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Theoretical Background and Present Status of the Stanford Relattvtty-Gyroscope Experiment

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Theoretical Background and Present Status of the Stanford Relattvtty-Gyroscope Experiment THEORETICAL BACKGROUND AND PRESENT STATUS OF THE STANFORD RELATTVTTY-GYROSCOPE EXPERIMENT C. W. F. Everitt, W. M. Fairbank and L. I. Setoff Institute of Theoretical Physics and Department of Physics Stanford University, Stanford, California 94305. ABSTRACT An earlier proposal that a measurement of the precession of the spin axis of a gyroscope moving through the Earth''s gravitational field would provide a test of Einsteiifs general theory of relativity is discussed in terms of the Brans-Dicke theory and other recent developments. It is shown that an accuracy of 0.001 arc-second per year in the orientations of two or more gyroscopes is required which would also provide a rough measure of the quadrupole effect. The work of the last seven years at Stanford University on this project, including the large-scale low-temperature physics techniques and advanced techniques in electronic instrumentation and control theory, is discussed in detail. Parts of the laboratory prototype are being tested and it is hoped that a preliminary test flight will take place in 1973 or soon afterwards. 1. THEORETICAL BACKGROUND Einstein's general theory of relativity has been of deep interest to scientists of several specialties for more than half a century. From the point of view of the physicist, a central problem consists in devising and then performing experiments that measure the rather subtle differences between the pre­ dictions of the relativistic and Newtonian theories of gravitation. This problem is far more difficult than the corresponding problem raised by the special theory of relativity. The differences between the predictions of the relativistic and Newtonian theories of mechanics are of order v2/c2, where v is the magnitude of the difference in velocity of two parts of the system and c is the speed of light; in recent times, many experiments have been performed in which this quantity is of order unity. In the case of the general theory of relativity, however, the corresponding parameter is <p/c2, where ç is the difference in gravitational potential of two parts of the system. For a red-shift measurement on the surface of the Earth, 9 = gh, where g is the local acceleration of gravity and h is the difference of height; gh/c2 is typically around 10~15. For the deflection or retardation of electromagnetic waves passing the Sun or other object of mass M at a minimum distance /•, or for the precession of a planetary orbit at this distance from the Sun, or for the red-shift of light originating at this point, 9 ~ GM/r, where G is the Newtonian gravitational constant. The quantity GM/c2r is roughly equal to 10-8 near the surface of the Sun, to 10~8 at the orbit of the planet Mercury, and to 10~9 close to the Earth where M is now the mass of the Earth. Interest in experimental tests of general relativity theory has recently been enhanced by an explicit formulation of the scalar-tensor theory* as a generalisation of the pure tensor theory of Einstein. This theory of Brans and Dicke expresses the amount of admixture of scalar field in terms of a positive parameter w which is infinite for the Einstein theory. On the basis of a measurement of the oblateness of the Sun and its interpretation in terms of a solar quadrupolar gravitational field, Dicke and Gol- denberg 2 have concluded that co is in the neighbourhood of 6, although this interpretation is by no means established. The value of <o has no effect on the predicted gravitational red-shift, and indeed it is well known that this prediction can be based on the equivalence principle without regard for the details of the theory of gravitation 3. For the other effects mentioned above, however, and for the gyroscope precession experiment to be discussed in detail below, this value of w decreases the predictions 33 GYRO 2 A0=7 sec t= I YEAR AÔ-Q.05 sec Figure 1. Gyroscopes with spin axes parallel to the Earth's axis or perpendicular to the plane of the orbit on board a polar-orbiting satellite. of the Einstein theory by 6 to 8 per cent. There is therefore considerable interest in performing expe­ riments of this kind to an accuracy of 0.1 to 1 per cent. It was suggested nearly a decade ago that a measurement of the precession of the spin axis of a gyroscope moving through the Earth's gravitational field would provide a test of Einstein's theory 4. —^. -> -> In that paper it was shown that the precession angular velocity vector is given by £2T + QG + ÛM, where -> 2 -*• •> QT = (l/2mc )(F X v) (Thomas precession) ->• 2 3 •> •> QG = (3 GM/2c r )(/- X v) (geodetic precession) -*. 