"Particle Creation by Black Holes", S. Hawking

[antaris]

link: "Particle Creation by Black Holes", S. Hawkinghttps://projecteuclid.org/journalArticle/Download?urlId=cmp/1103899181

Hello all,

I just want to know the meaning of the text followed by equotation 1.2.
Especially if the flat or asymptotic flat region could interpreted for Minkowski-Spacetime at an arbitary Point in M or just at the positive/negative asymptotic infinty?

Thanks in advance.

 

[Nugatory]

Especially if the flat or asymptotic flat region could interpreted for Minkowski-Spacetime at an arbitary Point in M or just at the positive/negative asymptotic infinty?

Did you intend to tag this thread as B-level? There is no really good answer to your question without moving deep into A-level math. But here's a try....

We're doing the calculation as if the curvature tensor is zero. There is no point anywhere in the manifold where they it is actually zero - "infinity" isn't a point, it's a way of saying that no matter how close to zero it at a point there will always be a point farther away where the curvature will be even smaller. That's asymptotic flatness.
Thus zero curvature is an approximation that we can make as accurate as we wish by starting farther away. And because the zero-curvature math is much more tractable than the zero-curvature-plus-insignificant-nonzero-bit math we do the calculation that way.

 

[antaris]

Thus zero curvature is an approximation that we can make as accurate as we wish by starting farther away.

Is this only valid for Hawking's formulation of QFT in curved spacetime or in general?
Regardless of Hawking's paper, can the starting point of a QFT also be any local Minkowski spacetime, such as the reference system of the Earth (in very good approximation -> asymptotic)?

 

[Nugatory]

Is this only valid for Hawking's formulation of QFT in curved spacetime or in general?

In general. It’s part of every piece of physics you’ve encountered since high school.

 

[antaris]

Ok, back to Hawking. Within the text below 1.2:

Is Hawking's interpretation valid in general in curved space time with dynamic solutions (time depent, rotating bodies, gravitational collapse, evaporation)? The described effect is realized everywhere in curved space time but would be a lot to small to measure at a approximately flat Region at distant position to any Horizon, e.g. of BH at center of the Milkyway?

 

[antaris]

The described effect is realized

I meant whether the effect is realized in Hawking's theory, not whether it is actually realized in nature.

 

[PeterDonis]

We're doing the calculation as if the curvature tensor is zero.

I don't think that quite gets at the key idea, although it's part of it. See below.

I just want to know the meaning of the text followed by equotation 1.2.
Especially if the flat or asymptotic flat region could interpreted for Minkowski-Spacetime at an arbitary Point in M or just at the positive/negative asymptotic infinty?

As @Nugatory has said, this can't really be answered properly without "A" level math, but I'll take a stab at it.

First, consider a simpler case: a quantum field on Minkowski spacetime, which is really globally flat everywhere. We can then pick any inertial frame we like, and use its time coordinate to decompose the quantum field into positive and negative frequency parts, or, equivalently, to define creation and annihilation operators, or, equivalently once more, to define a vacuum state, i.e., a state $\ket{0}$ that satisfies $a \ket{0} = 0$ for all annihilation operators. And this decomposition will be the same regardless of which inertial frame we choose.

Now, consider the following more complicated case, which is basically what Hawking is describing: we have a spacetime which starts out with a highly diffuse distribution of matter and/or radiation, so diffuse that its density is negligible, and therefore the spacetime in the far past can be approximated as flat. Then something happens to the matter and/or radiation that causes significant spacetime curvature to be present for a period of time (Hawking does not specify here what happens--he does not claim here that what happens is collapse to a black hole and then the hole evaporating away), but only for a finite time. Then the spacetime curvature decreases again, so that the spacetime in the far future again has a highly diffuse distribution of matter and/or radiation, so diffuse that its denstiy is negligible, and therefore the spacetime in the far future can be approximated as flat. But the region in between, where there is significant spacetime curvature, cannot be approximated everywhere as flat; it is only asymptotically flat.

Now, nobody knows how to do a fully rigorous calculation of the exact geometry of such a spacetime, or the exact state of a quantum field everywhere on such a spacetime. But Hawking realized that we could do a much simpler calculation that showed a significant result. The calculation works like this:

(1) In the far past of the spacetime, where it can be approximated as flat everywhere, define a vacuum state using the Minkowski time coordinate of any inertial frame, as described above.

(2) In the far future of the spacetime, where it can be approximated as flat everywhere, do the same.

(3) Show that the two vacuum states in #1 and #2 above are not the same. That is the result that Hawking showed in the paper you reference.

What #3 means is that, if the quantum field starts out in the vacuum state in the far past, it will not be in the vacuum state of the far future, because those are different states. That means that the presence of the middle region of the spacetime, where there is significant spacetime curvature, causes particle creation: there were no particles in the far past (vacuum state), but there now are particles in the far future (non-vacuum state).

Hawking then said that, since collapse of matter and/or radiation to a black hole is a process that causes significant spacetime curvature to be present where it didn't have to be present before (because the collapse could start from a highly diffuse distribution of matter and/or radiation whose spacetime curvature was negligible), such a process must also eventually result in the black hole evaporating away due to particle creation. That was the basis for the title of his paper.

 

[antaris]

Thank you very much. May I come back with new questions.