# Is there a good introduction to Hawking Radiation?

1. Aug 4, 2008

### JustStudent

I am just an undergraduate student of physics and I would like to know what would be the best (most pedagogical) introduction to Hawking Radiation.

2. Aug 4, 2008

### George Jones

Staff Emeritus
Do you have any experience with general relativity?

3. Aug 4, 2008

### JustStudent

I guess I have a reasonable knowledge of GR and QFT. At least, I hope so. Anyway, if it is not enough, I can always study more...

4. Aug 4, 2008

### George Jones

Staff Emeritus
Then maybe the book the book Quantum Effects in Gravity

http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521868341

by Mukhanov and Winitzki, which has exercises and solutions (not just answers). Take a look at the Table of Contents.

5. Aug 4, 2008

### DopplerDog

Also, try Wald's "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics", chapter 7.

6. Aug 5, 2008

### JustStudent

Thanks a lot. I will try both.

7. Aug 5, 2008

### George Jones

Staff Emeritus
This a standard and a very interesting book, but it has a fairly steep learning curve.

The other standard is Quantum Fields in Curved Space by Birrell and Davies.

A nice though less detailed introduction Chapter 9 Quantum Field Theory in Curved Spacetime from Spacetime and Geometry: An Introduction to General Relativity by Carroll.

8. Aug 5, 2008

### JustStudent

I looked at Wald. It seems that it is rather involved. I will try to get the other books. In any case, if you can indicate any material that will help, I will be very grateful.

9. Aug 5, 2008

### MeJennifer

I think the wikipedia article on Hawking Radiation is pretty nice.

10. Aug 5, 2008

### jnorman

i wish someone could explain this to me...
"In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole). "

why can only the negative energy particles fall in? why cant the positive energy particles fall in?

11. Aug 5, 2008

### George Jones

Staff Emeritus

The answer is at the end, but read the whole thing.

12. Aug 5, 2008

### jnorman

thanks george - i have read that article. the conclusion indicates:
"Note that this doesn't work in the other direction -- you can't have the positive-energy particle cross the horizon and leaves the negative- energy particle stranded outside, since a negative-energy particle can't continue to exist outside the horizon for a time longer than h/E. So the black hole can lose energy to vacuum fluctuations, but it can't gain energy."

now that might be correct, IF a BH gave a damn what it was absorbing. i do not see how an event horizon would be particular about what falls into it - it doesnt care if a positive particle falls in, and has no inherent knowledge of where or whence that particle arrived in that situation. the rest of the universe is not going to yell "stop it!! spit that particle back out here this instant, young man!!" and intercede on behalf of some lonely antiparticle.

for that matter, where the heck did all the antiparticles from the big bang go, anyway? why is there such a humongous imbalance in the amount of matter and antimatter observable in the universe today?

13. Aug 6, 2008

### JustStudent

It is quite nice indeed, but I am looking for the derivation of the effect. I did look at Hawking's original paper, but it is not very clear though, at least at my level. I was expecting something more digested once the topic is old enough to have filtered to textbooks.

14. Aug 6, 2008

### JustStudent

I've just checked Mukhanov and Winitzki. The book is awesome and very pedagogical, not only on Hawking radiation but all over the whole chapter 9. Exactly what I was looking for. Thanks a lot for the indication.

15. Oct 4, 2008

### stevebd1

While there seems to be plenty of info regarding Hawking radiation and static black holes, is there any info about how the various equations for T and P apply to rotating black holes?

Steve

16. Oct 4, 2008

### atyy

17. Oct 4, 2008

### George Jones

Staff Emeritus
18. Oct 6, 2008

### stevebd1

Thanks for the leads. The information I've managed to gather is-

Temperature-

$$T=\frac{\kappa}{2\pi}\left[\frac{\hbar}{k_bc}\right]$$

where $\kappa$ is the Killing surface gravity of the black hole, $\hbar[/tex] is the reduced Planck constant and [itex]k_b$ is the Boltzmann constant.

for a static black hole-

$$\kappa=\frac{c^4}{4Gm}$$

$$T=\frac{\hbar c^3}{8\pi Gk_bm}$$

for a rotating black hole-

$$\kappa=c^2\ \frac{(r_+-r_-)}{2\left(r_+^2+a^2\right)}=c^2\ \frac{(r_+^2-a^2)}{2r_+(r_+^2+a^2)}=c^2\ \frac{r_+-M}{2Mr_+}$$

all the above reduce to the static solution for $\kappa$ when a=0

where

$$r_+=M+\sqrt{M^2-a^2}$$

$$r_-=M-\sqrt{M^2-a^2}$$

and M is the gravitational constant (M=Gm/c^2) and a is the spin parameter in metres (a=J/mc)

various equations for T-

$$\tag{1}T=c^2 \frac{\sqrt{M^2-a^2}}{4\pi M\left(M+\sqrt{M^2-a^2}\right)}\left[\frac{\hbar}{k_bc}\right]$$

$$T=\frac{(r_+^2-a^2)\hbar c}{4\pi r_+(r_+^2+a^2)k_b}=\frac{(r_+-M)\hbar c}{4\pi Mr_+k_b}$$

