One Way Streets: Black Holes and Irreversible Processes

A lot of things in life are irreversible: they only ever happen in one direction.

If you drop an egg on the floor, it breaks open and the yolk comes out.  But eggs never reassemble themselves and leap into your hand.  If you mix hot water with ice water, you get lukewarm water.  But if you have a glass of water just sitting on the table, you’ll never see half of it freeze and the other half boil!  Why is this?

It turns out that the answer has to do with counting possibilities.  Take the example of the glass of water—let’s imagine there’s a specific glass of water in front of you.  It contains a great many water molecules: in a normal sized glass there are about 1025 of them.  There are a huge number of different ways to arrange these water molecules inside the glass (something roughly like 101025).  But it turns out there are only a finite number of possibilities that look similar to your glass of water.

That’s because you know the temperature of your glass of water.  If the molecules are moving around too quickly, they have a lot of kinetic energy and the water will be hotter than yours is.  So there’s a limit as to how quickly the molecules can be moving around.

You might think that there are an infinite number of possible positions of the water molecules in the glass, since space is continuous.  But if you try to measure the position of all the water molecules too precisely, you’ll run into trouble with quantum mechanics.  Heisenberg’s famous uncertainty principle says that the more accurately you can determine the momentum of an object, the less accurately the position can be determined, and vice versa.  Since we have a limit on how fast the water molecules can be moving, we also have a limit on how accurately their positions can be measured.*

When you work this out, you get a finite (but huge) number of different ways for the glass of water to be.  Physicists describe this using a number called the entropy.  Systems with a big entropy have many different possible ways they can be, while systems with smaller entropy have fewer possible ways they can be.**

The Second Law of Thermodynamics states that the entropy of the universe increases with time. Once you know that the entropy is just counting the number of ways things can be, it makes sense that it should always increase:

Imagine you have a jigsaw puzzle solved inside its box.  If you vigorously shake it, the pieces will break up and it will become unsolved.  However, if you shake an unsolved puzzle, it won’t solve itself.   There’s no law of physics which prevents the pieces from happening to fall in just the right way to solve the puzzle.  The reason is that there’sso many more different ways the puzzle can be unsolved than solved.  So you wouldn’t expect it to happen.

The same thing is true for things like mixing hot and cold water.  There are enormously more ways to arrange the molecules in a lukewarm glass than a glass which is half-hot and half-cold.  Suppose there are X different ways to arrange the hot/cold separated water, and a much bigger number Y different ways to arrange the lukewarm water.  If we start with the hot/cold water and let it sit for a few minutes, we nearly always get lukewarm water—this is possible because the X states of the hot/cold water turn into X out of the Y possible configurations of the lukewarm water.  However, practically speaking we can’t tell any more which of the Y configurations the lukewarm water is in—they all look the same to us since we’re too big to see the molecules—so we say that the entropy increases.

But if we start with the lukewarm water and wait a few minutes, we are starting with Y different possibilites, and at most X of these can turn into hot/cold water.  (The laws of physics require that possibilies which start out being different have to keep on being different.)  The rest of the Y configurations would have to do something else, like remain lukewarm.  So the probability of the water separating into hot/cold is at most X/Y.  Since Y is enormously bigger than X, this probability is very, very close to zero.

So things only ever go in one direction.  The odds of it going in the reverse direction are so tiny, that for all practical purposes you can bet on it as certain.

The Second Law only says that the entropy of the whole universe has to increase as time passes.  If two systems interact, it’s possible to transfer entropy from one system to the other.  For example, life on Earth is possible because the sunshine falling on the Earth has low entropy, while the heat radiation emitted to outer space has high entropy.***

Now let’s think about the past and the future.  For the most part, the basic laws of physics don’t distinguish between the past and the future, yet the Second Law makes them quite different.  Why is that?  There are some mysteries here which scientists still argue about.  But the best explanation seems to be that God must have started out the universe in some very special state with low entropy.  From that time onward, the entropy began to increase.  (If He had instead decreed that the universe would end in a low entropy state, then entropy would decrease and everything would run backwards!  But we wouldn’t be able to tell, since we would go backwards too, remembering the “future” and anticipating the “past.”)

After the Big Bang, the matter in the universe was very hot, uniform, and dense.  The universe expanded (increasing the range of possible positions for the particles), while the matter cooled off and slowed down (decreasing the range of possible velocities).  Eventually gravity caused the matter to become clumpy, producing galaxies, stars, and planets (decreasing the entropy stored in positions), but at the same time reheating the objects (increasing the entropy stored in velocities).  In all of these complicated changes, each decrease in entropy was more than compensated for by increases in the entropy of something else.If God were to continue guiding the physical universe along its current course, without any intervening changes to the regular pattern of events, it would eventually reach the maximum entropy state, which seems to be something like a universe with really tiny temperature in which all particles are separated by astronomical distances.  It would then remain that way forever (except for occasional random fluctuations into more special states).  Before that time, life would have to go extinct and all interesting processes come to an end.  As St. Paul says, “the creation was subjected to futility” by the will of God.   Although the Second Law is needed to support the beauty and life in the universe, it reminds us that this is only on the condition of inevitable decay and death.

