Swimming in Newtonian Spacetime: Motion by Cyclic Changes in Body Shape

Transcription

1 January 5, 004 Swimming in Newtonian Spacetime: Motion by yclic hanges in Body Shape Michael J. Longo University of Michigan, Ann Arbor, MI mlongoumich.edu In a recent article in Science, Jack Wisdom discusses the possibility of a machine "swimming" in spacetime by changing its shape. In his article Wisdom estimates that a swimmer near the surface of the Earth using meter-sized deformations can achieve a displacement ~10 - meter. He also states that this is a purely relativistic effect and there is no swimming effect in the analogous Newtonian problem. Here I show that in fact much larger displacements, ~10 7 meters, can occur for a Newtonian swimmer, as can be shown using concepts from elementary physics, carefully applied to a simple satellite in the curved spacetime (i.e., the gravitational field) near the Earth. These arguments are based only on energy and angular momentum conservation and Newton's laws. In a recent article in Science (1), Jack Wisdom discusses the possibility of a machine "swimming" in spacetime by changing its shape. In his words, "resuming spacetime is a curved manifold as portrayed by general relativity, translation in space can be accomplished simply by cyclic changes in the shape of a body, without any external forces." In his article Wisdom estimates that a swimmer near the surface of the Earth using meter-sized deformations can achieve a displacement ~10 - meter (). He also states that this is a purely relativistic effect and there is no swimming effect in the analogous Newtonian problem (). Here I show that in fact much larger displacements, ~10 7 meters, can occur for a Newtonian swimmer, as can be shown using concepts from elementary physics, carefully applied to a simple satellite in the curved spacetime (i.e., the gravitational field) near the Earth. These arguments are based only on energy and angular momentum conservation and Newton's laws. I note that there is no fundamental reason why a Newtonian swimmer is impossible. The Weak Equivalence rinciple prohibits local experiments for objects in "freefall" that can detect gravitational forces. However, the swimmers have non-negligible size, and it is perfectly possible for them to detect and use a gravitational force gradient (or curvature of space in general relativistic terminology) to continuously change the position of their center of mass by repeated cycling. As the simplest possible Newtonian "swimmer" I will consider a dumbbell-shaped satellite orbiting near the Earth's surface in a circular orbit (Fig. 1). The dumbbell consists of two point masses, each of mass m, connected by a rigid massless rod of length L. I suppose that the satellite is equipped with two small motors: one (the rotator) uses a reaction wheel to rotate the sat-

2 ellite about its center of mass and can be used to control the orientation of the dumbbell; the other (the translator) can move the masses closer together or farther apart along the rigid rod. The rod is assumed to be pointing radially as shown in Figure 1, position (1). The center of mass is initially at with one mass at ( + L), the other at ( L). The orbital angular velocity of both masses and the center of mass are initially w 0 counterclockwise, and the dumbbell is assumed to have an initial angular velocity w 0 about its center of mass so that its axis always points toward the center of the Earth as it goes around in a stable circular orbit. I consider the masses and reaction wheel as "point" masses () (1) m v m c L EATH Fig. 1. A dumbbell-shaped satellite in near-earth orbit (dimensions greatly exaggerated). In position (1), the center of mass is at In position (), the masses are pulled together, and the center of mass moves inward slightly due to the gravitational force gradient. The maneuver is assumed to be done quickly compared to the orbital period, and a radial orientation is maintained. We now use the translator to pull the two masses together to position (). For simplicity, I assume that this is done quickly, in a time much less than the orbital period, so that the satellite need not move appreciably along its orbit during the maneuver. I also assume that the rotator motor is used to maintain a radial orientation of the dumbbell axis throughout the closing maneuver. The center of mass remains at the center of the dumbbell as the masses are pulled together. When the masses are each a distance x from the center of mass, the net force on the outer mass is È 1 ù FO Í ú T( x) ÎÍ ( + x ) ûú + where T(x) is the tension in the connecting rod, G is the gravitational constant, M is the mass of the Earth, and I take positive inward (4). This is equal to the centripetal force in circular orbit,

