Review of Gravity Probe B


2
Scientific Motivation for GP-B


SIGNIFICANCE OF FRAME DRAGGING

Geometrical Viewpoint

Rotation and the Foundations of Physics. Rotation has played a central, if problematic, role in the foundations of mechanics and dynamics. Although natural philosophers from Galileo to Newton had a clear understanding of the invariance of physical law in reference frames in relative rectilinear motion, the same could not be said with respect to rotational motion. Newton's famous "bucket" thought experiment illustrates the problem. Water co-rotating with a bucket climbs the wall of the bucket. Is this caused by rotation relative to absolute space, or relative to distant matter? If the bucket did not rotate, while distant matter rotated around it, would the same behavior result? Newton's gravitational theory was incapable of answering this question.

Despite the success of Newtonian dynamics in accounting quantitatively for the details of planetary motion, the tides, and local gravity, this conceptual issue remained unresolved. Interestingly, Foucault's 1851 demonstration that the plane of swing of a pendulum maintained a relation to the fixed stars while the Earth rotated underneath it caused a public sensation, and Foucault pendula quickly appeared throughout Europe and the United States. And while few physics textbooks today discuss the success of Newtonian gravity in explaining such phenomena as the advance of the lunar perigee, they do tend to discuss Foucault's pendulum.

The conceptual relation between local dynamics and distant matter was a central theme of Ernst Mach's formulation of a natural philosophy. In 1872, in History and Root of the Principle of the Conservation of Energy, he wrote:

If we think of the Earth at rest and the other celestial bodies revolving around it, there is no flattening of the Earth, no Foucault's experiment, and so on—at least according to our usual conception of the law of inertia. Now one can solve the difficulty in two ways; either all motion is absolute, or our law of inertia is wrongly expressed . . . . I [prefer] the second. The law of inertia must be so conceived that exactly the same thing results from the second supposition as from the first.

Mach's thinking influenced Einstein's development of general relativity. Although he later grew disillusioned with Mach, Einstein's conception of the law of inertia was meant to embody the loose collection of ideas now called Mach's principle. The resulting theory, general relativity, was not completely successful in that regard, yet it did ultimately succeed in resolving the issue of Newton's bucket. Ironically, that fact was not demonstrated until 1966, as discussed below.

Geometry and Frame Dragging. General relativity describes gravitation as synonymous with the effects of curved space-time. A "test" body (an electrically neutral body small enough to be unaffected by tidal forces) moves on a geodesic, the straightest possible trajectory, in the space-time around a gravitating body. Thus a satellite in orbit around the Earth (assumed non-rotating for the moment) describes a helical path in space-time (a circle in space, while moving forward in time) that for a single orbit is, say, 7000 km in radius, and 1.5 light-hours or 1.8 billion km long in the "time direction." Any portion of that space-time curve can be regarded as straight to high approximation.

However, if the gravitating body also rotates, an additional geometrical effect, called frame dragging, should be present. There are a number of manifestations of this predicted effect. A particle released from infinity on the equatorial plane of a rotating body, moving initially in a radial direction (i.e., with zero angular momentum), will have it trajectory deflected away from a radial line so that it orbits the rotation of the body, all the while maintaining zero angular momentum. The period of a co-rotating body is longer than the period of a counter-rotating particle orbiting at the same radius. Light rays sent around the equatorial plane of rotating body (e.g., by the use of a ring of mirrors) take less time to return to a fixed point when they propagate with the sense of rotation of the body than when they propagate in the opposite direction. Finally, a gyroscope at rest outside a rotating body will precess relative to fixed objects at great distance. Since gyroscope axes define a local sense of non-rotation, local reference frames whose orientation is defined by gyroscopes rotate relative to frames fixed by distant object.

