Black holes are the most intriguing objects in the cosmos. It is broadly defined as the region in space-time which exhibits a strong gravitational pull , such that even light cannot escape. Black holes were first predicted in the theory of General Relativity as sufficiently compact masses that can deform the fabric of space time.In this article we will understand each of these terms : black holes, event horizon, Schwarzschild Radius , singularity, supermassive black holes, gravitional pull of black holes in depth.

Now before jumping directly into the further technicalities of the black hole , lets brush up on some definitions like , “escape velocity” .

Escape velocity
is the minimum speed required for an object to escape the gravitational influence of a massive body. Or its the minimum launching velocity for an object such that it never returns back. It is assumed that no extra energy is added along the way. A NASA rocket has fuel to continuously add energy, so it doesn’t need to be launched at the escape velocity. On the contrary , a ball if to be launched into space ,such that it never returns back needs to be launched at a specific velocity since there would be no fuel to provide energy through out its path.

   

For example , a cannon ball shot will escape the planet only if its launched  with a specific speed. In this case as we can see its the path E, in which the ball continues to move away rather than again coming under the earths gravitational influence unlike the other paths.

 

 

In order to define this escape velocity mathematically and determine the escape velocities of various planets , lets look at Newtons equations of gravity , in which he stated that the force of gravity between two objects of mass “m1″ and ‘m2” is,

where “G’ is the gravitational constant having a value of 6.67408 × 10-11 m3 kg-1 s-2

and “r” is the distance between their centers.

Also the Gravitational potential energy is

Therefore , a ball or a cannon when launched with the escape velocity is assumed to travel towards infinity when finally the velocity becomes 0 and thus its kinetic and potential energy as well.

 

and 

 

equating E(at surface) = E( at ∞) we get 

So, the escape velocity is

Now to calculate the escape velocity of earth we just nee to plug in the values of mass(M) , gravitational constant(G)  , and radius(R) in the equation. And it comes out to be 11.6 km/sec or 11184 m/s. Which means that if launch a cannon or a ball up into the air with a velocity of 11.6 km/sec that it would escape out the earths gravity and go out into space.

Why did we spent so much time knowing what escape velocity is?

It’s simply because ,  Black Holes can be defined as any object whose escape velocity is greater than the speed of light. And since we know nothing can travel more than the speed of light (including light itself) so nothing can escape from it.

Clearly the earth or in that case any other planets escape velocity is much much less than that of a black hole.

Before we proceed ahead i would want to clear this fact that , any object can in theory be transformed into a black hole (which includes you and I as well !!)if we can somehow manage to shrink down the object to a particular dimension(which is very very small compared to its actual size) keeping the mass constant.

HOW???

Well , here’s how,

we know,

If we reduce the radius (r) , then automatically the Velocity (v) will increase.

As we stated earlier that, in order to form a black hole , the escape velocity needs to be greater the speed of light. So if we can reduce “r”  to a certain value such that  velocity becomes ≥ the speed of light, then we can make the object a black hole ,ofcourse keeping the  mass constant throughout the process.

So if we wanted to convert the earth into a black hole then,

We would need to shrink it down to a marble sized object of radius 8 mm , keeping its mass constant.

Now lets assume we successfully did that ( Yes we do assume more bizarre things in Physics!) , then would the newly formed black hole would be able to sustain itself ?

Lets analyse it with an example.

suppose we keep a box of 1 kg on this newly formed black hole. Then the force that it would experience , would be

As we can see , this force is so immense that, the black hole would instantly collapse into itself under its own mass as it simply cannot sustain this incredible force, and in that process it would obviously take the box along with it as well. There is no such known forces in nature( electromagnetic , gravity, strong and weak nuclear forces) which can oppose this.

Now the question is how do we create a black hole that can sustain itself?

Well , we don’t need to worry about it as the universe has already being doing that since its beginning.

Lets see how this process takes place,

Formation of Black Holes

Stars as we know are nothing but huge glowing ball of fire, which is powered by nuclear fusion. We have discussed in details on how a star glows in our article on ‘The Standard Model of Particle Physics”.

