*[Guest post by **Noah Miller** – a Harvard Physics Ph.D student that took our seminar. Noah’s webpage contains wonderful and extensive notes that can be of interest to computer scientists. –Boaz]*

(The following blog post serves as an introduction to the following notes:)

Black Holes, Hawking Radiation, and the Firewall

There are many different types of “theoretical physicists.” There are theoretical astrophysicists, theoretical condensed matter physicists, and even theoretical biophysicists. However, the general public seems to be most interested in the exploits of what you might call “theoretical high energy theorists.” (Think Stephen Hawking.)

The holy grail for theoretical high energy physicists (who represent only a small fraction of all physicists) would be to find a theory of quantum gravity. As it stands now, physicists have two theories of nature: quantum field theory (or, more specifically, the “Standard Model”) and Einstein’s theory of general relativity. Quantum field theory describes elementary particles, like electrons, photons, quarks, gluons, etc. General relativity describes the force of gravity, which is really just a consequence of the curvature of spacetimes.

Sometimes people like to say that quantum field theory describes “small stuff” like particles, while general relativity describes “big stuff” like planets and stars. This is maybe not the best way to think about it, though, because planets and stars are ultimately just made out of a bunch of quantum particles.

Theoretical physicists are unhappy with having two theories of nature. In order to describe phenomena that depend on both quantum field theory and general relativity, the two theories must be combined in an “ad hoc” way. A so-called “theory of everything,” another name for the currently unknown theory of “quantum gravity,” would hypothetically be able to describe all the phenomena we know about. Just so we’re all on the same page, they don’t even have a fuly worked out hypothesis. (“String theory,” a popular candidate, is still not even a complete “theory” in the normal sense of the word, although it could become one eventually.)

So what should these high energy theoretical physicists do if they want to discover what this theory of quantum gravity is? For the time being, nobody can think up an experiment that would be sensitive to quantum gravitation effects which is feasible with current technology. We are limited to so-called “thought experiments.”

This brings us to Hawking radiation. In the 1970’s, Stephen Hawking considered what would happen to a black hole once quantum field theory was properly taken into account. (Of course, this involved a bit of “ad hoc” reasoning, as mentioned previously.) Hawking found that, much to everybody’s surprise, the black hole evaporated, realeasing energy in the form of “Hawking radiation” (mostly low energy photons). More strangely, this radiation comes out exactly in the spectrum you would expect from something “hot.” For example, imagine heating a piece of metal. At low temperatures, it emits low energy photons invisible to the human eye. Once it gets hotter, it glows red, then yellow, then perhaps eventually blue. The spectrum of light emitted follows a very specific pattern. Amazingly, Hawking found that the radiation which black holes emit follow the exact same pattern. By analogy, they have a temperature too!

This is more profound than you might realize. This is because things which have an temperature should also have an “entropy.” You see, there are two notions of “states” in physics: “microstates” and “macrostates.” A microstate gives you the complete physical information of what comprises a physical system. For example, imagine you have a box of gas, which contains many particles moving in a seemingly random manner. A “microstate” of this box would be a list of all the positions and momenta of every last particle in that box. This would be impossible to measure in pratice. A “macrostate,” on the other hand, is a set of microstates. You may not know what the exact microstate your box of gas is in, but you can measure macroscopic quantities (like the total internal energy, volume, particle number) and consider the set of all possible microstates with those measured quantities.

The “entropy” of a macrostate is the logarithm of the number of possible microstates. If black holes truly are thermodynamic systems with some entropy, that means there should be some “hidden variables” or microstates to the black hole that we currently don’t understand. Perhaps if we understood the microstates of the black hole, we would be much closer to understanding quantum gravity!

However, Hawking also discovered something else. Because the black hole is radiating out energy, its mass will actually decrease as time goes on. Eventually, it should disappear entirely. This means that the information of what went into the black hole will be lost forever.

Physicists did not like this, however, because it seemed to them that the information of what went into the black hole should never be lost. Many physicists believe that the information of what went into the black hole should somehow be contained in the outgoing Hawking radiation, although they do not currently understand how. According to Hawking’s original calculation, the Hawking radiation only depends on a parameters of the black hole (like its mass) and have nothing to do with the many finer details on what went in, the exact “microstate” of what went in.

However, physicists eventually realized a problem with the idea that the black hole releases its information in the form of outgoing Hawking radiation. The problem has to do with quantum mechanics. In quantum mechanics, it is impossible to clone a qubit. That means that if you threw a qubit in a black hole and then waited for it to eventually come out in the form of Hawking radiation, then the qubit could no longer be “inside” the black hole. However, if Einstein is to be believed, you should also be able to jump into the black hole and see the qubit on the inside. This seems to imply that the qubit is cloned, as it is present on both the inside and outside of the black hole.

Physicists eventually came up with a strange fix called “Black Hole Complementarity” (BHC). According to BHC, according to people outside the black hole, the interior does not exist. Also, according to people who have entered the black hole, the outside ceases to exist. Both descriptions of the world are “correct” because once someone has entered the black hole, they will be unable to escape and compare notes with the person on the outside.

Of course, it must be emphasized that BHC remains highly hypothetical. People have been trying to poke holes in it for a long time. The largest hole is the so called “Firewall Paradox,” first proposed in 2012. Essentially, the Firewall paradox tries to show that the paradigm of BHC is self-contradictory. In fact, it was able to use basic quantum mechanics to show that, under some reasonable assumptions, the interior of the black hole *truly* doesn’t exist, and that anyone who tries to enter would be fried at an extremely hot “firewall!” Now, I don’t think most physicists actually believe that black holes really have a firewall (although this might depend on what day of the week you ask them). The interesting thing about the Firewall paradox is that it derives a seemingly crazy result from seemingly harmless starting suppositions. So these suppositions would have to be tweaked in the theory of quantum gravity order to get rid of the firewall.

This is all to say that all this thinking about black holes really might help physicists figure out something about quantum gravity. (Then again, who can really say for sure.)

If you would like to know more about the Firewall paradox, I suggest you read my notes, pasted at the top of this post!

The goal of the notes was to write an introduction to the Black Hole Information Paradox and Firewall Paradox that could be read by computer scientists with no physics background.

The structure of the notes goes as follows:

- Special relativity
- General relativity
- Quantum Field Theory (in which I ambitiously tell you what QFT actually is!)
- Statistical Mechanics (this is the best part)
- Hawking radiation
- The information paradox

Because the information paradox touches on all areas of physics, I thought it was necessary to take “zero background, infinite intelligence” approach, introducing the all the necessary branches of physics (GR, QFT, Stat Mech) in order to understand what the deal with Hawking radiation really is, and why physicists think it is so important. I think it is safe to say that if you read these notes, you’ll learn a non-trivial amount of physics.

## One thought on “Why physicists care about the Firewall Paradox”