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Optimality of relativistic motions (*)

R. A. KRIKORIAN

Collège de France, Institut d’Astrophysique - 98bis Bd. Arago, 75014 Paris, France (ricevuto il 27 Settembre 1996; approvato il 20 Novembre 1996)

Summary. — By minimizing the elapsed time registered by an accelerated clock between two fixed events in time sequence, we determine a family of “suspected” time optimal motions. Because of the constraints imposed by the theory of relativity, this minimization requires the solution of an equivalent Lagrange problem with variable end point in a 9-dimensional space. Expressing the Euler necessary condition in terms of the components of the Frenet-Serret tetrad and curvatures of the trajectory it is shown that a) an optimal motion is a motion of constant proper acceleration, and b) a physically important realization of an optimal motion is the uniformly accelerated motion.

PACS 04.20 – Classical general relativity.

1. – Introduction

In the theory of relativity, the measurement of time along the world line of any material particle is provided by the ticking of a standard clock carried by the particle. According to the chronometric assumption, the time interval between two adjacent events in the history of an accelerated clock is assumed to be given by the space-time separation between these events [1-3]

ds 4

k

hrsdxrdxs, r , s 41, 2, 3, 4 , (1.1)

with hrs4 2 d4rd4s2 drs the metric tensor of special relativity. For finitely separated events the time registered by the clock is represented as

sA2 sB4



B A ds 4



v1 v2

o

hrs dxr dw dxs dw dw , (1.2)

where w is a parameter increasing from past to future along the world line of the clock. A direct consequence of the above assumption is the well-known result that the elapsed

(*) The author of this paper has agreed to not receive the proofs for correction.

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time between two events, as read by an accelerated clock, is less than that recorded by a clock moving with constant velocity. This path dependance raises the question of the existence of a privileged family of clock trajectories, those for which the elapsed proper time is minimum.

The determination of the class of accelerated motions, suspected of optimality, requires the solution of a special type of non-parametric Lagrange problem with variable end point in a 9-dimensional space. We shall see that the class of optimal motions may be characterized by the constancy of the proper acceleration or, equivalently, in geometrical terms, by the constancy of the first curvature of the trajectory. A particularly important physical realization of an optimal motion is the uniformly accelerated motion, characterized by the vanishing of the second and third curvatures of the trajectory. In sect. 2 we present the formalization of the variational problem and give the mathematical prerequisites for the treatment of the Lagrange multipliers. We derive the Euler and transversality equations for the problem under consideration and discuss the normality of the set of multipliers. In sect. 3 we formulate the Euler equation in terms of the components of the Frenet-Serret tetrad and derive a set of conditions which the candidates for optimality have to satisfy.

2. – Statement of the problem and mathematical prerequisities

2.1. Formalization. – The variational problem we study is that of finding in the class

M of space-time trajectories xr

4 xr(s) , s0G s G s1,

(2.1)

joining two fixed events E0, E1and satisfying the side conditions W14 hrsx .r x.s2 1 4 0 , (2.2a) W24 hrsx Or xOs 1 a2(s) 40 (2.2b)

and end conditions

Cm

(

s0, xr(s0), x .r (s0), xr(s1), x .r (s1)

)

4 0 , m 41 R p417, 2 GpG18 (2.3)

a trajectory E01which minimizes the definite integral I 4



s0

s1

ds . (2.4)

Dots denote differentiation with respect to s. Condition (2.2a) not only specifies the time-like nature of the trajectory, it also restricts the parameter s to be the proper time. Condition (2.2b) is a mathematical translation of the space-like nature of the 4-acceleration. The function a(s) appearing in eq. (2.2b) has the following physical meaning: it is equal to the magnitude of the 3-acceleration, in the instantaneous rest frame, divided by c2. Naturally, the minimizing trajectory E01is an extremal for which

the condition

x.4

D 1 is fulfilled.

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If we agree to measure the proper time s always from the event E0, without loss of

generality, we may take s04 0 . Evidently, the propert time s1 at the event E1 is

undetermined. For the problem under consideration, not only the end events E0, E1are

fixed but also the 4-velocities at these events, condition (2.3) may then be written in the form s04 x0r2 a0r4 x .r 02 b0r4 x1r2 a1r4 x .r 12 b1r4 0 (2.3a) with xr i (respectively x .r i) 4xr(si)

(

respectively x .r (si)

)

, i 40, 1.

