# Black Holes, how much energy do they give off?

1. Jun 9, 2008

### sollinton

Sorry for the dumb question but...

My friend and I were wondering how much energy is given off by an average black hole relative to its mass. (like X black hole gives off Y% of its mass as energy per second)

2. Jun 9, 2008

### Quisquis

3. Jun 10, 2008

### sollinton

Thanks alot, this really helps.

4. Oct 28, 2008

### Naty1

A black hole gets hotter as it radiates and gets smaller, so a smaller black hole dissipates more quickly than a massive one...but the lives in general are extremely long...I believe on roughly the order of the age of the universe....or at least ,really, really slowly...in other words, you'll have to watch really carefully to see one die....

5. Oct 28, 2008

### George Jones

Staff Emeritus
6. Nov 3, 2008

### stevebd1

It might be worth noting that black holes (particulaly rotating ones) can also loose energy via other processes-

According to one source, the total mass-energy of a black hole is-

$$\tag{1}M^2=\frac{J^2}{4M_{ir}^2}+\left(\frac{Q^2}{4M_{ir}}+M_{ir}\right)^2$$

where

$$\tag{2}M_{ir}=\frac{1}{2}\sqrt{\left(M+\sqrt{M^2-Q^2-a^2}\right)^2+a^2}$$

The first term (J) is rotational energy, the second term (Q) is coulomb energy and the third term (Mir) is irreducible energy (in all cases, M is the gravitational radius, a is the spin parameter in metres and J is angular momentum).

The first and second are extractable by physical means, such as the Penrose process, superradiance or electrodynamical process.

The irreducible part cannot be lowered by classical (e.g. non-quantum) processes and can only be lost through Hawking radiation. As high as 29% of a black holes total mass can be extracted by the first process and up to 50% for the second process (but realistically, charged black holes probably only exist in theory or are very short lived as they would probably neutralise quite quickly after forming).

Interestingly, the above makes it appear that kinetic rotational energy does contribute to the overall total mass of a black hole (or any rotating object for that matter) as the rest mass would be less than the rotating mass.

Another equation to calculate the rotational energy (which equals M-Mir) is-

$$\tag{3}E_{rot}=M-\sqrt{\frac{1}{2}M\left(M+\sqrt{M^2-a^2\right)}}$$

Also, the effects of rotation on Hawking radiation is equal to-

$$\tag{4}T_H=\frac{\hbar\kappa}{2\pi k_bc}=2\left(1+\frac{M}{\sqrt{M^2-a^2}\right)^{-1}} \frac{\hbar c^3}{8\pi Gk_bm}<\frac{\hbar c^3}{8\pi Gk_bm}$$

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.

The above effects of rotation on Hawking radiation imply that a maximal Kerr black hole (a/M=1) would give off no radiation.

(1),(2)
'Black Holes: A General Introduction' by Jeane-pierre Luminet
http://www.ece.uic.edu/~tsarkar/Goodies/Black Hole.pdf
pages 12 & 13

(3)
'Compact Objects in Astrophysics' by Max Camenzind
http://www.lsw.uni-heidelberg.de/users/mcamenzi/CObjects_06.pdf
page 271

(4)
'Black Hole Thermodynamics' by Narit Pidokrajt
http://www.physto.se/~narit/bh.pdf
page 10

Last edited: Nov 3, 2008
7. Nov 3, 2008

### Orion1

$$T_H = 2 \left(1 + \frac{M}{\sqrt{M^2 - a^2} \right)^{-1}} \frac{\hbar c^3}{8 \pi G k_b m} \leq \frac{\hbar c^3}{8 \pi G k_b m}$$

If that equation is correct under zeroth law, what would prevent a maximal Kerr quantum black hole from becoming completely stable?

Reference:
'Black Hole Thermodynamics' by Narit Pidokrajt - pg. 10

Last edited: Nov 3, 2008
8. Nov 4, 2008

### stevebd1

Initially I would say cosmic censorship but I found this in the black hole electron section of wiki-

The equation for the photon sphere that applies to both static and rotating black holes is-

$$\tag{1}R_{ph}=2M\left[1+cos\left(\frac{2}{3}cos^{-1}\frac{-a}{M}\right)\right]$$

which reduces to (what was) the event horizon radius for a maximal Kerr black hole. If the above statement is right, then the reduction of the photon sphere radius would appear to have some impact on the stability of a maximal Kerr quantum black hole (or maybe the photon sphere works differently at quantum level).

