Entropy and Singularities

[Perpetual Student]

Physicists describe a Black Hole, which is described as a singularity, as having high entropy or a high level of disorder. The singularity that gave birth o the universe is described as having low entropy or low disorder.

Two Questions:
1. Why does a black hole (a singularity) have high entropy when it appears to be so highly ordered?

2. Why does the primordial singularity have low entropy since it seems to be so similar to a black hole, which has high entropy?

Now. I am certain the above questions contain inherent misconceptions. Don't beat me up. I would like some help to to rid myself of those misconceptions.

 

[Reality Check]

Physicists describe a Black Hole, which is described as a singularity, as having high entropy or a high level of disorder. The singularity that gave birth o the universe is described as having low entropy or low disorder.

Two Questions:
1. Why does a black hole (a singularity) have high entropy when it appears to be so highly ordered?

2. Why does the primordial singularity have low entropy since it seems to be so similar to a black hole, which has high entropy?

Now. I am certain the above questions contain inherent misconceptions. Don't beat me up. I would like some help to to rid myself of those misconceptions.

Just off the top of my head (this is explained well in Roger Penrose's book The Road To Reality but I do not have my copy with me):

1. A black hole is not as highly ordered as you might think. Hawking radiation gives it a temperature and so a non-zero entropy. This is just as well otherwise we could violate the second law of thermodynamics by throwing matter into a black hole. The various calculations for the value of the entropy (see Black hole thermodynamics) are theoretical but agree that it is proportional to the area of the event horizon divided by the Planck length squared. The Planck length is really small so black hole entropy values are really high.
Note that a black hole's entropy is not really concerned with its singularity (or even whether a black hole has a singularity) but rather with its event horizon. This makes sense because nothing inside the event horizon is accessible to the outside universe and this includes any possible microstates that would contribute to the macrostate of a black hole.

In statistical thermodynamics the entropy is defined as being proportional to the logarithm of the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system.^WP

2. The primordial singularity is different from the singularity for a black hole - it does not have an event horizon since it is the entire universe. So its entropy may even be zero.

 

[Perpetual Student]

Just off the top of my head (this is explained well in Roger Penrose's book The Road To Reality but I do not have my copy with me):

1. A black hole is not as highly ordered as you might think. Hawking radiation gives it a temperature and so a non-zero entropy. This is just as well otherwise we could violate the second law of thermodynamics by throwing matter into a black hole. The various calculations for the value of the entropy (see Black hole thermodynamics) are theoretical but agree that it is proportional to the area of the event horizon divided by the Planck length squared. The Planck length is really small so black hole entropy values are really high.
Note that a black hole's entropy is not really concerned with its singularity (or even whether a black hole has a singularity) but rather with its event horizon. This makes sense because nothing inside the event horizon is accessible to the outside universe and this includes any possible microstates that would contribute to the macrostate of a black hole.

.

"General relativity describes a black hole as a region of empty space with a point-like singularity." (Wikipedia)
How can this be anything but a highly ordered system? How much more order can there be? -- just like the primordial singularity? Why would the event horizon change that? The event horizon is a result of the singularity; it is not part of it.

 

[Vorpal]

The short answer is that within GTR, a black hole has no observable structure besides a handful of parameters, and so "shouldn't" have entropy. However, certain results concerning stationary black holes have a resemblance to the laws of thermodynamics, e.g.,
--0th law: surface gravity κ is constant over the event horizon
--1st law: δM = κ δA + Ω δJ to first order in appropriate units (A area, Ω angular velocity, J angular momentum)
--2nd law: the area of the horizon is non-decreasing
--3rd law: reducing surface gravity to zero is impossible over with a finite sequence of operations
The resemblance was thought purely superficial until the discovery of Hawking radiation, in which κ (up to some scaling constant) acted as temperature in more than an analogy.

