arXiv:physics/0411233v1 [physics.class-ph] 25 Nov 2004 hoe I.1. Theorem celeration tv rngtv indpnigo h ieto.Let of arrival and direction. departure the the between on depending sign T negative or itive aetaclrto esrdby measured acceleration parent x of eration fteieta rm.Lt moreover, Let, frame. inertial the of ueyoeo h otatnsigrslso modern of see results introduction astonishing historical most an the (for physics of one surely aaerzto and parametrization igah ihodppr see papers old with liography ietplge a erfruae ihu n edof need any without observers reformulated accelerated be can space- of non-trivial topologies source in time paradox ultimate twin velocity. the the as not the aging velocity differential is changing relative however, acceleration since acceleration, does The effect; re- so dilation are there time time enters inertial the simple and in a proper lated receives that noticing aging consid- by differential be answer for could acceleration responsible whether ered to as question old eew att ics n pl tt oecsspre- cases some section. methods. to next elementary it more the apply with in and investigated discuss relation viously to that want prove we Here shall We mo- arbitrary, tion. otherwise observer but accelerated unidirectional, the an case undergoes the in aging differential and and ilfaeadcos oriae nawysc that Let such way suppressed. a be in can them coordinates of choose two and frame tial iladaclrtdosreshsbe discussed been iner- has of observers times accelerated proper and in difference tial role the the and between acceleration relationship of the particular in and bated, tdosre ihtmlk worldline timelike with observer ated ( τ = h ieeta gn mle yseilrltvt is relativity special by implied aging differential The hoeuissc that such units Choose acceleration relates that equation simple a give we Here T .T emr rcs,tequantity the precise, more be To ). 2 x x = 0 1 (¯ 0 = (0) τ  ) Z a − 0 O ( τ ¯ τ rtdadatra rirr rpcmsbc.W ics si a discuss during time observer We the accelerating between the back. relation by comes explicit measured acceleration an trip case, arbitrary dimensional an 1+1 after and erated hwn htis tlatnmrcl ouinwudalwt allow would solution numerical, least prob at reconstruction differen its, The the that non-decreasing. showing define is we it that observation show this and Using frame. inertial x e ihrsett h oa nrilfaeat frame inertial local the to respect with ) 0 R x o-lsdti ihrsett nieta rm per c appears frame inertial an to respect with trip non-closed A ecnie lc prdx rmwr hr nosre le observer an where framework “paradox” clock a consider We 0 ete(oiie nriltm interval time inertial (positive) the be (0) 0 y(iedlto-ceeainequation) dilation-acceleration (time by τ 1 h iedilation time The .INTRODUCTION I. a (¯ ( τ τ ,where 0, = ) ′ )d ieeta gn rmaclrto,a xlctformula explicit an acceleration, from aging Differential 4 x τ . ′ µ d , τ µ   c 0 = .Let 1. = 2 Z n NN izadiCpetr 0 -08 oa Italy Roma, I-00186 70, Caprettari dei Piazza INFN, and eatmnod ae´tcs nvria eSalamanca, de Matem´aticas, Universidad de Departamento .I a enlreyde- largely been has It ). 0 τ ¯ , O e lz el ecd14 -70 aaac,Spain Salamanca, E-37008 1-4, Merced la de Plaza r h coordinates the are 1 T τ − n oi a pos- a has it so and R srltdt h ac- the to related is stepoe time proper the is 0 O τ a a O ehave we , x ( ( τ µ τ K eteaccel- the be ) ′ ea acceler- an be − )d [0 : 1 eteiner- the be a τ o bib- a for , ′ d steap- the is , τ τ ¯ 11  ] 3 . → The . .Minguzzi E. (1) M R od nyif only holds fmroe h nlvlct of velocity final the moreover if a O K I.2. Theorem onciggoei,ta stetaetr fequation of trajectory the the that is noticing x that by spacetimes geodesic, flat) connecting in even hence (and functional se then ishes a h datg fgvn nepii oml o the for formula explicit an dilation. giving round-trip of inertial advantage and the spacetime Minkowski has 1+1 in property maximization gn ffc.I eto IAw hl ieaohrproof another inequality. give Cauchy-Schwarz shall the we use A not II does section that In effect. aging w nevl [0 intervals two of velocity final or initial the specify h eoditra tvelocity at interval second the efidteepce relation expected the find we K ihta fteoii of origin the