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Black Holes: So You *CAN* Divide By Zero

 “Black holes are where God (or the Flying Spaghetti monster, maybe) divided by zero.” That has to be one of my favorite math/science jokes. Surprisingly enough, weird things do happen to our equations when you work inside the event horizon of black holes – such as dividing everything by zero. So, other than being the source of a great joke, what are black holes?


Technically, a black hole is a region of spacetime where, by nature of its great mass, gravity prevents anything from escaping; this includes light. The term ‘black hole’ was coined by Dr. John Archibald Wheeler during his work in general relativity on gravitational collapse. However, Dr. Wheeler wasn’t the first person to hypothesize a black hole. Einstein wasn’t the first person to think up this phenomena either (even though his equations gave rise to the possibility). Rather, in the 18th century, John Michell and Pierre-Simon Laplace were the first to consider massive objects with gravity fields so great that even light couldn’t escape.
Later, in 1916, Karl Schwarzschild published the first modern characterization for a black hole based on Einstein’s general relativity. His work gave rise to the well-known Schwarzschild radius. This is the radius that extends from the center of an object, within which, if the object was compressed, the escape velocity would equal the speed of light. When the radius of an object is equal to or less than the Schwarzschild radius, it condenses to an infinite density and becomes a black hole. This radius is also called the ‘event horizon’.
If that did not make sense, consider this: If the radius of an object is equal to or less than it’s Schwarzschild radius, it turns into a black hole. For the mass of the sun, this radius is about 2 miles (meaning that if I were to compress the mass of the sun into a sphere with a two-mile radius, it would turn into a black hole). For the Earth, this radius is about 9 millimeters (or about the size of a marble).
How are black holes formed? As a massive star collapses it can turn into one of three things: A black hole, a neutron star, or a white dwarf. For stars with enough mass, as it collapses to the Schwarzschild radius, it turns into a black hole.

Here are some interesting fun facts about black holes:

1) Why can’t light escape a black hole? First, imagine that you have an object and you throw it into the air. If it is traveling at a speed less than the escape velocity, it will slow down and fall back to earth. However, if you throw it faster than the escape velocity, it will keep going forever (or until something else acts on it) – but it will still slow down. This happens because, as this object moves away from the Earth, it is pushing through a gravitational field and loses energy. Light is no different, as light exits a gravitational field, it too loses energy; but because the speed of light is constant, it can’t slow down and thus must lose energy some other way. A photon loses energy in its wavelength and undergoes a gravitational redshift. When light is emitted from inside the event horizon of a black hole, its wavelength gets infinitely long and the photon basically redshifts itself out of existence (think about it like the wavelength getting stretched really, really long).


2) The center of a black hole is also called a “singularity.” At this singularity, spacetime has an infinite curvature and the laws of physics break down. Because the singularity is within the Schwarzschild radius, it can never be observed. Physicists have also theorized the existence of a naked singularity, which is a singularity without the event horizon (meaning it could be observed); so far, no one has observed one. Personally, I have an extremely hard time understanding how such an object could exist, but the mathematics allows for this possibility.
3) Since the escape velocity inside a black hole is greater than the speed of light, it is probable we will never know what happens inside the event horizon since information cannot be communicated across the Schwarzschild radius.
4) When working inside the event horizon of black holes, we assume the laws of physics still hold true and our equations are still valid (even though they tend to blow up). Even though extremely weird things happen within black holes, according to our mathematics, there is no reason to believe the laws of physics work any differently, even though we have to work around things like dividing by zero.
5) If the sun were replaced with a stellar mass black hole (a black hole with the same mass as the sun), Earth would not be sucked in. In fact,Earth’s orbit would remain the same(however, with the lack of sunlight, the inability to have biological functions like photosynthesis would make life, shall we say… interesting).
6) Since black holes are not visible, scientists have to rely on other methods of detecting them. This includes: accretion of matter, x-ray binaries, gravitational lensing, and observing the gravitational interactions between visible matter and the black hole.

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About Mohammad Daeizadeh

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