2 3 •>-»--»•-2 > QM = (GI/c r )[(3/-/r )(û>r) — co] (mass-current precession). •»• •> -> Here, m, r and v are the mass, coordinate, and velocity of the gyroscope, respectively, and F is any ->• non-gravitational force exerted on it; M, I, and w are the mass, moment of inertia, and rotational angular velocity vector of the Earth respectively. The Thomas precession QT is a special relativistic rather than a general relativistic effect since it does not involve G, and was discovered in connection with atomic physics 8; it is zero for a gyroscope in free fall about the Earth since then no non-gravitational support force is exerted. The geodetic precession flG is a true gravitational effect; its contribution when 6 integrated over an orbital period was first found by de Sitter . The mass-current precession S5M makes the gyroscope experiment uniquely interesting since it is the only experiment thus far proposed that is likely to distinguish between the gravitational fields produced by matter at rest and by matter in motion; £2M is related to the effect of the mass current of the rotating Sun on the orbital motion of a planet, which was first calculated by Lense and Thirring 7 but is too small to observe. When the Brans-Dicke parameter w is not infinite, the geodetic precession given above is to be multipled 1.8,9,10,11 by (4 + 3o>)/(6 + 3co). In similar fashion, the mass-current precession is to be multiplied10 •u by (3 -f- 2o>)/(4 + 2<o). It has been pointed out quite recently that the general relativistic 34 effect of the mass quadrupole moment of the Earth may not be negligible. For the simplest case, in which the plane of the orbit is perpendicular to the quadrupole axis, this precession12>13 is D -3Q O where Q is the Earth's quadrupole moment. There is some advantage in using a polar orbit for the satellite that carries the gyroscope, since then the two principal precession vectors, Q.G and DM are perpendicular to each other. QR is perpendi­ cular to the plane of the orbit, and iiM is parallel to the Earth's axis. Thus, as shown in Fig. 1, gyro 1 with spin axis parallel to the Earth's axis measures only QG, while gyro 2 with spin axis perpendicular to the plane of the orbit measures only DM. The magnitude of the first precession is about 7 arc- second per year, and the time average of the second is about 0.05 arc-second per year. With this orbit, the time average of ÛQ is perpendicular to the plane of the orbit and has a magnitude of about 0.005 arc-second per year. In order to determine whether the Brans-Dicke parameter w is infinite, and its value if it is finite, it would be desirable to measure QG to an accuracy of 0.1 to 1 per cent. Then in order to establish the existence of the mass-current precession, it would be desirable to measure QM to a few per cent accuracy. Thus in both cases, an uncertainty of 0.001 arc-second per year is an ambi­ tious but appropriate objective, which would also provide a rough measure of the quadrupole effect. 2. EXPERIMENTAL STATUS Analysis and experimental work on the gyroscope programme has been proceeding during the past seven years in the Stanford University Department of Physics in conjunction with the Department of Aeronautics and Astronautics. A complete laboratory prototype has been designed, and parts oj it have been delivered to Stanford and are under test. We expect the evaluation of the laboratory hardware to be completed during the next two years. Meanwhile, plans for a preliminary test flight in 1973 and final flight as soon as possible thereafter, are under negotiation with the National Aero­ nautics and Space Administration. As discussed above, our aim in designing this experiment.has been to measure the orientations of two or more gyroscopes like those shown in Fig. 1 with respect to a star, with an accuracy of about 0.001 arc-second per year. While fully recognising the many practical obstacles that have to be over­ come to achieve this extreme accuracy, we believe that, at least in principle, it is within the possibilities of existing technology. In order to give some idea of the particular improvements called for, however, we may mention that the best gyroscopes currently available on Earth have residual drift rates about eight orders of magnitude greater than the design goal, and the available star-tracking telescopes have a long-term null stability about three orders of magnitude greater than this figure. Improvements of this kind in equipment can only be achieved by very precise matching of the design to the special requirements of the experiment. After prolonged thinking we have concluded that almost the only possible way of reaching these extremes of accuracy at present is by an extensive combination of three areas of technology that have opened up during the past ten or fifteen years: (1) space research, (2) large-scale low-temperature physics techniques, and (3) certain advanced techniques in electronic instrumentation and control theory.
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