of that with ( ml,ised h ceeainhsasnuaiya ¯ ex- at singularity this a (1) In has Eq. acceleration that the functions. is instead, acceleration problem ample, integrable first pre- The for it holds result. how the same see at the and look dicts to equation interesting dilation-acceleration is time it example elementary quite R 0 ( 1 τ ¯ τ sn h acyShazieult ( inequality Cauchy-Schwarz the Using . ehv also have We Often h ipeteapei hto nfr oinin motion uniform of that is example simplest The to need we place no In order. in are comments Some ( oe ihrsetto respect with moves f e ihzr eoiyi n nyif only and if velocity zero with τ .Thus 0. = ) − 2 norcs,lclymxmzstepoe time proper the maximizes locally case, our in 0 = ) d R τ h trip. the ecntuto fan of construction he 0 τ e fseilrltvt sas discussed also is relativity special of lem )( ilaiga ucino rprtime proper of function a as aging tial 5 a lpe nteieta rm n the and frame inertial the on elapsed oe ihrsett nte suitable another to respect with losed ( R h ieeta gn ffc spoe ncurved in proved is effect aging differential the τ R I ′ g )d 0 [ τ ¯ 2 γ peeuto htgvs nthe in gives, that equation mple f τ d a .Tesmls example simplest The A. = ] vsa nrilfae saccel- is frame, inertial an aves ′ τ ( d = τ , T ,with ), τ/ h ceeae bevrdprsfrom departs observer accelerated The τ ¯ )d > T = kg R n nti case this in and τ ]ad[¯ and 2] γ Z with , 0 = d 0 τ τ ¯ τ ¯ hoe mle h global the implies 1 Theorem . f e rtewrdieof worldline the or . ∗ ± K K = nrilclock inertial R τ/ k hspoe h differential the proves This . 0 τ tvelocity at a ∈ 2 g T ( , − τ O τ − ¯ R 1 ′ ≥ )d .I h rtinterval first the In ]. hti fadol if only and if is that , v ihrsetto respect with exp( = τ lhuhti sa is this Although . τ ¯ ′ R d hr h equality the where 0 O τ, τ ¯ . v e ihrsetto respect with R 0 d = R τ 0 τ a R ( O a τ x ( ′ fg )d 1 τ coincides / ′ τ d )d K d ′ τ x d τ ) 0 van- τ 2 ′ τ/ in , / (2) 2) = ≤ 2 2

(the initial and final singularities are not present if the motion of O is not forced to coincide with that of K’s T K origin for τ outside the interval). The reader can easily − check (or see next section), that if θ(τ) = tanh 1 v(τ) is dθ the rapidity then dτ = a (this follows from the additiv- ity of the rapidity under boosts and the fact that a small increment in rapidity coincides with a small increment in velocity with respect to the local inertial frame) and so

∆θ = adτ. Z A BC

If the acceleration causes, in an arbitrary small interval centered atτ ˜, a variation ∆θ in rapidity then we must FIG. 1: The textbook round-trip examples. write a = ∆θδ(τ τ˜) and generalize the time dilation- acceleration equation− with this interpretation. In pres- to K’s worldline. Moreover, we know that O starts with ence of such singularities, however, it is no longer true zero velocity so we can apply equation (2). First we have that T does not depend on the initial and final velocities of K. Indeed, we need to use this information to find the 1 τ gτ, τ [0, 4 τ¯], coefficient ∆θ. In the case at hand we have ′ ′ ∈ 1 3 a(τ )dτ =  gτ + gτ/¯ 2, τ [ 4 τ,¯ 4 τ¯], Z0 − ∈ 3 −1 −1 −1  gτ gτ,¯ τ [ τ,¯ τ¯]. ∆θ = tanh ( v) tanh v = 2 tanh v. − ∈ 4 − − −  −1 Integrating simple exponentials we arrive at T = Inserting a = 2 tanh vδ(τ τ˜) in Eq. (1) we find, 4 sinh gτ¯ . after some work− with hyperbolic− functions that, T = g 4 τ/¯ √1 v2 as expected. The reader should not be sur- − prised by the fact that this simple case needs so much II. THE RECONSTRUCTION PROBLEM IN work, as this is a rather pathological case. No real ob- SPECIAL RELATIVITY server would survive an infinite acceleration. The advan- tage of the time dilation-acceleration equation turns out in more realistic cases. In this section we consider the problem of reconstruct- ing the motion in the inertial frame starting from the knowledge of the acceleration. Similar mechanical prob- lems have been studied in10. It can be stated in full B. The constant acceleration case Minkowski spacetime as follows. Consider a timelike worldline xµ(τ) on Minkowski i This case has also been treated extensively in the spacetime and a Fermi-transported triad ei. Let a (τ) = 7,8 literature . The hypothesis is that in the interval [0, τ¯] (a(τ) ei) be the components of the acceleration vector − · i we have a = g with g R. Equation (1) gives immedi- with respect to the triad, a = a ei. Determine, starting ately − ∈ from the data ai(τ), the original curve up to an affine transformation of Minkowski spacetime. τ¯ τ¯ 2 −gτ gτ 2 Here the Fermi-transported triad represents gyro- T = e dτ e dτ = 2 (cosh gτ¯ 1) Z0  Z0  g − scopes. The components of the acceleration with respect to this triad are therefore measurable by O using three 2 gτ¯ orthogonal gyroscopes and an accelerometer. The solu- or T = g sinh . 