Black holes are one of the most mysterious and intriguing objects in the universe. The
name is appropriate because, first, it is black, meaning that it absorbs all the light coming
in, and does not reflect anything back. It’s a hole because it is as if it is a puncture
in the fabric of space-time. The space inside the event horizon does not behave
anything like empty space.
But in 1974, Stephen Hawking theorized that a black hole may not be so black after all.
His calculations showed that when you apply the laws of quantum mechanics to the classical
physics that had defined our understanding of black holes, you find that they shine.
They emit radiation. They give off photons.
But, how is this possible given that black holes can only absorb light. And if it can't
reflect light, then where are these photons coming from? How did they get created?
Surely these photons could not be coming from nothing. Or can they?
The explanation is coming up right now…
The gravity around a black hole is so strong that it had been thought that nothing could escape
from it, including light.
If we ignore quantum mechanics, then in classical physics, the mass of a black hole cannot decrease,
it can either stay the same or get larger, because nothing can escape a black hole. But
things can fall in it, so it can gain mass that way. But things can’t escape from it.
If mass and energy are added to a black hole, then its radius should get bigger. If the
radius gets bigger, then its surface area will get larger as well according to the equation:
A = 4piR^2. This is just the formula for the surface area of any sphere.
For a black hole, the R in this equation is called the Schwarzshield radius. And this
radius is proportional to the mass of the black hole according to this equation:
To Stephen Hawking and others, this idea of the surface area staying the same or increasing
looked very similar to the 2nd law of thermodynamics. The second law of thermodynamics states that
“In any natural process, the entropy of a closed system always increases or remains
constant, it never decreases.”
So Hawking postulated an analogous theorem for black holes, and it is called the second
law of black hole mechanics. And it says: “In any natural process, the surface area
of the event horizon of a black hole always increases, or remains constant. It never decreases."
So now you can see the parallels with the 2nd law of thermodynamics regarding entropy.
Similar to the 2nd law, there are also ways to state the other 3 laws of thermodynamics
in a way that are true for black holes as well.
The analogy with the laws thermodynamics suggest that perhaps black holes are physically a
thermal body.
In thermodynamics, there is something called a black body. A black body is something that
doesn’t transmit or reflect any radiation, it only absorbs radiation. Analogously, a
black hole is something also doesn’t transmit or reflect any radiation, it only absorbs
it. It absorbs photons.
If a black hole can be thought of as a black body, then it must have a temperature associated
with it, because a black body in thermodynamics always has a temperature.
But if it has a temperature, it must shine in some way.
But now we have a conundrum, because according to classical physics, a black hole is not
supposed to release anything. Stuff only goes in. No stuff is supposed to come out. So how
do we reconcile these two thoughts?
When Stephen Hawking saw these ideas, he found the idea of shining black holes to be preposterous.
He set out to prove why they would NOT shine.
But when he applied the laws of quantum mechanics to general relativity, he found the opposite
to be true. He realized that stuff can come out near the event horizon. In 1974, he published
a paper where he outlined a mechanism for this shine.
So what was the mechanism he outlined that would allow black holes to emit photons?
So the simplest explanation is this: All of space is teaming with virtual particles that
come in and out of existence all the time and everywhere. This is based on the Heisenberg
uncertainty principle.
One version of the Uncertainty Principle can be written as the following:
Delta E times Delta T is less than or equal to Planck’s constant over 4 pi. So basically,
what this equation says is that the uncertainty in energy and uncertainty in time are inversely
proportional to each other, because the product of the two is equal to a constant. In other
words, if you know very precisely the energy of a system, then you can’t know the time
over which you made that measurement very well. Or visa versa, you can know the time
very well, but not the energy.
But what this equation also tells you is that you can get particles with an energy delta
E and if it occurs for a very short period of time, delta t, such that the product of
the two is less than Planck’s constant over 4 pi. That is, particles can exist that violate
this uncertainty principle.
How is this possible? Well, this is one of the crazy things about quantum mechanics.
Violations are allowed. But it’s as if by not obeying this Heisenberg uncertainty principle,
the universe really doesn’t register or record its existence because no measuring
device would ever be able to measure this directly.
A particle with some finite energy, as long as the change in time is very small, can exist.
So what's happening is particle/antiparticle pairs borrow temporary energy from the present,
and give it right back in the future by annihilating themselves.
This is how virtual particles are formed in empty space. And space is teeming with them.
This is called also the quantum foam.
You might ask, if we can’t measure it, how do we know it's actually happening.
Well, it does affect the universe in ways that are measurable, for example, it manifests
as a force in something called the Casimir effect, in which the quantum foam outside
a set of two plates is greater than the pressure inside the plates, and this creates a force
pushing the plates together.
So this virtual particle creation and annihilation does exist, and is a central part of quantum mechanics
The severe curvature of space-time near the event
horizon of a black hole disturbs this quantum foam in ways that you don’t see in normal
empty space. As Neutrinos and antineutrinos, or electrons and positrons, and other particle-antiparticle
pairs can get created, sometimes when two of these particles are close to the horizon,
one particle can get sucked into the black hole before the two particles have a chance
to annihilate each other.
This kind of capture and release by the black hole can happen anywhere in the space around
the event horizon - outside it as well as inside it.
