## How do black holes destroy information and why is that a problem?

Today I want to pick up a question that many

of you asked, which is how do black holes destroy information and why is that a problem. I will not explain here what a black hole

is or how we that know black holes exist, for this you can watch my earlier video. Let me instead get right to black hole information

loss. To understand the problem, you first need

to know the mathematics that we use for our theories in physics. These theories all have two ingredients. First, there is something called the “state”

of the system, that’s a complete description of whatever you want to make a prediction

for. In a classical theory, that’s one which

is not quantized, the state would be, for example, the positions and velocities of particles. To describe the state in a quantum theory,

you would instead take the wave-functions. The second ingredient to the current theories

is a dynamical law, which is also often called an “evolution equation”. This has nothing to do with Darwinian evolution. Evolution here just means this is an equation

which tells you how the state changes from one moment of time to the next. So, if I give you a state at any one time,

you can use the evolution equation to compute the state at any other time. The important thing is that all evolution

equations that we know of are time-reversible. This means it never happens that two states

that differ at an initial time will become identical states at a later time. If that was so, then at the later time, you

wouldn’t know where you started form and that would not be reversible. A confusion that I frequently encounter is

that between time-reversibility and time-reversal invariance. These are not the same. Time reversible just means you can run a process

backwards. Time reversal invariance on the other hand

means, it will look the same if you run it backwards. In the following, I am talking about time-reversibility,

not time-reversal invariance. Now, all fundamental evolution equations in

physics are time-reversible. But this time-reversibility is in many cases

entirely theoretical because of entropy increase. If the entropy of a system increases, this

means that it if you wanted to reverse the time-evolution you would have to arrange the

initial state very, very precisely, more precisely than is humanly possible. Therefore, many processes which are time-reversible

in principle are for all practical purposes irreversible. Think of mixing dough. You’ll never be able to unmix it in practice. But if only you could arrange precisely enough

the position of each single atom, you could very well unmix the dough. The same goes for burning a piece of paper. Irreversible in practice. But in principle, if you only know precisely

enough the details of the smoke and the ashes, you could reverse it. The evolution equation of quantum mechanics

is called the Schroedinger equation and it is just as time-reversible as the evolution

equation of classical physics. Quantum mechanics, however, has an additional

equation which describes the measurement process, and this equation is not time-reversible. The reason it’s not time-reversible is that

you can have different states that, when measured, give you the same measurement outcome. So, if you only know the outcome of the measurement,

you cannot tell what was the original state. Let us come to black holes then. The defining property of a black hole is the

horizon, which is a one-way surface. You can only get in, but never get out of

a black hole. The horizon does not have substance, it’s

really just the name for a location in space. Other than that it’s vacuum. But quantum theory tells us that vacuum is

not nothing. It is full of particle-antiparticle pairs

that are constantly created and destroyed. And in general relativity, the notion of a

particle itself depends on the observer, much like the passage of time does. For this reason, what looks like vacuum close

by the horizon does not look like vacuum far away from the horizon. Which is just another way of saying that black

holes emit radiation. This effect was first derived by Stephen Hawking

in the 1970s and the radiation is therefore called Hawking radiation. It’s really important to keep in mind that

you get this result by using just the normal quantum theory of matter in the curved space-time

of a black hole. You do not need a theory of quantum gravity

to derive that black holes radiate. For our purposes, the relevant property of

the radiation is that it is completely thermal. It is entirely determined by the total mass,

charge, and spin of the black hole. Besides that, it’s random. Now, what happens when the black hole radiates

is that it loses mass and shrinks. It shrinks until it’s entirely gone and

the radiation is the only thing that is left. But if you only have the radiation, then all

you know is the mass, change, and spin of the black hole. You have no idea what formed the black hole

originally or what fell in later. Therefore, black hole evaporation is irreversible

because many different initial states will result in the same final state. And this is before you have even made a measurement

on the radiation. Such an irreversible process does not fit

together with any of the known evolution laws – and that’s the problem. If you combine gravity with quantum theory,

it seems, you get a result that’s inconsistent with quantum theory. As you have probably noticed, I didn’t say

anything about information. That’s because really the reference to information

in “black hole information loss” is entirely unnecessary and just causes confusion. The problem of black hole “information loss”

really has nothing to do with just exactly what you mean by information. It’s just a term that loosely speaking says

you can’t tell from the final state what was the exact initial state. There have been many, many attempts to solve

this problem. Literally thousands of papers have been written

about this. I will tell you about the most promising solutions

some other time, so stay tuned.

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