This post is a continuation of the series of posts containing sections of my research paper titled ” An Introduction to Relativity, Black Holes, and Hawking Radiation”

I recommend reading the previous posts of this series for a better understanding of this post. The links for the same are given below.

Black Holes:

The material in this chapter borrows extensively from chapter 12 of [3].

Black holes are formed by the collapse of a sufficiently large star. The process of this collapse is quite an interesting story.

A star is originally formed by the gravitational collapse of a lot of Hydrogen and Helium gas. Due to this collapse, the temperature increases exponentially thus causing the fusion of Hydrogen molecules to release an enormous amount of energy. This released energy counters the gravitational collapse and maintains a stable state.

Eventually, this Hydrogen runs out and the gravitational collapse continues until the temperatures have risen enough again to burn Helium and other elements.

However, this too shall eventually run out.

When all the thermonuclear fuel runs out in the star, two things can happen:

  1. Either an equilibrium star is formed like a white dwarf or a neutron star.
  2. Or a black hole is formed.

When the star is relatively small, a white dwarf star is formed. These are formed by the Fermi pressure of electrons. This pressure is created due to the fact that more than 2 electrons cannot occupy a single orbital (Pauli’s exclusion principle). For smaller stars, this pressure is enough to prevent further gravitational collapse. However, these white dwarfs are considerably smaller than their original stars. Our Sun will form a white dwarf star. Even bigger stars form neutron stars to counter gravitational collapse. These are stable due to the fermi pressure of neutrons. These are much smaller and denser. If the star has a mass even more than the mass of stars that form neutron stars, then these fermi pressures are not enough to cancel the gravitational collapse. An ongoing gravitational collapse takes place that leads to the formation of a black hole.

Defining new coordinates:

In this section, we will define a new set of coordinates called Eddington-Finkelstein Coordinates.

We have seen in the previous section that the Schwarzschild geometry outside a spherically symmetric star is independent of the time coordinate.

From the function we defined for Schwarzschild geometry we see that there are singularities (values of r where the function tends to infinity), at r = 2M and r = 0.

It is noticed that the singularity at r = 2M is not actually a singularity in spacetime. Rather, it is a singularity due to the coordinate system that has been chosen.

Eddington-Finkelstein Coordinates attempt to remove the singularity created at this point by redefining the time coordinate according to the following relation:

Thus, the line element of the Schwarzschild geometry in these coordinates is given by:

The r = 0 coordinate is a singularity in both systems. This is a point of infinite spacetime curvature. It is a physical singularity.

Light rays in black holes:

Let us see how light rays behave in a black hole.

We know that light rays move on null lines that have ds2 = 0, dθ = dφ = 0 we get that:

There are some light rays that move along v = constant lines. These are known as ingoing because as t increases, r decreases to keep v constant.

We solve that for dv/dr and get:

Light rays that travel along these curves are known are radial light rays.

Radial light rays are known as outgoing if r>2M and ingoing if r<2M.

r = 2M is a special solution of the above equation which describes neither outgoing nor ingoing rays. These rays are completely stationary.

These rays form a surface called the horizon of the black hole that divides spacetime into two regions. Outside the horizon, the light rays can overcome the force and escape to infinity. However, inside the horizon, the gravity is so strong that no light can escape from it, thus, justifying the name “black hole”.

The Horizon:

The horizon is a stationary, three-dimensional surface made of light that neither escapes, nor falls into the singularity. Once the horizon is crossed, one cannot escape the gravitational force.

The horizon continues inside the star also. It grows from the center and stops at r = 2M

The area of the horizon is given by A = 16πM2.

When a mass of say M0 falls into the black hole the radius of the horizon increases to 2(M+M0).

This shows that the area of the horizon of a black hole increases when a mass of sufficient energy falls in it by a factor proportional to the mass of the falling object.

Note: The next section is not required for a good understanding of Hawking radiation and it has been bounded by two lines. However, I have decided to add it in as it is a very interesting concept!

Falling into a Black Hole:

Let us consider two observers Alan and Joe:

Alan is stationary at a distance rR from the surface of a star collapsing into a black hole.

Joe is riding the surface of this star down to r = 0.

Let us first look at Joe.

Suppose that Joe sends light pulses at equal intervals according to a clock that is traveling with him.

These light pulses will reach Alan as long as Joe has not crossed the r = 2M mark (remember we defined this radius as the Schwarzschild radius!). After crossing this r = 2M mark, there is no way that the light rays emitted by Joe will reach Alan. They will curve to smaller values of r and finally end up at the singularity at r = 0.

This is the final fate of the entire star as well once it crosses r = 2M. It turns out that whether the star is spherical or non-spherical inside the horizon, it will end up in a singularity.

This is due to the fact that every timelike worldline in this situation will lead to r=0, because of the strong gravitational force.

Thus, there is no way Alan can receive Joe’s signals and there is also no way that Joe can escape with his life. Even if he is able to leave the surface of the star there is no possible timelike worldline he can follow that won’t end up at r = 0.

Now, let us look at Alan. What does he observe?

As the star collapses, the light rays which are being sent at equal intervals according to Joe are actually coming with longer and longer intervals according to Alan. Thus, the light received is becoming increasingly redshifted.

Also, no light is received after the star crosses r = 2M. Therefore, he doesn’t ACTUALLY see the star collapse. The last signal is sent out just before r = 2M.

For a quantitative idea of this one can look at example 12.2 of chapter 12 of [3].

As the collapse progresses, Alan receives increasingly redshifted light with lesser and lesser energy. He just sees the star lose luminosity and finally go to zero. The time period of this, however, is very small, of the order of 10-5 seconds.

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