Our problem is thus a non-parametric problem with variable end point since the end value s1 is undetermined. However, it differs from the usual type of

non-para-metric problem by the fact that first and second derivatives of xrenter in the end and side conditions. In order to recover the usual non-parametric form, we transform it into an equivalent problem by introducing four further variables jrrelated to the xrby the differential equations [4]

x.r

(s) 4jr(s) (2.5)

and we replace the side and end conditions (2.2a), (2.2b) and (2.3a) by F14 hrsjrj .s 4 0 , (2.6a) F24

y

2hrs j.rj.s a2

z

1 2 2 1 4 0 , (2.6b) s04 x0r2 a0r4 jr02 b0r4 x1r2 a1r4 jr12 b1r4 0 . (2.7)

We have written F2in the form (2.6b) to render F21 1 homogeneous of the first degree

in j.r, a property which will be useful.

Differentiating eq. (2.6a) with respect to s and taking into account eq. (2.6b) we obtain the following relation:

hrsjrj O s

4 a2 (2.8)

which we shall also use later. With this formulation, the equivalent problem is a non-parametric Lagrange problem with variable end point, in the space of points (s , xr, jr), of a special type since the integrand in (2.4) is equal to unity. Such problems are known as time optimal problems.

2.2. The Euler and transversality equations. – For the classical Lagrange problem with variable end point, the multiplier rule is the combined result of two theorems, the Euler necessary condition and the transversality condition. The results of this rule may be stated with the help of two functions F and G [5-7]. For the problem under consideration we shall need constant multipliers l0, e0, er, er, er, er with multipliers

pr(s), l1(s), l2(s). The functions F and G can be taken in the form

(2.9) F 4l1(s) hrsjrj .s 1 pr(s)[jr2 x .r ] 1l2(s)

y

u

2h rsj . rj.s a2

v

1 2 2 1

z

, (2.10) G 4l0s11 e0s01 er(x0r2 a0r) 1er(j0r2 b0r) 1er(x1r2 a1r) 1er(jr12 b1r)

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repeated indices in (2.9) and (2.10) are understood to indicate summation from 1 to 4. Relativity appears only in the demand that F and G should be invariant under Lorentz transformations. This requirement fixes the relativistic variance of the multipliers. The function F defined by eq. (2.9) possesses the following important property. Since F 2

prjr1 l2is homogeneous of the first degree in x

.r

and j.rit follows that the relation

F(k x.r, k j.r

) 2prjr1 l24 k[F(x

.r , j.r

) 2prjr1 l2]

holds, If we differentiate this expression with respect to k and put k 41 we have the useful identity x.rF x.r1 j .r Fj.r4 F 2 prjr1 l2. (2.11)

From the multiplier rule we know that [5-7]:

1) For any s at which X. 4 (x.r, j.r) and hence the multipliers are continuous, the extremals of the equivalent problem must satisfy the Euler equation

d dsFx

.2 F

x4 0 , X 4 (X1, R , X8) 4 (x1, R , x4, j1, R , j4) . (2.12)

2) At the end values X04 (x0r, jr0) and X14 (x1r, jr1) the following transversality

equations hold:

.

`

/

`

´

¯G ¯Xs j 4 (21 )j[Fs]j, j 40, 1, s41, R, 8 , ¯G ¯sj 1 X .s j ¯G ¯Xs j 4 (21 )j[F]j summed on s , (2.13)

where we have set [F]jfF

(

sj, Xj, X .

j, pr(sj), li(sj)

)

, j 40, 1, i41, 2, r41, R, 4 and use [Fs]j for the components of FX. evaluated at s0 and s1. Moreover, the multipliers

cannot vanish simultaneously.

