A Simple Derivation of the Lorentz Transformation and of the Related Velocity and Acceleration Formulae. Jean-Michel Levy

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A simple derivation of the Lorentz transformation and of the related velocity and acceleration formulae. Jean-Michel Levy

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Jean-Michel Levy. A simple derivation of the Lorentz transformation and of the related velocity and acceleration formulae.. American Journal of Physics, American Association of Physics Teachers, 2007, 75 (7), pp.615-618. 10.1119/1.2719700. hal-00132616

HAL Id: hal-00132616 https://hal.archives-ouvertes.fr/hal-00132616 Submitted on 22 Feb 2007

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. hal-00132616, version 1 - 22 Feb 2007 in ietewl nw ih lc rvrat thereof variants or clock e.g. light from or known below contraction (see well devices length the demon- conceptual using like to and by postulates order two dilation Einstein’s in time experiments strate thought use (SR) h ocpulydmnigsbeto SR. of with subject contact first demanding con- direct conceptually very these group the a with for of allows dispensing use siderations physics, are the they fundamental though bypass as in to Important linearity. example, and beginners. structure for for advantages allows, effects obvious It these has from and possible directly LT is not the certainly with. deriving is begin However, which to one but easiest texts the the advanced which derive in route to usual the is postulates taking (LT), the transformation to Lorentz return authors most tm nevl.As,a nms aeso the ’stan- on so-called papers the in most configuration’, frames dard in reference transformations two as to between limited was Also, derivation the the of subject, contrived extension interval’. transformation rather is artificial ’time a time an think through the introducing we it argument, derive what obtained to missed and way formula paper easiest this the of author the ulse nti ora ogtm ago. paper time a long in a already Journal used this was in in derivation published which of and LT, type velocity the the This of new of derivation tranformations. easy part possibly acceleration This an space and for allows the simple frames turn writing very motion. two of a relative way in to uniform leads formula and reasoning addition rectilinear vector expressing in to basic purely amounts a the which through argument contraction length geometrical from derived be pass. second principles a for fundamental left be from can derivations elaborate along ipedrvto fteLrnztasomto n fthe of and transformation Lorentz the of derivation simple A ayitoutr ore nseilrelativity special on courses introductory Many ntepeetatce eso htteL can LT the that show we article, present the In 2 oee,oc hs w ffcsaeestablished, are effects two these once However, ) OX etrfruafo lmnaygoer.Tersl sfurt is result equations. 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CONTRACTION T 0 = eae eoiyand velocity related 2 L v c 0 nadrcinparallel direction a in c the (1) 2 a variation of the pegs length when they are moving. time dilation derivation. For example, we can imag- By the first postulate, this moving clock must have ine that the rims of the mirrors are fitted with skates the same period in its rest frame than its twin at rest gliding perpendicularly to the mirror planes in two in the laboratory. parallel straight grooves. On the other hand, the length traveled by the signal in the lab is longer than the length it travels in the clock rest frame (see Fig.1 right) If T is the interval III. LORENTZ TRANSFORMATION ALONG between two ticks in the lab, then by Einstein’s sec- THE x AXIS ond postulate and the Pythagorean theorem, we have that Let us now envision two frames in ’standard ′ v 2 2 2 configuration’ with K having velocity with respect (cT/2) = L0 + (vT/2) (2) to K and let x, t (resp. x′,t′) be the coordinates of event M in the two frames. Let O and O′ be the from which spatial origins of the frames; O and O′ co¨ıncide at T time t = t′ =0 T = 0 = γT (3) v 2 0 1 − ( c ) p Here comes the pretty argument: all we have to do follows, showing that the moving clock runs more is to express the relation slowly in the lab than its stationnary twin. The sec- ′ ′ ond equal sign defines the ubiquitous Lorentz γ factor. OM = OO + O M (5)

between vectors (which here reduce to oriented C. Length contraction segments) in both frames.

