ∞ of the translation as a pa- rapidity ϕ or its “” 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 , 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. Indeed, such a λ(x)= t−1dt = log x 1 constant could be included in the expression Z (I.5) without altering its property (I.4). Of course, this is but the definition of the Na- In conclusion, what we have proven is that pierian logarithm function, the main property of under the hypothesis of essential parametriza- which precisely is to realize the isomorphism of tion, any one-dimensional connected differen- the multiplicative group of positive real numbers tiable group is isomorphic to the additive group with the additive group of real numbers. of real numbers. In particular, this result im- B. The 2 2 real symmetrical and unimodular plies that the group is abelian, which is far from matrices form× a group ; the generic element may 3 evident a priori . be written : Now there are important groups without a diffe- a b 2 2 M = , a b =1, a 6 1 rentiable essential parametrization. The b a − group in the is such a case ; we may choose   Considering b as the parameter of the group with an additive parameter, the angle, but either we 2 1/2 restrict its range to [0, 2π] and lose differentia- a =(1+ b ) , the group law may be written bility, or take the entire set of real numbers but b b′ =(1+ b2)1/2b′ + b(1 + b′2)1/2. lose uniqueness of the parametrization. Howe- ∗ One has ver, when the parametrization is not essential ′ 2 1/2 2 1/2 but if fx(e) differs from zero in a neighborhood fx(y)=(1+ x ) y + x(1 + y ) , of the identity, the isomorphism exists in such a e =0, f ′(e)=(1+ t2)1/2. neighborhood. The general result, which we shall t not prove, is that any connected one-dimensional An additive parameter exists and is given by b differentiable group is isomorphic either to Ê −

ϕ(b)= (1 + t2) 1/2dt = arg sinh b. (II.1)

Ê Z (the real line) or to Ì = / (the circle, as 0 the plane rotation group). One may even aban- Z We have then don the hypothesis of differentiability and only ask for local measurability of the function fx. b = sinh ϕ, a = cosh ϕ, A nice counterexample exhibiting the limits of so that the group elements are written the theorem is the following. Consider a two- cosh ϕ sinh ϕ parameter group, not necessarily abelian, for ins- M(ϕ)= . (II.2) sinh ϕ cosh ϕ tance the one-dimensional translation-dilatation   American J. of Physics, Vol. 47, No. 12, December 1979 3 ′ This abstract group can be considered as the Lo- (It is clear that for t = 1, ft (e) = 0 indeed.) rentz group for one space dimension, ϕ being The additive parameter is given by now the rapidity ; we will investigate this group b from a more physical point of view in the next θ(b)= (1 t2)−1/2dt = arcsin b. 0 − section. Z There is another way to derive the expres- We have thus sion (II.2), avoiding for students the integration (II.1). It suffices to assume the existence of the b = sin θ, a = cos θ, additive parameter ϕ and to express the group law in terms of it : and ′ ′ cos θ sin θ M(ϕ)M(ϕ )= M(ϕ + ϕ ). M(θ)= , sin θ −cos θ This equation leads to :   ′ ′ ′ so that the group is identified with the rotation a(ϕ + ϕ )= a(ϕ)a(ϕ )+ b(ϕ)b(ϕ ) group in the plane, in accordance with the gene- ′ ′ ′ (b(ϕ + ϕ )= b(ϕ)a(ϕ )+ a(ϕ)b(ϕ ) ral theorem. One could also postulate the existence of θ Let us define and derive all known properties of the functions u(ϕ)= a(ϕ)+ b(ϕ) a (cosine) and b (sine) from the group law satis- v(ϕ)= a(ϕ) b(ϕ) fied by the M matrices.  − These functions then obey : ′ ′ u(ϕ + ϕ )= u(ϕ)u(ϕ ) III RELATIVITY GROUPS FOR ONE- ′ ′ DIMENSIONAL SPACE (v(ϕ + ϕ )= v(ϕ)v(ϕ ) so that they are exponential functions. The uni- We will now use the additive parameter theo- modularity condition a2 b2 = 1 writes rem to establish from first principles the theory − of relativity. The discussion here is limited to u(ϕ)v(ϕ)=1, a one-dimensional space ; Sec. IV will generalize so that u and v are inverse exponentials : it to three-dimensional space. We look for space- u(ϕ) = exp(kϕ) time transformations, connecting two inertial re- ference frames, which satisfy the following requi- v(ϕ) = exp( kϕ)  − rements : (i) they preserve the of yielding in turn space-time ; (ii) they form a group ; (iii) they are a(ϕ) = cosh(kϕ) compatible with space reflexion ; and (iv) they b(ϕ) = sinh(kϕ) allow for some notion of . It has already  been shown elsewhere4 that such hypotheses are in agreement, of course, with (II.2) (up to the sufficient to derive the Lorentz transformations allowed constant factor k). (and their degenerate cousins, the Galilei trans- C. The 2 2 real antisymmetrical unimodular × formations). In particular there is no need for matrices any postulate dealing with the constancy of the a b 2 2 of . The reader is referred to previous R = , a + b =1 b− a work for the discussion of the physical signifi-   4 form a group as well, the group of plane ro- cance of the new postulates . tations. Now, because of the sign ambiguity, The homogeneity condition (i) is equivalent a = (1 b2)1/2, the parametrization by b is not to the linearity of the transformations in space- an essential± − one. However, in a neighborhood of time. We thus may write these transformations the identity (characterized by b = 0, a = 1), the in matrix form group law reads t′ a(ϕ) b(ϕ) t t ′ = = M(ϕ) b b = (1 b2)1/2b′ + b(1 b′2)1/2. x c(ϕ) d(ϕ) x x ∗ − −       (III.1) One has 2 1/2 2 1/2 fx(y)=(1 x ) y + x(1 y ) , where we suppose the transformation to depend − − upon a single parameter (see a discussion of this e =0, f ′(e)=(1 t2)1/2 t − point in the first paper of Ref. 6) which we American J. of Physics, Vol. 47, No. 12, December 1979 4 choose as an additive one, according to the fun- this relationship also yields the unimodularity damental theorem, so that the matrices M(ϕ) property of the M matrices obey det M(ϕ)=1. (III.9) ′ ′ M(ϕ + ϕ )= M(ϕ)M(ϕ ) (III.2a) Therefore, (III.6) becomes −1 M (ϕ)= M( ϕ) (III.2b) a(ϕ) b(ϕ) d(ϕ) b(ϕ) − = . M(0) = I. (III.2c) c(ϕ) −d(ϕ) c(ϕ) −a(ϕ) − −    (III.10) If we change the direction of the space axis, the From (III.10) we infer transformation labeled by ϕ becomes represen- 2 ted by the matrix a = d, a bc =1. (III.11) − a(ϕ) b(ϕ) Let us now consider the multiplication law Mˆ (ϕ)= . (III.3) c(ϕ) −d(ϕ) (III.2a). In particular we can write −  ′ ′ ′ The set of these matrices of course form a group a(ϕ + ϕ )= a(ϕ)a(ϕ )+ b(ϕ)c(ϕ ). isomorphic to the original one. The condition of Since this law is commutative, we must have the under space reflexion now requires the equality two groups to be identical (and not only isomor- ′ ′ b(ϕ)c(ϕ )= b(ϕ )c(ϕ) phic) ; namely, there must exist some parameter ϕˇ, depending on ϕ, such that or c(ϕ) c(ϕ′) Mˆ (ϕ)= M(ˇϕ) = = Cst, b(ϕ) b(ϕ′) with so that b and c are two proportional functions— ϕˇ = χ(ϕ). (III.4) unless one is zero. The proportionality constant can be adjusted to 1 or 1. Indeed, changing the Now since ϕ is an additive parameter for the space