from Nemiroff, R. J., American Journal of Physics, 61, 619 (1993)


[1] Although these arguments don't work for a particle orbiting inside the photon sphere! See M. A. Abramowicz and A. R. Prasanna, "Reversed Sense of the Outward Direction for Dynamical Effects of Rotation Close to a Schwarzschild Black Hole," submitted.

[2] This point is raised in several introductory books on astronomy and gravitation. See, for example, W. J. Kaufmann III, The Cosmic Frontiers of General Relativity (Little, Brown, and Company, Boston, 1977), pp. 120-150.

[3] R. L. Forward, Dragon's Egg (Ballantine, New York, 1980).

[4] L. Niven, Neutron Star (Ballantine, New York, 1968), p. 9.

[5] C. Ftaclas, M. W. Kearney, and Pechenick, K. R., "Hot Spots on Neutron Stars. II - The Observer's Sky," Astrophys. J. 300, 203-208 (1986).

[6] H.-P. Nollert, H. Ruder, H. Herold, and U. Kraus, "The Relativistic `Looks' of a Neutron Star," Astron. Astrophys. 208, 153-156 (1988).

[7] C. T. Cunningham, "Optical Appearance of Distant Observers near and Inside a Schwarzschild Black Hole," Phys. Rev. D. 12, 323-328 (1975).

[8] J. Schastok, M. Soffel, H. Ruder, and M. Schneider, "Stellar Sky as Seen From the Vicinity of a Black Hole," Am. J. Phys. 55, 336-341 (1987).

[9] H. C. Ohanian, "The Black Hole as a Gravitational `Lens'," Am. J. Phys. 55, 428-432 (1987).

[10] J.-P. Luminet, "Image of a Spherical Black Hole with Thin Accretion Disk," Astron. Astrophys. 75, 228-235 (1979).

[11] W. M. Stuckey, "The Schwarzschild Black Hole as a Gravitational Mirror," Am. J. Phys., submitted (1992).

[12] L. Palmer and W. Unruh, shown at a Texas Symposium on Relativistic Astrophysics in the late 1970s.

[13] B. R. Iyer, C. V. Vishveshwara, S. V. Dhurandhar, "Ultracompact (R less than 3 M) Objects in General Relativity," Class. Quant. Grav. 2, 219-228 (1985).

[14] K. Schwarzschild, "Ueber das Gravitationalsfeld einer Massenpunktes nach der Einsteinschen Theorie," Sitzunsgsber. dtsch. Akad. Wiss. Berlin, 189-196 (1916).

[15] A. Einstein, "Die Grundlage der allgemeinen Relativitatstheorie," Ann. Phys. 49, 769-822 (1916).

[16] See, for example, S. Chandrasekhar, "The Mathematical Theory of Black Holes," (Clarendon, Oxford, 1983).

[17] See, for example, G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics (Wiley, New York, 1979), p. 7.

[18] S. Refsdal, "The Gravitational Lens Effect," Mon. Not. Roy. Astron. Soc. 128, 295-306 (1964).

[19] S. Liebes Jr., "Gravitational Lenses," Phys. Rev. B. 133, 835-844 (1964).

[20] A discussion for large r/R_S is given by Y. Avni and I. Shulami, "Flux Conservation by a Schwarzschild Gravitational Lens," Astrophys. J. 332, 113-123 (1988).

[21] A. Einstein, "Lens-like Action of a Star by the Deviation of Light in the Gravitational Field," Science 84, 506 (1936).

[22] The discovery paper for the first "radio ring" is: J. N. Hewitt, E. L. Turner, D. P. Schneider, B. F. Burke, G. I. Langston, and C. R. Lawrence, "Unusual Radio Source MG1131+0456 - A Possible Einstein Ring," Nature 333, 537-540 (1988).

[23] R. D. Blandford and R. Narayan, "Cosmological Applications of Gravitational Lensing," Ann. Rev. Astron. Astrophys. 30, (1992) in press.

[24] R. D. Blandford and C. S. Kochanek, "Gravitational Lenses," in Dark Matter in the Universe, Volume 4, Jerusalem Winter School for Theoretical Physics , eds: J. Bahcall, T. Piran, and S. Weinberg (World Scientific, Singapore, 1987), pp. 133 - 205.

[25] F. H. Shu, The Physical Universe, An Introduction to Astronomy (University Science Books, Mill Valley, 1982), pp. 137-138.

[26] D. Hoffleit, "The Bright Star Catalog, 4th Revised Ed.," (Yale, New Haven, 1982).

[27] See discussions in: S. L. Shapiro and S. A. Teukolski, Black Holes, White Dwarfs, and Neutron Stars (Wiley, New York, 1983).

[28] But black holes can emit radiations as they evaporate, for a good discussion on this, see S. W. Hawking, "The Quantum Mechanics of Black Holes," Sci. Am. 236, 34-40 (1977).

[29] There has been some discussion as to whether a neutron star can exist in a state this compact. For some discussion on this see C. E. Rhoades Jr. and R. Ruffini, "Minimum Mass of a Neutron Star," Phys. Rev. Lett. 32, 324-327 (1974). An equation of state for a neutron star that allows a neutron star to be this compact is given in S. Bahcall, B. W. Lynn, and S. B. Selipsky "New Models for Neutron Stars," Astrophys. J. 362, 251-255 (1990). A discussion including the relevant physical principles involved is given in A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolski, Problem Book in Relativity and Gravitation (Princeton, Princeton, 1975).

[30] For a lively discussion of extreme energy conditions see M. S. Morris and K. S. Thorne, "Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity," Am. J. Phys. 56, 395-412 (1988).

[31] J. van Paradijs, "Possible Observational Constraints on the Mass-Radius Relation of Neutron Stars," Astrophys. J. 234, 609-611 (1979).

[32] This possibility was pointed out to me by Kent Wood in 1990.

[33] R. J. Nemiroff, "Trip to a Neutron Star: The Movie," Bull. Am. Astron. Soc. 23, 1418 (1991).

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