There is simply no answer to your question in pure GTR, except the re-affirmation that the singularity itself is irrelevant. There was a half-hearted suggestion by Penrose to identify entropy of a black hole with Weyl curvature in some manner, which would make qualitative sense--black holes have lots (Ricci curvature represent stress-energy-momentum, of which they have none, but Weyl curvature represents tidal forces), big bang has little, etc., but quantitatively it never really went anywhere as far as I'm aware of (but I'm far from an expert in this field). At some point there was a string-theory based calculation that explicitly derived the entropy of a black hole in the limiting case of κ→0, if I recall correctly (which unfortunately has a decent chance of not being the case).

 

[Reality Check]

"General relativity describes a black hole as a region of empty space with a point-like singularity." (Wikipedia)
How can this be anything but a highly ordered system? How much more order can there be? -- just like the primordial singularity? Why would the event horizon change that? The event horizon is a result of the singularity; it is not part of it.

Definition of entropy from Wikipedia:

In statistical thermodynamics the entropy is defined as being proportional to the logarithm of the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system:
...
This definition is considered to be the fundamental definition of entropy (as all other definitions can be mathematically derived from it, but not vice versa). In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

For a black hole

• The 'observed macroscopic description of the thermodynamic system' is a very simple and ordered system described by 10 parameters ("these being a, m, , the direction of the spin axis, the position of the mass-centre, and its 3-velocity" - The Road to Reality by Roger Penrose, page 715).
• The 'number of microscopic configurations' is every possible way that mass can collapse to form a black hole. This is a very large number.

The Bekenstein-Hawkin formula for the entropy that can be attributed to a black hole is a combination of GTR and quantum mechanics - both G and Planck's constant appear in the equation.
In his book, Roger Penrose gives the magnitudes of the entropy of:

• The 2.7K microwave radiation is 10⁸ or 10⁹ per baryon in natural units (roughly speaking this is the number of photons per baryon left over from the Big Bank).
• The entropy of the supermassive black holes in galaxies is about 10²¹ per baryon (taking our galaxy as typical).

 

[G O R T]

"General relativity describes a black hole as a region of empty space with a point-like singularity." (Wikipedia)
How can this be anything but a highly ordered system? How much more order can there be? -- just like the primordial singularity? Why would the event horizon change that? The event horizon is a result of the singularity; it is not part of it.

First we need to work on your definition of "order".
Order in thermodynamics concerns the segregation of energy in a way that makes that energy available to do work. Evenly distributed energy in a closed system (no matter how much energy there is) has no way to perform work and so has low order. Black holes absorb everything and the energy absorbed is no longer available to do any work, thus high entropy and low order.

 

[sol invictus]

Two Questions:
1. Why does a black hole (a singularity) have high entropy when it appears to be so highly ordered?

Things evolve naturally towards forming black holes (i.e., given a collection of mass in flat space it will spontaneously collapse due to gravitational attraction). By the second law of thermodynamics, the entropy must increase during such a process. It turns out it increases to a kind of maximum - black holes are the most entropic objects possible given their size.

As for where all that entropy is "stored", it's near the horizon. The horizon appears to all measurements made outside it as a very hot surface which happens to be at the bottom of a deep well of gravitational potential. Hot surfaces naturally have high entropy densities.

2. Why does the primordial singularity have low entropy since it seems to be so similar to a black hole, which has high entropy?

That's actually an extremely good question.

First, not all cosmological singularities have low entropy. For example if the universe re-collapses and crunches, its entropy will increase throughout that process, and hence that future singularity must have high entropy. On the other hand the one in our past must have had a small entropy. How can this be?

One answer is that there is no horizon cloaking the singularity (in a sense, we're inside it). In a black hole it was the horizon that stored the entropy, not the singularity. This also explains why future singularities have high entropy: as you approach one, a horizon forms around you which cuts off most of the space, and it's that horizon which carries the entropy.

But fundamentally no one can answer that - unlike in the case of black holes, where the origin of the entropy is understood at least at a rough level, the smallness of the entropy of the big bang remains one of the most profound mysteries in science. There are various hypotheses about it, but they are all deficient in one way or another.

 

[Perpetual Student]

Thanks everyone. It is clearer to me now why a black hole is an extremely high entropy object.
So, the low entropy of the primordial singularity remains a mystery. It appears, from what I have read from several sources, that accepting the cosmological singularity's low entropy is similar to accepting the cosmological singularity in the first place. It simply was!