Maths Of Black Holes by Boris Kramar

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What is a black hole? - The basic idea of a black hole is simply an object whose gravity is so strong that light cannot escape from it. It is black because it does not reflect light, nor does its surface emit any light. The center of a black hole is infinitely small and infinitely dense, hence laws of physics work differently for them. - They are formed, when stars turn into supernovae. The “detonation” of the star causes lots of small black holes to form. They eventually consume each other, as well as matter (gas, dust) around them, growing in size. In the end we get a really big black hole, which continues to grow…frightening. - Even though they may attract matter, usually they do not. If our Sun suddenly turned into a black hole, the Earth would orbit it as usual. However, there are exceptions, which lead to supermassive black holes creating. - There is a supermassive black hole in the center of our galaxy, Milky Way. It’s mass is about 3 million suns, which is about 6*10 to the power of 36 kilograms.

Schwarzschild Black Holes - The simplest black holes. Perfectly spherical, they possess the so-called “event horizon” – the outer edge. In other words, the inside of such a black hole is cut out from outside by an exact edge – and everything inside works differently.

Schwarzschild Equation §Sr = 2 Gm/c² * Where G and c are constants (Gravitational constant and speed of light); and Sr is the Schwarzchild Radius. The Equation allows us to determine radius of event horizon for any black hole, knowing its mass…or vice versa! Now let us try to figure out, which radius should our Earth have to be able to become a black hole? So Sr=2 x 6. 67428 x 10^-11 x 5. 9742 × 10^24 and divide by (3 x 10^8)^2 …which gives us Sr of 0. 0088 metres…about 9 mm. So if the Earth was a ball of radius 9 mm but the same mass, it would be a black hole. That’s all about density.

But let us go further… Let us imagine that I’ve decided to become a black hole, so that I can attract other people!!! Now, which radius should I have for it? Sr=2 x G x 65 and divide by c squared… The result is 9. 64 x 10 to the power of -26. Microscopic – but black holes of such size do in fact exist in the Universe. By putting any mass value into this equation you can find the Schwarzschild Radius for any black hole.

The Hawking Radiation - The gravitational field around the black holes causes pairs of particles and anti-particles to appear. One of them returns to the black hole, another escapes. The mass lost is bigger and the black hole eventually depletes. However, it is unknown yet what actually happens to the particles attracted back to the black hole…this is known as The Hawking Paradox.

How long may it take? § - Let us do this for our Earth-mass black hole. G is the Gravitational constant, M is the mass in kg; h is the Planck constant and c is the speed of light… Numbers are really big and long at this point, so I will simply tell you the result, which is… … 2. 694 x 10 to the power of 50 seconds. This is app. 8. 5 x 10 to the power of 42 years. Yes, a tiny 9 mm-radius ball would deplete for that long…what I tried to show you is that only the smallest black holes (like the “human-mass” one) actually deplete like this. The one in the center of our galaxy will probably exist for a nearly infinite amount of time – and remember, it still can eat something, so it is probably immortal.

Time Dilation - Near the event horizon something really weird happens to time and space. If you were steadily falling into the black hole (let’s imagine you somehow survived it), you would feel the time differently. It is difficult to understand because we did not have any chance yet to test such a situation. §

Into the Darkness! Time for some action! Fasten your belts, guys, we are approaching the Messier-31 Black Hole, center of Andromeda Galaxy, which has a mass of 45000000 Suns!

Closing to the Event Horizon! § R is our starting distance, Sr is the Schwarzschild radius, M is the mass, c is the speed of light. Let us take R of 10 km. Calculations…result is 266360000 seconds or 8. 4 years. OK.

Horizon Reached! That was a long journey, but we did it. Practically we entered the black hole. Now I want to ask you a question. CAN WE RETURN?

The answer is… - By no means. Remember, that light does not have an opportunity to leave the black hole, that’s why it is black. And we certainly can not go faster than the light.

How much time do we have? . . Now we can only wait. Our next destination is the Singularity - center of Messier-31. But for how long will we travel through the darkness? (I promise, this is the last formula!) §

THANKS FOR LISTENING.

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