Relativistic glider

Eduardo Gueron*Centro de Matematica, Computacao e Cognicao, Universidade Federal do ABC, 09210-170, Santo Andre, SP, Brazil

Ricardo A. Mosna†

Instituto de Matematica, Estatıstica e Computacao Cientıfica, Universidade Estadual de Campinas, C.P. 6065, 13083-859,Campinas, SP, Brazil.

(Received 16 November 2006; published 16 April 2007)

We present a purely relativistic effect according to which asymmetric oscillations of a quasirigid bodyslow down or accelerate its fall in a gravitational background.

DOI: 10.1103/PhysRevD.75.081501 PACS numbers: 04.20.�q, 04.25.�g, 45.40.�f

The motion of extended bodies in general relativity hasbeen a source of unexpected phenomena up to the present.Although the problem in its full generality is notoriouslycomplicated [1], approximated analysis of quasirigidbodies already discloses interesting effects. A remarkableexample and quite surprising result was recently presentedby Wisdom, who demonstrated that cyclic changes in theshape of a quasirigid body may lead to net translation incurved spacetimes [2]. Since this takes place in the absenceof external forces, one can portray the body as swimmingin spacetime. Later, this effect was shown to dominate overthe so-called ‘‘swinging’’ (Newtonian) effect, whose for-mulas are well known, for high enough oscillation frequen-cies [3]. The latter is a direct expression of the fact thatexpanding and contracting a body in a nonuniform gravi-tational field gives rise to a net work on it (due to tidalforces). However, this effect is increasingly suppressed forhigher frequencies, since the gravitational potential felt bythe body is then nearly the same through its complete cycle(see Eq. 5 of [3]).

Here we present a purely relativistic effect which (likethe swimming and unlike the swinging effect) is not sup-pressed for high frequencies, but (like the swinging andunlike the swimming effect) is not related to a gravitationalgeometric phase. It comes about by letting a quasirigidbody perform asymmetric oscillations in a gravitational

background. We show that, as a result, the body slowsdown or accelerates its fall depending on how asymmetricits internal oscillations are.

We propose a simple model to investigate this effect:two test particles of mass m connected by a massless strutwhose length varies periodically, i.e., an oscillating dumb-bell. The strut is mathematically implemented as a con-straint between the particles. Physically, and followingWisdom’s idea (see last section of [2]), the active strutshould be constantly and locally monitored in order toguarantee that the constraints are maintained.

A test particle under the influence of a sphericallysymmetric gravitational field moves, in the context ofgeneral relativity, in Schwarzschild spacetime. Its equa-tions of motion follow from the Lagrangian

L p � �mc

���������������������������������������������������������������������������������1�

2GM

c2rp

�c2

�dtds

�2�

1

1� 2GMc2rp

�drpds

�2

vuut ;

(1)

wherem is the mass of the test particle and s is an evolutionparameter (we assume that the body falls vertically towardsthe gravitational source). For the dumbbell, we choose thecoordinate time t as the evolution parameter and take therelevant Lagrangian as

L d � �mc

���������������������������������������������������������������������1�

2GM

c2r1

�c2 �

1

1� 2GMc2r1

�dr1

dt

�2

vuut �mc

���������������������������������������������������������������������1�

2GM

c2r2

�c2 �

1

1� 2GMc2r2

�dr2

dt

�2

vuut��������r2�t��r1�t��l�t�

; (2)

where l�t� is the constraint as seen by the observer atinfinity. In the Newtonian limit, l�t� represents the lengthof the strut connecting the particles at time t. This imple-mentation of the constraint clearly embodies considerablesimplification (for instance, it fails to be covariant) [4].However, it is important to note that, by choosing a peri-

odic l�t� such that l�0� � l�T� � 0, all observers will agreethat the worldlines of both particles meet at the end of eachperiod. In the following, we numerically solve the equa-tions of motion coming from (2) for a family of constraintfunctions l��t�whose members always meet this condition.Of course, different observers will disagree on the symme-try and shape of a given function l�t� as measured by them.

