Why are Black Holes black? (Gravitational Red Shift)

Why are Black Holes black?


We all know: black holes are black. No light escapes a black hole. But, why? 

We come across the answer for this while studying gravitational red shift.

Gravitational Red Shift

Gravitational red shift is similar to the concept of Doppler red shift, but different. Doppler red shift is a change (lowering) in frequency of a wave due to its velocity. But, gravitational red shift is the lowering of frequency of a wave due to gravitational force in the space-time fabric.
The derivation that explains this shift happens o be simple, but we shall leave that for now. At the end of the derivation we conclude that the relative frequency change of a wave (due to gravitational force) equals to GM/c2R; where G and c are our familiar constants - Newton's 'G' and the light's 'c', and M and R are the mass and radius of the star respectively. The star here could be the star of any phase - a main sequence star, a super giant or even a black hole! G and c being constants, we shall worry about the M and R only.

For most of the stars, including our dear Sun, the M/R ratio is too small to make any noticeable gravitational red shift. However, as a typical white dwarf contains the mass as that of our Sun in the size as that of our Earth, the M/R ratio is big enough to make an apparent gravitational red shift. And when we say 'big enough', its just big 'enough' - its just on the limit of measurement.

Black holes

Now, what if a star is more and more denser. So denser that the M is too big than the R. Extremely dense that it is a black hole and its radius is zero. Zero? The R we take here is the radius enclosing the mass M. The mass M of a black hole is contained in the singularity and hence R=0. The Schwarzschild radius of a black hole, the usual radius we would talk about regarding a black hole, is the radius to the event horizon and we shall not confuse it with the R here.

In this case, the red shift stretches the photon's wavelength to infinity. For a photon to come out of that point in space-time, it necessitates an energy higher than its initial energy. Hence, no photon can ever leave that star - the star emits no radiation at all - it remains black.

Defining a Black Hole

This calls for defining a black hole in terms of the red shift equation.
Applying General Relativity and complex mathematics, we see that GM/c2R should be greater than or equal to 1/2 to call a star a black hole.

The Escape Speed

At the Schwarzschild radius, the escape speed would be light-speed c, and inside it greater. 



All these interpretations makes black holes black. We shall not feel racist while calling them black holes; we are just being scientific.

Love,
ED

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