Tuesday, March 5, 2013

The Schwarzschild Radius

In 1915, German astronomer Karl Schwarzschild obtained a solution to Einstein’s equations for the gravitational field outside a non-rotating, spherically symmetric body. This solution produced a singularity at, what is known as, the Schwarzschild Radius (this is a coordinate singularity, not to be confused with the gravitational singularity at the centre of a black hole-ed). This became the subject of debate for decades on its physical significance or whether it could ever possibly occur in nature. It was not until the latter half of the 20th century that a possibility of a ‘black hole’ was accepted.

So what is the Schwarzschild radius exactly? It is the radius that an object of a certain mass would need to be to create an event horizon from which light cannot escape. Essentially it is the size that the object must reach to create a black hole.
The equation itself to calculate this size is:

rs = 2Gm/c^2


rs is the Schwarzschild radius
G is the gravitational constant
m is the mass of the object
c is the speed of light

Black holes can be classified by their Schwarzschild radius:

A supermassive black hole has an average density inside its Schwarzschild radius that is very low (103 kg/m3, for example, the density of water). This is as for objects of constant density, the Schwarzschild radius increases more quickly than the radius. At around 150,000,000 times the mass of the Sun such an accumulation will fall inside its own Schwarzschild radius and thus it would be a supermassive black hole of 150,000,000 solar masses. Supermassive black holes up to 18 billion solar masses have been observed.

A stellar black hole is one which has an average density (inside of its Schwarzschild radius) which is simlair to that of an atomic nucleus. An object of such density would form a black hole at about 3 solar masses.

Primordial black holes are miniature black holes, comparable in mass to much lighter objects. For instance Earth has a Schwarzschild radius of 9mm, about the size of a peanut. A Schwarzschild radius for a 120 lb. human would be 8.082e-26m. Due to such high density at these sizes, no known mechanism could form an object this compact. They may have existed when densities were extremely high, during the earliest times of evolution of the universe shortly after the Big Bang. This type of black hole is purely hypothetical though.

Wanna know how big the Schwarzschild radius would be for your mom? Your car? Your cat? Your annoying neighbor across the street? Click here for some Schwarzschild calculating fun: http://physics.unl.edu/~klee/flash_astro/bhole_sim010.swf

References and resources:

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