Relativistic Tidal Forces and the Possibility of Measuring Them

Abstract

The relativistic corrections to the Newtonian tidal accelerations generated by a rotating system are studied. The possibility of testing the relativistic theory of gravitation by measuring such effects in a laboratory in orbit around the Earth is considered. A recent proposal to measure a rotation-dependent tidal acceleration as an alternative to the Stanford gyroscope experiment is critically examined and it is shown that such an experiment does not circumvent the basic difficulties associated with the gyroscope experiment.PACS numbers: 04.20.Cv, 04.80. + z, 95.30.Sf In the relativistic theory of gravitation, as well as in Newton's theory, a local system of coordinates ("laboratory") may be chosen which is inertiai except for the presence of tidal forces. A general theory of tides has been developed based on an extension of the concept of a local Fermi frame. 1 In strong fields, the relativistic tidal effects are predicted to lead to interesting phenomena such as the emission of tidal gravitational radiation. 1 Relativistic corrections to the Newtonian tidal accelerations caused by a massive rotating source (such as the Earth) will be considered in this paper and the possibility of measuring these effects will be critically examined.To interpret the results of measurements in a laboratory frame in terms of a local clock and locally determined spatial directions, it is necessary to refer the (covariant) equations of motion to a local tetrad frame consistent with the measurement procedure. The simplest possibility is to carry along a set of three (orthogonal) gyroscopes and to characterize all local events by a Fermi coordinate system. Let 2 X (a) M be such a tetrad system so that A (0 / =X M is the tangent vector of a representative path (e.g., the center of mass) and X (i) M are the spatial ("gyroscope") directions. The scalar tidal accelerations take the form -kijX 1 under the conditions of interest here (see Ref. 1 for details). Here x l are the local spatial coordinates and the symmetric tidal matrix k is given by K i3 zR fivpo X (i) fi^V^( j) P^°-(1/ Thus the local "Newtonian" equations of motion (in terms of the proper time r) should be supplemented by this tidal force. It usually proves convenient to use spatial axes A (i / =M i 3 (r)x u / , where M is an orthogonal matrix. The local equations of motion are now simply those in a ro-tating system with the tidal matrix given byThe tidal matrix depends on the moments of the source; for a rotating mass distribution, one may separate the contributions of the mass and angular momentum. The latter effect may be attributed to a gravitational "magnetic" field which is anticipated on the basis of a certain analogy with electrodynamics. 3 The nature of such a field was first elucidated (to linear order in angular momentum) in the framework of Einstein's theory by Thirring and Lense. 4 Efforts aimed at obtaining observational evidence for this field have concentrated on an experiment to measure the cumulative effect of the precession of a gyroscope in orbit aroun...

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