**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 d*4*rd*4*s2 drs* the metric tensor of special relativity. For finitely separated
events the time registered by the clock is represented as

*sA2 sB*4

*B*

*A*

*ds 4*

*v*1

*v*2

## 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.

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) ,* *s*0*G s G s*1,

(2.1)

*joining two fixed events E*0*, E*1and satisfying the side conditions
*W*1*4 hrsx*
.*r*
*x*.*s*_{2 1 4 0 ,}
*(2.2a)*
*W*2*4 hrsx*
O_{r}*x*O*s*
*1 a*2*(s) 40*
*(2.2b)*

and end conditions

C*m*

### (

*s*0

*, xr(s*0

*), x*.

_{r}*(s*0

*), xr(s*1

*), x*.

_{r}*(s*1)

### )

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

*a trajectory E*01which minimizes the definite integral
*I 4*

*s*0

*s*1

*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 c*2*. Naturally, the minimizing trajectory E*01is an extremal for which

the condition

*x*.4

D 1 is fulfilled.

*If we agree to measure the proper time s always from the event E*0, without loss of

*generality, we may take s*0*4 0 . Evidently, the propert time s*1 *at the event E*1 is

*undetermined. For the problem under consideration, not only the end events E*0*, E*1are

fixed but also the 4-velocities at these events, condition (2.3) may then be written in the
form
*s*0*4 x*0*r2 a*0*r4 x*
.* _{r}*
0

*2 b*0

*r4 x*1

*r2 a*1

*r4 x*.

*1*

_{r}*2 b*1

*r*4 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 s*1 is undetermined. However, it differs from the usual type of

*non-para-metric problem by the fact that first and second derivatives of xr*_{enter 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 jr _{related to the x}r*

_{by 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*
F1*4 hrsjrj*
.* _{s}*
4 0 ,

*(2.6a)*F24

### y

*2hrs*

*j*.

*rj*.

*s*

*a*2

### z

1 2 2 1 4 0 ,*(2.6b)*

*s*0

*4 x*0

*r2 a*0

*r4 jr*0

*2 b*0

*r4 x*1

*r2 a*1

*r4 jr*1

*2 b*1

*r*4 0 . (2.7)

We have written F2*in the form (2.6b) to render F*21 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 a*2
(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 _{, j}r*

_{), 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 l*0*, e*0*, er, er, er, er* with multipliers

*pr(s), l*1*(s), l*2*(s). The functions F and G can be taken in the form*

(2.9) * _{F 4l}*1

*(s) hrsjrj*.

_{s}*1 pr(s)[jr2 x*.

*r*

*] 1l*2

*(s)*

### y

### u

*2h*

*rsj*.

*r*.

_{j}*s*

*a*2

### v

1 2 2 1### z

, (2.10)*0*

_{G 4l}*s*1

*1 e*0

*s*0

*1 er(x*0

*r2 a*0

*r) 1er(j*0

*r2 b*0

*r) 1er(x*1

*r2 a*1

*r) 1er(jr*1

*2 b*1

*r*)

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 l*2*is homogeneous of the first degree in x*

._{r}

*and j*.*r*_{it follows that the relation}

*F(k x*.*r _{, k j}*.

*r*

*) 2prjr1 l*2*4 k[F(x*

._{r}*, j*.*r*

*) 2prjr1 l*2]

*holds, If we differentiate this expression with respect to k and put k 41 we have the*
useful identity
*x*.*r _{F}*

*x*.

*r1 j*.

_{r}*Fj*.

*r4 F 2 p*

_{r}jr1 l_{2}. (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}

*x*4 0 , *X 4 (X*1*, R , X*8*) 4 (x*1*, R , x*4*, j*1*, R , j*4) .
(2.12)

*2) At the end values X*0*4 (x*0*r, jr*0*) and X*1*4 (x*1*r, jr*1) 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]j*f*F*

### (

*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 s*0

*and s*1. 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*.*r*4 0 ,
*(2.12a)*
*2l*2*hrs*
*j*
O
*s*
*a*2

### u

*l*. 2

*2 2 l*2

*a*.

*a*

### v

*hrs*

*js*

*a*2

*1 l*. 1

*hrsjs2 pr*4 0 ,

*(2.12b)*

First end point 0 Second end point 1

*a) e*0*1 x*
.* _{r}*
0

*er1 j*.

*0*

_{r}*er*4 0

*d) l*0

*1 x*.

*1*

_{r}*er1 j*.

*1*

_{r}*er*4 0

*b) 2pr4 er*

*e) pr4 er*(2.138)

*c) l*1

*( 0 ) hrsjs*0

*2 l*2

*( 0 ) hrs*

*j*.

*s*0

*a*2

*4 er*

*f ) l*1

*( 1 ) hrsj*

*s*1

*2 l*2

*( 1 ) hrs*

*j*.

*s*1

*a*21

*4 2er*.

*From eq. (2.12a) it is clear that*

*pr*4 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 _{, j}r_{) it must be normal, i.e.}*

*l*0c0 . For the deduction of this conclusion we shall find the following two relations

*connecting l*2*, l*1*and prconveniently. Multiplying eq. (2.12b) by jr*and using eqs. (2.2),
*(2.6a), (2.8) we obtain*

*2l*2*1 l*

.

1*4 prjr*.
(2.15)

*The second relation, connecting l*2*and 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*.2*4 prj*
._{r}

. (2.16)

Equations (2.15) and (2.16) imply

*22 l*.2*1 l*

O

14 0 .

