Distortion in the Metric of a Small Center of Gravitational Attraction due to its Proximity to a Very Large Mass

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Abstract
The problem of two bodies in general relativity in an especially restricted mode is analyzed. One body is of small mass, and, under the influence of gravitational attraction, moves toward a much larger mass whose field produces ``tidal'' deformations in the geometry of the smaller one. To evaluate these deformations, we treat the Schwarzschild metric of the particle by a perturbation analysis similar to that performed by Regge and Wheeler. The boundary conditions for this analysis are obtained from the metric of the background field expressed in a novel set of comoving coordinates, here called Fermi normal coordinates. These coordinates have been described in the previous paper. In this paper we use the Schwarzchild metric given there in comoving coordinates as an asymptotic approximation for the full solution which is evaluated here. A perturbation analysis using the ``tidal deformation'' (background curvature) as an expansion parameter is performed on the metric of a small Schwarzschild particle. The solution so obtained satisfies Einstein's equations for empty space for small deviations from a nonflat metric. This solution also reduces to the Fermi metric in the appropriate limit, namely, when the ``distance'' from the geodesic is large compared to the radius of the small mass but small compared to the separation of the two masses. Thus only the quadratic terms in the distance are important. This solution is then used to find the deformations in shape of the throat of the wormhole and to locate this region of symmetry (the throat) by finding a coordinate transformation which leaves the metric invariant.