LOG#192. Bits on black holes (I).

Hi, doctorish ladies and gentlemen!

I am going to teach some bits of black hole today. For the simplest static (non-rotatory) black hole, the whole space-time is fully specified by mass $M$ , and some constants, like $G_N, \hbar, c, k_B, \ln (2),\ln (10)$ . The critical radius for a black hole is the Schwarzschild radius

$R_S=\dfrac{2G_NM}{c^2}$

The black hole area, assuming $D=4=3+1$ space-time, reads

$A_{BH}=\dfrac{16\pi G_N^2M^2}{c^4}$

And the surface gravity at the event horizon is written as follows

$\kappa=g\vert_S=g=\dfrac{c^4}{4G_NM}$

It is very interesting that this surface gravity is, in fact, the maximal force guessed by the maximal force follower, divided by the black hole mass, i.e., $g=F_M/M$ . Surface gravity creates tides with units $m/s^2/m$ equal to:

$d_\kappa=\dfrac{c^6}{4G_N^2M^2}=2\kappa/R_S$

The celebrated Bekenstein-Hawking area formula for the entropy is, as you already know if you follow my blog:

$S_{BH}=\dfrac{k_B c^3 A}{4G_N\hbar}=\dfrac{k_B \cdot 4\pi G_NM^2}{\hbar c}$

with units in $J/K$ . A note I have never done before: entropy from Boltzmann formula $S=k_B\ln \Omega$ has dimensions of $J/K$ , energy divided by absolute temperature. Using Shannon definition, you get

$H=-\sum_i p_i\ln p_i$

using units of nats. Nats are equal to $1/\ln 2$ shannons (Sh) or $1/\ln 10$ hartleys, bans or dits. And 1 hartley is $\log_2 (10)$ bit = $\ln (10)$ nats. Therefore, you can express the BH entropy in terms of $J/K$ , hartleys, or shannons (i.e., bits or dits as well!). Hawking’s biggest discovery was the black hole temperature, that fixed the $1/4$ factor in the area law from the Bekenstein’s biggest discovery, the analogy between black holes and thermodynamics:

$T_{BH}=\dfrac{\hbar c^3}{8\pi G_N M k_B}$

And, moreover, since black holes behave as blackbodies, they radiate! As they radiate, they become smalles (I am neglecting accretion, of course, from the macro-world) and they explote. The evaporation time for $D=4$ black holes is

$t_{ev}=\dfrac{5120}{\hbar c^4}M^3$

and the black hole luminosity from a pure blackbody reads

$L=A\sigma T^4=\dfrac{\hbar c^2}{3840\pi R_S^2}=\dfrac{\hbar c^6}{15360\pi G_N^2M^2}$

from a BH flux

$\phi=\dfrac{L}{4\pi R_S^2}=\dfrac{\hbar c^6}{61440\pi^2G_N^2R_S^2M^2}$

As the luminosity is power, or rate of change of energy:

$L=-\dfrac{dE}{dt}=\dfrac{\hbar c^6}{15360\pi G_N^2M^2}$

with $E=Mc^2$ becomes

$-\dfrac{dM}{dt}=\dfrac{\hbar c^4}{15360\pi G_N^2 M^2}$

$-M^2dM=\dfrac{\hbar c^4}{15360\pi G_N^2 M^2}dt$

$\int_0^{t_{ev}}dt=-\int_{M_0}^0M^2 \dfrac{15360\pi G_N^2}{\hbar c^4 }dM$

$t_{ev}=\dfrac{15360\pi G_N^2}{\hbar c^4 }\int_0^{M_0}M^2dM$

and thus ( $M_P$ is the Planck mass, and $M_\odot=2\cdot 10^{30}kg$ is a solar mass)

$\boxed{t_{ev}=\dfrac{5120\pi G_N^2}{\hbar c^4}M_\star^3=\dfrac{5120\pi G_N^2 M_\odot^3}{\hbar c^4}\left(\dfrac{M}{M_\odot}\right)^3=\dfrac{5120\pi\hbar M^3}{c^2 M_P^4}\approx 2.1\cdot 10^{67}\left(\dfrac{M}{M_\odot}\right)^3yrs}$

$\boxed{t_{ev}=8.410\cdot 10^{-17}\dfrac{M}{kg}s}$

The black hole power or luminosity reads

$P_{BH}=L_{BH}=\dfrac{\hbar c^6}{15360\pi^2G_N^2}$

for a solar mass black hole becomes

$P_{\odot BH}=L_{\odot BH}=9.007\cdot 10^{-29}W$

A Planck mass BH evaporates in about $10^{-39}s$ . For a solar mass BH, you need about $10^{75}s$ . Primordial BH, born in the early universe withe evaporating time about $2.667Gyrs$ , evaporating by now, has to be about $10^{11}kg$ . However, taking the cosmic microwave background (CMB) temperature, and equating it to the Hawking temperature, you obtain a bound about $5\cdot 10^{22}kg\sim M_{Moon}$ . Black holes as dark matter (primordial black holes. PMB) have been proposed. The interesting window of mass is

