PHYSICAL REVIEW D 67, 062003 ͑2003͒ Covariant calculation of general relativistic effects in an orbiting gyroscope experiment Clifford M. Will* McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, Missouri 63130 ͑Received 17 December 2002; published 26 March 2003͒ We carry out a covariant calculation of the measurable relativistic effects in an orbiting gyroscope experi- ment. The experiment, currently known as Gravity Probe B, compares the spin directions of an array of spinning gyroscopes with the optical axis of a telescope, all housed in a spacecraft that rolls about the optical axis. The spacecraft is steered so that the telescope always points toward a known guide star. We calculate the variation in the spin directions relative to readout loops rigidly fixed in the spacecraft, and express the variations in terms of quantities that can be measured, to sufficient accuracy, using an Earth-centered coordi- nate system. The measurable effects include the aberration of starlight, the geodetic precession caused by space curvature, the frame-dragging effect caused by the rotation of the Earth and the deflection of light by the Sun. DOI: 10.1103/PhysRevD.67.062003 PACS number͑s͒: 04.80.Cc I. INTRODUCTION by the on-board telescope, which is to be trained on a star IM Pegasus ͑HR 8703͒ in our galaxy. One important feature of Gravity Probe B—the ‘‘gyroscope experiment’’—is a this star is that it is also a strong radio source, so that its NASA space experiment designed to measure the general direction and proper motion relative to the larger system of relativistic effect known as the dragging of inertial frames. astronomical reference frames can be measured accurately The experiment will place into Earth orbit a spacecraft con- using very long baseline radio interferometry ͑VLBI͒.Inor- taining four gyroscopes and a telescope ͑for details of the der to reduce telescope errors and biases, the spacecraft will experiment see ͓1,2͔͒. The gyroscopes are 4 cm diameter be ‘‘steered’’ by attitude control forces to keep the centroid fused quartz or single crystal silicon balls, machined to be as of the stellar image centered on the telescope axis. spherical as possible ͑to tolerances better than one ppm͒, and However, because the spacecraft orbits the Earth and the coated with a thin film of niobium. At the low temperatures Earth orbits the Sun, the apparent direction to the star will provided by a dewar of superfluid helium surrounding the vary in a periodic manner because of stellar aberration, gyroscope assembly, the niobium is superconducting, and which amounts to five arcseconds from the orbital motion, each spinning gyroscope has a London magnetic moment and to 20 arcseconds from the annual motion of the Earth. parallel to its spin axis ͑stray magnetic fields, such as those These are both substantially larger than the frame-dragging due to the Earth, will have been suppressed by many orders effect being measured. But, far from being an annoying an- of magnitude before the niobium goes superconducting͒. The cillary effect getting in the way of testing relativity, the ab- orientations of the gyros’ magnetic moments are measured erration of starlight is in fact central to the success of the via changes in the magnetic flux through superconducting experiment ͓3͔. The reason is that the superconducting quan- current loops that encircle each gyro, and that are attached to tum interference device ͑SQUID͒ magnetometers that are the gyroscope housing. The four gyros will initially have connected to the current loops surrounding the gyros do not their spins aligned parallel to the symmetry axis of the space- measure angle, they measure voltage, induced by the varying craft, which coincides with the optical axis of the telescope magnetic flux threading the loops. They must therefore be mounted on the end of the spacecraft. In order to average out calibrated to provide a scale factor that converts voltage to numerous unwanted torques on the gyros, the spacecraft will angle. Because the orbit of the spacecraft, the location of the rotate about its symmetry axis at a rate of between one and star and the speed of light can all be established to high 10 times per minute. precision, the aberration of starlight can be predicted accu- According to general relativity, a perfect gyroscope in or- rately. It can also be separated from the relativistic preces- bit about the Earth will precess relative to distant inertial sions because of its unique temporal variability over the 12ϩ frames because of two effects. The first, and most important, months of the science phase of the mission. Through this is the dragging of inertial frames, caused by the rotation of means, the experiment can be calibrated with an accuracy the Earth, also called the Lense-Thirring effect. Over a time sufficient to achieve