In K, OM = x, OO′ = vt and O′M seen from K Now the moving clock is traveling in a direction per- ′ x ′ ′ ′ pendicular to the plane of its mirrors relative to the is γ since x is O M as measured in K Hence a ﬁrst lab observer. In this case, no check can be kept of the relation: inter-mirror distance. To make sure that (for the same x′ v) the clock period hasn’t changed, we can imagine it x = vt + (6) accompanied by an identical second clock oriented as γ before with respect to its lab velocity. Both clocks In K′, OM = x since x is OM as measured in K, have the same period in their common rest frame and γ by the argument already given, the clock moving par- OO′ = vt′ and O′M = x′. Hence a second relation: allel to its mirrors has period T in the lab (cf. above); x = vt′ + x′ (7) therefore we can be sure that the clock moving perpen- γ dicularly to its mirrors also has period T in the lab frame. Anticipating the result which will be forced Relation (6) yields immediately upon us, let L be the inter-mirror distance as measured in the laboratory frame. Now consider the time x′ = γ(x − vt) (8) taken by the light signal to make its two-way travel in the laboratory frame; starting from the rear mir- which is the x-axis ’space’ part of the LT and relation ror (which was the ’lower’ mirror before the clock was (7) yields the inverse rotated), it will reach the front mirror after a time t ′ ′ given by ct = L+vt and will need a further time lapse x = γ(x + vt ) (9) t′ given by vt′ = L−ct′ for the return leg, which makes L of this ’space part’. Eliminating x′ between (8) and a total of 2 Equating this expression with the c(1− v ) c2 (9) quickly leads to the formula for the transformed one already obtained for T , one is forced to conclude time: that t′ = γ(t − vx/c2) (10) v 2 L0 L = L0 1 − ( ) = (4) r c γ the inverse of which could easily be found by a similar elimination of x. That the distances in the directions orthogonal to the Coordinates on the y and z axes are unchanged for motion are not changed can be demonstrated by in- the already stated reason that distances do not vary voking grooves arguments like the one we used for the in the directions perpendicular to the velocity. 3

IV. THE CASE OF AN ARBITRARY and in frame K′: VELOCITY Op(γ−1)r = vt′ + r′ (19) In the following, v will denote the velocity vector of K′ w.r.t. K and r (resp. r′) the position vector of Using (16) relation (18) yields immediately: the event under consideration as measured in frame r′ = Op(γ)(r − vt) = (1 + (γ − 1)u ⊗ u)(r − vt) (20) K (resp K′). We further define v which is probably the simplest way to write the space u = (11) |v| part of the rotation free homogenous LT. The usual γ factor of the one dimensionnal transformation is sim- the unit vector parallel to v. ply replaced by the operator Op(γ) From our findings of section 2, we see that only the By substituting (20) into (19), we find: component of r parallel to v is affected when looking at it from the other frame, while the normal compo- Op(γ−1)r = vt′ + Op(γ)(r − vt) (21) nents are unchanged. We resolve r into parallel and perpendicular components according to or, using

r = uu.r + (1 − u ⊗ u)r = rk + r⊥ (12) Op(γ)v = γv (22) where the dot stands for the 3-space scalar product, and with the explicit form of Op: 1 is the identity operator and u ⊗ u is the dyadic 1 − γ vv.r which projects out the component parallel to u from ( − (γ − 1)) + γvt= vt′ (23) the vector it operates upon, viz γ v2

(u ⊗ u)V = (u.V)u (13) Using now v 1 − γ2 = −( )2γ2 (24) The operator which contracts the projection on u by c γ while leaving the orthogonal components unchanged and crossing away v on both sides, (23) yields: must yield: ′ v.r u.r 1 − γ t = γ(t − ) (25) u + (1 − u ⊗ u)r = (1 + u ⊗ u)r (14) c2 γ γ i.e. the time transformation equation. Let us therefore deﬁne 1 − γ Op(γ−1)= 1 + u ⊗ u (15) V. VELOCITY AND ACCELERATION γ TRANSFORMATIONS The inverse operator must correspond to multiplication of the longitudinal part by γ and is therefore A. Velocity

−1 −1 Op(γ)= Op(γ ) = 1 + (γ − 1)u ⊗ u (16) The two formulas thus obtained for the L.T. are so simple that they can readily be used to yield the as can also be checked by multiplication of the velocity transformation equation without the need of right-hand sides of (15) and (16) . Note that these complicated thought experiments and algebraic ma- u operators are even in and therefore independent of nipulations. Diﬀerentiating (20) and (25) w.r.t. t and v the orientation of . taking the ratio of the equalities thus obtained yields,