$100kg=10^5g<M_{PMB}<10^{17}g=10^{14}kg$

Even more,…If you add extra dimensions of space, e.g. $n$ extra space-like dimension, and you define the fundamental scale of gravity as $M_D$ and not $M_P$ , then you have that the evaporating time for a higher dimensional BH scales as follows

$t_{ev}(n)\sim \dfrac{1}{M_D}\left(\dfrac{M_{BH}}{M_D}\right)^{\frac{n+3}{n+1}}$

Interesingly, the limit $n=0$ and $n=\infty$ are “the same”, excepting by the diffusion of gravitational flux through $M_D$ .

Black hole species have sizes:

Micro BH. Mass about the moon mass. Radius about 0.1mm.
Stellar BH. Mass up to tens of solar masses. Radius about 30km or a few hundreds of km.
Intermediate mass BH. Yet to be discovered but hinted since LIGO GW detections and other clues. Masses since hundreds of solar masses up to almost a million of solar masses. Radii are variable. It is about Earth radius for a thousand solar mass BH.
Supermassive BH. From millions to billions or more (there are people arguing about an upper bound on BH mass) of solar masses. Radii are between 0.001 and 200 AU (astronomical units).

There are other types of black holes: extremal (superextremal), type D, with cosmological constant, primordial, … Mechanisms for BH production in laboratory and/or astrophysical scenarios are also interesting. Even their simulation via analogue fluid systems or quantum computing! For Kerr or Kerr-Newmann black holes, the following bound is known

$Q^2+\left(\dfrac{J}{M}\right)^2\leq M^2$

in Planck units. The equivalence between the Hawking process, the Unruh radiation and the Schwinger mechanism is also curious. Another interesting radius for BH systems are:

$R_{PS}=\dfrac{3}{2}R_S=\dfrac{3G_NM}{c^2}$

$R_{ISCO}=3R_S=\dfrac{6G_NM}{c^2}$

the photon sphere and the inner stable circular orbit radius, respectively. And

$T_H=\dfrac{\kappa}{2\pi}$

$T_U=\dfrac{\hbar a}{2\pi k_Bc}$

are the Hawking and Unruh temperatures in natural units. Check what is the condition to both formulae to give the same number. Do you know what is the fastest way to get the correct BH entropy formula using basic thermodynamics. Know the Hawking temperature in natural units $T_H=1/8\pi M$ . Knowing this, you can fix the number in BH from thermodynamics:

$dS=dQ/T=8\pi M dQ=8\pi M dM$

$dS=8\pi M dM=d(4\pi M^2)$

$S=4\pi M^2=\pi R_S^2$

You only get the right formula if you put by hand an extra $1/4$ factor. Indeed, in extra dimensions, you also get:

$R_S\sim \dfrac{1}{M_D}\left(\dfrac{M_{BH}}{M_D}\right)^{1/1+n}$

$T_{BH}=\dfrac{n+1}{4\pi R_S}$

$M_D^{n+2}=\dfrac{(2\pi R)^n}{8\pi G_{4+n}}=M_4^2/V_n$

$S_{BH}=\left(\dfrac{4\pi}{n+2}\right)\left(\dfrac{M_{BH}}{M_D}\right)^{\frac{n+2}{n+1}}\left(\dfrac{2^n\pi^{(n-3)/2}\Gamma\left(\dfrac{n+3}{2}\right)}{n+2}\right)^{\frac{1}{n+1}}$

More on extra dimensional p-branes, not directly black holes but alike. The tension for a p-brane reads

$T_{p}=\dfrac{1}{g_s(2\pi)^p(\alpha^{'})^{(p+1)/2}}$

and the YM coupling

$g_{YM,p}^2=2(2\pi)^{p-2}g_sl_s^{p-3}$

The Dirac-Nambu action

$S_{Dp}=-T_{Dp}\int d^{p+1}\sqrt{\eta+ \left(\partial X\right)^2+2\pi \alpha^{'}F}$

Note, that M-theory fixes $R_{11}=l_sg_s$ in some way. Black branes are BH-like solutions in superstring/M-theory. In extra dimensions, the newtonian gravitational law reads off as follows

(1) $\begin{equation*} \boxed{F_D=G_D\dfrac{Mm}{R^{D-2}}=\left(\dfrac{D-3}{D-2}\right)\dfrac{8\pi \overline{G_N}}{\Omega_{D-2}}\dfrac{Mm}{R^{D-2}}} \end{equation*}$

where the omega is the surface area of the unit $D$ sphere

$\Omega_D=\dfrac{2\pi^{(D+1)/2}}{\Gamma \left(\frac{D+1}{2}\right)}$

Kerr (rotating, uncharged) BH are interesting. Collide two of these Kerr BH. They will emit energy in form of electromagnetic, gravitational or any other form of radiation. The maximal efficiency is known to be (due to Hawking himself):

