gravitation and cosmology bell

ISSN 0202-2893, Gravitation and Cosmology, 2010, Vol. 16, No. 4, pp. 307–312. c© Pleiades Publishing, Ltd., 2010.

A Study of the Motion of a Relativistic Continuous Medium

S. A. Podosyonov1, J. Foukzon2, and A. A. Potapov3

1Aprelevka, Naro-Fominsky rayon, Moscow region 143360, Russia2Israeli Institute of Technologies

3V.A. Kotel’nikov Institute of Radioelectronics of the Russian Academy of SciencesReceived March 31, 2010; in final form, June 11, 2010

Abstract—The main purpose of the present paper is to give an exact and correct expression describing theproperties of the proper length in arbitrary relativistic translationally moving media in Minkowski space.We show, in particular, that the standard solution of Bell’s well-known problem [1] must be revised. A newsolution has been found, describing the behavior of a finite physical length in the Lagrangian non-inertialreference frame comoving to the medium. This solution is absent in the existing literature. We concludethat, in the case of large enough accelerations a0 and initial distances between some points of the medium,i.e., under the condition u ≡ a0L0/c2 � 1, where c is the speed of light, the calculations presented in somewell-known papers (namely, [1, 2, 10–12]) are incorrect and should be revised. For the velocity valuesu � 1, our results and those of all the enumerated papers coincide.

DOI: 10.1134/S0202289310040080

1. INTRODUCTION

In a moving continuous medium, e.g., a bunch ofelectrons in a constant electric field, with zero initialvelocity in an inertial reference frame (IRF), where theinteraction between the electrons is regarded small ascompared with their interaction with the external field,the bunch length does not change. This is shownin J. Bell’s problem [1], where the role of a bunch isplayed by a thread which connects identical pointlikerockets in linearly accelerated motion. The solution[1] is also used in calculations of bunch motion inlinear colliders [2]. However, in the non-inertial ref-erence frame (NRF) comoving with the bunch, orwith the thread in Bell’s problem, there is no correctexpression for a finite instantaneous length in specialrelativity (SR). (We will use the Minkowski spacesignature (+−−−), the Greek indices will vary from0 to 3 and Latin ones from 1 to 3.) The standardexpression for finding an element of physical distancedL2 obtained with the aid of the spatial metric tensor,

γik = −gik +g0ig0k

g00(1)

is used incorrectly. The claim [3] that in generalrelativity (GR) integration of the element dL is mean-ingless, is true. Unlike that, in SR, integration alonga hypersurface orthogonal to the world lines doesmake sense. In such cases it is necessary to take intoaccount that, according to the Gauss and Peterson-Codazzi equations [4] in SR, a hypersurface orthog-onal to the world lines (the Lagrangian comoving

NRF) is curved, and the curvature is entirely ex-pressed in terms of the second fundamental tensor ofthe hypersurface. In [5–8] it has been shown that thistensor is identical to the strain rate tensor. Therefore,in calculating the instantaneous lengths between anytwo particles of the continuum it is necessary to takeinto account the curvature of the spacelike hyper-surfaces orthogonal to the world lines of a particleof the continuum. In the Euler coordinates xµ, thecurvature tensor of the hypersurface has the form

∗Rµν,γσ = ΣνγΣµσ − ΣµγΣνσ, (2)

where the strain rate tensor Σαβ is orthogonal to the4-velocity V α normalized to 1,

Σµν = ∇(µVν) − V(µFν), (3)

where Fµ is the 4-acceleration.In the Lagrangian coordinate yk, the strain rate

tensor is related to the second fundamental tensor ofthe hypersurface bkl:

bkl =12

∂γkl

∂s= Σ̂kl. (4)

Equation (4) establishes a relationship between thefirst and second fundamental tensors on the relevanthypersurface.

Due to orthogonality of the 4-velocity to the ex-pression (3), the 4-dimensional tensor lies in the 3-dimensional curved spatial hypersurface orthogonalto the basic particles’ world lines, which describesthe physical 3-dimensional space of the Lagrangian

307

308 PODOSYONOV et al.

comoving NRF. The existence of such a hypersurfacehas been proved by J.L. Synge [16]. The integrabilitycondition for its existence is the opportunity to solvethe Pfaff equation

Vµdxµ = 0. (5)

