APS/123-QED

Acceleration in : What is the meaning of ”uniformly accelerated movement” ?

Julien Baglio∗ and Matthieu Bethermin† Physics Department, ENS Cachan 61 avenue du Pr´esidentWilson, 94235 Cachan, France (Dated: May 26, 2007) In Newtonian mechanics, no problems arise while thinking about the meaning of an uniformly accelerated movement. Thus, if the movement is viewed in a frame R, considering a movement with a constant causes no particular problem. In particular, it is equivalent to view it as a movement due to a constant force, according to Newton’s second law. How this can be seen in Einstein’s special relativity ? This article shows that the first consideration is no longer valid, while that of the second still holds. We will also present another kinematic point of view which gives exactly the same results as the dynamical one, but is unnatural considering the common sense.

PACS numbers: unclassified - internal note

I. INTRODUCTION If we boost to the frame, associated with the system, where its speed is null, the acceleration is not the same. Until the beginning of the XXe century, the paradigm Adding the inertia force −ma and using the second 0 underlying mechanics had been that of Newton : Newtonian law, we obtain ma = F − ma. Thus, as a 0 was absolute and switching between two inertial frames is a constant, a remains constant and the movement should follow . In a modern point is still uniformly accelerated. Applying well a Galilean 0 of view, laws of physics were to follow an invariance under transformation, we even obtain that a = 0, which is the Galilean group. not a surprise after all. But it was not the case of Maxwell’s laws of elec- trodynamics. Einstein, following the work of Poincar´e and Lorentz, developped the special relativity to cure the paradox : laws of physics are invariant under II. ACCELERATION IN SPECIAL RELATIVITY Lorentz-Poincar´egroup, which means in particular that Galilean transformation have to be replaced by that How can the work done above can be generalised in of Lorentz. The most important consequence is the special relativity ? vanishing of absolute time.

In this article, we come back to a familiar notion : the 1. A naive point of view which fails uniformly accelerated movement. In Newtonian mechan- ics, an uniformly accelerated movement in a frame noted The first idea to give sense to the idea of an uniformy R means that the acceleration mesured in that frame is accelerated movement would be holding our definition constant : (1). Consider a mobile in an inertial frame R. If we con- sider the meaning of ”uniformly accelerated movement” a = cst (1) as a = cst, we therefore obtain exactly the solution (2). But a problem is immediately raised, if the speed Integrating this equation it is easiest to obtain of the mobile is considered : nothing prevents it to be greater than the , which is inconsistant with special relativity itself ! 1 2 x(t) = at + v0t + x0 (2) 2 Thus we have proved that the definition (1) has to be abandonned in special relativity : the kinematic notion where v0 and x0 stand for the initial velocity and posi- tion of the system. By Newton’s second law of mechanics, of ”uniformly accelerated movement” seems not to hold. this means that if R is an inertial frame, a constant ac- tion is applied on the system. In what follows, R will 2. The dynamical meaning always be assumed as an inertial frame. Before coming back to the kinematic point of view, let’s investigate the behaviour of that of the dynamical. ∗Also at SLAC Theoy Group, Stanford University. In special relativity, the impulsion is defined through †Also at SLAC Babar Group B, Stanford University. a quadrivector : 2

Definition 1 A mobile has a ”uniformly accelerated movement” when a’ = cst dU µ E  P µ = = , γmv (3) dτ c It means that at every moment, while boosting to the 0 µ proper frame of the mobile, the value of a is the same. where U is the speed quadrivector, γ is the Lorentz Using (A.2) with v = V it gives factor and τ is the proper time of the mobile studied. We then define p = γmv as the relativistic impulsion, which is equivalent to the newtonian impulsion as v → 0 a 1 (i.e. v << c). a0 = A = γ  v2 2 1 − Then using c2

dp = F Using the expression of the γ-factor, it reduces to dt

and using the dynamical definition of ”uniformly ac- dv celerated movement”, i.e. F = cst, we then obtain a0 = A = γ3a = γ3 (7) dt dγv m = mA (4) Integrate equation (7) is done with the separation of dt variables method, and recalling that F where we have used A = . As A is a constant, the   m integration of (4) is easiest : dv v = d   3/2 r   v2   v2  1 − 1 − γv = At + γ(0)v(0) (5) c2 c2

In order to compare this calculus with that will follow, We then obtain we assume that the initial speed is null, and we rotate our axis, in order to have v = vzˆ (without loosing gen- erality with the last restriction). We then reorganize the v At = r + cst (8) equation, to obtain v2 1 − c2 At v(t) = r (6) With the initial condition v(0) = 0 and reorganizing A2t2 1 + the equation : c2 At Thus, the dynamical meaning of ”uniformly acceler- v(t) = r (9) A2t2 ated movement” still holds in special relativity. 1 + c2

III. HOW TO BUILD A KINETICAL MEANING which is the same as (6). WITHOUT BEING IN CONFLICT WITH SPECIAL RELATIVITY Thus we have demonstrated that it is possible to hold a kinematical point of view in the meaning of ”uniformly We have seen in the previous section that the only accelerated movement”. But it is necessary to boost at definition which seems to hold in special relativity is every moment to the proper frame in order to have a the dynamical one. Here we present another kinetical correct definition. definition which will be consistent with special relativity, and is equivalent to that of Newtonian in the v << c limit. Acknowledgments Consider R as our inertial frame, and R0 as the proper frame of the mobile studied. Recall that R0 is at each We wish to acknowledge Landau and Lifshitz [1] for time the tangential frame of the mobile, and thus is con- having pushed us in those thinkings, Mederic Pigeard stantly changing. We will restrict ourselves to a privi- for his grateful discussion and FuturaSciences [2] for its leged axis zˆ. Then helping and confirmation of our discussion. 3

APPENDIX: LORENTZ TRANSFORMATION OF inertial frame R0 in uniform rectilign translation at con- THE ACCELERATION stant speed V along the z − axis in R-frame. Then with 1 γ = r , we have We present there the Lorentz transformation of the V 2 1 − acceleration. Let’s take an inertial frame R and another c2

0 0 dvx0 a 0 = = a x dt0 x 0 0 dvy0 a 0 = = a (A.1) y dt0 y  

0 0 dvz0 az  1 V (vz − V )  az0 = =  +  dt0 γ  V v 2  V v 3   1 − z c2 1 − z  c2 c2

[1] L. D. Landau and E. Lifshitz, Classical Field Theory (1964). [2] FuturaSciences (2007), forums.futura- sciences.com/thread147381.html.