Black Holes

Ever wondered what Black Holes are? What is the Maths behind it and why is it so interesting????

Well in this blog, you’ll get to know about basic mathematical aspect of black holes.

Black hole is defined as a region in spacetime where the spacetime is highly curved, meaning that the gravitational force is so large that even EM radiation is unable to escape it’s gravitational pull.

Black holes are usually found in centre of galaxies or near the same.

Black holes are formed when stars having 8-13 or more solar mass exhaust their fuel I.e. Hydrogen, which then implode.

When star has used up the hydrogen and fused it into Helium, then the star uses the helium in turn as a fuel, producing elements with higher and higher Z, like Oxygen, Neon, Aluminium etc.

Since the temperature of particles isn’t enough to maintain outward pressure, the core undergoes gravitational collapse. Temperature is defined as average kinetic energy per particle.

Note: This can be avoided by star only in creation of a electron degeneracy pressure, as described by Freeman Dyson in 1967 to explain stability of matter

The elements continue to fuse and when they reach till iron or nickel, iron fuses, but heat is absorbed by the iron, and henceforth, the heat cannot be evolved by the process, which leads to loss of the internal thermal energy. This NiFe core then undergoes an implosion, and the gravitational potential energy is released as Supernova.

Their are various types of Supernovae, and they depend upon the mass of remnants of the stars. If the mass of remnant is above TOV limit, then the star will turn to a black hole, but if it isn’t then it will turn to a neutron star.

                      Image credit   - Google

New TOV limit is ≈ 2.3MΘ (Solar Masses)

But why are Black Holes black? Sir John Wheeler coined the term Black Hole. We know that even Light cannot escape from Black Hole once it has entered Event Horizon. This is because of the Escape Velocity. 

Escape Velocity is the minimum Velocity required by an object to escape the gravitational pull or sphere of influence. 

Escape Velocity is described as:                                                        

                               V = √(2GM/R)

                                     

Black hole possesses only mass, angular momentum and charge according to no hair conjecture and Kerr-Newman Metric.

You must be wondering what exactly a metric is? Well a metric is a well proven Fact. Various Physicists researched upon various conditions and proved them, and came to a conclusion. It is metric.

Schwarzschild Metric explains a black hole with M≠0,Q=0,J=0

Kerr-Newman Metric describes Black Holes where M≠0,Q≠0,J≠0

 Kerr Metric describes BH ( Black hole) with (Q = 0, J ≠ 0), and Reissner–Nordström metric describes BH with (Q ≠ 0, J = 0) where Q is Electric Charge and J is angular momentum

In fact, Black Holes' physical Quantities must satisfy following equation:                                                                    

                               Q²+(J/M)² ≤ M²

Where M is Black Hole’s mass, Q is charge and J is angular momentum.

In simplest case, the BH possess mass but neither charge nor angular momentum, and such Black Holes are called Schwarzschild Black Holes.        Schwarzschild Radius

The angular momentum limit for black holes is given by the following Formula:

                                     J ≤ GM/c²

Event Horizon is a boundary around black hole. When this boundary is crossed by a particle, it is no longer capable of escaping out. Event Horizon is more commonly described as a Boundary on Space Time. According to no-hair conjecture, no information can flow out of the Event Horizon except the Three Physical Quantities- Angular Momentum, Mass and Electric Charge are explained by the Kerr-Newman metric.   Kerr matric

The escape velocity of Black Hole at event horizon is the speed of light. This is described as follows.

Contemplating formula for escape velocity, let V = c

Therefore:

                                   c = √(2GM/R)

                                   c² = 2GM/R

                                   R = 2GM/c²

This Radius is called Schwarzschild Radius, the radius of event horizon, and it is given by the formula

                                    2GM/c² = R

Interestingly, Black Hole with a non zero spin or non zero charge has a smaller radius.

We call event horizon so because any event, which occurs within the boundary can't be observed from out, and hence forth no observations can be made by observers. Then how the hell do we predict if something has gone into black hole?

The answer is that anything, that is attracted to a black hole releases X-Rays, which when detected also detects the presence of black holes. X-Rays are released because the tidal forces acting around the black hole cause a rise in matter's temperature, releasing various radiations, among which X-Rays are quite prominent.  EM Radiation

Black holes are usually classified on basis of their mass.

Smallest type of black holes on basis of their mass are micro-black holes, which have mass equal to 7 x 10²² kg And have a radius up to one millimetre!

Second type of black holes are Stellar size black holes, with mass up to 2 x 10³¹, with radius up to 30 kilometre

Intermediate –Black Holes have a mass up to 2 x 10³³ kilograms and radius of about 10³ kilometres. Super Massive Black Holes are largest types of black holes, with a mass about 10,000 ΜΘ (solar mass) to 50 billion ΜΘ (theoretically).

They are formed by collision of various stellar black holes with other types of Black Holes

They have a really large radius (about 0.001-400 AU or even more!!)

Supermassive Black hole

So now, your misconceptions about Black Holes must be clear, and you now know the basics of Black Holes and Basic Mathematical Aspect about it, in terms of Newtonian Gravity and various metrics!

.                         Image credit- Google

Article by- @Ojjas