As remarked by Synge, “It follows that, in the caseof non-rotational motion, the current lines form anormal congruence; in other words, there is a family ofthree-dimensional surfaces to which the current linesare orthogonal. It has allowed for introducing the lawof translational motion of a continuum in Lagrangevariables in the form [17]

xµ = fµ(yν), (6)

where the Euler variables xµ are the Cartesian(Galilean) coordinates in Minkowski space, and yk

are the Lagrange coordinates which are constantalong each world line of a particle of the continuum(numbers of the world lines), while y0 is a parameterthat enumerates the hypersurfaces orthogonal to theworld lines. The parameter y0 is equivalent to theglobal time of the Lagrangian comoving NRF. Atconstant y0, the proper times τ of the continuumparticles are, in general, different, as are different thetimes t in the IRF as well. If we choose the parametery0 as a time parameter, then, after transition to thecomoving NRF, the cross-terms g0k will be absent inthe metric, which substantially simplifies all proofs.However, one can choose any time parameter in theabove law of motion. One often chooses the propertime τ or the IRF time t. In this case, there emergenonzero cross-terms g0k .

Let usmake clear themeaning of the spatial metrictensor (1), choosing in (6) an arbitrary time parametery0. In [3], the spatial metric (1) has been obtainedusing the method of radar signal propagation be-tween two infinitely close world lines of particle ofthe medium. We wish to show that the spatial dis-tance between two close world lines from [3] preciselycoincides with the spatial distance between theseworld lines calculated in the hypersurface orthogonalto these world lines. We start from the interval inthe Cartesian coordinates of Minkowski space, rep-resented in the form [18]

dS2 = ηµνdxµdxν = VµVνdxµdxν

− (−ηµν + VµVν) dxµdxν . (7)

Using the law of motion (6), the obvious relations

V µ = Θ∂xµ

∂y0, ηµνV

µV ν = 1 = Θ2g00,

gµν = ηαβ∂xα

∂yµ

∂xβ

∂yν, (8)

γµν ≡ −(ηµν − VµVν), γµνV µ = 0, (9)

and the conditions that the Lagrangian NRF is co-moving,

V̂ 0 =1

√g00

, V̂0 =√

g00,

V̂ k = 0, V̂k =g0k√g00

, (10)

we obtain the following expression for the squaredinterval (7) in the Lagrangian comoving NRF:

dS2 =(√

g00dy0 +g0k√g00

dyk

)2

−(−gkl +

g0kg0l

g00

)dykdyl. (11)

The squared interval (11) coincides withZel’manov’s [13]. One should stress that in SRthis splitting has a direct physical meaning. In GR,the physical meaning of (11) is not evident sincethe initial coordinates, like the Cartesian ones inSR, are absent. The relation (7) for an elementof spatial length in Cartesian coordinates containsthe metric tensor γµν (9), and it coincides with theprojection operator which is everywhere orthogonalto the particle world lines. It means that it is thehypersurface orthogonal to the world lines that formsthe physical three-dimensional space.

A very important circumstance is that the radarmethod [3] has led us to the expression (1) which isidentical to the spatial interval in (11). Thus the radarmethod, being applied to two infinitely close particleworld lines, has led to the fact that the element ofa spatial physical interval lies on the hypersurfaceorthogonal to the world lines, hence, the whole lengthmust be equal to a sum of length elements on this hy-persurface. If the motion of the continuum is differentfrom rigid motion in Born’s sense, then the physicalspace is curved (2). A neglect of this circumstanceleads to errors in calculations of an instantaneousphysical distance.

2. THE SUGGESTED METHODOF CALCULATION

The calculations are performed in the simplestmanner for one-dimensional motion. Let us consider,for instance, Bell’s well-known problem where twoidentical pointlike rockets simultaneously (by IRFclocks) begin to move in the same direction, onefollowing the other, with constant and equal accel-erations (in the astronauts’ reference frame). Sup-pose that these rockets are connected by a rubber(a thread) which does not affect their motion. It isrequired to determine how the instantaneous distancebetween the rockets changes from the viewpoint ofthe astronauts. In the Minkowski plane, i.e., the

GRAVITATION AND COSMOLOGY Vol. 16 No. 4 2010

A STUDY OF THE MOTION 309

two-dimensional section of Minkowski space thatcontains the world lines of the rockets and those ofall particles of the thread, all these world lines fill acertain plane curvilinear band. The world line of eachparticle of the thread is obtained from another parti-cle’s world line by a shift along the axis of motion inthe laboratory IRF. However, the congruence of suchworld lines will not be rigid in Born’s sense. Speakingof the physical comoving space of the medium (thethread), we mean the spacelike line in the Minkowskiplane which is everywhere orthogonal to the worldlines of the thread and the rockets. Mathematically,it means that this line is everywhere orthogonal to the4-velocity field V µ of the bunch, i.e., holds the Pfaffequation

Vµdxµ = V̂νdyν = 0, µ, ν = 0, 1. (12)

The squared length element on the spacelike curvethat connects the two world lines, of which one be-longs to the backward rocket (at an arbitrary butfixed time instant t2 = t) and the front rocket (at thesought-for instant t1 of theworld time), obtained from(12), is

dL2 = γµνdxµdxν , (13)

taking into account that the metric tensor of thehypersurface is a projection operator to the surfaceorthogonal to V µ and γµν is taken from (9).

