<![CDATA[Black Holes and the Golden Ratio]]>

The golden ratio shows up in the physics of black holes!

Or does it?

Most things get hotter when you put more energy into them. But systems held together by gravity often work the other way. For example, when a red giant star runs out of fuel and collapses, its energy goes down but its temperature goes up! We say these systems have a negative specific heat.

The prime example of a system held together by gravity is a black hole. Hawking showed—using calculations, not experiments—that a black hole should not be perfectly black. It should emit ‘Hawking radiation’. So it should have a very slight glow, as if it had a temperature above zero. For a black hole the mass of the Sun this temperature would be just 6 × 10^-8 kelvin.

This is absurdly chilly, much colder than the microwave background radiation left over from the Big Bang. So in practice, such a black hole will absorb stuff—stars, nearby gas and dust, starlight, microwave background radiation, and so on—and grow bigger. But if we could protect it from all this stuff, and put it in a very cold box, it would slowly shrink by emitting radiation and losing energy, and thus mass. As it lost energy, its temperature would go up. The less energy it has, the hotter it gets: a negative specific heat! Eventually, as it shrinks to nothing, it should explode in a very hot blast.

But for a spinning black hole, things are more complicated. If it spins fast enough, its specific heat will be positive, like a more ordinary object.

And according to a 1989 paper by Paul Davies, the transition to positive specific heat happens at a point governed by the golden ratio! He claimed that in units where the speed of light and gravitational constant are 1, it happens when

$\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }$

Here $J$ is the black hole’s angular momentum, $M$ is its mass, and

$\displaystyle{ \frac{\sqrt{5} - 1}{2} = 0.6180339\dots }$

is a version of the golden ratio! This is for black holes with no electric charge.

Unfortunately, this claim is false. Cesar Uliana, who just did a master’s thesis on black hole thermodynamics, pointed this out in the comments below after I posted this article.

And curiously, twelve years before writing this paper with the mistake in it, Davies wrote a paper that got the right answer to the same problem! It’s even mentioned in the abstract.

The correct constant is not the golden ratio! The correct constant is smaller:

$\displaystyle{ 2 \sqrt{3} - 3 = 0.46410161513\dots }$

However, Greg Egan figured out the nature of Davies’ slip, and thus discovered how the golden ratio really does show up in black hole physics… though in a more quirky and seemingly less significant way.

As usually defined, the specific heat of a rotating black hole measures the change in internal energy per change in temperature while angular momentum is held constant. But Davies looked at the change in internal energy per change in temperature while the ratio of angular momentum to mass is held constant. It’s this modified quantity that switches from positive to negative when

$\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }$

In other words:

Suppose we gradually add mass and angular momentum to a black hole while not changing the ratio of angular momentum, $J,$ to mass, $M.$ Then $J^2/M^4$ gradually drops. As this happens, the black hole’s temperature increases until

$\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }$

in units where the speed of light and gravitational constant are 1. And then it starts dropping!

What does this mean? It’s hard to tell. It doesn’t seem very important, because it seems there’s no good physical reason for the ratio of $J$ to $M$ to stay constant. In particular, as a black hole shrinks by emitting Hawking radiation, this ratio goes to zero. In other words, the black hole spins down faster than it loses mass.

Popularizations

Discussions of black holes and the golden ratio can be found in a variety of places. Mario Livio is the author of The Golden Ratio, and also an astrophysicist, so it makes sense that he would be interested in this connection. He wrote about it here:

• Mario Livio, The golden ratio and astronomy, Huffington Post, 22 August 2012.

Marcus Chown, the main writer on cosmology for New Scientist, talked to Livio and wrote about it here:

• Marcus Chown, The golden rule, The Guardian, 15 January 2003.

Chown writes:

Perhaps the most surprising place the golden ratio crops up is in the physics of black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies such as the sun have a “negative specific heat”. This means they get hotter as they lose heat. Basically, loss of heat robs the gas of a body such as the sun of internal pressure, enabling gravity to squeeze it into a smaller volume. The gas then heats up, for the same reason that the air in a bicycle pump gets hot when it is squeezed.

