The radius of gyration is a familiar concept in Newtonian mechanics and
a suitably defined relativistic generalization of it turns out to be
very useful for analyzing rotational effects in strong gravitational
fields. The present paper contains a discussion of the properties of
this quantity and of its level surfaces (the von Zeipel cylinders) and
also of its connection with the effective potential for photon motion
and with ideas of centrifugal force. The direction of increase of the
radius of gyration gives a preferred determination of the local outward
direction relevant for the dynamical effects of rotation, but this
direction becomes misaligned with the global outward direction in
strong-field situations. This misalignment underlies some apparently
counterintuitive behavior of the centrifugal force in strong fields
which has recently been the subject of considerable interest.
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The radius of gyration is a familiar concept in Newtonian mechanics and a suitably defined relativistic generalization of it turns out to be very useful for analyzing rotational effects in strong gravitational fields. The present paper contains a discussion of the properties of this quantity and of its level surfaces (the von Zeipel cylinders) and also of its connection with the effective potential for photon motion and with ideas of centrifugal force. The direction of increase of the radius of gyration gives a preferred determination of the local outward direction relevant for the dynamical effects of rotation, but this direction becomes misaligned with the global outward direction in strong-field situations. This misalignment underlies some apparently counterintuitive behavior of the centrifugal force in strong fields which has recently been the subject of considerable interest.
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We develop a method of calculating the X-ray signal from a 'spot'
corotating with an accretion disk around a black hole and the emission
line from the X-ray-illuminated disk. The disk self-eclipse is
considered, and all effects of general relativity on the motion of
photons are included with no approximation. The eclipse strongly affects
the shape of the X-ray light curve of the spot and the K fluorescent
line irradiated by the disk. The FWHM and strength of the line profile
are reduced significantly if the eclipsed region of the orbit is very
deep. For a given radius of a ring in the disk, this occurs when the
eclipsed region of the orbit is deeper than pi/2. The centroid of a line
profile is also subject to the eclipse effect.
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