Because geometry underlies all gravitational dynamics in GR, one can think of the effect just described as a "dragging" of the space-time geometry around the rotating body, much as a rotating cylinder causes a viscous fluid in which it is immersed to be dragged around in a whirlpool-like fashion. It is important to emphasize that this geometric effect associated with rotation is conceptually different from the static space-time curvature produced by a non-rotating body. The latter effect imprints itself on the external, far field of the source via the mass M, a scalar quantity (as in the limiting gravitational acceleration at large distances, given by GM/R2). By contrast, frame dragging imprints itself via the angular momentum of the source, a pseudo-vector quantity J.

Frame Dragging and Newton's Bucket. The existence of the frame-dragging effect suggests that rotation is not strictly absolute, but can be relational, that is, defined relative to other masses, just as is rectilinear motion. Although approximate solutions of the equations of general relativity for rotating bodies were obtained as early as 1918 (by Lense and Thirring, whence the alternative terminology "Lense-Thirring effect" for frame dragging), it was not until 1966 that an indication of this relational property of rotation was found. This result came from a theoretical analysis of the space-time in the interior of a slowly rotating, approximately spherical shell of matter. A hypothetical gyroscope at the center of the shell was shown to precess, and in the limit that the shell's gravitational radius 2GM/c2 tends to its physical radius (a condition corresponding loosely to cosmological values), the precession angular velocity tends to that of the shell itself. In other words, in that limit, gyroscope axes are locked to the distant matter constituting the shell. In 1985, further extensions of this work showed that, at the center of the shell, the requisite centrifugal forces would be induced by frame dragging, sufficient to cause water to climb the side of a "non-rotating" bucket, exactly in accord with Mach's stated preference. Consequently, within GR, rotation really is a relational concept, defines with reference to distant matter.

Thus frame dragging within general relativity has significant conceptual and philosophical implications concerning the relationship between local physics and the distant cosmos and the possibility of "absolute" space.


Gravitomagnetic Viewpoint

Another viewpoint on frame dragging exploits a similarity, in the weak-field, slow-motion limit, between general relativity and electrodynamics. Specifically, the space-time metric component g00 - 1 - 2/c2 +. . ., which contains the Newtonian gravitational potential , is analogous to the scalar potential V of electromagnetism. The component g0i, which has no correspondence in Newtonian gravitation, is analogous to the vector potential Ai (i varies over the spatial dimension). Associated with these potentials are a "gravitoelectric" field Eg, a "gravitomagnetic" field Bg, and equations of motion that approximately parallel the corresponding Maxwell equations and Lorentz force equation of electrodynamics. The spatial part of the metric gif, which relates to spatial curvature, has no counterpart in electromagnetism. It affects some of the equations but plays no direct role in frame dragging. This viewpoint also arises from treating general relativity at lowest order as a tensor (spin-2) field theory, analogously to treating electromagnetism as a vestor (spin-1) theory.

In this approach, static matter generates a gravitoelectric potential g00 and space curvature gif, while moving matter generates in addition a gravitomagnetic potential g0i. A rotating mass generates a gravitomagnetic dipole field, analogous to the magnetic dipole field of a rotating charge (apart from a numerical factor), and a rotating matter current (a gyroscope) external to the source experiences a torque ("spin-spin" interaction) analogous to that of a current loop in a magnetic field (apart from a sign change that reflects the attractive nature of gravity).

Gravitomagnetism and Lorentz Invariance. In electrodynamics there is an intimate connection between electric and magnetic fields, resulting from Lorentz invariance. What appears to be pure electric field in one reference frame can be combined electric and magnetic field as seen in a reference frame moving relative to the first. General relativity is compatible with Lorentz invariance at its foundational level, and thus there should be analogous connections between gravitoelectric and gravitomagnetic effects. the field of a mass moving with uniform velocity v relative to an observer should be equivalent to that of a static mass as seen by an observer moving with velocity -v. The field of the moving mass contains a gravitomagnetic field generated by its mass current (g0i = -4viGM/Rc3). The field of the static mass contains only the gravitoelectric field g00, and the spatial curvature gij gsij. Under a Lorentz transformation to the frame of an observer with velocity -v, there results, to first in v/c, g0i = -vi(g00 + gs)/c = -4viGM/Rc3. Thus, gravitomagnetism can be said to be related to gravitoelectrostatics through Lorentz invariance.