From (12), the equation for the spacelike curvereduces to

dx1

dx0=

√1 + β2

β, β =

a0 t

c. (14)

Its solution gives

x1(t) =c2

a0

(√1 + β2

− ln∣∣∣∣1 +

√(1 + β2)β

∣∣∣∣)

+ A. (15)

It is required to determine the times from the equa-tions of intersection of the spacelike curve (15) withthe world lines of the back and front particles. Thecurved world lines can be found from the law of mo-tion of the medium for the globally hyperbolic case [9]

x1(y1, t) = y1 + (c2/a0)[√

1 + a02t2/c2 − 1],

x2 = y2, x3 = y3, x0 = y0, (16)

or

x1(y1, τ) = y1 + c2/a0[cosh(a0τ/c) − 1],

x2 = y2, x3 = y3,

t = (c/a0) sinh(a0τ/c). (17)

which leads to the corresponding expressions for theinterval

dS2 =c2dt2

1 + a02t2/c2

− 2a0tdtdy1

(1 + a02t2/c2)1/2

− (dy1)2 − (dy2)2 − (dy3)2, (18)

dS2 = c2(dτ)2 − 2 sinh(a0τ/c)cdτdy1

− (dy1)2 − (dy2)2 − (dy3)2, (19)

From (16), y1 = 0 for the back particle and y1 = L0

for the front one. The quantity A is a parameter thatenumerates the curves orthogonal to theworld lines ofthe thread particles. The quantity A is constant alongeach curve and is excluded from the final result.

Excluding A from the equations and supposingthat, for theworld line of the back particle, an arbitrarytime instant t or β has been chosen, we find anequation for finding t1 or β1 at the intersection pointof the spacelike curve orthogonal to the thread worldlines with the world line of the front particle:

L0 =c2

a0ln

∣∣∣∣∣(1 +

√(1 + β2))β1

(1 +√

(1 + β21))β

∣∣∣∣∣ . (20)

Omitting simple but rather bulky algebraic transfor-mations, we find

β1

β= cosh

(a0L0

c2

)

+ sinh(

a0L0

c2

)√1 + β2, (21)

where L0 is the initial length of the thread. Let uscalculate the thread length L(t) as a function of theworld time. From (12)–(14) we find the expressionfor the length dL

dL =c2

a0

dt

t=

c2

a0

dβ

β. (22)

Integrating (22), we obtain

L(t) =c2

a0ln

(cosh

(a0L0

c2

)

+ sinh(

a0L0

c2

)√1 + β2

). (23)

Using the Pfaff equation (12) in the Lagrangian co-moving NRF, we arrive at the same result as givenby (22), while the spatial “physical” metric (1) is ob-tained in a natural way using the projection operator(9), equivalently to the radar method. Indeed,

dy1

dy0= − V̂0

V̂1

=1β

, (24)

GRAVITATIONAND COSMOLOGY Vol. 16 No. 4 2010


where we have used (10) and (18). From (14) itfollows

dy1

dx0=

dy1

dx1

dx1

dx0=

1

β√

1 + β2. (25)

Eqs. (9) and (16) yield

γµν∂xµ

∂yi

∂xν

∂yk= γik = −gik +

g0ig0k

g00,

which coincides with (1). From (1) and (25) we havefor one-dimensional motion

dL =√

1 + β2dy1

dx0dx0 =

c2

a0

dβ

β,

whence follows (23). A more general formula on thebasis of Zel’manov’s theory of chronometric invari-ants [13] has been suggested by J. Foukzon [14],and it also leads in a special case to the result (23).The coincidence follows from the fact that, in the La-grangian comovingNRF, the Pfaff equation is entirelyequivalent to the requirement that the chronometri-cally invariant time interval turns to zero. Indeed,from (10) and (11) we have(

√g00dy0 +

g0k√g00

)dyk

= V̂0dy0 + V̂kdyk = 0. (26)