scale by a factor −h : group of matrices Mˆ , it is clear that χ must be x hx, an additive function of ϕ. Indeed, from (III.4) → one derives corresponds, after (III.1) to the following trans- ′ ′ ′ formations of the functions b and c and their M[χ(ϕ + ϕ )] = Mˆ (ϕ + ϕ )= Mˆ (ϕ)+ Mˆ (ϕ ) ′ quotient = M[χ(ϕ)] + M[χ(ϕ )] −1 c 2 c ′ b h b, c hc, h . = M[χ(ϕ)+ χ(ϕ )]. → → b → b Four cases therefore can be distinguished It results that 1 b = c : The M matrices are unimodular ma- ϕˇ = κϕ. (III.5) trices of the type a(ϕ) b(ϕ) Now, changing twice the orientation of the space M(ϕ)= . axis, we must recover the original parametriza- b(ϕ) a(ϕ) 2   tion, so that κ = 1. The case κ = 1 is tri- We recover the Lorentz group (see Sec. II B) ˆ vial, since the equation M(ϕ) = M(ϕ) leads to 2 b = c : The M matrices are unimodular ma- b = c = 0, that is, no relationship between space trices of− the type and time. We are now left with the condition a(ϕ) b(ϕ) −1 M(ϕ)= . Mˆ (ϕ)= M( ϕ) (= M (ϕ)). (III.6) b(ϕ) a(ϕ) − −  According to (III.3), the last relationship yields The group (see Sec. II C) is a rotation group—in the parity properties of the functions a, b, c, and space time ! d : 3 b = 0 : The modularity property (III.11) im- plies that a = 1. The M matrices are of the type a( ϕ)= a(ϕ) b( ϕ)= b(ϕ) (III.7) − − − 1 0 c( ϕ)= c(ϕ) d( ϕ)= d(ϕ) (III.8) M(ϕ)= . − − − c(ϕ) 1   Taking into account the equalities The group law yields ˆ det M(ϕ) = det M(ϕ) and det M( ϕ) = ′ ′ [det M(ϕ)]−1, − c(ϕ + ϕ )= c(ϕ)+ c(ϕ ) American J. of Physics, Vol. 47, No. 12, December 1979 5 or where ′ ′ c(ϕ)= γϕ. β = b (0), γ = c (0) The corresponding group is the Galilei group (do not forget that β and γ are 1 3 and 3 1 × × with a dimensionless parameter ϕ (and γ an ar- matrices, that is, a “line” and a “column” vec- bitrary velocity unit). tor, respectively). Relationship (IV.1) now yields 4 c = 0 : Proceeding as in 3 we obtain a set of differential equations ′ ′ 1 βϕ a = bγ, b = aβ M(ϕ)= , ′ ′ 0 1 c = dγ, d = cβ   which is the definition of the Carroll group. from which we deduce the equation 4 We now introduce a requirement of causality ′′ a = βγa. according to which there exist certain pairs of events such that their time interval ∆t keeps (βγ is the scalar product of the vectors β and the same sign in all reference frames, i.e., re- γ.) The function a therefore is of exponential mains under transformations with any type. The specific solution to be selected is dic- ϕ. This condition suffices to dismiss cases 2 and tated by the parity property (III.7) and the ini- 4 . The Lorentz and Galilei groups then remain tial condition (III.2c). Similar argument apply as the only possible physical groups of relativity to the initial computation of b, c, d. The nature in space time. of the solutions depends on the sign of βγ, or its vanishing. Since a scale change in the additive IV RELATIVITY GROUPS FOR parameter multiplies βγ by a positive number, THREE-DIMENSIONAL SPACE we are led to consider the following cases. 