On the other hand, the gravitomagnetic field of a rotating mass cannot be obtained from the static field of a non-rotating mass by a simple rotation of coordinates, first, because such a rotating frame contains centrifugal and coriolis pseudoforces that distinguish it from a non-rotating coordinate system cannot be defined globally, indeed can be defined only out to a radius at which the rotational velocity equals the speed of light. Thus, although some aspects of gravitomagnetism can be related directly to static gravity, frame dragging cannot be related to it so simply.

This result is consistent with the idea that frame dragging imprints the angular momentum J of the source on the distant space-time. A linearly moving source imprints both its mass M and its linear momentum p on the distant space-time; however, the latter can always be eliminated by a global Lorentz transformation to a frame in which the body is at rest (p = 0). On the other hand, the angular momentum, like the mass, cannot be changed or eliminated by a global transformation.

Gravitomagnetism and Astrophysical Processes. The precession and forces associated with frame dragging have found important applications in astrophysical processes. Models for relativistic jets of matter ejected from the cores of quasars and active galactic nuclei invoke such frame-dragging forces acting on the matter and magnetic fields associated with accretion disks around rapidly rotating, supermassive black holes.

Frame-dragging effects also play an important role in the late-time evolution (the final few minutes) of in-spiraling binary systems of compact stars (neutron stars or black holes). That role includes precessions of the spins of the objects and of the orbital plane and contributions to the emitted gravitational radiation and the evolution of the orbital phase. These effects are potentially detectable in gravitational wave signals received in the worldwide array of laser inferferometric gravitational wave observatories currently under construction, including LIGO in the United States and a similar project called VIRGO in Europe.


SIGNIFICANCE OF GEODETIC PRECESSION

Geometrical Viewpoint

The geodetic effect is most simply viewed as a combination of a precession resulting from gravitoelectrostatics, and a precession related to curved space-time. A gyroscope in motion in the gravitoelectric field of a body experiences a precession that is described by the interaction of special relativistic corrections to the basic equations of motion with the external gravitoelectrostatic field, completely analogous to the effect in electrodynamics. This piece amounts to one-third of the total effect.

The remaining two-thirds of the effect comes from the curvature of space around the source. It can be understood by a two-dimensional analogy: on the surface of the Earth, transport a vector (a stick with an arrowhead lying on the surface) locally parallel to itself (i.e., not moving to the right or to the left) around a closed curve. If, for example, the curve consists of following the 0° line of longitude from the equator to the North Pole, following the 90° line of longitude from the Pole to the equator, and then following the equator back tot he starting point, the vector will be found to have rotated by 90° relative to its initial orientation. This failure of a parallel-transported vector to return to its initial state on completing a closed path is the hallmark of curvature (indeed, this process is used in differential geometry to define the Riemann curvature tensor). thus a gyroscope, whose axis can be shown to undergo parallel transport (provided that the gyroscope is in free fall), will undergo a change in its spin direction on completing each orbit in the curved space-time around the Earth. The precise amount turns out to be twice that of the gravitoelectric precession.


Gravitomagnetic Viewpoint

An alternative, purely gravitomagnetic, viewpoint works in the co-moving frame of the gyroscope, in which there is an apparent gravitomagnetic field of the source in linear motion (-4viGM/Rc3), resulting in a precession analogous to that of a spin in a magnetic field. However, the net effect is reduced by 25 percent by the Thomas precession, which results from the fact that the co-moving frame of the gyroscope is actually a sequence of Lorentz frames with different instantaneous directions of the velocity, and whose axes therefore are rotated relative to each other. (The relative effect of Thomas precession here is smaller than in the electromagnetic case because of the factor of 4 that appears in the gravitomagnetic potential.)


