Quantum Physics and Black-Hole Evaporation
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 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 a black hole can only absorb light. And if it can not reflect it, then where are these photons coming from? How are they created? What is making these black holes shine?
Surely these photons could not be coming from nothing. Or can they?
The gravity around a black hole is so strong that it had been thought 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 seconds 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 is also ways to state the other 3 laws of thermodynamics in a way that are true for black holes as well.
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.
In 1972, theoretical physicist Jocob Bekenstein recognizing the similarities between the 2nd law of thermodynamics and the 2nd law of black hole dynamics, proposed that the surface area of a black hole is proportional to its entropy.
So he proposed essentially that the greater the surface area, the greater the entropy of a black hole.
You might ask why is there any correspondence with thermodynamics and black hole dynamics?
Why should there be any disorder in a black hole, because can’t really see any disorder from the outside?
The larger a black hole is, meaning the larger its surface area is, the more stuff it has consumed. Since everything that the black hole consumes has information. For example if you throw a computer or book into a black hole, the black hole will consume that information. This information ends up inside the black hole. The more information a system contains, the greater the entropy of the system.
Why does entropy increase with information? Think of it this way, the more disordered a system is, the more information is required to describe it. So it follows that the opposite is true as well. if you have 100 coins and only the first one is heads, then there is just one way to describe it. But if 50 of them are heads and 50 are tails, distributed at random, it takes more information to describe that. This is higher entropy.
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.
And 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?
Beckenstein didn’t pursue this idea any further because he presumed that this is where the analogy between the laws of thermodynamics and the laws of black hole dynamics must fail.
However, 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 1975, he published a paper where he outlined a mechanism for this shine.
The remarkable thing is that quantum mechanics was not Hawking’s area of expertise, he was a general relativity expert. But he taught himself quantum mechanics. And because of his physical challenges, he did most of the calculations in his head. This is truly amazing.
So what was the mechanism he outlined that would allow black holes to emit photons?
I will first explain this in the simplest visualization, which you’ve probably read about and seen in other videos. This has some problems and may not quite an accurate model of what’s probably really going on.
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 this can be written as the following.
Delta-E*Delta-t >= h/4*pi
That’s 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.
Put more simply particle/antiparticle pairs borrow temporary energy from the present, and give it right back in the future by annihilating themselves. This occurs over a shorter time than can be measured.
This is how virtual particles are formed in empty space, and space is teeming with them. This is called the quantum foam. It’s like tiny bubbles of energy constantly forming and then snapping out of existence almost instantly. I showed this in my quantum field theory video as well.
You might ask, if we can’t measure it, how do we know it happens.
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.
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. The result of this is the one particle goes inside, and the other one escapes to the outside.
These kind of capture and release can happen with particles 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 ultimately 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. Even though this is still an approximation, it is 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. So think of it this way:
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 blackhole 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 the Hawking Radiation.
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But is Hawking radiation real? Can we measure it?
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:
t=yM^3 y=5120*pi*G^2/(h(bar)C^4)
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 small 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 from what we can observe.
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 physicist dispute its existence.
Here’s what I find incredible about black hole entropy.
Beckenstein showed that the entropy of a black hole is defined by this equation:
S=zA z=C^3*k/(4h(bar)G)
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 hole is negligible in comparison. It’s not in burning wood or heat dissipated from the sun, or your body or other living things.
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 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.
