Technical question about Hawking radiation

[Dmitry67]

I see how it is derived for the BH and in case of Unruh effect.
In Unruh effect it is a function of acceleration
In case of BH distance to the event horizon plays important role

What I can't find, however, how tidal forces should be taken into account. For example, in the universe with DE (accelerated expansion) the temperature never falls to 0, because there will be always hawking-like radiation from the cosmological horizons

I don't see how it is derived. The metric is not schwarzschild so formula for Hawking radiation can not be used. Formula for Unruh radiation uses 'a', acceleration... I am blocked.

 

[Finbar]

Your question is about de-sitter space right? Or more generally one with a cosmological horizon. I believe the temperature is given by the surface gravity at the horizon.
So for a de-sitter space with positive Lambda.


$$ds^2= -(1-\Lambda r^2/3) dt^2+ \frac{1}{1-\Lambda r^2/3} r^2 + r^2 d \Omega$$

the horizon is

$$r_H= \sqrt{3/ \Lambda}$$

and the temperature is

$$T= \frac{1}{4 \pi} \frac{\partial g_{tt}}{\partial r}(r_H) = \frac{1}{2 \pi r_H}$$


As you point out the distance to the horizon plays a role. The temperature derived this way is the one measured by an observer who's proper time is t in this case an observer at r=0 i.e. where the metric is flat minkowski.

 

[Naty1]

What I can't find, however, how tidal forces should be taken into account.

Just what do you mean by this? Tidal forces at black holes stretch in the radial direction and compress in the transverse direction...The radial acceleration is what separates the virtual particles in one interpretation...are you wondering how the compression affects Hawking radiation?? Am unsure if it has any effect.

If so interesting question...I see no reason compression would necessarily be a factor in the Unruh effect...in fact depending on the horizon it might be expansive? Anybody know?

 

[Dmitry67]

finbar, thank you!

Naty1, now I have formula, let me make some calculations. I have an idea, I will try to show how it can be used.

 

[Naty1]

I'm checking Leonard Susskind's THE BLACK HOLE WAR and will post items relevant to this discussion. He mentions Hawking tour de force calculation of black hole evaporation was in 1975 in a paper called PARTICLE PRODUCTION BY BLACK HOLES

Did Hawking utilize tidal forces in any way??

Susskind notes that no black holes are currently evaporating, hence there is nothing to "observe" since they are all colder than intergalactic space and hence are absorbing heat. So for those who subscribe to the philosophy "nothing unobservable is part of science" we might have an issue...

(unrelated to this thread, but interesting)
Susskind mentions a string theory explanation of Hawking radiaton: sometimes a portion of a string, a string loop, extends above the black hole horizon and crosses itself...about 10% of the time it forms a loop and breaks off...Susskind says that's Hawking radiation...

 

[Dmitry67]

Did Hawking utilize tidal forces in any way??

He did not use it. I did :)

 

[Dmitry67]

Another related question.
Say, tidal forces are not ripping apart but compressing.
So Universe is contracting.
Is there an analog of some "horizon" in such case?
As in expanding universe Rh is a distance of infinite redshift, then in contracting universe, is there a distance of infinite blueshift? (I doubt it)

 

[Finbar]

Another related question.
Say, tidal forces are not ripping apart but compressing.
So Universe is contracting.
Is there an analog of some "horizon" in such case?
As in expanding universe Rh is a distance of infinite redshift, then in contracting universe, is there a distance of infinite blueshift? (I doubt it)

Just take the cosmological constant as negative( Anti-de sitter space) and then you have the conformal boundary at r= infinity for which gtt= - infinity.