3. ANALYSIS OF THE PREVIOUS STUDIES

To find the metric corresponding to the motion of alinearly accelerated continuous medium in an instan-taneously comoving inertial reference frame (ICIRF),one usually [9] proceeds from a pseudo-Euclideaninterval given in the Euler variables xµ expressed interms of the Lagrange variables yν in the form ofthe laws of motion (16), (17). From [9] we have thefollowing expressions for the “physical distance” atthe IRF time t:

dL2 = (1 + a02t2/c2)(dy1)2

+ (dy2)2 + (dy3)2, (27)

and for the proper time τ ,

dL2 = cosh2(a0τ/c)(dy1)2 + (dy2)2 + (dy3)2. (28)

The nonzero component of the strain rate tensor ofthe Lagrangian comoving system has the form (4),

Σ̂11 =a0

2c2sinh(2a0τ/c). (29)

The calculations on the basis of finding the threadlength in Bell’s problem using the ICIRF lead tosignificant errors for u � 1. The essence of sucherrors is that the straight line drawn in theMinkowskiplane along the spacelike unit vector of the ICIRF

on the world line of the back rocket in the directionof the front rocket, does not coincide in directionwith the spacelike unit vectors of the ICICR locatedat other points of the thread, including the pointwhere the front rocket is attached [11]. This erroris a consequence of neglecting the curvature of thespacelike line connecting the rocket world lines in theMinkowski plane. The spacelike straight line alongthe ICIRF spatial unit vector, connecting the rocketworld lines, does not belong to the physical space butis only tangent to it at one point (the ICIRF origin).Similar calculations of the thread length are pre-sented in [10], where the ICIRF origin was located onthe front rocket. Since the unit vectors of the ICIRFbases considered in [10] and [11] were not parallel,the calculation results were different. Another way ofcalculating the thread length was used in [2, 12]: thestandard metric (1) was used, and its length element,as we have shown above, lies in the curved spacelikeline, orthogonal to the world lines of the thread andthe rockets. However, for finding the length usingEq. (27), the spacelike line (the integration path) waschosen to be the straight line t = const, dy2 = dy3 =0. This led to the result

2L(t) = L0

√1 + a0

2t2/c2 =L0√

1 − v2(t)/c2. (30)

In other words, an infinitesimal Lorentzian length-ening of the thread in the Lagrangian comovingNRF was wrongly replaced with a finite lengthening,which is only justified for u � 1. For comparison,we present the relative lengthening expressions, K =L/L0, according to [2, 10–12]:

K[10] =(1 + u cosh(γ) −

√1 + u2 sinh2(γ))

u, (31)

K[11] =(−1 + u cosh(γ) +

√1 + u2 sinh2(γ))

u,

(32)

K[2,12] = cos γ. (33)

where γ = a0τ/c. We see that the solution of Bell’sproblem has led to three radically different results.This indicates that there is no proper understandingin the problem of finding a finite length of the threadin Bell’s problem.

4. COMPARISON OF THE RESULTS

For comparison with the above enumerated stud-ies, let us rewrite Eq. (22) in a dimensionless form:

K =L(t)L0

=ln(cosh(u) + sinh(u) cosh(γ))

u,


A STUDY OF THE MOTION 311

cosh(γ) =√

1 + β2. (34)

For u � 1 and β of the order of unity we obtain theexpression

K = cosh(γ) = K[2,12] = K[10] = K[11]. (35)

connected with the ordinary Lorentz transformation.Quite a different situation emerges in the case of anultrarelativistic motion of a continuous medium (e.g.,for a bunch of charged particles in a linear collider).Since this problem has been treated in detail in [15](where Eqs. (16), (18) and (44) contain misprints bythe authors’ fault) and is not a subject of the presentpaper, we will only note that, in the ultrarelativisticcase, the difference in the expressions becomes verysignificant. If u � 1 and γ ∼ 10, which correspondsto a real case, then K[2,12] ∼ 104 and do not dependon u. For u ∼ 103, K[10] ∼ 10−3, K[11] ∼ 2 × 104. Inour case, for the specified values of the parameters uand γ, K ∼ 1.01. Since u is large and γ ∼ 10, theexpression for K is simplified and takes the form

K = 1 + (2/u) ln cosh(γ/2). (36)

Thus the length of an electronic bunch in our case,at the end of the acceleration process, taking into ac-count the curvature of the spacelike line, is practicallypreserved. The standard calculation according to [2,12], which neglects both the curvature of the worldline of the thread front end and the curvature of thespacelike curve, predicts a thread lengthening by afactor of 10 000 at the end of the acceleration in theLagrangian comoving NRF. The improved approachof [11] yields for this case a lengthening by a factorof 20000, and lastly, the approach of [10] predictspractical vanishing of the thread length at the end ofthe acceleration. Let us note that, for observers inan IRF, the globally hyperbolic motion [9] in Bell’sproblem leaves invariable both the thread length andthe length of an electronic bunch in a collider.