3 FO È 1 ù Í ú + T( x) mw ( + Î ( + x ) û x) ( 1) Similarly for the inner mass, FI È 1 ù Í ú - T( x) mw ( - Î ( - x ) û x) Subtracting Eq. () from Eq. (1) and simplifying gives È x FO - FI Í -4 Í Î - x ù ú T( x) m x ú + w û Equating the total gravitational force to the centripetal force on the center of mass, È 1 1 ù GM Í + ú m 4 ÎÍ ( x x + ) ( - ) ûú w ; so w Substituting Eq. (4) into Eq. (), T( x) È x Í Î Í - x x ù ú û ú + x since x <<< L.(5) To move the masses the translator motor must pull on either one with a force T(x), and each moves through a distance L, since the center of mass is always midway between the masses. As a result, the translator motor must do positive work W T on the masses, W T 0 Ú Tdx Ê L ˆ Á Ë L L In addition, the rotator motor must do work to prevent the masses from spinning about the center of mass as they are pulled together. When the masses are together, the rotational kinetic energy 0, so this work is equal to the rotational energy the dumbbell initially had about its center of mass, 1 1 L W Iw0 ( ml ) w0 ( 7) where I ml is the rotational inertia of the dumbbell about its center of mass. Thus the total work done by the motors is ( 5) ( 6)

4 L W WT + W 4 ( 8) This work will increase the orbital energy of the satellite, while its angular momentum remains constant at its initial value mv I ( L) + mv O ( +L) mv, where v I and v O are the velocities of the inner and outer masses respectively and v is the velocity of its center of mass in the initial circular orbit. The total orbital energy (i.e., kinetic + potential) of either mass is (6) E - ( 9) a where a is the semimajor axis of the orbit. For a circular orbit of radius this becomes E /(). The initial total orbital energy of the two masses is then After the masses are pulled together, E 1 Ê 1 1 ˆ - Á + - Ë + L - L E G( m) M - where ' c is the semimajor axis of the new (elliptical) orbit. Therefore, the change in total orbital energy is Ê 1 DE E - E ˆ - Á Ë d 1 where d -. We equate this to the work done by the motors given by Eq. 8, or d DE W 4L This gives a change in the semimajor axis of the orbit after the masses are pulled together of d From Eq. 11, the center of mass moves slightly inward when the dumbbell is closed. The gravitational potential energy becomes more negative; the kinetic energy increases; and the total energy becomes more positive due to the energy expended by the motors. The angular momentum 4L ( 10) ( 11) 4

5 remains constant. The velocity acquires a slight radial (inward) component. This causes the initially circular orbit to become slightly elliptical. From Eq. 9, the semimajor axis increases slightly as E becomes more positive. From Eq. 11, for a large dumbbell with L 10,000 meters, at approx. 1.0 E 6.9 x 10 6 m, d m (inward) For a satellite with L 1 m, similar to the dimensions of Wisdom's general relativistic swimmer example (), d 4 ( 1 m) m Though this is quite small, it is still more than 16 orders of magnitude larger than that in Wisdom's general relativistic calculation! Wisdom only analyzed a single step(shape change) for his swimmer, rather than a complete cycle, but he notes (7) that it would be interesting to consider whether the swimmer could gradually increase the radius of its orbit by repeated full cycles. While it might seem the Newtonian swimmer in Fig. 1 would only return to its original center-of-mass radius if the masses were returned to the original extended position, this is not the case. From Eq. 6 it is apparent that if is replaced by ( d ) and the integration is done from x0 to xl, the (negative) work done in reopening the dumbbell will be slightly larger than that for the closing step, and the outward movement will be slightly larger than the inward first step. The net step d for a fast close/open cycle can be calculated from Eq. 11 as, (5,8) -7 m ª - Ê d 1 1 ˆ 4L 4L L Á L Ë L 4 ( 1) where the minus sign means the center of mass moves slightly outward each cycle. For the L meter example above, the net step d is ª6 x 10 4 m. For the L 1 m swimmer with dimensions like Wisdom's, d is ª6 x 10-0 m. While very small, this step for a complete cycle is still ~6000 times larger than Wisdom's general relativistic calculation for a single step. In principle, the Newtonian swimmer can continuously "swim" by opening and closing the dumbbell, while maintaining its radial orientation, in complete analogy to Wisdom's general relativis- 5