With these considerations in mind, we compute the fallof the oscillating dumbbell and compare it with that of thesingle particle [the particular case when l�t� � 0], both

*Electronic address: eduardo.gueron@ufabc.edu.br†Electronic address: mosna@ime.unicamp.br

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with the same initial conditions. Given that the cyclicchanges considered here affect a single degree of freedom,we can anticipate that the swimming effect will not arise inthis case [5]. Therefore the sought-after relativistic effectwill be isolated once we work in a regime for which(Newtonian) tidal effects are negligible.

We numerically solved the Euler-Lagrange equationsassociated with (2) for the family fl��t�g portrayed inFig. 1(a). The asymmetry parameter �, ranging from �1to 1, quantifies how much l��t� fails to be symmetric aboutits peak: the function corresponding to � � 0 has its peakat t � T=2, while a generic l��t� attains its maximum att � 1��

2 T [6]. For each value of �, the equations of motionof the oscillating dumbbell and of the point mass wereintegrated, with rp�0� � r1�0� � R and _rp�0� � _r1�0� � 0.Their separation �r � r1�T� � rp�T� after one period thenyields the net displacement of the dumbbell. Repre-sentative results, for R � 120 and �l � 5� 10�3, in unitsof GM=c2, are shown in Fig. 1(b), where �r� 109 was

plotted against the dumbbell oscillation frequency ! �1=T. The relativistic case, corresponding to (1) and (2),is represented by solid lines, while its Newtonian counter-part is depicted by dashed lines.

Figure 1(b) shows that the Newtonian �r is alwaysnegative and that it becomes negligible when the frequencyis large enough (as expected from our earlier discussion). Itis important to note that this happens irrespective of thevalue of the asymmetry parameter �. This is in markedcontrast to the relativistic case. Indeed, while the relativ-istic curve for the symmetric case (� � 0) is surrounded byNewtonian curves, those corresponding to positive valuesof � lead to a positive net displacement, a purely relativ-istic effect. Therefore, a relativistic robot who cannot swimmay still soften its fall by repeatedly stretching and shrink-ing its body in an asymmetric fashion. Note that therelativistic �r does not become negligible for large valuesof !, and that its sign and magnitude in this regime aredetermined by �.

It is worth noting that the beginning of the plateaus inFig. 1(b) occurs in a frequency domain which is fairlyindependent on the oscillation amplitude �l [7]. By keep-ing �l small one can thus obtain the effect in hand forreasonably small velocities. It is interesting to note that, byTaylor expanding the Lagrangian (2) in _r1 and _r2 whilekeeping the remaining (position-dependent) terms unper-turbed, one needs to retain terms at least up to fourth orderin v=c to capture the effect under consideration. Thisshould be compared to Wisdom’s swimming effect, whichalready takes place when the relevant Lagrangian is ex-panded up to second order in v=c. The effect presentedhere thus arises from an intricate coupling of the gravita-tional field to the velocities of the particles.

From some numerical analysis, we estimate that the netdisplacement after one period, in the large frequency re-gime (corresponding to the plateaus in Fig. 1), is given by

0 0.02 0.04 0.06−3.0

−1.5

0.0

1.5

3.0

0.25

1.00

0.75

0.50

0.00 0.2 0.6 0.8 1.0 0.4 0.0

(b)

(a)

r10

9(G

Mc2

)

oscillation frequency (c3 GM )

l(t

)l

t T

= –0.15

= + 0.4

= –0.4

= + 0.15= 0

FIG. 1 (color online). (a) Family of constraint functions l��t�[6]. (b) Net displacement �r � r1�T� � rp�T� of the dumbbell,relative to the falling point mass, after one period. The initialconditions are set by _rp�0� � _r1�0� � 0, rp�0� � r1�0� � 120and �l � 5� 10�3 (in units of GM=c2). The relativistic(Newtonian) case is represented by solid (dashed) lines. Theplot shows that the Newtonian �r is always negative and that it isnegligible for ! * 0:03c3=GM, irrespective of �. In contrast,we see that the relativistic �r crucially depends on the asym-metry parameter �, and does attain positive values (for positive�). Causality problems (due to large velocities) only start toappear at much higher frequencies, not shown above.