A star continues to glow throughout its lifetime, because the Gravitational force of the star due to its own mass is counteracted by the radiation pressure due the nuclear reaction going on in its core.

 

This keeps the star stable and prevents the star from getting collapsed due to its own weight.

But at the end of star’s life , when all the hydrogen gets over and there is no more “fuel’ left to continue the fusion, the radiation pressure slowly goes away and the gravity starts to dominate.

Consequently the star starts to collapse on itself. Now the obvious question is whats going to finally stop this collapse? or will this collapse ever come to a stop?

The answer is, yes the collapse will stop at a point and there arises two cases :

  1. The atomic forces stops the collapse
  2. or Pauli’s Exclusion principle comes into action, which states that no two electrons can be in the same state.

Sometimes when the star is big enough the gravity completely knocks of the Pauli’s Exclusion Principle and the protons and the electrons are squashed together to form neutrons, and we get whats known as the neutron star. But on the other hand when the star is much much bigger say a million times bigger than our sun , then the whole mass of the stars core collapses and a black hole is formed.

So black holes are the by-product of star, that has ended its journey.

But as we mentioned earlier, theoretically any mass can be converted into a black hole if we could shrink the object down to a much smaller scale , keeping its mass constant.

Now that we know how a black hole is formed and all its definitions, its time to familiarize ourselves with few terms  related to a black hole.

Event Horizon –

 a boundary in spacetime through which matter and light can only pass inward towards the  mass of the black hole.Nothing, not even light, can escape from inside the event horizon. The event horizon is referred to as such because if an event occurs within the boundary, information from that event cannot reach an outside observer, making it impossible to determine if such an event occurred.The shape of the event horizon of a black hole is always approximately spherical.

Schwarzschild Radius –

The boundary surrounding the singularity within which the escape velocity is greater than the speed of light(c)  is called the Schwarzschild Radius. It is a physical  parameter that shows up in the Schwarzschild solution to Einstein’s field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole.It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.

The Schwarzschild radius is given as

where G is the gravitational constant, M is the object mass in kilograms and c is the speed of light in meters per second.
Singularity-

 a region where the space-time curvature becomes infinite. For a non-rotating black hole, this region takes the shape of a single point and for a rotating black hole, it is smeared out to form a ring singularity that lies in the plane of rotation. It is the singular region that contains all the mass of the black hole and has an infinite density.The appearance of singularities in general relativity is commonly perceived as signaling the breakdown of the theory. We are still not sure what singularity exactly is! .Its like the “something divided by zero’ error.  

There is a big myth which goes around as , “black holes suck everything like a giant vacuum cleaner’.

This is however not true. Black holes don’t suck anything unless something is not at its vicinity and falls past its event horizon. Yes when an object comes very close to it and falls under its strong gravitational field, it does get pulled towards it , and once it crosses the event horizon, it simply cannot be brought back again. So black holes don’t go sucking things randomly , until something unfortunately comes really close to it.

Therefore if we replaced our sun with a black hole of the same size, then nothing would be affected , except the universe getting super cold. The trajectories of the planets and the way the solar system functions wouldn’t change much.

What if you fell into a black hole? Would you die ? or survive as shown in the movie “Interstellar” ?

Well unfortunately sooner or later, depending on the size of the black hole we do die. However it is not the strong gravity that is main and direct cause of our death its the very high tidal forces , which arises due to the way gravity acts at different parts of our body. The difference in the gravitational attraction at the Schwarzschild Radius and say 6 feet above the Schwarzschild Radius will be huge. So If we fall feet first then, our head would experience much much less force than our feet does and the consequence is you are going to get stretched out of existence  by a process called spaghettification. But you would keep accelerating towards it until you cross the Schwarzschild Radius and then finally  collapse into the black hole.