With reference to eqs. (2.9) and (2.10) one finds that the Euler and transversality equations take the following forms:

p.r4 0 , (2.12a) 2l2hrs j O s a2

u

l . 22 2 l2 a. a

v

hrs js a2 1 l . 1hrsjs2 pr4 0 , (2.12b)

First end point 0 Second end point 1

a) e01 x .r 0er1 j .r 0er4 0 d) l01 x .r 1er1 j .r 1er4 0 b) 2pr4 er e) pr4 er (2.138) c) l1( 0 ) hrsjs02 l2( 0 ) hrs j.s 0 a2 4 er f ) l1( 1 ) hrsj s 12 l2( 1 ) hrs j.s 1 a21 4 2er.

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From eq. (2.12a) it is clear that

pr4 const (2.14)

which we have taken into account in writing down eqs.

(

2.138b

)

,

(

2.138e

)

.

We note that if there is a minimizing function X 4 (xr, jr) it must be normal, i.e.

l0c0 . For the deduction of this conclusion we shall find the following two relations

connecting l2, l1and prconveniently. Multiplying eq. (2.12b) by jrand using eqs. (2.2), (2.6a), (2.8) we obtain

2l21 l

.

14 prjr. (2.15)

The second relation, connecting l2and pr, may be obtained as follows. We differentiate eq. (2.11) with respect to s and use the Euler equation (2.12); this yields

l.24 prj .r

. (2.16)

Equations (2.15) and (2.16) imply

22 l.21 l

O

14 0 .

(2.17)

Now if l04 0 , it results from the transversality equations (2.138) that er4 er4 pr4 er4 l1( 1 ) 4l2( 1 ) 40 .

(2.18)

With pr4 0 , eqs. (2.15), (2.16) reduce to

l.24 0 , 2l21 l

.

14 0 .

(2.19)

Since l2( 1 ) 4l1( 1 ) 40, eqs. (2.19) entail

l2(s) 4l1(s) 40 .

(2.20)

The vanishing of l2 and l1 implies the vanishing of the multipliers er and e0 in

eqs.

(

2.138a

)

,

(

2.138c

)

. As a consequence of l04 0 , all the multipliers are equal to zero,

but this is contrary to the multiplier rule. Hence l0c0 and we can accordingly set l04 1 .

All the useful information coming from the multiplier rule having been obtaianed, we henceforth return to the space-time viewpoint and shall associate with each point of the clock trajectory a particularly interesting tetrad

(

the so-called Frenet-Serret (FS) frame

)

. The reformulation of the Euler equation (2.12b) in terms of the tetrad components will permit to derive a set of equations satisfied by the multipliers l1, l2

and the curvatures of the world line. A recent application of the method of moving frames to the study of the Euler equations, with Lagrangians depending on the acceleration, is presented in ref. [8].

3. – Optimal motions

On the basis of Newtonian analogy, the dynamical state of a particle with 4-velocity

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4-hyperacceleration prdefined by [9] gr 4 j.r4 anr, jrn r4 0 , nrnr4 21 , (3.1a) sr 4 g.r4 a2jr 1 a.nr 1 abbr, jrb r4 brnr4 0 , brbr4 21 , (3.1b) pr 4 s.r4 [ 3 a a.] jr 1 [a31 aO2 ab2] nr 1 [ 2 a.b 1ab.] br 2 abgcr, (3.1c) jrc r4 nrcr4 brcr4 0 , crcr4 21 , where nr, br, crare defined by the FS formulas [1]

n.r4 ajr1 bbr, b.r4 2bnr2 gcr, c.r4 gbr. (3.18)

At each event on the world line L of the particle the vectors jr, nr, brand crform an orthonormal tetrad or FS frame. By analogy with the FS formulas for a curve in an

n-dimensional space, the scalars a, b, g are called the curvatures of the world line.

Resolving pr, as given by eq. (2.12b), into its components along the FS frame we get [l.12 l2] jr1

y

l2 a. a 2 l . 2

z

nr a 2 l2b a b r 4 pr. (3.2) Comparison of prn r4 l.2 a 2 l2 a.

a2 with eq. (2.16) gives l2a

.

a2 4 0

(3.3)

with a c 0 . Since the vanishing of l2implies the constancy of the 4-velocity jr, we infer

from eq. (3.3) that a time optimal motion must satisfy the condition

a 4const .