$\varepsilon_m=1-\dfrac{1}{\sqrt{2}}$

Charged BH are worst for efficiency process in BH thermodynamics. Hawking knows this from his paper, Gravitational radiation from colliding black holes. Rotating BH has a Hawking temperature

$T_{BH}(Kerr)=\dfrac{\hbar c^3}{4\pi k_B G_N M}\frac{\sqrt{1-(a/M)^2}}{1+\sqrt{1-(a/M)^2}}$

where $a=Jc/G_NM$ is the Kerr parameter. Take yourself a few Planck absements, $A_P=L_PT_P\sim 10^{-79}ms$ , in order to guess that the right formula for the power of Kerr BH is

$P_{BH}=\dfrac{\hbar c^6}{1920}\dfrac{\left(1-(a/M)^2\right)^2}{\pi G_N^2M^2\left(1+\sqrt{1-(a/M)^2}\right)^3}$

For a Kerr-Newmann BH (rotating, charged), the power is

$P_{BH,KN}=\dfrac{\hbar c^6}{240\pi G_N^2M^2}\dfrac{\left(1-\dfrac{K_CQ^2}{G_NM^2}-\left(\dfrac{a}{M}\right)^2\right)^2}{\left(2+2\sqrt{1-\dfrac{K_CQ^2}{G_NM^2}-\left(\dfrac{a}{M}\right)^2}-\dfrac{K_CQ^2}{G_NM}\right)^3}$

To end this post, and to prepare you for the follow up post, let me speculate a little bit about BH “are” or “we think they are”:

Black holes are spacetime, rotating or not, charged or not, but they are highly curved space-time fully specified by some parameters or numbers.
Quantum BH or at least semi-classical BH has a temperature and they evaporate. If you take this as serious, the own space-time does decay, at least, highly curved space-time.
BH have microstates, but we don’t know for sure what they are.

That entropy of any BH seems to scale like area and not like volume, is sometimes refferred as the holographic principle. BH entropy seems to be non-extensive. Indeed, this suggests a link with condensed matter. Or even with solid state theory. Are BH “materials”? Of course, since matter and energy are equivalent when you are using relativity, they should. That charges of BH are on the boundary, it seems, and not on the full volume seems to be something like topological insulators or superconductors. The quantum theory of space-time is yet to be built. Is it a world crystal like P. Jizba and collaborators suggest? A crystal is any highly ordered microscopic structure with lattices extending in all directions of space. However, solids are generally much more complex. Polycrystals are many crystals bonded or fused together. Is space-time or a BH a polycrystal? The classification of solids in crystalline, polycrystalline and amorphous could be also useful in BH physics! Polymorphism implies many crystals or phases. There are allotropy and polyamorphism as well. Furthermore, if you extend these thoughts to quasi-crystals, you get a bigger picture of black holes. Quasicrystals are non periodic “ordered” arrays of atoms. Could BH be quasicrystals? The International Union of Crystallography (IUC) defines crystal in a very general fashion. Its definition contains ordinary periodic discrete crystals, quasicrystals, and any other system showing some periodic diffraction diagram/pattern. Crystallinity itself is any structural order material, solid of material system. This definition paves the way towards topological ordered systems. It yields and correlates with hardness, density, transparency, diffusion and other material features. Crystalites or grains are the basic pieces (atoms) of polycrystals or polycrystalline matter. There are also materials “in between” crystals and amorphous materials. They are called paracrystals. More precisely, paracrystals are short medium range ordering lattice material, similar to liquid crystal phases, but lacking ordering in one direction at least. Geologists admit, today, four levels of crystallinity:

Holocrystalline.
Hypocrystalline.
Hypohyaline.
Holohyaline.

Open question: could you guess a way to classify BH solutions with certain geological dictionary? I did it. And it is lot of fun! Let me know if you arrive to some conclusion like me.

Open question (II): the Hagedorn temperature is the temperature beyond string theory ceases to have sense. The degrees of freedom have to be redefined beyond that point. Dimensionally, check that

$T_H=\dfrac{1}{2\pi\sqrt{\alpha^{'}}}=\dfrac{1}{l_s}=\dfrac{\hbar c}{2\pi l_sk_B}$

When is Hagedorn temperature equal to Planck temperature? Could they be different? Following the same arguments, calculate the temperature of a gas of $p$ branes and its Hagedorn temperature.

See you in my next blog post!

Epilogue:

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LOG#192. Bits on black holes (I).

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