I think I didn’t use the right terminology when I referred to narrowing the state space. What I really meant to ask was: If the radiation is anisotropic, doesn’t that exclude a huge number of microstates that could apply to the macrostate, and doesn’t that in turn reduce the entropy?

Knowing absolutely nothing about the Hawking equations, I’m having to reason this out without the math, but it seems to me that any virtual particle that finds itself outside the event horizon does so with a particular energy, some of which is carried as velocity. Isn’t it the case then that all radiation that succeeds in not being re-gobbled by the hole has to have a velocity whose magnitude is not only greater than the escape velocity of the event horizon, but has a positive normal direction component, with respect to the event horizon, as well? (Anything with a negative normal component is going to hit the event horizon again, right?)

In the case of One Big Hole, this seems to generate radiation that is anisotropic. Isn’t that a big no-no? And, more importantly, doesn’t that significantly narrow the state space, and therefore reduce the entropy?

John– no real impact at all. The fact that the universe is homogeneous on large scales means that you can pick any comoving “tube” of spacetime, stretching from the Big Bang into the future, and treat it as an approximately closed system.

What I mean is, every year, even if space itself weren’t expanding, the radius of the observable universe would be expanding by one light year, as light that hadn’t had time to reach us before is now reaching us.

“Information is conserved” means that if you know the entire state of the system (namely positions and velocities) then you can find the state of the system at any earlier or later time.

The caveat for GR is that we have a lot of freedom for how we label points in spacetime, so to know the “position” it is not enough to just give coordinates but we also have to have some idea of what the coordinates mean. So in addition to the normal coordinates for position and derivatives for velocity, we must also impose some constraints to be able to interpret the coordinates known as gauge-fixing conditions.

In so far as I know, it is a semantic issue whether you consider “positions” to mean the coordinates + gauge fixing to interpret them, or if you think of the positions as the coordinates. In the former case the information is simply the state of the system (x, v) for all the degrees of freedom, while in the latter case the information is (x,v) for the state of the system and the appropriate gauge fixing conditions.

But of all the concepts in this chapter; general relativity, quantum field theory, black holes, etc.; the most bizarre is that of eternity. We can say some things will never happen because they are too improbable, but then eternity turns that on its head and says such things must happen.