If the partner is left outside, it will no longer have a partner with which to annihilate,
so it will remain and escape from the black hole. This particle will be carrying energy
with it. This is what we perceive as Hawking radiation outside the black hole. This is
how a black hole shines.
Where did this energy of the escaped particle come from? From our perspective outside the
black hole, the particle we got is positive, but this means that the black hole got negative
energy. In other words it lost energy. This is the same thing as losing mass because of
mass energy equivalence of Einstein’s famous equation, E=MC^2.
So the virtual particles are created in space by borrowing energy, but ultimately, so that
nothing violates the law of energy conservation, the energy of the shine is really coming
from the mass of the black hole.
So this is a popular way to think of Hawking Radiation but it has some problems. I think
the biggest problem with this is that the radiation from black holes is not in all wavelengths,
as would be expected with this mechanism. The radiation actually has a wavelength equal
to the size of the black hole. So smaller black holes emit shorter wavelengths, or more
energy, than larger black holes.
So a more accurate way to look at this the following. Now this is still an approximation,
but it is a probably a closer approximation.
In reality, there really are no particles, only fields. This is the crux of quantum field
theory.
The actual Hawking calculations considered waves coming in from infinity and being scattered
or disrupted because of the black hole event horizon, as it was forming.
Certain vibrations of waves are deflected by the gravitational field of the black hole
as it forms in the past. Some of these get distorted or even absorbed by the event horizon.
Some waves do not get deflected at all.
He showed that the wave entering the event horizon was disrupted in a way that the wave
on the other side, carried away energies corresponding to the size of the black hole.
Particles with waves as large as the event horizon get lost within the event horizon,
so the energy we see are about as large as the event horizon.
The quantum fields that have wavelengths the size of the black hole get out with more energy
than they came in with, because waves that get absorbed by the black hole have to be negative
energy in order for us to see positive energy in our universe.
This corresponds to an energy spectrum analogous to a black body at a certain temperature.
So this is why Black holes have a temperature and this is what we perceive as
Hawking Radiation.
But is Hawking radiation real? Can we measure it? Not directly but
Hawking found a formula for the temperature of a black hole.
Note that the temperature is proportional to the reciprocal or inverse of the mass.
As the black hole evaporates over time, the M in the equation becomes smaller and smaller.
This means that the temperature rises as the black hole evaporates. As the black hole evaporates,
its mass decreases.
So the hottest black holes are the smallest ones. This is why they lose energy faster.
Now here’s the interesting part, as the mass goes to zero, the evaporation rate goes
to infinity. So this tells us that near the end of the evaporation process, we would see
an explosion of the black hole as the mass is quickly used up. This would be seen as
a burst of high energy photons or gamma rays.
The lifetime of a black hole is calculated using this equation:
if you do the calculations, it means that anything with mass less than 10^15 grams would
have evaporated by now. These would be tiny black holes about as massive as Mount Everest.
They would only be about the size of a proton by the way. Hawking theorized that such tiny
black holes could have existed at the time of the big bang.
But it also means that black holes slightly larger than 10^15 grams, would be evaporating
around this time in our universe. And if this is happening, it means that we should see
a bunch of Gamma Ray bursts.
So do we detect Gamma Ray bursts?
We absolutely do. In fact about one gamma ray bursts or GRB occurs per day.
However the pattern of gamma rays do not fit with what we would expect to see in a black
hole explosion. What we see are bursts with variations in brightness, from bright to dim
to bright again.
The black hole evaporation should look like a steady increase in luminosity from a low
value to a high value until a final explosion.
So these gamma ray bursts are attributed to another phenomenon – probably colliding
neutron stars, or explosions of supermassive stars, not evaporating black holes.
So the data does not support the idea that very small black holes exist. But despite
the fact that no direct evidence of Hawking Radiation exists, it perfectly fits within
the laws of quantum mechanics, and few if any physicists dispute its existence.
Here’s what I find incredible about black hole entropy.
First let’s clarify what entropy is – it is a measure of the amount of disorder in
a system. You can scramble an egg, but you can’t unscramble it. A more disordered system,
like the scrambled egg, has greater entropy. There is only one way to assemble an egg,
but many ways to scramble it. You need more information to describe the scrambled or disordered
state. So information is also proportional to entropy.
Beckenstein showed that the entropy of a black hole is defined by this equation:
A is the area of a black hole. And Z is very large constant. This means entropy of black
holes is a huge number. A black hole of the size at the center of our Milky Way galaxy,
has an entropy on the order of about 10^91
If you take all the entropy in the universe, ignoring gravity and other black holes, I
mean, take the entropy of all the matter, stars, burning fossil fuels, all the dark matter,
it would only be about 10^88.
So our black hole at the center of just our galaxy has almost 1000 times the entropy of
the entire universe. And there are at least 100 billion other such black holes in the
universe.
So almost all the entropy of the universe is contained in black holes. Anything outside
black holes is negligible in comparison.
And if you equate entropy with information, this should tell us that most of the information
of the universe also lies within black holes. Why is this the case? What the heck is going
on inside these things? Since no one can ever go inside and come back out to tell us, it’s
hard to say.
But before you get too excited, this doesn’t mean that black holes are a giant computer
or brain. It’s not information like in books, or hard drives. But it’s information that
defines the various microstates of particles in a system. It’s mind bending thoughts
like this that makes science really interesting.
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