GP-B AND OTHER TESTS OF GENERAL RELATIVITY

Experimental Gravity and General Relativity

Prior to 1960, the empirical basis of general relativity consisted the Eötvös experiment, which verified the underlying equivalence principle, and two experiments that checked the theory itself: the deflection of light and Mercury's perihelion advance. The latter two experiments were regarded as being good only to 20 to 50 percent, and 10 percent, respectively.

Since 1960, however, significant progress has been made, both in improving the precision of existing tests and in performing new high-precision tests. This progress was enabled by the rapidly evolving technology of high-precision, high-stability, quantum-governed measuring tools, such as atomic clocks, lasers, and radio telescopes, together with progress in space exploration.

Improved tests were made of the Einstein equivalence principle, the foundation for the geometric viewpoint of gravitational theory. This principle is satisfied by general relativity and by all theories called "metric theories." these tests included improved tests of the composition-independence of free fall (Eötvös experiment: null tests to 10-12), tests of spatial isotropy (local Lorentz invariance of non-gravitational interactions: null tests to 10-22), and tests of the gravitational redshift (to 10-4). It is worth noting that a satellite test of the equivalence principle (STEP) has been proposed that could improve the test of composition-independence of free fall to the level of 10-17.

The "classic tests" of general relativity were substantially improved: light defection (using Very Long Baseline Interferometry, or VLBI) to 0.1 percent, and Mercury's perihelion advance to 0.1 percent. New tests were performed: Shapiro time delay in signal propagation (using Viking spacecraft tracking) to 0.1 percent equality of acceleration of Earth and Moon toward the Sun (Nordtvedt effect) to 10-12 (translated to a 10-2 null test of relevant theoretical parameters). the Hulse-Taylor binary pulsar provided a test of the existence of gravitational waves in agreement with general relativity to 0.4 percent. Because the system contains neutron stars with strongly relativistic, nonlinear internal gravitational fields, the observations also provided indirect support for the theory in strong-gravitational-field regimes, through its prediction that such internal structure is effaced in the orbital and gravitational wave dynamics (by contrast with most alternative theories).

No previous experimental tests of general relativity directly probe the effect of frame dragging. Some effects of gravitomagnetism associated with translational motion of matter are present in such tests as the Nordtvedt effect, and in the orbital dynamics and gravitational wave emission of the binary pulsar, and some authorities have argued that the gravitomagnetic field has already been confirmed by indirect measurements. However, the gravitomagnetic effects in question occur in complicated combination with other effects, and so the gravitomagnetic contributions cannot be cleanly separated. No gravitomagnetic effects associated with rotation have ever been detected directly, in isolation from other relativistic gravitational effects.


Alternative Metric Theories of Gravity

Within a restricted class of alternative theories of gravity called metric theories, a useful framework has been developed, called the parameterized post-Newtonian (PPN) framework. It characterizes the weak-field, post-Newtonian limit of substantial, though not complete, range of metric theories by a set of 10 parameters, , , , 1, 2, . . . , whose values vary from theory to theory. Such theories generally contain, in addition to the basic space-time metric, auxiliary fields (scalar, vector, tensor, and so on) that mediate the gravitational interaction. The Jordan-Fierz Brans-Dicke scalar-tensor theory is the most famous example; recently, extensions of that theory have become popular in inflationary cosmological model building and in superstring-inspired gravitational theories.

In general relativity, = = 1, while the other parameters vanish. Observations of the Shapiro time delay and of light deflection place the bound | - 1| < 2 x 10-3, and measurements of Mercury's perihelion advance combined with measurements of yield | - 1| < 3 x 10-3.

