Simple Black Hole Paradox
By Wolfgang G. Gasser
In General Relativity the units of space and time are manipulated in such a way that the speed of light remains constant despite [acceleration due to] gravitation. That leads to obvious contradictions in the case of black holes. Albert Einstein himself had overlooked the derivability of black holes from his theory and argued that they are outside the range of application of General Relativity.
Black holes arise in GR in situations where classical physics would result in escape velocities higher than the speed of light, or more exactly, where the difference of gravitational potential between the emission point of electromagnetic radiation and an observer is bigger than:
ΔVblack_hole = 0.5 c2
If the potential difference ΔV is less than ΔVblack_hole, electromagnetic radiation can reach the observer with a frequency reduced by the following factor (time dilation):
√ [1 - ΔV 2 / ΔVblack_hole2]
If the sun shrank to the size needed for an escape velocity of exactly c, it would be a black hole for observers outside our galaxy, because ΔV > ΔVblack_hole, however not for observers on the earth, because ΔV < ΔVblack_hole. Radiation from the sun could reach the earth with a strong frequency decrease. But independently of origin and frequency, radiation from the earth could reach observers outside our galaxy. The contradiction is obvious. An analogous contradiction arises in Special Relativity, if observers drifting apart at a velocity higher than c are assumed.