2 tion to this problem may be relevant for future space travellers. Indeed, although the twin ‘paradox’ has been studied mainly assuming the possibility of some commu- C. A more complicated example nication by light signals, it is more likely that when dis- tances grow communication becomes impossible. Sup- This example was considered by Taylor and Wheeler9. pose the space traveller does not want to be lost but It has the advantage that the acceleration has no Dirac’s still wants the freedom to choose time by time its tra- deltas and O departs from and arrives at K with zero jectory, then he/her should find some way to know its velocity. The interval is divided into four equal parts inertial coordinates. The only method, if no references of proper time durationτ/ ¯ 4. The acceleration in these in space are given, is to solve the reconstruction prob- intervals is successively g, g, g and g. lem. Keeping track of the acceleration during the jour- − − One can easily convince him/herself that since the ac- ney the observer would be able to reconstruct its inertial celeration in the second half interval is opposite to the coordinates without looking outside the laboratory. In one in the first half interval the observer indeed returns particular he would be able to construct (merging an ac- 3 celerometer, three gyroscopes, and an ordinary clock) an observer K(τ) moving along that geodesic sees the mo- 0 µ inertial clock i.e. a clock that displays x (τ). tion of the accelerated observer as a round trip. If xK(τ) The solution to the reconstruction problem gives also 1 1 are its coordinates xK(τ)(τ) = xK(τ)(0) = 0, thus the to O the advantage of knowing its own position even be- previous invariant reads fore K knows it. Indeed, O can know xµ(τ) immediately 0 0 while K has to wait for light signals from O. In case of T (τ)= xK(τ)(τ) xK(τ)(0), perturbations in the trajectory, O can immediately ap- − ply some corrections while, for great distances, a decision that is, T (τ) is the travel duration with respect to an from K would take too much time. inertial observer that sees the motion of the accelerated In 1+1 dimensions the reconstruction problem can be observer as a round trip that ends at τ. Using the relation 2 2 solved easily. For higher dimensions it becomes much a b = (a b)(a + b) we have from Eq. (6) − − more complicated and numerical methods should be τ ′ τ ′ τ ′′ ′′ ′ − τ ′′ ′′ ′ used. Let us give the solution to the 1+1 case. We use T 2(τ)= e 0 a(τ )dτ dτ e 0 a(τ )dτ dτ . Z R Z R  the timelike convention η00 = 1. 0 0 µ µ 0 1 0 µ µ If v = dx /dx , v = dx /dx and u = dx /dτ, then Remarkably the dependence on v(0) disappears. This fol- we have lows from the fact that contrary to x0(τ) and x1(τ), the duµ d vµ quantity T (τ) is a Lorentz invariant and as such should aµ = = . not depend on the choice of frame (i.e. the choice of dτ dτ √1 v2 − v(0)). Let (0,a) be the components of the acceleration in the In order to prove the second theorem note that if, with respect to K, O departs with zero velocity then from (5), local inertial frame (the first component vanishes because 1 1 u a = 0). Since the square of the acceleration is a Lorentz after imposing the round-trip condition x (¯τ) = x (0), · 2 µ we have invariant we have a = a aµ or − τ¯ τ ′ ′ 2 d 1 2 d v 2 sinh[ a(τ )dτ ]dτ =0. a = ( ) ( ) Z0 Z0 − dτ √1 v2 − dτ √1 v2 − − 2 1 dv that is, the two factors in the formula for T coincide. = ( )2, −(1 v2)2 dτ Finally, if O departs and returns with zero velocity we − τ¯ have 0 adτ = 0 as it follows from the already derived dθ relationR a = dθ/dτ. but a has the same sign as dv/dτ and hence a = dτ , − where θ = tanh 1 v is the rapidity, or

