The Bhagavata Purana carries the story of King Kukudmi who had a beautiful daughter called Revati. Several young men aspired for her hand and the anxious father wanted to make the right choice. At last he decided to consult a wiser brain, no less than Lord Brahma. He took his daughter to see Brahma. As Brahma was busy with some work, he asked the king to wait for a while. And sure enough, after waiting for a few moments, Kukudmi was admitted to the presence of Brahma. When Brahma heard of Kukudmi’s problem he laughed and said: “While you waited a few moments here, thousands of years have passed on Earth. The young men you were talking about are no more. But I will tell you of a suitable match from those who will be there by the time you get back.” And Brahma recommended the name of Balram, brother of Krishna.
In modern times this story has similarities with the phenomenon of time dilatation induced by strong gravity. When Einstein proposed his general theory of relativity, he had a built-in concept of how space and time measurements get modified in regions of strong gravity. Before Einstein came on the scene, physicists were accustomed to rather naïve concepts of space and time — the Euclidean geometry that we all learn at school and controlled spatial measurements where time seemed to flow uniformly at the same rate for everybody. This simplistic notion was changed once the theory of relativity took over.
Take, for example, the familiar theorem of geometry that the three angles of a triangle add up to two right angles. Consider the thought experiment wherein we station three space observers around the Sun so that light signals exchanged by them follow tracks grazing the solar limb. This vast triangle in space will have its three angles adding up to a sum slightly higher than two right angles. This happens because the geometry in the vicinity of the Sun is not that of Euclid: it follows slightly different rules because of the Sun’s powerful gravity.
In the somewhat protective environment of the Earth, gravity is not very large and so its effect on space-time measurements is negligible. But in astronomy we encounter several scenarios that lead to massive objects that have such strong force of gravity that under its continuous attraction these objects have a powerful urge to contract. To gauge the power of this urge imagine that a star with the mass and radius of the Sun would shrink to a point in 29 minutes! In practice this disaster does not overtake the Sun, because it has thermal forces within to oppose the collapsing tendency of gravity. Indeed most stars that the astronomer observes maintain their equilibrium under the opposing forces of gravity and thermal pressures.
However, a star cannot rely on its thermal energy for ever. Its source, the nuclear fusion reactions in its central core, gives way when the nuclear fuel runs out and gravity finds itself unopposed in its control of the star. In such a case the star may go on shrinking and shrinking. It is now well established that a star with more than three solar masses will have this future in store.
Let us now visualise two observers, A and B, who are in communication via light signals. A is located far away from the shrinking star whereas B is sitting right on it. The geometry around A is like that of Euclid because there is no significant gravitational presence there. The observer B, however, is in a place where the star’s gravity is strong and it is going to get stronger and stronger as the star shrinks. Suppose A arranges with B that B shall send him a light signal every hour on the hour. In reality what happens?
Initially A will notice a slight delay in receiving B’s signals. The delay occurs because the time measurement around B is different from that around A, thanks to the strong gravity near B. And, as time proceeds, this delay will get longer and longer. A will wonder why B is so slow in sending signals. On the other hand, B is keeping strictly to the schedule and sending signals hourly by his watch. Eventually a stage would come when A may have to wait several years to get the next signal from B. We have a situation exactly like that involving Kukudmi and Brahma. And, ultimately, the shrinking star crosses a critical barrier from inside which no signal can ever come out. This state is popularly known as the black hole. No matter how long A waits he is not going to get another signal from B.
Black holes with this information barrier came as a solution of Einstein’s equations of relativity. Karl Schwarzschild was the first to do this work, although black holes were thought to be too esoteric to be real. As the expert on stars, Arthur Stanley Eddington, observed, nature should not permit such strange situations to develop. But he had underestimated what nature would permit. Today astronomers are discovering black holes of various masses, ranging from a few times the solar mass, to billions of solar masses.
While Kukudmi was in Dwapar Yuga, we have a modern example of this phenomenon right here on Earth. Let A represent an applicant and B stand for a bureaucrat. A has applied to B’s office for some urgent information. He waits and waits for a reply from B, cursing his dilatoriness while sending him reminders. B, however, runs his schedule by a different time and would be surprised if he is accused of being slow. A has to appreciate the fact that he has run into one of the many bureaucratic black holes that exist in our country.
The writer, a renowned astrophysicist, is professor emeritus at Inter-University Centre for Astronomy
and Astrophysics, Pune University Campus
