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There is a point in the original derivation of Hawking's radiation that I don't see clearly.

Following two null geodesics in the background of a collapsing star, one that will generate the future horizon, and the other starting at an affine parameter distance $-\epsilon$ from it. Why would the second geodesic stay at an affine parameter distance $\epsilon$ from the first along its propagation getting out of the star ? I don't really see how the geodesic deviation equation could be of some help while working with this affine parameter distance.

I would appreciate any help.

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  • $\begingroup$ Geodesics are used in Hawking's analysis as they describe the trajectories of constant phase surfaces of the different modes of the field. You can show that modes that reach the horizon are highly blue shifted when they are passing through the star. Therefore they don't scatter but retain their affine distance. This justification of Hawking's ray optics analysis is provided within his paper. $\endgroup$ – BoundaryGraviton Aug 20 '16 at 6:45
  • $\begingroup$ @BoundaryGraviton : You mean that using the geometric optics in curved spacetime for null rays correspond to keeping the affine parameter distance between the rays constant ? $\endgroup$ – faero Aug 20 '16 at 7:42
  • $\begingroup$ Yes because otherwise there is no justification for following geodesic curves. Field modes are waves, so to speak. They can scatter inside the star and not get out of the star at all. But because the modes that reach the future null infinity with their constant phase surfaces at an affine distance $\epsilon$ from the future horizon are highly blue shifted when they are traced back to the star, they travel through the star. $\endgroup$ – BoundaryGraviton Aug 20 '16 at 8:38
  • $\begingroup$ @BoundaryGraviton : All right, following a null ray (which in any case must be geodesic), the constant phase surface must come out of the star. But why compared to nearby null rays, is the affine parameter distance between them constant ? Shouldn't we use geodesic deviation equation, Raychaudhuri's equation or something similar to compare behaviour of nearby geodesics ? Even in geometric optics ? Spacetime is curved, so in a way, this distance may change, the rays still staying null. $\endgroup$ – faero Aug 20 '16 at 9:25
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Hawking radiation was calculated by Hawking in an idealized collapsing star, really an evolving Schwarzschild solution where the star moves into its horizon. But it is true and perhaps easier to understand in an already collapsed one, or really more accurately a perennial stationary black hole. Either way, he calculates what happens to the virtual particles in the vacuum. One way of interpreting it is that statistically some will follow geodesics away from the horizon and others towards it. The one directed outwards may escape the gravitational field, the other one will get pulled in. Since the escaping one has to become a real particle (it escapes and we may physically detect it), it is as if the other one going into the horizon has negative energy, and can be interpreted as a positive energy particle leaking out of the black hole.

There is a more accurate way to do this and it is to calculate it as photons tunneling out from inside the horizon. Tunneling can be explained quantum mechanically. The two explanations are equivalent. Hawking had to do this actually using a quantum field calculation in the background black hole general relativistic gravitational field.

See the Wikipedia article on Hawking radiation to get an idea.

EDIT after OP comment right below:

So, more on Hawking's calculation. His calculation is rather complicated, and is done in a number of other reviews and treatments. One that has his calculations, but also has material for the Unruh effect and Rindler coordinates in flat spacetime and other details that make it easier to follow, is openly available at https://arxiv.org/pdf/gr-qc/0010055.pdf

But a similar calculation I found a little easier to follow is here: http://kiso.phys.se.tmu.ac.jp/thesis/m.h.kuwabara.pdf

Kuwabara's pdf is a review of the Hawking radiation, and explains the approach and calculations. He first explains QFT in curved spacetime, with the difference from Minkowski (and thus standard QFT) that the metric is the GR metric, not Minkowski. The metric (i.e., gravitational field) is a fixed classical GR background, and is not quantized. The field is quantized. He does it for a free scalar field in a Schwarzschild metric (also does it for a sandwich expanding universe, but Hawking and others have done it with the same thermodynamic results for more general charged and rotating stationary black holes, and other quantum fields). He does not treat the Unruh or Rindler coordinates, the other online link I gave above does.

In both links noted above it is noted that one can change coordinate systems, and actually needs to in order to be able to solve and interpret the equations in the curved spacetime, and eventually get to a wave equation with a potential. It can be solved as an expansion in terms of the standard mode operators $a_k$ and its adjoints. It is also noted that the vacuum in diferent reference frames can look different, and the mode operators need to be transformed using Bogoliubov transformations in order to interpret the out states in terms of the in states. This is true also in flat spacetimes in an accelerated reference frame, where one gets Unruh radiation. In Kuwabara he then goes on to calculate the particle number for the radiation, in terms of the transformed mode operators, and gets the expression for the blackbody radiation that Hawking found. He discusses and you have to treat carefully the in and out states.

That's not exactly what you are asking about inbound or outbound geodesics, but this represents the QFT approach taken by Hawking. Maybe this is what you meant. I am not sure, so hope it may help.

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    $\begingroup$ Actually, I was just referring to his own calculation to understand the approximation ; my question was not on other methods to obtain the radiation. $\endgroup$ – faero Aug 19 '16 at 21:48
  • $\begingroup$ I got this idea from the thesis of Kuwabara, thank you. I was hoping to find a more precise argument with something concerning congruence of geodesics, but I think that it is good enough. $\endgroup$ – faero Aug 20 '16 at 16:22
  • $\begingroup$ Understood. Really, it was calculated that way by Hawking also. It is not a model of classical particles following geodesics (though I'd like to see that also), but of quantum fields in the background, and what happens to the particle number as function of frequency. The quantum field model on the background GR field is on sort of firm ground. $\endgroup$ – Bob Bee Aug 20 '16 at 19:08
  • $\begingroup$ Nevertheless, as I understand it, the geodesic trick is useful in order to extract the evolution of the (late time) modes. $\endgroup$ – faero Aug 20 '16 at 19:39

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