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The structure of Killing horizons in the static vacuum C metric which represents the gravitational field of a uniformly accelerating Schwarzschild-type particle is studied. It is shown that for A 'm < 1/27 there exist two physically meaningful horizons. One horizon is analogous to the Schwarzschild surface and the other is similar to a flat surface in Euclidean space traveling at the speed of light along the axis of symmetry. This secor~d surface is called the Rindler surface because of its analogy with the Rindler surface in the limit that geometry becomes Euclidean. As the acceleration increases, the Schwarzschild surface distorts from its original spherical shape. Its shape becomes teardroplike with the pointed end oriented in the direction of the acceleration. In the forward direction the Schwarzschild surface moves outward from the origin as the acceleration continues to increase in accordance with the principle of equivalence. In the backward direction the surface shrinks from its original Schwarzschild surface as the acceleration increases for relatively small values of the acceleration. This is also expected from the principle of equivalence. As the acceleration reaches A = l/v'gm the Schwarzschild surface in the backward direction reaches a minimum distance from the origin at r = v'5m and as the acceleration further increases it reverses its direction of motion and grows outward until it reaches the original Schwarzschild surface at r = 2m. This behavior is an apparent violation of the principle of equivalence. As the acceleration increases, the Rindler surface moves inward approaching the Schwarzschild surface, and finally when A = l/v%rn the two surfaces unite and produce a naked singularity. Radial geodesic and nongeodesic motions are also investigated. It is shown that for small accelerations the results are in agreement with the principle of equivalence and the effects of dragging of the inertial frame due to the rectilinear acceleration.

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