Hawking's black hole paradox explained - Fabio Pacucci
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Following the definition of the authoritative Oxford English Dictionary, a paradox is “a statement or proposition which, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems logically unacceptable or self-contradictory”. The word comes from the Greek “paradoxon” which means “contrary to the opinion”. Throughout the history of Science, paradoxes have many times sparkled new ideas, or even new fields. For example, in ancient times the paradox of Achilles and the tortoise triggered interest in infinitesimals.
In this dig deeper section, I will describe three very famous scientific paradoxes that have helped in advancing scientific knowledge in the fields of physics, mathematics and astronomy. I will describe each paradox and clarify why it is (at least apparently) self-contradictory. The readers are very encouraged to try and find a solution to these paradoxes on their own, or look for it on the online resources provided here.
Achilles and the Tortoise Paradox – This is likely one of the best known paradoxes of all times, although it is just one representation of a more general paradox of classical mechanics, usually named “Zeno’s paradox of motion”, from the name of the Greek philosopher and mathematician born around 495 BCE in the current south of Italy. In Greek mythology, Achilles was the strongest warrior of the Greek army of Agamemnon in the famous Trojan war. We all know that Achilles must be way faster than a slow-moving, peaceful tortoise. Now, assume that Achilles and a tortoise are up for a race, and the tortoise is given a little, finite, advantage ahead of the starting line. Both Achilles and the tortoise begin to run at the same time. By the time Achilles reaches the position that was previously occupied by the tortoise, the latter will have advanced by a small step. Once Achilles reaches this new position, the tortoise would have advanced further. Hence, the separation between the tortoise and Achilles would constantly decrease, but never reach zero. So the tortoise will eventually… win! Of course, we all know this is not true (otherwise, following Zeno, all motion would be impossible!) and Achilles will, alas, always win. So, where is the trick?
Infinite Hotel Paradox – This paradox, or thought experiment, deals with the counter-intuitive properties of infinite sets and was devised by the German mathematician David Hilbert in 1924. Assume the existence of a hotel named “Infinity”. This hotel has the remarkable property of having an infinite number of rooms, numbered 1, 2, 3… up to infinity. Luckily for the owner of this hotel, all its rooms are occupied by an infinite number of guests. An additional individual shows up at the reception, asking for a room. Can we accommodate the newly arrived person? What happens if one, infinitely long, bus with an infinite number of prospective new guests arrives? Can we accommodate all of them? And what if an infinite number of buses, each of them with an infinite number of prospective new guests, arrives? Can we still accommodate them? The answer to all these questions is yes. The real problem is… how?
The Olbers’ Paradox – This paradox was formulated by the XVIII century German astronomer Heinrich W. Olbers and proved to be fundamentally important for the development of the study of the Universe as a whole, or Cosmology. The paradox originates from the very simple observation that the night sky is dark. We all know this for a fact, but this observation is troubling in case we assume the Universe to be infinite, thus containing an infinite amount of stars. If this were to be true, then any line of sight would encounter, sooner or later, the surface of a star, close-by or far away. Hence, the night sky should not be dark at all, rather should be as bright as the surface of our Sun. Quite surprisingly, it is not even sufficient to assume that the Universe is finite to solve the paradox. So, where is the trick? The final solution was firstly hinted at by the American writer Edgar Allan Poe (!) in his poem Eureka and then completely solved by the development of the Big Bang Theory.
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