Now interestingly , if someone were to see this entire fate of you falling into a black hole, he would observe an entirely different story. He would see that you, instead of accelerating towards the Schwarzschild Radius, would get slower and slower and never actually reaching the Schwarzschild Radius. So actually it is of a false hope, because he will think that you haven’t yet met your doom and there is still time to save you ! but the reality is you have long since crashed into the black hole and been completely annihilated.

The reason for this is that the gravitational forces here are so strong that light is having an enormous difficulty travelling away and consequently the observer simply sees this space craft apparently travelling and taking longer and longer and getting slower and slower to get to the Schwarzschild Radius but never crossing it.

There is a mathematical way to demonstrate it.

From special relativity there is an invariable quantity called the proper time. The point about special relativity is that all observers if they are measuring a particular distance or time,depending on their relative velocities they will measure different distances at different time. Nobody agrees on the measurement.

But there is a quantity called proper time, which everybody can agree on.

you can calculate proper by :

where c is the speed of light, T is tau or proper time , t is the time that the observer measures and x is the distance that the observer measures.

So if you measure time and distance and somebody else measures time and distance, then you will get different answers compared to them, provided you are moving with related velocities. But if we plug it into this formula

both parties will get the same T , which is the proper time. The proper time wont vary for you both.

There is a special equation called the Schwarzschild metric, which was derived from Einsteins field equation, which has a striking similarity with the above equation. The Schwarzschild metric says (in polar coordinates) :

where c is again the speed of light, T is the proper time, rs is the Schwarzschild radius, r is measured distance, t is the time measured and dΩ is the combination of ϴs and Фs. and rs is 

Now two peculiar cases arise , when;

  1. r = rs
  2. r= 0

1.1  when r= rs : (at the Schwarzschild radius)

The proper time is time measured by you  travelling in the spaceship towards the Black Hole. The R.H.S of the equation shows the time (t) that is measured by the observer who is seeing you falling towards the black hole. And in this case when r = rs ,i.e when you hit the Schwarzschild radius , the equation shows us that for an infinitesimally small amount of proper time(T) measured in the rocket, the time measured by the observer will be infinity. Which means that for you even a tiny fraction of time in the spaceship will be an eternity for the person observing you. Which means you will be frozen in space near the black hole, according to the observer.

2.1 when r=0 (when you hit the black hole)

 

This means you have hit the black holes singularity. As we can see, the equation here makes no sense at all. The singularity doesn’t obey any laws of physics. We till have no explanation for it.

Energy and Entropy of a Black Hole

To understand the energy and entropy of a black hole, lets assume

we add one unit of energy to fall into the black hole , i.e say one photon.

we know the energy of the photon is hv , where h = plank’s constant. v is the frequency of the light of which the photon is a part. E=hv can also be written as :

Now if we assume that the wavelength of the light  λ= Rs is of the order of the schwarzschild radius, which it needs to be . If lambda is larger than the schwarzschild radius then it will pass away without ever effecting it.  But if lambda is broadly of the order of the schwarzschild radius then we can have this situation:

we take the famous Einsteins formulae  ,    E = mc2   

 

which means that a change in mass :  

 

so this is the change in mass of a consequence of one photon falling into the black hole.

Now we have said that lambda λ  is Rs and we know that  

now according to this formula the radius Rs i.e the schwarzschild radius varies with change in mass by,

so a small increase in mass leads to a small increase in the schwarzschild radius. Now substituting the above calculated value of Δm we get:

 

the surface area associated with the schwarzschild radius is going to be :

 

This shows us that area increases by dA as we throw in one photon of energy, or one element of entropy.

If we put in several photons of entropy , then the increase in surface are will be:

Rearranging the equation we get:

This is the Bekenstein formula, it shows us how the entropy of the black hole, changes as the area increases.  

According to thermodynamics,

were dE is the change in energy , T is the temperature and , dS is the change in entropy.

This is an interesting result, that shows that the temperature of the black hole is inversely proportional to its mass. As the mass of the black hole increases its temperature decreases , and as the mass decreases the temperature increases.

So a smaller black hole will be hotter than a bigger black hole. This is contrary to usual concepts, because we reckon that as mass increases its energy also does increase ( E=mc^2) and we usually associate increase in energy as increase in temperature. But black holes as we can see are quite the reverse.

