How massive does an object have to be in order to be transformed into a black hole? There is a very simple way to find this out using a standard algebraic equation.

A black hole is simple to understand in principle, but nearly impossible to truly picture in one's mind. The best way that scientists have found to truly comprehend these strange objects are by way of creating mathematical definitions of their composition and behavior.

While most such equations have evolved from the famously complex mathematics of general relativity (it was said upon the publication of Einstein's theory that there were only a dozen people in the world who could understand it), and are therefore prone to causing nothing but trouble to a novice mathematician, there are a few exceptions which can be very interesting (and even entertaining).

What is a Black Hole?

To use the simplest possible definition - a black hole is any object which is so massive (that is, it has associated with it such a great force of gravity) that even light itself is unable to escape it.

The central idea here is what is known as "escape velocity." Every massive object has associated with it a particular escape velocity - that is, the speed at which an object must travel in order to break free from the gravitational force. In other words - if you were to fire a cannon straight up into the air, how fast would the cannonball have to travel in order to make it into outer space?

Now, the only way an object could be massive enough that light would be unable to escape its gravity, in theory, is if its mass was so great that its escape velocity is greater than 300,000 km/s (that is, the speed of light). With an escape velocity greater than this, any object can, potentially, be a black hole.

Determining the Size of a Black Hole

In 1916, German physicist Karl Schwarzschild was finally able to determine once and for all the exact radius of any black hole's "event horizon" (which signifies the outer edge of a black hole, the "point of no return" for light and any other object falling into the black hold), which can be found using only the knowledge of an object's mass.

The formula Schwarzschild created is this: Sr = 2Gm/c²

Where Sr is the Schwarzschild radius, G is Newton's gravitational constant (about 6.67428 x 10^-11), m is any object's mass, and c is the constant speed of light (300,000,000 m/s).

Thus, by knowing only a single variable - m - one can, using simple algebra, discover the Schwarzschild radius of any given object.

Applications

Unless one is a physicist or astronomer, the practical applications to knowing the equation for Schwarzschild radius are very few. It can, however, be a source of mathematical amusement.

This equation tells, simply, how small a given object's mass would have to be compacted to in order to be considered a black hole.

Consider the following: If the Earth's mass is plugged into the equation, the result becomes Sr = 0.0088, which can be translated to about 9 mm. This means that if the entire mass of the Earth were suddenly to be compacted into a tiny ball just 9mm across, it would have a great enough mass density to be considered a black hole, and the gravity within this 9mm radius would be sufficient that light would be unable to escape.

Likewise, the Schwarzschild radius of the sun can be similarly estimated to a radius of about 2.5 km. Much larger than the Earth, to be sure, but still rather insignificant when compared to the largest black hole yet "discovered," which exceeds the mass of the sun by about eighteen billion times.

And what size black hole would the mass of an average human being translate to?

The reader is encouraged to explore this question on their own.

References:

Hawking, S. (1988). A Brief History of Time. New York, NY: Bantam Books.

Hawking, S. (1993). Black Holes and Baby Universes and Other Essays. New York, NY: Bantam Books.

Carr, B. J., & Giddings, S. B. (2005, May). Quantum Black Holes. Scientific American , pp. 48- 55.