October 11 - The Schwarzchild Radius
Meaning of Equation
The Schwarzchild Radius represents the ability of a mass to cause curvature in space and time. If an object is crushed to a sphere of Schwarzchild radius, its escape velocity will equal that of light and the object will become a black hole.
Every massive object is associated with five physical quantities: Inertial mass, rest mass energy, Compton wavelength, Schwarzchild radius and the standard gravitational parameter. The Schwarzchild radius actually represents the ability of a mass to curve the space and time. This quantity actually originated from Einstein's field equations of general relativity. Before turning to the physical significance of this radius, let us have a quick look at Einstein's picture of the Universe.
According to Albert Einstein, the universe is a fabric of space and time. Newton developed his laws by considering the fact that space and time are absolute. They are same for everyone, no matter where the person is in the Universe. Einstein said No! Time is not absolute (See this why it isn't absolute). Space and time are affected by mass. To get a picture of it, look at this image below.
Sun warps the fabric of space and time around it. Earth does that too but the magnitude is very less as compared. The difference between the two? Their mass. Thus, massive objects bend more space-time than the lighter ones. Another fascinating feature of this beautiful theory is that even light is affected by this curvature. So the light coming from stars behind the Sun will follow a curved path when that light passes in the vicinity of the Sun (That's how General Relativity was proven experimentally).
Now the amount of bending of light will depend on the space-time curvature which will eventually depend on the mass distribution of that object. Remember, a star of 5 solar masses might not be a black hole but a black hole of 5 solar masses can exist. The difference is that the star has its mass entirely distributed over a large volume but in case of a black hole, the entire mass is distributed in a point like singularity. Thus the space-time curvature for a black hole is enormously large.
Now what if I have a mass distribution that bends the space-time curvature so much that even the fastest entity in this universe, light, gets trapped around that object? For this to happen, the escape velocity of that object (the minimum velocity required to break through the gravitational potential of an object) will equal that of light. Thus the kinetic energy (1/2 mv^2) will be equal to gravitational potential energy (GMm/R). Equating v = c, the velocity of light, we get the above formula for the Schwarzchild radius. Thus, if anything in this universe is suppressed into a sphere of this radius, it will cause such a tremendous curvature in space time that even light will not be able to escape and thus the object will become a black hole.
Okay so now when we know the concept and the formula, let us have some fun by plugging the values of different objects in it. For Sun, R is 3 km. So if Sun is compressed into a sphere of radius 3 Km, it will become a black hole. For Earth it is 9 mm. For a human.... I leave that to you: plug in your mass M and find out!
The Schwarzchild radius arises due to singularity term in the Schwarzchild metric of Einstein's field equation. For advanced study, see Schwarzchild Metric.