(1)
'Compact Objects in Astrophysics' by Max Camenzind
http://www.lsw.uni-heidelberg.de/users/mcamenzi/CObjects_06.pdf
page 259

Last edited: Nov 4, 2008
9. Nov 6, 2008

### Orion1

In reference 1 - pg. 253, the equation stated in relativistic units for the dual outer and inner horizons:
$$r_{\pm} = M_H \pm \sqrt{M_H^2 - a_H^2}$$

Where the key is:
$$M_H = r_g = \frac{Gm}{c^2}$$
$$a_H = a \cdot r_g$$

Therefore, the equation for the dual outer and inner horizons in physical natural units (S.I.) is:
$$\boxed{r_{\pm} = r_g \pm \sqrt{r_g^2(1 - a^2)}}$$

Is this equation translation from relativistic to natural units correct?

Reference:
Compact Objects in Astrophysics - Max Camenzind

Last edited: Nov 6, 2008
10. Nov 6, 2008

### stevebd1

Last edited: Nov 6, 2008
11. Nov 6, 2008

### Orion1

Kerr metric...

The effects of angular momentum on Hawking radiation in physical units (S.I.):
$$\boxed{T_H = 2 \left(1 + \frac{1}{\sqrt{(1 - a^2)} \right)^{-1}} \frac{\hbar c^3}{8 \pi G k_b m} \leq \frac{\hbar c^3}{8 \pi G k_b m}}$$

Kerr metric angular momentum:
$$\alpha = \frac{J}{mc}$$

Radii for the dual outer and inner horizons in physical units (S.I.):
$${r_{\pm} = r_g \left[1 \pm \sqrt{(1 - a^2)}\right]$$

$$r_{e} = r_g \left[1 + \sqrt{(1 - a^2 \cos^{2} \theta)}\right]$$

Radii for the dual outer and inner horizons in physical units (S.I.):
$$r_{\pm} = \frac{r_{s} \pm \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}$$

$$r_{e} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}$$

$$r_s = 2 \cdot r_g$$

Establishing the equations between the Kerr metric angular momentum terms $$\alpha$$ and $$a$$ for the inner horizon radius:
$$r_{-} = \frac{1}{2} \left(2 r_g - \sqrt{(2 r_g)^{2} - 4 \alpha^{2}} \right) = r_g \left[1 - \sqrt{(1 - a^2)}\right]$$

Solving for $$a$$ for the inner horizon radius:
$$\boxed{a = \frac{\alpha}{r_g}}$$

Reference:
Kerr Metric - Important surfaces - Wikipedia

Last edited: Nov 7, 2008
12. Nov 7, 2008

### stevebd1

For some reason, wikipedia do not recognise the inner Cauchy horizon of Kerr black holes, so when they talk about the outer horizon, they refer to the static limit or the ergosphere and when they talk about the inner horizon, they refer to the outer event horizon (I don't know if it's still recorded under discussion but there was a request to remove any mention of the inner Cauchy horizon which went ahead). This would mean the following would apply-

ergosphere-

$$r_{e} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} cos^{2}}}{2} = r_g \left[1 + \sqrt{(1 - a^2 cos^{2} \theta)}\right]$$

(wiki has this down as the outer event horizon)

outer event horizon-

$$r_{+} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2} = r_g \left[1 + \sqrt{(1 - a^2)}\right]$$

(wiki has this down as the inner event horizon)

And to extrapolate-

inner event horizon-

$$r_{-} = \frac{r_{s} - \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2} = r_g \left[1 - \sqrt{(1 - a^2)}\right]$$

(which wiki has ignored)

In regard of your solution expressing $a$ as a unitless spin parameter between 0 and 1, the following would apply-

$$a=\frac{\alpha}{r_g}$$

It's worth noting that while wikipedia use $\alpha$ (alpha) to represent the spin parameter in metres, I have seen $\alpha$ represent redshift in frame-dragging equations (which derive from Kerr metric) and $a$ is used to represent the spin parameter in metres. To represent the spin parameter as a unitless number between 0 and 1 (or in natural units), I've seen the following terms used- $|a|$, $a_\ast$ or even simply $a/M$ as in the 'black hole parameters' section on scholarpedia.

Last edited: Nov 7, 2008
13. Nov 8, 2008

### Orion1

The effects of angular momentum on photon sphere radius in physical S.I. units:
$$r_{ps} = 2 r_g \left[1 + \cos \left(\frac{2}{3} \cos^{-1} -a \right) \right]$$

$$\boxed{r_{ps} = 3 r_g \; \; \; a = 0}$$
$$\boxed{r_{ps} = r_g \; \; \; \; a = 1}$$

Does anyone have the radius equations for the two photon spheres?