Hawking Radiation and Black Holes

In the year 1975 Prof. Hawking published a shocking result: if one takes quantum theory into account, it seems that black holes are not quite black! 
 Instead, they should glow slightly with “Hawking radiation”, consisting of photons, neutrinos, and to a lesser extent all sorts of massive particles.  This has never been observed,  and since the only black holes we have evidence for are those with lots of hot gas falling into them, whose radiation would completely swamp this tiny effect.  Indeed, if the mass of a black hole is M solar masses, Hawking predicted it should glow like a black-body of temperature    6 × 10-8/M kelvins. 

so only for very small black holes would this radiation be significant . As we have already seen, smaller the black holes the more hotter they are. The most drastic consequence is that a black hole, left alone and unfed, should radiate away its mass, slowly at first but then faster and faster as it shrinks, finally dying in a blaze of glory like a hydrogen bomb. 

The principle behind this is quantum fluctuations. Virtual particle pairs are constantly being created near the horizon of the black hole, as they are everywhere.  Normally, they are created as a particle-antiparticle pair and they quickly annihilate each other.  But near the horizon of a black hole, it’s possible for one to fall in before the annihilation can happen, in which case the other one escapes as Hawking radiation.

However this is not the whole story. There is literally a lot more to Hawking radiation and we will definitely study it in details in some future article( which would come really soon !! )

Black Holes are the most efficient way to convert Mass into Energy.

Black holes have an unreasonable efficiency . They are great at extracting energy from mass.

This is weird , because we have seen that, nothing can escape a black hole once it is inside its schwarzschild radius.

But the efficiency of a black hole comes from what stuff does while falling towards them. Anything that falls in a gravitational field speeds up gaining kinetic energy and when it crashes with something  this kinetic energy is converted into heat. This heat can then radiate away as infrared radiation , slightly decreasing the mass of the object.

For example , when a meteor comes in the gravitational field of a planet , it gains kinetic energy and when it collides with the air in the atmosphere this kinetic energy gets converted into heat which is radiated away , and in this process the mass of the meteor decreases.

for planets and stars this conversion of mass into energy is pretty pathetic. An object falling through the earths atmosphere and crashing into it only converts 0.000001% of its mass into energy!!! . This is bad as any ordinary chemical reaction.

But black holes have something special going on for them. Black holes have incredible gravity and it completely bends the spacetime so much that any object that comes under its gravitational influence, accelerates so fast and gains so much kinetic energy that it almost converts almost 50% of its mass into energy. However if the object keeps falling past the event horizon, then all the energy will be stuck in the black hole.

∴ Therefore the we can let black holes to convert mass into energy is  by , letting the object to slowly spiral into the black hole, crashing into other stuffs on the way, heating up , radiating that energy away and thereby reducing its speed , slowing down more , spiraling to yet to a lower orbit. This process continues until it reaches the innermost possible orbit.

This is exactly what goes on in accretion disks around Black holes

All the matter in the accretion disk , which includes space debris mostly , is converted into energy. And for rotating black holes , this mass to energy conversion rate is around 42% !!!. Which is way more than even nuclear reactions and any chemical reaction. All the energy gets shot out into space from the poles of the black holes.
Black Hole Information Paradox and its possible solution(MECOs)

Black Holes are engines of destruction that removes from our universe anything that crosses out the event horizon.

But mass and energy aren’t removed from existence , rather they add up to the mass of the black hole as we say earlier.

And we also know that this mass can escape as it gradually leaks away by Hawking Radiation, over long scales of time. This same Hawking Radiation maybe more destructive than the black hole itself. It may destroy the complete information, and also information is no way can escape out the black hole if it happens to cross its event horizon. This complete destruction of the information violates a major principle in physics i.e the ” Law of Conservation of Information”. Which cant be done at any cost.

So the only way it can be in the Hawking radiation (naively) is if it creates a copy of that information while it is inside. Having two copies of the information, one inside, one outside, also violates quantum theory and the ” Law of Conservation of Information”.