τ A. Differential aging − v(τ) = tanh[ a(τ ′)dτ ′ + tanh 1 v(0)]. (3) Z 0 We give now a different proof that T (¯τ) > τ¯ unless 0 2 1 2 a(τ) = 0 for all τ [0, τ¯] in which case T (¯τ)=¯τ and O From dx = dτ/√1 v and dx = dτ v/√1 v we ∈ have − − is at rest in K. The idea is to define the differential aging even for ′ τ τ proper times τ < τ¯ as the differential aging between 0 0 ′′ ′′ −1 ′ x (τ) x (0)= cosh[ a(τ )dτ + tanh v(0)]dτ ,(4) K(τ) and O. The differential aging at τ is therefore by − Z0 Z0 ′ definition ∆(τ) = T (τ) τ, that is the difference be- τ τ − − x1(τ) x1(0)= sinh[ a(τ ′′)dτ ′′+ tanh 1 v(0)]dτ ′, (5) tween the proper time elapsed for an inertial observer − Z0 Z0 that reach x(τ) from x(0) and that elapsed in the accel- erating frame. Roughly speaking if at proper time τ the Note that v(0) is also easily measurable by O since at τ = accelerating observer asks “What is the differential aging 0, K and O are crossing each other. Without knowing now?” the answer using this idea would be: it is the dif- v(0) the inertial coordinates are determined only up to ferential aging between you and an imaginary twin who a global affine transformation. Indeed, we may say that reached the same event where you are now, but moving the knowledge of v(0) specifies, up to translations, the along a geodesic. This definition has the advantage of inertial coordinates and frame with respect to which we avoiding conventions for distant simultaneity. describe O’s motion. Theorem II.1. Now, consider the invariant under affine transforma- The differential aging ∆(τ) is a non- tions decreasing function d∆ T 2(τ) = [x0(τ) x0(0)]2 [x1(τ) x1(0)]2. (6) 0, (7) − − − dτ ≥ Since x(τ) is in the chronological future of x(0) there is where the equality holds for all τ ′ [0, τ] iff a(τ ′)=0 for a timelike geodesic passing through them. The inertial all τ ′ [0, τ]. ∈ ∈ 4

τ Proof. Let Θ(τ)= 0 a(τ)dτ. The derivative of T (τ) is III. CONCLUSIONS R dT = cosh A(τ), dτ We have discussed the reconstruction problem in spe- cial relativity showing its relevance for the construction where of inertial clocks and in general for the positioning of the

τ − ′ ′ space traveller. We have given a simple formula that re- 1 e Θ(τ )dτ A(τ)=Θ(τ)+ ln 0 . lates the round-trip inertial time dilation with the accel- R τ Θ(τ ′) ′ 2 { 0 e dτ } eration measured by the non-inertial observer and have R applied it to some well know cases to show how it works Since cosh A 1 this proves Eq. (7). Now, suppose even in the presence of singularities. We believe that it d∆ ≥ dτ (τ) = 0 then A(τ)=0or could be useful in order to explain clearly the relation- ship between acceleration and differential aging T (τ) τ. τ ′ e−Θ(τ )dτ ′ Indeed, the differential aging effect is obtained easily− by e−2Θ = 0 . (8) R τ Θ(τ ′) ′ applying the Cauchy-Schwarz inequality. 0 e dτ R Although there is a section on the twin paradox in al- Assume d∆ (τ ′) = 0 for all τ ′ [0, τ] then the previ- most every textbook on special relativity, examples with dτ ′ ous equation holds for all τ [0∈, τ]. Differentiating we singularities are not always completely satisfactory, while obtain ∈ more refined examples require a lot of work. On the ′ ′ contrary the derivation of the time dilation-acceleration −Θ τ Θ(τ ) ′ Θ τ −Θ(τ ) ′ − e e dτ e e dτ 2a(τ)e 2Θ = 0 − 0 =0, formula is quite elementary needing only some concepts R τ Θ(τ ′) R′ 2 − ( 0 e dτ ) from calculus. Its derivation as a classroom exercise would probably convince students of the reality of the ′ R that is a(τ) = 0 for all τ [0, τ]. differential aging effect. ∈ Acknowledgments Since ∆(0) = 0 this theorem implies that ∆(τ) > 0 for τ > 0 unless a(τ ′) = 0 for all τ ′ τ. This proves again the differential aging effect. However,≤ the theorem says I would like to acknowledge useful conversations with: something more. It proves that the definition of differen- D. Amadei, A. L`opez Almorox, C. Rodrigo and C. tial aging we have given is particularly well behaved. It Tejero Prieto. I am also grateful to A. Macdonald for allows us to say that, as proper time passes, the imagi- his comments and suggestions. This work has been nary twin is getting older and older with respect to the supported by INFN, grant n◦ 9503/02. accelerating observer.

