A harmonic space approach to quantum gravity of stationary space-times with SO(3) symmetry A rest frame in a stationary, axially symmetric space-time is defined as a synchronizable frame which is as nearly Killing as possible. This is a possible generalization of the Newtonian rest frame. A kinematical theorem giving the condition for the existence of a rest frame whose time vector is a linear combination of the Killing vectors is proved. The frame is also unique. The condition is shown to be weaker than the assumption of orthogonal transitivity. The surfaces of simultaneity of the rest frame are the surfaces of constancy of a particular Killing time coordinate, and its time vector is the component of the time Killing vector orthogonal to the angular Killing vector. Some properties of the rest frame are then discussed; it is shown that the frame is well-behaved down to the event horizon, where its time vector becomes null. Under a suitable condition on the event horizon, the time vector coincides with a Killing vector there. The gravitational redshift relation in the rest frame is derived. There is a dependence on the angular momentum of the geodesic. Furthermore, the event horizon is shown to be an infinite redshift surface for the rest frame observers. Finally, the three-vector potential of Landau and Lifshitz is interpreted and shown to be closely related to the rest frame, and a corresponding four-vector potential is invariantly defined.Lemma: If the vector field w a is never null in some region, then w a = (w B W B)-l w a satisfies W[a,Bl = 0if and only if
Magnitude of the Delboeuf (concentric circle) illusion was determined by the up-and-down (staircase) method. Variables were (a) diameter ratio between concentric circles, (fe) circumference remaining after dotting one of them, and (c) reflectance contrast, the reflectance of the circumference relative to the white background. Previous evidence is supported: Assimilation occurred, the test (judged) circle being displaced toward the inducing circle; the distance paradox was evident with maximum displacement near diameter ratios (test: inducing circle) of 2:3 and, less consistently, 3:2. Dotting or reducing the reflectance contrast of the test circle increased illusory effects. Dotting or reducing the reflectance contrast of the inducing circle diminished them. Reducing equally the reflectance contrast of both circles had little effect on the size of the illusion.
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