Reference:
Photon sphere - Wikipedia
Inside a black hole - the Kerr black hole

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14. Nov 10, 2008

### stevebd1

$$R_{ph2}=2M\left[1+cos\left(\frac{2}{3}cos^{-1}\frac{a}{M}\right)\right]$$

which puts it at 3M at a/M=0 and 4M at a/M=1

'Spherical Photon Orbits Around a Kerr Black Hole'
http://www.physics.nus.edu.sg/~phyteoe/kerr/

15. Nov 10, 2008

### Orion1

The effects of angular momentum on photon sphere radius in physical S.I. units:
$$r_{ps \mp} = 2 r_g \left[1 + \cos \left(\frac{2}{3} \cos^{-1} \mp |a| \right) \right]$$

$$-|a|$$ - prograde photon orbit
$$+|a|$$ - retrograde photon orbit

$$\boxed{r_{ps-} = 3 r_g \; \; \; a = 0}$$
$$\boxed{r_{ps+} = 3 r_g \; \; \; \; a = 0}$$
$$\boxed{r_{ps-} = r_g \; \; \; \; \; a = 1}$$
$$\boxed{r_{ps+} = 4 r_g \; \; \; \; a = 1}$$

Reference:
Spherical Photon Orbits Around a Kerr Black Hole

Last edited: Nov 11, 2008
16. Nov 11, 2008

### stevebd1

There seems to be a problem with the equation at the poles. As a increases, the prograde orbit passes through the event horizon. (EDIT: Equation in post #15 removed)

Below is a link to images from the 'Spherical Photon Orbits' website of example orbits (which look fairly spherical)-

http://www.physics.nus.edu.sg/~phyteoe/kerr/table.html

Last edited: Nov 12, 2008
17. Nov 11, 2008

### Orion1

The graphic model shown in attachment #1 of post #13, displays the dual photon spheres as oblate spheroids, this graphic model appears to be incorrect.

Last edited: Nov 11, 2008
18. Nov 11, 2008

### stevebd1

That seems to be the case.

Last edited: Nov 12, 2008
19. Nov 11, 2008

### stevebd1

Regarding the OP's question of how much energy do black holes give off, there is also a process called Penrose Pair Production (PPP)-

key to Fig. 3.14-

'Magnetohydrodynamics on the Kerr Geometry' by A Mueller
pages 47 & 48

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20. Nov 11, 2008

### Orion1

Kerr metric quantum black hole evaporation time...

Effects of angular momentum on a (4+n)-dimensional Kerr metric quantum black hole Hawking radiation evaporation time:
$$\boxed{t(n, a)_{ev} = \frac{320 G^2 m_p^3}{\hbar c^4 \sqrt{\pi}} \left[ \frac{E_{BH}}{E_p} \left( \frac{8 \Gamma\left(\frac{n+3}{2} \right)}{n+2} \right) \right]^{\frac{3}{n+1}} \left(1 + \frac{1}{\sqrt{(1 - a^2)} \right)^{4}}} \; \; \; a < 1$$

$$\boxed{t_0 = \frac{320 G^2 m_p^3}{\hbar c^4 \sqrt{\pi}}}$$

$$\boxed{t_0 = 9.733 \cdot 10^{-42} \; \text{s}}$$

$$E_{BH} = 14 \; \text{Tev}$$

$$\boxed{t(10, 0.999)_{ev} = 3.275 \cdot 10^{-40} \; \text{s}}$$

Note:
$$\boxed{\lim_{a \to 1} t(n, a)_{ev} = \infty}$$

Reference:
Black Hole - LHC - simulation video
Hawking radiation black hole evaporation - Wikipedia
Black Hole Thermodynamics - Narit Pidokrajt - pg. 10
The effects of angular momentum on Hawking radiation #11 - Orion1
Stable micro black holes - Wikipedia
Quantum Black Holes: the Event Horizon as a Fuzzy Sphere
Quantum black hole Hawking radiation evaporation time #454 - Orion1
Naked singularity - Wikipedia
Cosmic censorship hypothesis - Wikipedia
Existential logic #428 - Orion1
Compact Objects in Astrophysics - Max Camenzind
CERN - micro black holes pose planetary risk - Otto E. Rossler