This was a major paradox that kept theoretical physicist puzzled till date.

However Indian physicist Dr. Abhas K.Mitra believes that the problem with the black hole information paradox lies in the problem itself. He tells us that maybe black holes  aren’t that complicated as it seems.

In the year 2000, he wrote a long paper  on ” Non-occurrence of trapped surfaces and Black Holes in spherical gravitational collapse” and attacked the problem from various sites. He came to the conclusion that:

  1. there cannot be no exact black holes
  2. no exact event horizon
  3. and no apparent horizon

Immediately before the formation of a black hole, the outer radiation pressure must counteract the inward pull of the gravity. So we have a quasi static state, which has the same size of a black hole. But this object is quasi static and is still contracting and radiating maybe at an infinitesimally slow rate and trying to attain the perfect black hole state.

But in doing so , in its journey it must radiate its entire mass energy. It has to become a zero mass black hole. He also proposed that such objects should be strongly magnetized and thus called it , “Magnetospheric Eternally Collapsing Object” or MECO.

This prediction of the MECO was verified by his American colleagues in 2006. In a famous Quasar, the central object appears more to be ultra-magnetized as MECOs rather than black holes, as predicted by him.  Incidentally black holes themselves don’t have any magnetic field, they only have accretion disks which gives weak magnetic fields.Since then almost a 100 black holes have been found to have ultra strong magnetic field, that cannot be explained by present black hole paradigms.

In 2016 a  NASA report on a quasar revealed that it was emitting corona, which is fire, and this was also predicted long before by Dr. Mitra as he said these MECOs were like ultra compact suns, which emit fire.

Dr. Mitra doesn’t deny the fact that there are no exact black holes or rather goes against General Relativity

Actually there is a thin line.

The mass  of all objects in General Relativity is a result of an integration constant. This value is large for galaxies and entire solar systems, smaller for individual stars and much smaller for planets and moons. What Dr. Mitra has shown is, since the mass of the black hole resides in a point, its mass-energy is zero and zero mass energy in relativity doesn’t mean the absence of matter, it means that all positive energies are counteracted by the negative gravitational energy. This means that whatever we are thinking as event horizon being a sphere is nothing but a point.

So to sum it up, according to Dr Mitra , whatever we have called black holes for so long may might not be so. Rather they are quasi static black holes or MECOs.

And back to the information paradox , when there is not exact black hole, then nothing is trapped, hence there is no information paradox.

However there exists many other theories which we will discuss later.

CONCLUSION

We are left with no choice but to agree that indeed black holes are fascinating objects of the cosmos. They definitely are still not fully understood yet. Black holes are full of wonder and mystery which is yet to be discovered , and probably that’s why they fascinates me the most. Scientists are continuously researching on these amazing “space creatures” even though they are “invisible”. 

Regardless of everything I believe that black holes might be key to understanding the nature of reality itself.

More articles on black hole will be coming soon.

please leave a comment below or ask any question you want to regarding the article “Black Holes”.

I would highly recommend few books that would really help you to know  in depth about black holes  and much more regarding the cosmos:

  1.  THE THEORY OF EVERYTHING
  2. A BRIEF HISTORY OF TIME
  3. GEORGE AND THE BIG BANG
  4. A Brief History of the Universe: From Ancient Babylon to the Big Bang (Brief Histories)
  5. The Physics Book: From the Big Bang to Quantum Resurrection, 250 Milestones in the History of Physics (Sterling Milestones)
  6. THE BIG BANG THEORY
  7. Cosmos
  8. Relativity: The Special and the General Theory (Routledge Classics)
  9. Black Holes: The Reith Lectures
  10. The Oxford Companion to Cosmology (Oxford Quick Reference)

Cheers!

 


Mayukh

I am Mayukh Bagchi. 3rd year Undergraduate Engineering student at SRM University , physics and in general Astrophysics is my passion and I want to pursue the same. This blog is dedicated to serve the community with honest and clear insights to the cosmos.

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