GRAVITATIONAL PRINCIPLES AND MATHEMATICS

Three cases of photon orbits near a gravitating body

Stated differently, the three cases of photon orbits near a gravitating body can be classified as: "always outside the photon sphere," "crossing the photon sphere," and "always inside the photon sphere." The first is the case of a photon passing the neutron star or black hole, reaching a critical radius R_c, and then escaping again toward infinity. In this case the photon does not reach or cross the photon sphere. Its distance from the star decreases monotonically until R_c, and then increases monotonically thereafter. The second case is that of a photon continuing to come toward the neutron star (or black hole) until it impacts the surface (or falls through the event horizon). Here its distance decreases monotonically. The third case is that of a photon emitted from the surface of a strong gravity neutron star, reaching a critical radius R_c, and then falling back down and again impacting the neutron star surface. This critical radius is given by the cubic equation solution

where n = 0 is for the first case and n = 2 is for the third case. Photons climbing out of a gravitating object become less energetic. This loss of energy is known as a "redshifting", as photons in the visible spectrum would appear more red. Similarly, photons falling into a gravitational field become more energetic and exhibit a blueshifting. The observed energy E_observed at radius r_observed of a photon emitted at radius r_emitted with energy E_emitted is

Note that the magnitude of the redshifting (blueshifting) effect is not a function of the emitted angle or the received angle of the photon - it depends only on how far radially the photon had to climb out of (fall into) the potential well. Also note that the power received from a continuously emitting source would have an additional factor of [(1 - R_S/r_emitted) / (1 - R_S/r_observed)]^(1/2) caused by the relative differences in the perceived rate of the number of photons emitted per unit time.

The effect a gravitational field would have on the actual perceived color of an object is more complex, however, as it depends on the distribution of photons emitted from the source at different energies relative to the sensitivity of the observer to measuring photons of different energies. For example, an object that would be described as green might be very bright in the ultra-violet - but this would not normally be perceived, as people cannot see the ultra-violet. Were this object put in a strong gravitational field and viewed from far away, so that the photons would be significantly redshifted, the strong ultra-violet emission could be shifted into violet emission and the object would look more blue, even though its light has been redshifted. This is an exceptional case, however, and redshifted objects may indeed appear more red.