1 βγ =1. We obtain The additive parameter theorem allows for an cosh ϕ (sinh ϕ)β easy generalization of the preceding derivation M(ϕ)= . (sinh ϕ)γ 1 + (cosh ϕ 1)γβ to three-dimensional space. We write the space  −  transformation as (IV.2) t′ a(ϕ) b(ϕ) t t 2 βγ = 1. It suffices to replace in (IV.2) the ′ = = M(ϕ) x c(ϕ) d(ϕ) x x hyperbolic − functions by ordinary sine and cosine         functions. where a, b, c, d now are, respectively, 1 1, 1 3, × × 3 βγ =0. Integration yields : 3 1 and 3 3 matrices. All expressions of Sec. III×up to (III.9× ) then remain valid. However, the 1 ϕβ M(ϕ)= 1 2 . previous algebric derivation, which needs no dif- ϕγ 1+ 2 ϕ γβ ferentiation technique, becomes cumbersome for   The causality condition now eliminates case 2 three-dimensional space and we prefer an infi- and imposes β = 0 in case 3 . We are left with nitesimal approch, which indeed is closer to ad- two cases only : vanced methods relying on Lie algebras (this me- a β = 0, leading to the three-dimensional Ga- thod, of course, also works in the preceding one- lilei transformations dimensional case). 1 0 Consider the multiplication law (III.2a). De- M(ϕ)= , ′ ′ ϕγ 1 riving with respect to ϕ and putting ϕ = 0, we   obtain the following equality where the vector γ defines the direction of the M ′(ϕ)= M(ϕ)M ′(0), (IV.1) Galilean boost. b case 1 . To put the matrix (IV.2) in a more where the derived matrix is familiar form, let us perform an arbitrary change ′ ′ ′ a (ϕ) b (ϕ) of space coordinates through some matrix T . M (ϕ)= ′ ′ . c (ϕ) d (ϕ) The transformation matrix M now becomes :   1 0 1 0 For ϕ = 0, according to (III.7), wee see that M(ϕ) −1 0 T 0 T     ′ 0 β −1 M (0) = , cosh ϕ (sinh ϕ)βT γ 0 = −1 .   (sinh ϕ)Tγ [1 + (cosh ϕ 1)]TγβT American J. of Physics, Vol. 47, No. 12, December 1979  −  6 It is easily seen that a matrix T always exists purely spatial transformations, namely rotations such that βT −1 is the transposed of the matrix in three-dimentional space, exist as well, and Tγ. With convenient units we now obtain should show up in our derivation of relativity groups. The answer is to be found in the dismis- cosh ϕ (sinh ϕ)nt M(ϕ)= , sal, after equation (III.5), of the solution κ = 1. (sinh ϕ)n [1 + (cosh ϕ 1)]nnt  −  V ACKNOWLEDGMENTS which is a standard expression for the Lorentz transformation with rapidity ϕ, in the direction One of the author (J.-M. L.-L.) wishes to of the unit vector n. thank the D´epartement de Physique, Universit´e The question can be asked why we only de Montr´eal, where part of the work was done, found genuine space-time tranformations (Lo- for its hospitality. Prof. J.-P. Serres graciously rentz or Galilei) as possible transformations bet- furnished us with the counter-example at the end ween reference frames. Indeed, it is clear that of Sec. I.

a. Present address : Laboratoire de Physique Th´eorique et Hautes , Universit´eParis VII (Tour 33), 2 Place Jussieu, 75221 Paris Cedex 05, France. b. Present address : Laboratoire de Physique Th´eorique, Universit´ede Nice, Parc Valrose, 06034 Nice Cedex, France. 1. See, for instance, J.-M. L´evy-Leblond, Riv. Nuovo Cimento 7, 187 (1977) and J.-M. L´evy-Leblond (to be published). 2. When v > 1, formula (.2) leads to complex and indeed undetermined values of ϕ. Another way to understand the restriction v < 1 for velocities of reference frames, whithout invoking the reality of ϕ, is to remark that the addition law (.1) is a group law only if the v’s are less than one. The importance of the group law hypothesis will be developed in Sec. III. 3. Let us also note that the existence of an inverse has not been invoked in the demonstration. Therefore, a one- dimensional differentiable connected semi-group is isomorphic to the additive semi-group of positive real numbers. 4. J.-M. L´evy-Leblond, Am. J. Phys. 44, 271 (1976), A. R. Lee and T. M. Kalotas, Am. J. Phys. 43, 434 (1975), and additional references in these papers. American J. of Physics, Vol. 47, No. 12, December 1979 7