Things are not so simple, however, for a spinning black hole, since there is an outward “centrifugal force” acting to prevent any shrinkage of the hole. The force depends on how fast the hole is spinning. It turns out that at a critical value of the spin, a black hole flips from negative to positive specific heat—that is, from growing hotter as it loses heat to growing colder. What determines the critical value? The mass of the black hole and the golden ratio!

Why is the golden ratio associated with black holes? “It’s a complete enigma,” Livio confesses.

Extremal black holes

As we’ve seen, a rotating uncharged black hole has negative specific heat whenever the angular momentum is below a certain critical value:

$\displaystyle{ J < k M^2 }$

where

$\displaystyle{ k = \sqrt{2 \sqrt{3} - 3} = 0.68125003863\dots }$

As $J$ goes up to this critical value, the specific heat actually approaches $-\infty$ ! On the other hand, a rotating uncharged black hole has positive specific heat when

$\displaystyle{ J > kM^2}$

and as $J$ goes down to this critical value, the specific heat approaches $-\infty.$ So, there’s some sort of ‘phase transition’ at

$\displaystyle{ J = k M^2 }$

But as we make the black hole spin even faster, something very strange happens when

$\displaystyle{ J > M^2 }$

Then the black hole gets a naked singularity!

In other words, its singularity is no longer hidden behind an event horizon. An event horizon is an imaginary surface such that if you cross it, you’re doomed to never come back out. As far as we know, all black holes in nature have their singularities hidden behind an event horizon. But if the angular momentum were too big, this would not be true!

A black hole posed right at the brink:

$\displaystyle{ J = M^2 }$

is called an ‘extremal’ black hole.

Black holes in nature

Most physicists believe it’s impossible for black holes to go beyond extremality. There are lots of reasons for this. But do any black holes seen in nature get close to extremality? For example, do any spin so fast that they have positive specific heat? It seems the answer is yes!

Over on Google+, Robert Penna writes:

Nature seems to have no trouble making black holes on both sides of the phase transition. The spins of about a dozen solar mass black holes have reliable measurements. GRS1915+105 is close to $J=M^2.$ The spin of A0620-00 is close to $J=0.$ GRO J1655-40 has a spin sitting right at the phase transition.

The spins of astrophysical black holes are set by a competition between accretion (which tends to spin things up to $J=M^2$ ) and jet formation (which tends to drain angular momentum). I don’t know of any astrophysical process that is sensitive to the black hole phase transition.

That’s really cool, but the last part is a bit sad! The problem, I suspect, is that Hawking radiation is so pathetically weak.

But by the way, you may have heard of this recent paper—about a supermassive black hole that’s spinning super-fast:

• G. Risaliti, F. A. Harrison, K. K. Madsen, D. J. Walton, S. E. Boggs, F. E. Christensen, W. W. Craig, B. W. Grefenstette, C. J. Hailey, E. Nardini, Daniel Stern and W. W. Zhang, A rapidly spinning supermassive black hole at the centre of NGC 1365, Nature (2013), 449–451.

They estimate that this black hole has a mass about 2 million times that of our sun, and that

$\displaystyle{ J \ge 0.84 \, M^2 }$

with 90% confidence. If so, this is above the phase transition where it gets positive specific heat.

The nitty-gritty details

Here is where Paul Davies claimed the golden ratio shows up in black hole physics:

• Paul C. W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989), 1909–1914.

He works out when the specific heat vanishes for rotating and/or charged black holes in a universe with a positive cosmological constant: so-called de Sitter space. The formula is pretty complicated. Then he set the cosmological constant $\Lambda$ to zero. In this case de Sitter space flattens out to Minkowski space, and his black holes reduce to Kerr–Newman black holes: that is, rotating and/or charged black holes in an asymptotically Minkowskian spacetime. He writes:

In the limit $\alpha \to 0$ (that is, $\Lambda \to 0$ ), the cosmological horizon no longer exists: the solution corresponds to the case of a black hole in asymptotically flat spacetime. In this case $r$ may be explicitly eliminated to give