−0.75 −0.5 −0.25 0 0.25 0.5 0.75−7.5

−5

−2.5

0

2.5

5

7.5

asymmetry parameter

asym

met

ryfa

ctor

FIG. 2. Adimensional scale factor �� as a function of theasymmetry parameter �.

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�r � ���l2

R2

GM

c2 ; (3)

where �l is the amplitude of l��t� and �� is a dimension-less factor associated with the asymmetry parameter �, seeFig. 2. This expression was obtained from numerical simu-lation employing a large range of values for R and �l,going from 50 to 107 and 10�7R to 10�2R, respectively.Equation (3) shows that the effect under consideration canbe orders of magnitude higher than the swimming effect,which is proportional to �l2=R3 [8]. This result is some-what surprising since, as discussed above, the swimmingeffect is already present when the Lagrangian is expandedup to second order in the velocities, while the effectpresented here needs at least a fourth-order expansion inv=c.

It is instructive to calculate the net displacement �r interms of the coordinate distance D traveled by the fallingpoint mass during the period T. Using the rough estimateD gT2, with g � GM=R2, we immediately see that

�r=D ��l!=c�2 (4)

in this regime. In this way, if the dumbbell is engineered sothat �l! 0:01c, we get �r=D 10�4. Therefore, suchresult may be within reach of experimental verification.This would open the door to new tests of general relativityby means of local experiments [9].

We close by noting that, if �r=D were to reach 1, thedumbbell would be able to float in space. However, this isapparently not possible, since numerical simulations indi-cate that the value attained by �r=D is typically limited to15%. Therefore, it seems that, although this mechanismcannot be employed to fly, it might be conceivably used asa relativistic paraglider to slightly soften the landing on anatmosphereless planet.

The authors thank G. E. A. Matsas, S. R. Oliveira andAlberto Saa for fruitful discussions. E. G. is grateful to P. S.Letelier for the technical support. R. A. M. acknowledgesfinancial support from FAPESP.

[1] W. G. Dixon, Proc. R. Soc. A 314, 499 (1970); NuovoCimento 34, 317 (1964).

[2] J. Wisdom, Science 299, 1865 (2003).[3] E. Gueron, C. A. S. Maia, and G. E. A. Matsas, Phys. Rev.

D 73, 024020 (2006).[4] A related (and also simplified) quasirigidity condition

could be defined by imposing that the geodesic length ofconstant t slices connecting the worldlines of the particlesbe given by the constraint function l�t�. The obtainedresults are essentially the same for either of these choices.

[5] It is well known that any geometric phase effect (includingthe swimming effect, Berry’s phase, Hannay angle, etc.)can only happen if at least two parameters are varied alonga (nontrivial) loop in the appropriate parameter space. See,e.g., the book A. Shapere and F. Wilczek, GeometricPhases in Physics (World Scientific, Singapore, 1989).

[6] The functional form of the constraint functions in Fig. 1(a)

is given by l��t� � �l exp �1���2!t�2

�1��2�!t��1�!t��, where ! �

1=T and the overall normalization was chosen so thatthe amplitude of l��t� is �l. Other families fl��t�g withdiverse shapes were also tested, with no appreciablechanges in the results.

[7] Numerical simulations indicate that a variation in �l of 5orders of magnitude leads to a change of less than 1% inthe characteristic frequency for which the correspondingplateau begins.

[8] See eq. (20) of Ref. [2].[9] We note that, if M is the Earth’s mass, then R

109GM=c2 at Earth’s surface. Although we couldnumerically confirm Eq. (3) only for 50GM=c2 & R &

107GM=c2 (due to limitations of our integration routine),there is no indication that it would break down at suchscale.

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