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