(3.4)

Consequently, in view of the fact that p.r4 0 , we find, by differentiation of eq. (3.2) with respect to s, that the multipliers l1 and l2 must satisfy the following equations:

lO12 l . 2

g

1 1 1 a

h

4 0 , (3.5a) l.2b a 1 (l2b) . 4 0 , (3.5b) l.12 l2

g

1 2 b2 a2

h

2 lO2 a3 4 0 , (3.5c) l2bg a 4 0 . (3.5d)

It is clear from eq. (3.5d ) that for b c 0 we must have

g 40 .

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We go further and suppose that condition (3.6) holds also for the limiting value

b 40. Thus, for a fixed value of a, we have a one-parameter family Fa(b) of suspected optimal motions. Let us now determine the expressions of sr and pr corresponding, respectively, to b 40 and bc0.

Case 1. If b 40, eqs. (3.5) read l O 22 l . 24 l.2 a , (3.58a) l.12 l24 l O 2 a3 . (3.58b)

The solution of eqs. (3.58a) (3.58b) is either

l24 const

(3.7a) or

l24 K Q e6as.

(3.7b)

We retain only the solution (3.7a) since it is one which represents a physical interest. From eq. (3.58b) we get

l.14 l24 const .

(3.8)

If we put eq. (3.8) into eq. (3.2) we obtain, on account of the constancy of a,

pr4 0 .

(3.9)

Returning to the Euler equation (2.12b), one can write

sr 4 a2jr (3.10)

and the hyperacceleration prtakes the form

pr

4 a3nr. (3.11)

We recognize in eq. (3.10) the covariant definition of the uniformly accelerated motion first established by Hill. We note that for such a motion sr is timelike and pr spacelike [10].

Case 2. If b c 0 , from eq. (3.5b) we obtain l2ba4 const , a 4

a

1 1a . (3.12)

To state some definite result we suppose b 4const; this implies the constancy of l2

and eq. (3.5c) simplifies by the vanishing of the last term so that eq. (3.2) reads 2l2 b2 a2j r 2 l2 b ab r 4 pr. (3.13)

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Now, for a and b constant, we have for srand prthe following expressions: sr 4 a2jr 1 abbr, (3.14a) pr 4 [a32 ab2] nr. (3.14b)

From these equations we see that if a 4b, a) the suspected optimal motion may be characterized by the vanishing of the hyperacceleration pr and the fact that the suracceleration sris a null vector, b) the multiplier pr is proportional to sr, c) equality is achieved for l24 2 a2.

Concluding remarks.

1) Accelerated motions competing for optimality are motions of constant proper acceleration whose space-time trajectories have a constant first curvature a and a vanishing third curvature g (the analogue of torsion).

2) A particularly important physical realization of a time optimal motion, with a timelike suracceleration sr and a spacelike hyperacceleration pr, is the uniformly accelerated motion which is characterized by the vanishing of the second curvature b of the trajectory.

3) Accelerated motions with srs

rG 0 and b differing from zero are mathematically admissible candidates for optimality; however, the physical interest of such optimal motions must be established.

* * *

We do thank S. KICHENASSAMY for valuable discussions.

R E F E R E N C E S

[1] SYNGEJ. L., Relativity: The General Theory (North-Holland) 1960.

[2] SYNGEJ. L. and GRIFFITHB. A., Principles of Mechanics (McGraw-Hill) 1960. [3] ROHRLICHF., Classical Charged Particles (Addison-Wesley) 1965.

[4] CARATHEODORYC., Calculus of Variations and Partial Differential Equations of the First Order (Chelsea) 1982.

[5] BLISSG. A., Am, J. Math., 52 (1930) 673; Lectures on the Calculus of Variations (University of Chicago Press) 1946.

[6] EWINGG. M., Calculus of Variations with Applications (W. W. Norton) 1969 (subsequent reprint by Dover, 1985).

[7] ALEXEEV V., GALEEV E. and TIKHOMIROV V., Recueil de problèmes d’optimisation (Mir) 1987.

[8] NESTERENKOV. V., FEOLIA. and SCARPETTAG., J. Math. Phys., 36 (1995) 5562.

[9] KICHENASSAMYS., C. R. Acad. Sci. Paris, 260 (1965) 3001; Symposia on Theoretical Physics, Vol. 107 (Plenum Press) 1967, p. 137.

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