Non-zero values for either of the parameters 1 or 2 signal the presence of auxiliary fields whose coupling to the distant universe produces local gravitational effects dependent on the local velocity relative to a preferred universal frame. Such effects appear as violations of local Lorentz invariance in gravitational interactions, and they produce anomalies in geophysics (Earth tides) and in orbital dynamics. Assuming that the solar system moves relative to the cosmos with the velocity 350 km/s, as determined from the dipole anisotropy of the cosmic background radiation, several bounds have been placed on the parameters, specifically |1| < 4 x 10-4.

In the PPN framework, the frame-dragging effect depends on the combination 1 + + 1/4. The 1 + part comes from the connection between gravitomagnetism and gravitoelectrostatics via Lorentz transformations (in the PPN framework, g00 + gs = 2(1 + ) GM/Rc2; see the section "Gravitomagnetic Viewpoint," pp. 6-8), and the 1 indicates a possible violation of that local Lorentz invariance. Thus from this point of view, frame dragging tests the local Lorentz invariance of gravity. The bounds that have been placed on and 1 are tighter in their implications for frame dragging than those GP-B can hope to achieve. It should be noted, however, that those bounds come from experiments whose conceptual basis is completely different from that of frame dragging and rely on an assumption about the relevant velocity that controls preferred frame effects. GP-B measures frame dragging directly.

The geodetic effect depends on the combination 1 + 2. the first term corresponds to the gravitoelectric precession, the second term to the effect of spatial curvature; equivalently, 2 + 2 comes from gravitomagnetic precession viewed from the gyroscope's frame, with a reduction of -1 from Thomas precession (despite the use of Lorentz transformations in this latter argument, 1 does not appear). With a projected accuracy of 75 ppm in its measurement of the geodetic effect, GP-B offers a factor-of-20 improvement in the accuracy of the measurement of , from 2 x 10-3 to 10-4. This is at the level where deviations from the exact unity value of GR could occur in a class of well-motivated, cosmologically important scalar-tensor alternative theories (generalizations of the Brans-Dicke theory), in which cosmological evolution following inflation naturally drives such theories toward but not all the way to equivalence with GR. Depending on the specific model, deviations from = 1 could lie between 10-3 and 10-7. A bound from GP-B could constrain such models.


Wider Classes of Gravitational Theory

Metric theories of gravity whose post-Newtonian limits fit within PPN framework represent only a portion of the "space" of alternative theories. This space includes metric theories that do not fit the PPN model, and the relatively poorly explored class of non-metric theories of gravity. It is fair to say that, should a breakdown of general relativity at the classical (non-quantum) level occur, it is likely to involve non-metric gravity and would lead to a radical conceptual revision of our view of gravity.

There is strong reason to suspect, from a number of different quarters, that non-metric revisions of GR at some level will be necessary. Unlike the other fundamental interactions, GR has a dimensional coupling constant and is not renormalizable in quantum field theory. The theory stands as a major stumbling block in the way of the unification of the interactions. In other words, physicists devoted to unification believe that GR must break down at some level. This is one of the greatest challenges of modern theoretical physics. It is generally assumed, though not proven, that the failure of GR will occur at the level of quantum gravity, far from the regime of observable effects that can be tested by local experiments. On the other hand, examples exist of unification-induced modifications of GR (in superstring-inspired theories, for instance), in which residual effects do occur at the classical, detectable level of cosmology.

Non-metric modification of GR could still be viable, provided they are compatible with the high-precision experiments that check the Einstein equivalence principle underlying metric gravity. (One motivation for proposing experiments such as STEP is to provide dramatically improved tests of this principle and thereby to test for the effects of such modifications.) Within this broader class of theories, no conclusion can be drawn about prior bounds on frame-dragging effects from other experiments such as light deflection, time delay, or tests of local Lorentz invariance. On the other hand, there are currently no examples of non-metric theories that agree with all local observations and yet predict a detectably different frame dragging.


