Observable Properties of Large Relativistic Masses

Interstellar space may be full of very dense and very faint stars (supernova remnants, very old white dwarfs, etc.), on whose surfaces the gravitational field intensity is very high. What, according to general relativity, are their observable properties, and what are the maximum gravitational effects as a matter of principle? To answer this, the exact test particle orbits (geodesics) of the Schwarzchild metric are derived, classified and described, on the basis of an exact classical model. The latter is of considerable help in making the properties of the orbits immediately evident, and represents the principal advantage of the present derivation over previous ones. Also, as radial coordinate, instead of the usual largely arbitrary r, the metric coefficient g00 = A2(r) is used. A2 has immediate physical significance as the red shift ratio v(r)/v(∞), and its use also simplifies the formulas. The test-particle scattering angle (generalizing the results of Darwin), and the differential scattering cross section, as well as the capture cross section of a sphere of radius A2(r) = A2(R) = AR2, are calculated as a function of test-particle energy, and presented graphically. The observed, augmented, angular diameter of a sphere is calculated in terms of AR2, and some peculiar lenslike effects of masses, first discussed by Einstein, are reviewed. It is then pointed out that an important result of Curtis for the interior field, implies AR2 > 0.514, so that the exterior field extends at most down to A2 = 0.514, and the region A2 < 0.514—which would be the most pathological region of the exterior—cannot in fact exist, if Curtis' arguments are assumed valid. As a result, quasihyperbolic test-particle orbits exist with pericentrum equal to the radius of any conceivable spherical mass, and the radius can therefore in principle always be determined by an asymptotic scattering experiment. As a further result, the maximum photon-scattering angle can be no more than about 110° (and larger for slower particles), providing a cutoff at this point in the photon-scattering cross section, and allowing a massive star to produce at most one secondary image of another star. As a practical matter, none of the effects discussed here seems large enough to be measurable today, with the sole well-known exception of the red shift, which for the Curtis limiting sphere would be large enough to shift a visible spectrum into the near infrared.