6 tic swimmer. A small amount of net chemical or electrical energy is required to complete the cycle. There is, however, a much more efficient strategy for a swimmer to gain or lose energy if we drop the assumption that the maneuvers are made in a time small compared to the orbital period. If it is initially in a very elliptical orbit, the closing maneuver can be done at perigee to give an energy gain and inward step comparable to those in Eqs. 8 and 11 above, while the opening maneuver can be done near apogee where the gravitational field is much smaller and the corresponding work almost negligible. A simple case to analyze is a satellite in a parabolic orbit initially with E 0. The initial orbital energy is E 1 1 mv - 0, so mv 6 where v and are the initial velocity and perigee. In this case we can neglect the rotational energy, and the work done in closing the dumbbell is given by Eq. 6. After closing, the energy is 1 E mv - W T L From angular momentum conservation, v v ( 15) Substituting Eqs. 1 and 15 into Eq. 14, we get after simplifying d L - - L - Thus, a satellite in an elliptical orbit with E 0 can change its perigee L / and its orbital energy ª L by closing the dumbbell or rotating it 90 at perigee. The change in energy for a satellite in an elliptical orbit can be large enough to make E > 0 and the satellite will escape. Similarly, a satellite in an open orbit can move into a closed orbit with the inverse maneuver. A more practical strategy for a swimmer would be to simply rotate the satellite as shown in Fig.. Thus a dumbbell-shaped satellite with dimensions ~100 meters, comparable to the International Space Station, in a highly elliptical orbit can decrease its perigee by approx. 5 mm each time it passes through perigee(9). For a near-earth orbit, this corresponds to several centimeters per day (10). This gives the possibility of efficiently converting solar cell power to orbital energy without the need for any propellant. Though this effect may be too small to be useful for satellites orbiting the Earth, there may be special situations where it could be useful. Note that the steps given in Eqs. 11 and 16 do not ( 1) ( 14) ( 16)

7 depend on the masses of either body. Thus, a dumbbell-shaped satellite in orbit around a small asteroid could gradually change its orbit by rotating about its center of mass. A satellite with L 0 m in a very elliptical orbit with periaster 1000 m, for example, could change its periaster in ~ m steps. In summary, I conclude that a Newtonian swimmer is indeed possible and perhaps even practical. This analysis follows simply from conservation of energy and angular momentum. A swimmer in orbit can change its orbit by rotating about its center of mass and trading chemical, solar, or other energy for orbital energy, without the need for any propellant. The Newtonian swimming effect is completely analogous to the general relativistic one(1), but is many orders of magnitude larger(11). Wisdom's general relativistic calculation therefore appears to be incorrect in that it does not contain the much larger Newtonian effect. Figure. A dumbbell-shaped satellite can increase its orbital energy by rotating about its center of mass, so that as it passes through apogee it assumes a radial orientation, and as it passes through perigee it rotates so that both ends of the dumbbell are at the same radius. The perigee will move slightly inward each time it performs this maneuver. EATH 7

8 eferences and Notes 1. J. Wisdom, "Swimming in Spacetime: Motion by yclic hanges in Body Shape", Science 99, 1865 (00). See also,. Seife, in "ecipe for rocket-free space travel", Science, 8 Feb., p. 195).. J. Wisdom, loc. cit., last paragraph.. J. Wisdom, loc. cit., footnote 16. Note that Wisdom talks about cyclic changes on a curved manifold, without any external forces. The Newtonian equivalent to the curved manifold is the gravitational field, which can be considered as an internal force in the Earth-satellite system. 4. I use mks units throughout. 5. The approximation that L<<< is not necessary but is done for simplicity. The exact expression differs from the approximate one by terms ~ L /, or <10 14 for L1 meter. 6. D. Halliday,. esnick, and J. Walker, Fundamentals of hysics, 6 th Edition, Wiley & Sons, NY (00), hap J. Wisdom, loc. cit., footnote A net step d 0 would occur only if the gravitational force were a linear function of radius. 9. A somewhat analogous situation is a boy on a swing who can gradually increase his mechanical energy (and angular momentum) by "pumping" his legs in the direction of travel as the swing passes through the lowest point. 10. This is much less than the normal orbital decay of the International Space Station which is ~00 m per day. 11. Of course, the Newtonian Equations 1 and 1 do not contain the velocity of light, c. Wisdom's general relativistic calculation therefore appears to be incorrect in that it does not contain the much larger Newtonian effect. 8