OTHER TESTS OF FRAME DRAGGING
OR GEODETIC PRECESSION

There has been no prior, direct test of general relativistic frame dragging. Apart from GP-B, the leading current proposal for a possible future test is LAGEOS III, a third laser-ranged geodynamics satellite launched into an orbit whose inclination is supplementary to that of LAGEOS I or II. The frame dragging induced by the rotation of the Earth causes a precession of the orbital planes of both satellites (the orbits are in effect gyroscopes); the use of two satellites with accurately supplementary inclinations permits the cancellation of the 107 times larger, but equal and opposite precessions induced by the Earth's Newtonian multipole moments. At best, this proposed experiment would yield a 10 percent test of frame dragging. It has not been approved for launch by any space agency at present.

Other less promising or less fully developed proposals include detecting the gravitomagnetic contribution to gravity gradients, as measured by orbiting superconducting gravity gradiometers; measuring the precession of the plane of a Foucault pendulum erected at the South Pole; and measuring the precession of orbiting non-cryogenic gyroscopes by optical means. A recently published proposal (B. Lange, Phys. Rev. Lett. 74, 1904 (1995)) based on the latter idea would use an autocollimator to sense the orientation of an unsupported gyro, thus giving it the working name AC-USG. The design of such a project is still at the conceptual stage, but it is claimed that it could be much more accurate than the present GP-B design. The natural angular sensitivity of an optical autocollimator is far better than that of a readout based on the superconducting London moment; the single gyro in AC-USG would be in a drag-free environment, with a much larger spacing between gyro and housing than in GP-B; the spacecraft would roll around the gyro axis rather than around the direction to the reference star, thereby minimizing a certain class of spurious torques; and two counter-orbiting satellites could be used to largely cancel some other kinds of errors. Despite these apparent advantages, it is too soon to say whether the AC-USG could work as claimed. The error analysis of the GP-B is the result of decades of work, many Ph.D. theses, and detailed engineering designs, and a similarly thorough and cautious approach would be needed for AC-USG. Consequently, the task group could not assess its claims quantitatively or discuss the budget for such a project; but if future scientific developments require a better measurement of gyro precession, this approach could be a promising one.

One test of geodetic precession has been reported, namely that of the lunar orbit (viewed as a gyroscope) in the field of the Sun, measured using lunar laser ranging combined with VLBI data (see B. Bertotti, I. Cinfolini, and P.L. Bender, Phys. Rev. Lett. 58, 1062 (1987) and I.I. Shapiro et al., Phys. Rev. Lett. 61, 2843 (1988)). The result agrees with general relativity to about 2 percent. In the Hulse-Taylor binary pulsar, the effect of frame dragging of the pulsar's spin axis caused by the spin of its companion is too small to be detected. There is, however, a potential precession of the pulsar's spin caused by a combination of the gravitomagnetic field generated by the companion's orbital motion (relative to the center of mass), together with the companion's gravitoelectric field and the resulting space curvature, through which the pulsar moves. Although a very significant secular change in the radio pulse shape has been observed (an effect not observed in other pulsars), given the uncertainties in the structure of the emitting region of pulsars, it seems unlikely that such measurements will ever yield results better than the results of the lunar test of the geodetic effect, much less those of GP-B.

Geodetic precession is sensitive to the value of the PPN parameter . VLBI measurements of the deflection of light are unlikely to reach below the GP-B level of 10-4 in (1 - ). No planned or proposed interplanetary probes will have the requisite tracking capability to measure the Shapiro time delay to higher accuracy than has been done. Planning for orbiting optical interferometers with microarc-second accuracy and the capability to improve light deflection measurements by 2 or more orders of magnitude appears to have halted. The European Space Agency has plans for a successor to the Hipparchos mission, the Global Astrometric Interferometer for Astrophysics (GAIA), with 20-microarc-sec accuracy, which could measure light deflection and to 10-4. Although this accuracy would be comparable to that of GP-B, this mission is unlikely to fly before 2006. Thus on the 1999 to 2000 time frame of GP-B, there is unlikely to be a competitive measurement of space curvature via the parameter .




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