∗ Electronic address: [email protected] E. A. Desloge and R. J. Philpott. Comment on “The case 1 P. Pesic. Einstein and the twin paradox. Eur. J. Phys. 24, of the identically accelerated twins” by S.P.Boughn. Am. 585–590 (2003). J. Phys. 59, 280–281 (1991). 2 G. Holton. Resource letter SRT1 on special relativity the- T. A. Debs and L. G. Redhead. The twin “paradox” and ory. Am. J. Phys. 30, 462–469 (1962). the conventionality of simultaneity. Am. J. Phys. 64, 384– L. Marder. Time and the space-traveller (University of 392 (1996). Pennsylvania Press, 1971). R. P. Gruber and R. H. Price. Zero time dilation in an E. Sheldon. Relativistic twins or sextuplets? Eur. J. Phys. accelerating rocket. Am. J. Phys. 65, 979–980 (1997). 24, 91–99 (2003). H. Nikoli`c. The role of acceleration and locality in the twin 3 G. D. Scott. On solutions of the clock paradox. Am. J. paradox. Found. Phys. Lett. 13, 595–601 (2000). Phys. 27, 580–584 (1959). C. E. Dolby and S. F. Gull. On radar time and the twin G. Builder. Resolution of the clock paradox. Am. J. Phys. “paradox”. Am. J. Phys. 69, 1257–1261 (2001). 27, 656–658 (1959). 4 E. A. Milne and G. J. Whitrow. On the so-called“clock- R. H. Romer. Twin paradox in special relativity. Am. J. paradox” of special relativity. Philos. Mag. 40, 1244–1249 Phys. 27, 131–135 (1959). (1949). H. Lass. Accelerating frames of reference and the clock J. Kronsbein and E. A. Farber. Time retardation in static paradox. Am. J. Phys. 31, 274–276 (1963). and stationary spherical and elliptic spaces. Phys. Rev. W. G. Unruh. Parallax distance, time, and the twin “para- 115, 763–764 (1959). dox”. Am. J. Phys. 49, 589–592 (1981). C.H. Brans and D.R. Stewart. Unaccelerated-returning- S. P. Boughn. The case of indentically accelerated twins. twin paradox in flat space-time. Phys. Rev. D 8, 1662–1666 Am. J. Phys. 57, 791–793 (1989). (1973). E. Eriksen and Ø. Grøn. Relativistic dynamics in uniformly T. Dray. The twin paradox revisited. Am. J. Phys. 58, accelerated reference frames with application to the clock 822–825 (1990). paradox. Eur. J. Phys. 11, 39–44 (1990). R. J. Low. An acceleration-free version of the clock para- 5

dox. Eur. J. Phys. 11, 25–27 (1990). 252–261 (1987). J. R. Weeks. The twin paradox in a closed universe. Amer. 8 R.H. Good. Uniformly accelerated reference frame and Math. Monthly 108, 585–590 (2001). twin paradox. Am. J. Phys. 50, 232–238 (1982). J. D. Barrow and J. Levin. Twin paradox in compact 9 E. F. Taylor and J. A. Wheeler. Spacetime Physics (W. spaces. Phys. Rev. A 63, 044104 (2001). H. Freeman and Company, San Francisco, 1966). J. P. Luminet J. P. Uzan, R. Lehoucq, and P. Peter. Twin 10 R. Montgomery. How much does the rigid body rotate? paradox and space topology. Eur. J. Phys. 23, 277–284 A Berry’s phase from the 18th century. Am. J. Phys. 59, (2002). 394–398 (1991). O. Wucknitz. Sagnac effect, twin paradox and space-time M. Levi. Geometric phases in the motion of rigid bodies. topology - Time and length in rotating systems and closed Arch. Rational Mech. Anal. 122, 213–229 (1993). Minkowski space-times. gr-qc/0403111, 2004. M. Levi. Composition of rotations and parallel transport. 5 C. Møller. The Theory of Relativity (Clarendon-Press, Nonlinearity 9, 413–419 (1996). Oxford, 1962). 11 These theorems hold also in a non-topologically trivial 6 S. W. Hawking and G. F. R. Ellis. The Large Scale Struc- Minkowski spacetime but there O should move along a ture of Space-Time (Cambridge University Press, Cam- worldline which is homotopic to K’s origin worldline; this bridge, 1973). follows from their validity in the covering Minkowski space- 7 E. A. Desloge and R. J. Philpott. Uniformly accelerated time. reference frames in special relativity. Am. J. Phys. 55,