(2.17)

*Now if l*04 0 , it results from the transversality equations (2.138) that
*er4 er4 pr4 er4 l*1*( 1 ) 4l*2( 1 ) 40 .

(2.18)

*With pr*4 0 , eqs. (2.15), (2.16) reduce to

*l*.24 0 , *2l*2*1 l*

.

14 0 .

(2.19)

*Since l*2*( 1 ) 4l*1( 1 ) 40, eqs. (2.19) entail

*l*2*(s) 4l*1*(s) 40 .*

(2.20)

*The vanishing of l*2 *and l*1 *implies the vanishing of the multipliers er* *and e*0 in

eqs.

### (

*2.138a*

### )

,### (

*2.138c*

### )

*. As a consequence of l*04 0 , all the multipliers are equal to zero,

*but this is contrary to the multiplier rule. Hence l*0c0 and we can accordingly set
*l*04 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 l*1

*, l*2

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

*4-hyperacceleration pr*_{defined by [9]}
*gr*
*4 j*.*r4 anr*, *jr _{n}*

*r*4 0 ,

*nrnr*4 21 ,

*(3.1a)*

*sr*

*4 g*.

*r4 a*2

*jr*

*1 a*.

*nr*

*1 abbr*,

*jr*

_{b}*r4 brnr*4 0 ,

*brbr*4 21 ,

*(3.1b)*

*pr*

*4 s*.

*r4 [ 3 a a*.

*] jr*

*1 [a*3

*1 a*O

*2 ab*2

*] nr*

*1 [ 2 a*.

*b 1ab*.

*] br*

*2 abgcr*,

*(3.1c)*

*jr*

_{c}*r4 nrcr4 brcr*4 0 ,

*crcr*4 21 ,

*where nr*

_{, b}r_{, c}r_{are defined by the FS formulas [1]}

*n*.*r _{4 aj}r_{1 bb}r*,

*b*.

*r*,

_{4 2bn}r_{2 gc}r*c*.

*r*. (3.18)

_{4 gb}r*At each event on the world line L of the particle the vectors jr _{, n}r_{, b}r_{and c}r*

_{form 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*.1*2 l*2*] jr*1

### y

*l*2

*a*.

*a*

*2 l*. 2

### z

*nr*

*a*2

*l*2

*b*

*a*

*b*

*r*

*4 pr*. (3.2)

*Comparison of pr*

_{n}*r*4

*l*.2

*a*

*2 l*2

*a*.

*a*2 with eq. (2.16) gives
*l*2*a*

.

*a*2 4 0

(3.3)

*with a c 0 . Since the vanishing of l*2*implies 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*.*r*_{4 0 , we find, by differentiation of eq. (3.2)}
*with respect to s, that the multipliers l*1 *and l*2 must satisfy the following equations:

*l*O1*2 l*
.
2

### g

1 1 1*a*

### h

4 0 ,*(3.5a)*

*l*.2

*b*

*a*

*1 (l*2

*b)*. 4 0 ,

*(3.5b)*

*l*.1

*2 l*2

### g

1 2*b*2

*a*2

### h

2*l*O2

*a*3 4 0 ,

*(3.5c)*

*l*2

*bg*

*a*4 0 .

*(3.5d)*

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

*g 40 .*

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 p}r*corresponding, respectively, to b 40 and bc0.*

*Case 1. If b 40, eqs. (3.5) read*
*l*
O
2*2 l*
.
24
*l*.2
*a* ,
*(3.58a)*
*l*.1*2 l*24
*l*
O
2
*a*3 .
*(3.58b)*

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

*l*24 const

*(3.7a)*
or

*l*2*4 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*.1*4 l*24 const .

(3.8)

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

*pr*_{4 0 .}

(3.9)

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

*sr*
*4 a*2*jr*
(3.10)

*and the hyperacceleration pr*_{takes the form}

*pr*

*4 a*3*nr*_{.}
(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 p}r*
spacelike [10].

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

*a*

*1 1a* .
(3.12)

*To state some definite result we suppose b 4const; this implies the constancy of l*2

*and eq. (3.5c) simplifies by the vanishing of the last term so that eq. (3.2) reads*
*2l*2
*b*2
*a*2*j*
*r*
*2 l*2
*b*
*ab*
*r*
*4 pr*.
(3.13)

*Now, for a and b constant, we have for sr _{and p}r*

_{the following expressions:}

*sr*

*4 a*2

*jr*

*1 abbr*,

*(3.14a)*

*pr*

*4 [a*3

*2 ab*2

*] 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 sr _{is a null vector, b) the multiplier p}r*

_{is proportional to s}r_{, c) equality}*is achieved for l*2

*4 2 a*2.

*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 p}r_{, is the uniformly}
*accelerated motion which is characterized by the vanishing of the second curvature b of*
the trajectory.

3) Accelerated motions with *sr _{s}*

*r*G 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] SYNGE*J. L., Relativity: The General Theory (North-Holland) 1960.*

[2] SYNGEJ. L. and GRIFFITH*B. A., Principles of Mechanics (McGraw-Hill) 1960.*
[3] ROHRLICH*F., Classical Charged Particles (Addison-Wesley) 1965.*

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

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

[6] EWING*G. 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 SCARPETTA**G., J. Math. Phys., 36 (1995) 5562.**

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