Mimicking what has been done in section 3, let us dr′ dr (with V′ = and V = ) (26) write again dt′ dt OM = OO′ + O′M (17) but for vectors now, taking care of the invariance of ′ 1 (1 + (γ − 1)u ⊗ u)(V − v) V = v.V (27) the orthogonal parts. We get in frame K: γ 1 − c2 r = vt+ Op(γ−1)r′ (18) which is the general velocity transformation formula. 4

B. Acceleration frame in newtonian physics. Setting V = v and taking v parallel to A we also The compact Op notation helps to keep the algebra retrieve another known fact: a particle in rectilinear tidy when diﬀerentiating (27) w.r.t. t; dividing the motion undergoes a proper acceleration which is larger 3 derivative of (27) by that of (25) one ﬁnds than its lab acceleration by a factor γ .

v.V v.A These two examples are but special cases of a general 1 Op(γ)A(1 − 2 )+ Op(γ)(V − v) 2 A′ = c c formula connecting proper acceleration and accelera- 2 V.v 3 γ (1 − c2 ) tion in the laboratory frame, which can be obtained (28) by setting v = V in (29), viz. Expliciting Op, simplifying and regrouping terms, one ′ 2 obtains after a page of algebra: A = γ Op(γ)A (30)

γ v.Av v×(V×A) Here γ and Op(γ) are calculated using the labora- A − 2 + 2 A′ = γ+1 c c (29) tory velocity of the accelerating body that is also the 2 V.v 3 γ (1 − c2 ) velocity of the inertial frame in which it is instanta- neously at rest. Equation (30) can be readily inverted V u′ By making the necessary substitutions: → , to yield the laboratory acceleration given the proper V′ u v V V → , → − and specializing to parallel to acceleration, if needed. Ox, one can easily check that the component equations derived from (29) agree with those published 2 in. VI. SUMMARY AND CONCLUSION As an example of use of this acceleration transformation, we take V = v and v.A = 0, and obtain ′ 2 We have shown that the general rotation free ho- A = γ A retrieving the known result that a particle mogenous LT can be derived once length contrac- in a circular storage ring undergoes a proper (A′) ac- tion has been established by writing the elementary celeration that is a factor γ2 larger than the lab (A) vector relation (sometimes dubbed ’Chasles’ relation) acceleration.(7) Moreover, the two accelerations are OM = OO′ + O′M in the two frames considered.8 parallel, which is far from obvious a priori. Observe The extension from the special one dimensional case that all the terms which can make A′ and A diﬀer- to the 3-dimensional case is completely straightfor- ward. The relation we have obtained allows for a sim- ent in direction as well as in magnitude vanish in the ple derivation of the velocity and acceleration trans- c → ∞ limit, consistent with the fact that accelera- formations without the need for complicated thought tion is an invariant quantity under a change of inertial experiments and algebraic manipulations.

a Electronic address: [email protected] from Gedanken Experiments,” Am. J. Phys. 42, 909- 1 Kenneth Krane, Modern Physics, John Wiley & sons, 910, (1974) New-York, 1983 p. 23 5 Wolfgang Rindler, Relativity, Special, General and Cos- 2 W.N. Mathews Jr. ”Relativistic velocity and accelera- mological Oxford University Press, New-York, 2001, p.5 tion transformations from thought experiments,” Am. 6 See e.g. ref. 5 p. 57 J. Phys. 73, 45-51 (2005) 7 See e.g. ref. 5 p. 100 3 For the sake of completeness, it must be mentionned 8 There are much faster derivations than the one pre- that the LT can in fact be derived by using the sole rel- sented here. See e.g. Alan Macdonald, ”Derivation of ativity principle (ﬁrst Einsein postulate) and dispensing the Lorentz transformation,” Am. J. Phys.49, p. 493, with the second (invariance of the speed of light) This (1981). Our purpose here was to be simple, as stated in is certainly most appealling from a puristical viewpoint, the title and abstract, and to go beyond the ’standard but it demands more abstract work which is what we are conﬁguration’ trying to avoid here, having in mind beginning students. 4 David Park, ”Derivation of the Lorentz transformations