Power-

derived from-

$$P=A\epsilon \sigma T^4$$

where $\epsilon$ is 1 and $\sigma$ is the Stefan–Boltzmann constant

$$A=4\pi (r_+^2+a^2)$$

$$P=\left(4\pi (r_+^2+a^2)\right)\left(\frac{\pi^2 k_b^4}{60\hbar^3 c^2}\right)\left( \frac{(r_+^2-a^2)\hbar c}{4\pi r_+(r_+^2+a^2)k_b}\right)^4$$

$$P=\frac{(r_+^2-a^2)^4\hbar c^2}{3840\ \pi r_+^4(r_+^2+a^2)^3}$$

Alternative equation for P using different $\kappa$-

$$P=\frac{(r_+^2+a^2)(r_+-M)^4\hbar c^2}{3840\ \pi r_+^4M^4}$$

This is about as far as I got. It appears in order to calculate time, mass has to be extracted from the P equation. Does anyone see a way?

regards
Steve

(1)- page 6
'Hawking radiation of Dirac particles via tunneling from Kerr black holes'
http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.1410v2.pdf

more reference-
http://arxiv.org/find/all/1/all:+AN.../1/0/all/0/1?skip=0&query_id=35869ea8e9349689

Last edited: Oct 6, 2008
19. Nov 13, 2008

### stevebd1

I found the following equation which goes towards isolating mass in order to progress towards calculating time in respect of a rotating black hole-

$$\tag{1}T=\frac{\hbar\kappa}{2\pi k_bc}=2\left(1+\frac{M}{\sqrt{M^2-a^2}\right)^{-1}} \frac{\hbar c^3}{8\pi Gk_bm}$$

Based on the above, the Killing surface gravity for a rotating black hole can be expressed as-

$$\kappa=c^2\ \frac{(r_+-r_-)}{2\left(r_+^2+a^2\right)}=2\left(1+\frac{M}{\sqrt{M^2-a^2}\right)^{-1}}\frac{c^4}{4Gm}$$

(while the second equation is good for a/M<1 it doesn't work at exactly a/M=1 where the results should produce zero due to the outer and inner event horizons converging at the gravitational radius, leaving a naked singularity (as does the first equation). While the gradient of decrease is good up to a/M=1, zero at a/M=1 is implied. It does, however, work for a Schwarzschild solution at a=0).

The second equation can be simplified to-

$$\kappa=2\left(1+\frac{1}{\sqrt{(1-a_{\ast}^2)}\right)^{-1}}\frac{c^4}{4Gm}$$

where a=a/M

A solution for Hawking radiation that applies to both static and rotating black holes is-

$$T = 2\left(1+\frac{1}{\sqrt{(1-a_{\ast}^2)}\right)^{-1}} \frac{\hbar c^3}{8 \pi G k_b m}$$

Equations can be formulated for power (P) and time (t)-

$$P=A\epsilon \sigma T^4$$

where for a rotating black hole-

$$\tag{2}A=8\pi M\left(M+\sqrt{M^2-a^2}\right)$$

which can be rewritten relative to the Schwarzschild equation as-

$$A=\frac{1}{2}\left(1+\sqrt{(1-a_\ast^2)}\right) \frac{16 \pi G^2m^2}{c^4}$$

Meaning the equations for power (P) and time (t) for a rotating black hole would be-

$$P=\frac{1}{2}\left(1+\sqrt{(1-a_\ast^2)}\right)\frac{\hbar c^6}{15360 \pi G^2 m^2}\ 16\left(1+\frac{1}{\sqrt{(1-a_{\ast}^2)}\right)^{-4}}$$

$$t=2\left(1+\sqrt{(1-a_\ast^2)}\right)^{-1}\ \frac{5120 \pi G^2 m^3}{\hbar c^4}\ \frac{1}{16}\left(1+\frac{1}{\sqrt{(1-a_{\ast}^2)}\right)^{4}}$$

(1)
'Black Hole Thermodynamics' by Narit Pidokrajt
http://www.physto.se/~narit/bh.pdf
page 10

(2)
'Black Holes: An Introduction' by Derek J. Raine & Edwin George Thomas