5. CONCLUSION

In conclusion, let us note that a calculation of theproper length of a thread in Bell’s problem, usingan ICIRF instead of the Lagrangian comoving NRF,leads to an absurd result in the ultrarelativistic case.A reasonable result is obtained by taking into accountthe curvature of a spacelike curve according to (23).At small values of u, u → 0, K = K[2,12] = K[10] =K[11] = cosh γ.

What is the essence of the errors in [1, 2, 10–12]? In [1, 2, 12], the calculations of a spatial phys-ical distance use Eq. (1), which, as we have proved,determines the physical spacelike metric of the hy-persurface orthogonal to the world lines. This metriccoincides with the spatial metric [3] obtained by the

radar method for infinitely close world lines of mediumparticles. The use of the radar method for particles atfinite distances and for continuum motions which arenot rigid in Born’s sense leads to errors. It is shownin a figure in [14]. Integration in [2, 12] between theworld lines takes place along the line t = const orτ = const′, which is the same for a globally hyperbolicmotion. Thus, during integration, the authors leavethe physical space comoving with themedium. In [10]and [11], one calculates the length along a spacelikevector of a certain ICIRF from the world line of Bell’sfirst rocket up to intersection with the world line of thesecond rocket [10], and in [11], vice versa, from thesecond to the first one. In both cases the authors leavethe Lagrangian comoving NRF. The spacelike unitvectors of the ICIRF do not lie on the same space-like straight line at their intersection points with theworld lines of thread elements which connects Bell’srockets up to its rupture. In our paper, unlike all theenumerated ones, the length elements are integratedalong the same line in Minkowski space for which thespatial metric tensor (1) is defined, remaining insidethe physical space. A more detailed considerationhas been performed by the authors in [14, 15]. Therelation (23) is quite new and has not been used in thescientific literature by anybody but the authors.

REFERENCES1. J. S. Bell, Speakable and Unspeakable in Quan-

tum Mechanics (Cambridge University Press, 1993),p. 67.

2. S. S.Gershtein andA. A. Logunov, J. Bell’s Problem.Particle and Nuclear Physics 29, 5th issue (1998).

3. L. D. Landau and E. M. Lifshitz, Field Theory (Nau-ka, Moscow, 1973).

4. P. K. Rashevsky, Riemannian Geometry and TensorAnalysis (GITTL, Moscow, 1953).

5. S. A. Podosenov, Tetrad Formulation of Motionof an Elastic Medium in Special Relativity, Izv.Vuzov, Fiz. № 4, 45 (1970).

6. S. A. Podosenov, Relativistic Kinematics of a De-formable Medium in Special Relativity. In: Prob-lemy Terii Gravitatsii. Teor. i Mat. Fiz., 1st issue(VNIIOFI, Moscow, 1972), p. 60–72.

7. S. A. Podosenov, Space, Time and Classical Fieldsof Bound Structures (Sputnik publishers, Moscow,2000).

8. S. A. Podosenov, A. A. Potapov, and A. A. Sokolov,Impulse Electrodynamics of Wide-Band RadioSystems and the Fields of Bound Structures (Ra-diotekhnika, Moscow, 2003).

9. A. A. Logunov, Lectures on Relativity and Gravi-tation. Modern Analysis of the Problem (Nauka,Moscow, 1987).

10. D. V. Redzic, Note on Devan-Beran-Bell’s space-ship problem, Eur. J. Phys. 29, 11 (2008).

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11. D. V. Peregudov, Comments to the paper by Redzic[10] Eur. J. Phys. 29 (2008).

12. V. L. Ginzburg and Yu. N. Eroshenko,Once Again onthe Equivalence Principle, Uspekhi Fiz. Nauk 165,2 (1995).

13. A. L. Zel’manov, in: Proc. of the 6th Meeting onCosmogony (AN SSSR publishers, Moscow, 1959).

14. J. Foukzon, S. A. Podosenov, and A. A. Potapov, Rel-ativistic length expansion in general acceleratedsystem revisited, ArXiv: 0910.2298.

15. S. A. Podosenov, J. Foukzon, and A. A. Potapov,Bell’s Problem and a Study of Electronic Bunchesin Linear Colliders, Nelineinyi Mir 7 (8), 612 (2009).

16. J. L. Synge, Relativity: the General Theory(NHPC, Amsterdam, 1960).

17. S. A. Podosenov, Relativistic Mechanics of a De-formable Medium in Tetrad Formulation, PhD the-sis (Peoples’ Friendship University, Moscow, 1972).

18. H. Dehnen, in: Einstein Proceedings 1969–1970(Nauka, Moscow, 1970), p. 140.


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