the desired overall accuracy. of one year, for a polar orbit of about 640 km altitude, this In such a complex and delicate experiment, there are nu- causes the gyroscopes to precess in an East-West direction merous sources of error, including anomalous, nonrelativistic by around 42 milliarcseconds ͑mas͒. The second is the geo- torques on the gyros, variations in electronics responses, er- detic precession, caused by the curved spacetime around the rors in guide star position and proper motion, effects of ac- Earth. This effect results in an annual precession in the cumulated charges on the rotors from cosmic rays and other North-South direction of about 6600 milliarcseconds. charged particles, and so on. Nevertheless, the team that is The reference direction to the ‘‘distant stars’’ is provided mounting this experiment—Stanford University, Lockheed- Martin Space Systems, and NASA—has proposed that a measurement of the relativistic precessions at the level of 0.4 *Email address: [email protected] mas/yr is feasible; this would provide a test of relativistic 0556-2821/2003/67͑6͒/062003͑7͒/$20.0067 062003-1 ©2003 The American Physical Society CLIFFORD M. WILL PHYSICAL REVIEW D 67, 062003 ͑2003͒ frame dragging at the one percent level and of the geodetic and Sec. V gives concluding remarks. We use the conven- effect at the level of 6 parts in 105. tions of ͓19͔ and units in which Gϭcϭ1. The precession of gyroscopes according to general rela- tivity was first calculated independently by Pugh and by II. SPACECRAFT BASIS VECTORS AND GYROSCOPE Schiff ͓4–6͔, who calculated the precession relative to a SPINS given inertial coordinate system. Wilkins ͓7͔ analyzed the We consider a spacecraft with four-velocity u, whose ori- precession in more detail, attempting in particular to ascer- entation is described by three spacelike basis vectors or- tain how much of the geodetic effect could be regarded as thogonal to u that are rigidly tied to the spacecraft ͓23͔. The Thomas precession and how much was due to the curvature vector ␰ points along the spacecraft symmetry axis, while ␳ of space. He was also the first to attempt a more-or-less and ␩ are orthogonal to ␰ and to each other. These four coordinate-free calculation by comparing the spin directions vectors e␣ form an orthonormal basis, with e␣ e␤ϭ␩␣␤ , with the directions of incoming light rays from distant • where ␩␣␤ is the Minkowski metric, valid in the local iner- sources. tial frame of the orbiting spacecraft. The spacecraft moves on Recent discussions of gyroscope precession generally fall a geodesic ͑we ignore atmospheric drag and translational at- into three groups. One group provides formal calculations of titude control forces͒, with ٌ uϭ0. Because the spacecraft the precession of a gyroscope relative to a local tetrad or u is rotating about its symmetry axis, it acts as a gyroscope, so congruence of geodesics, sometimes with applications to that its spin axis ␰ is parallel transported, except for attitude- specific spacetimes such as Schwarzschild, Kerr, or de Sitter ͓ ͔ control forces that keep the telescope pointed toward the but seldom refers to specific aspects of Gravity ٌ ␰ϭ , 12–8 guide star, so that u t, where t denotes the effect of those Probe B. Another group makes coordinate-dependent calcu- ␰ ͓ ͔ lations of specific precession terms ͑generally in the post- forces on 24 . This, together with the orthonormality of Newtonian approximation͒, and discusses details of the vari- the basis and the geodesic equation, are sufficient to establish ous terms, frequently with reference to Gravity Probe B ͓13– that ͔ ,This group also includes unpublished calculations by the ٌ ␩ϭ␻␳Ϫ͑␩ t͒␰ . 16 GP-B team in the course of developing the data analysis u • procedures for the experiment. A third group mixes calcula- ٌ ␳ϭϪ␻␩Ϫ͑␳ t͒␰, ͑1͒ tions of precessions referred to a formal tetrad ͑sometimes u • including comparisons with the direction to a distant star͒ where with applications to Gravity Probe B; these include the stan- ͒ ͑ ␻ϵ␳ ٌ ␩ϭϪ␩ ٌ ␳ ͔ ͓ dard textbook presentations 17–21 , as well as papers de- • u • u 2 signed in part to sort out the meaning of the various preces- sion terms—geodetic, frame-dragging and Thomas is the locally measured roll angular frequency of the space- precession ͓7,22͔. craft. Defining the spacecraft roll phase ␪ϵ͐␻d␶, where ␶ To date, we know of no calculation that, on the one hand, is spacecraft proper time, we define ‘‘fixed’’ roll reference takes a fully covariant approach, and that on the other hand, axes which do not rotate with the spacecraft: incorporates critical operational aspects of the GP-B experi- ␩ ϵ␩ ␪Ϫ␳ ␪ ment, and performs an end-to-end calculation in terms of 0 cos sin , observable quantities. Such a covariant calculation is of in- ␳ ϵ␩ sin ␪ϩ␳ cos ␪. ͑3͒ terest for several reasons.