$(\beta + \gamma)^3 + \beta^2 -\beta - \frac{3}{4} \gamma^2 = 0. \qquad (2.17)$

Here

$\beta = a^2 / M^2$

$\gamma = Q^2 / M^2$

and he says $M$ is the black hole’s mass, $Q$ is its charge and $a$ is its angular momentum. He continues:

For $\beta = 0$ (i.e. $a = 0$ ) equation (2.17) has the solution $\gamma = 3/4$ , or

$\displaystyle{ Q^2 = \frac{3}{4} M^2 } \qquad (2.18)$

For $\gamma = 0$ (i.e. $Q = 0$ ), equation (2.17) may be solved to give $\beta = (\sqrt{5} - 1)/2$ or

$\displaystyle{ a^2 = (\sqrt{5} - 1)M^2/2 \cong 0.62 M^2 } \qquad (2.19)$

These were the results first reported for the black-hole case in Davies (1979).

In fact $a$ can’t be the angular momentum, since the right condition for a phase transition should say the black hole’s angular momentum is some constant times its mass squared. I think Davies really meant to define

$a = J/M$

This is important beyond the level of a mere typo, because we get different concepts of specific heat depending on whether we hold $J$ or $a$ constant while taking certain derivatives!

In the usual definition of specific heat for rotating black holes, we hold $J$ constant and see how the black hole’s heat energy changes with temperature. If we call this specific heat $C_J,$ we have

$\displaystyle{ C_J = T \left.\frac{\partial S}{\partial T}\right|_J }$

where $S$ is the black hole’s entropy. This specific heat $C_J$ becomes infinite when

$\displaystyle{ \frac{J^2}{M^4} = 2 \sqrt{3} - 3 }$

But if instead we hold $a = J/M$ constant, we get something else—and this what Davies did! If we call this modified concept of specific heat $C_a$ , we have

$\displaystyle{ C_a = T \left.\frac{\partial S}{\partial T}\right|_a }$

This modified ‘specific heat’ $C_a$ becomes infinite when

$\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5}-1}{2} }$

After proving these facts in the comments below, Greg Egan drew some nice graphs to explain what’s going on. Here are the curves of constant temperature as a function of the black hole’s mass $M$ and angular momentum $J:$

The dashed parabola passing through the peaks of the curves of constant temperature is where $C_J$ becomes infinite. This is where energy can be added without changing the temperature, so long as it’s added in a manner that leaves $J$ constant.

And here are the same curves of constant temperature, along with the parabola where $C_a$ becomes infinite:

This new dashed parabola intersects each curve of constant temperature at the point where the tangent to this curve passes through the origin: that is, where the tangent is a line of constant $a=J/M.$ This is where energy and angular momentum can be added to the hole in a manner that leaves $a$ constant without changing the temperature.

As mentioned, Davies correctly said when the ordinary specific heat $C_J$ becomes infinite in another paper, eleven years earlier:

• Paul C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978), 1313–1355.

You can see his answer on page 1336.

This 1978 paper, in turn, is a summary of previous work including an article from a year earlier:

• Paul C. W. Davies, The thermodynamic theory of black holes, Proc. Roy. Soc. Lond. A 353 (1977), 499–521.

And in the abstract of this earlier article, Davies wrote:

The thermodynamic theory underlying black-hole processes is developed in detail and applied to model systems. It is found that Kerr-Newman black holes undergo a phase transition at an angular-momentum mass ratio of 0.68M or an electric charge (Q) of 0.86M, where the heat capacity has an infinite discontinuity. Above the transition values the specific heat is positive, permitting isothermal equilibrium with a surrounding heat bath.

Here the number 0.68 is showing up because

$\displaystyle{ \sqrt{ 2 \sqrt{3} - 3 } = 0.68125003863\dots }$

The number 0.86 is showing up because

$\displaystyle{ \sqrt{ \frac{3}{4} } = 0.86602540378\dots }$

By the way, just in case you want to do some computations using experimental data, let me put the speed of light $c$ and gravitational constant $G$ back in the formulas. A rotating (uncharged) black hole is extremal when

$\displaystyle{ c J = G M^2 }$

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