easy or formal proof of Hawking effect … that anyone can understand?

where could i find an explanation (appart from Wikipedia) of how the Hawking's effect is obtained from quantum field theory GR and thermondynamic :D

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1 Answer

You can try this approach which is only superficially different from Wikipedia:

Consider a black hole metric

$$(1 - {2m\over r} ) dt^2 + {dr^2\over 1 - {2m\over r}} + r^2 d\Omega^2$$

Continued to imaginary time (I flipped the sign of $dt^2$). Now write $r=2m + u$, and expand the metric to leading order:

$${ u \over 2m} dt^2 + {2m\over u} du^2 + 2m^2 d\Omega^2$$

and notice that after a coordinate change ${u = 8m v^2}$, the du part turns into $dv^2$ and the $dt$ part becomes the angular form of a polar metric

$${ v^2 {dt^2\over 16m^2} + dv^2 }$$

Which means that ${t\over 4m}$ needs to be periodic with period $2\pi$, just like $\theta$ in polar coordinates, or else you have a curvature singularity at $v=0$. The variable t is periodic at infinity with the same period, and this means that the inverse temperature for a path-integral on this background is $8\pi m$. That tells you the temperature of doing quantum field theory on this background is the Hawking temperature, assuming there is no singularity on the Euclidean horizon. This is Hawking's argument of the late 1970s.

The physical justification for all this formal stuff is just Wikipedia's argument--- taking a near horizon limit turns this into Rindler space. Assuming nothing funky happens on the horizon is equivalent to using the equivalence principle to get Unruh radiation in the Rindler space, and making the periodicity of t constant over the whole space is just continuing the radiation with Einstein redshift.

Hawking's original calculation of 1974-1976 performs a Bogoliubov transformation on the outgoing and incoming modes, assuming that the horizon-crossing modes look like vacuum. The calculation uses an eikonal approximation in the near horizon limit, and this means Hawking is really considering only the rate of geodesic separation near the horizon. So you might as well do it only in the near-horizon limit, and this is tantamount to just doing Unruh's calculation. The rest of Hawking's calculation just matches Unruh's result to the far-away modes by consistently redshifting them, and this you might as well do using the local Unruh temperature.

Further, Hawkings assumption that the infalling modes are not occupied, that they are vacuum, is equivalent to the statement that an infalling observer sees nothing special, and this is equivalent to the statement that the near-horizon radiation is Unruh. So all these superficially different calculations are exactly the same in physical content and in assumptions.

I am just going through this to convince you that the Wikipedia presentation is exactly the same as all others, except made so that everyone can understand it. There is nothing more in any of the pre-1980 demonstrations. There are some more recent ones involving tunneling of particles, associated with Wilczek,, which I never went through.

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