However, Paul goes on to add that this was done in hope; hope that the creation will be “set free from  slavery to decay, into the glorious freedom of God’s children.”  As Christians, we expect important supernatural events (such as the Second Coming of Jesus and the Resurrection of the Dead) to take place before life goes extinct. It’s interesting to speculate on what the laws of physics might be like in the New Heaven and New Earth, but obviously we simply don’t have enough information to say.

However, in the currently existing Heavens, there are already peculiar objects which challenge our ideas about how to count possibilities: black holes.

In our current best theory of gravity (Albert Einstein’s general relativity, the gravitational field consists entirely of distortions in space and time. Spacetime is an actual physical entity.  It not only influences matter, it is also influenced by matter.  Amounts of distance and time vary from place to place, similar to how the strength of the electric or magnetic field depends on where you are.

Any object with mass, such as the Earth, will slightly distort space and time in its vicinity.  For example, time runs about one part in a billion times slower on the surface of the Earth than it does in outer space.  (Believe it or not, this is what causes things to fall down!)

When a big enough star dies, one of the possible outcomes is that it collapses into a black hole.  (There are also enormous black holes which form in the center of galaxies, which can have millions or even billions of times the mass of the sun!)  Black holes are objects whose gravity is so strong that not even light can escape.

Because the paths of lightrays are distorted, anything that falls in past a certain surface (the event horizon), must inevitably fall in towards the center of the black hole (the singularity).  You see, time is distorted so much, that once you cross the event horizon, progress towards the center is just as inevitable as progress from the past to the future.

All of the matter that used to be in the star gets crunched down to zero size in the singularity!  As far as we know, time comes to an end for anything that falls into the singularity.  But we don’t really know for sure, since our best theories of physics don’t work when you get too close to the singularity.

So you shouldn’t think of a black hole as being composed of any kind of ordinary matter “stuff.”  It is composed entirely out of distortions in spacetime, i.e. the gravitational field.  Similarly, the event horizon isn’t made out of anything tangible; it is just an imaginary surface in empty space marking the point of no return.  If you fell across the event horizon, you wouldn’t hit any object as you crossed; it’s just empty space.

Now let me clear up a common misconception.  A lot of people think that once a black hole forms, it would start sucking everything nearby into it like a vacuum cleaner.  This isn’t quite true.  A black hole doesn’t have any more gravity than anything else of the same mass at the same distance.  If the sun collapsed into a black hole next Tuesday (which can’t really happen—it’s too small a star—but let’s just suppose) then the orbit of the Earth would remain exactly the same.

But if you drop things in close enough, they will get eaten.  Like people, black holes grow bigger when they eat things.  In fact, the radius of their event horizon is proportional to their total mass.  This is where the irreversibility is going to come in—black holes get bigger, but do they ever get smaller?

In 1971, Stephen Hawking mathematically proved an “Area Increase” theorem based on the equations of classical general relativity.  (In physics lingo, “classical” means not taking into account quantum mechanics.)  Hawking showed that the surface area of a black hole’s event horizon can only increase as time passess—it can’t get any smaller.  Similarly, if you merge two black holes into each other to make a single black hole, the total area of the result has to be greater than the sum of the original two.

This led Jacob Bekenstein to propose that black holes actually have an entropy which is proportional to their event horizon.  Remember that the Second Law says that the entropy always has to go up.  Bekenstein’s proposal was based on an analogy between the behavior of matter systems (whose entropy increases) and the behavior of black holes (whose area increases).

At first everyone thought that Bekenstein’s idea was ridiculous.  After all, if black holes really work the same way as ordinary thermodynamical systems, then they’d have to have a nonzero temperature.  If you put a hot thing next to a cold thing, heat flows from the hot object to the cold object.  That means that a black hole in empty space (which is at absolute zero) would have to emit radiation.  But everyone knew that nothing comes out of a black hole.

It turns out that everyone was wrong.  Once you take into account quantum mechanics, black holes do radiate particles just like a thermal object.  This was shown by Stephen Hawking in 1977 just a few years later.  Although nothing can escape from inside the event horizon, this “Hawking radiation” emerges from just outside the event horizon.  The larger the black hole, the colder its temperature.  In principle, any kind of particle could come out of a sufficiently small, hot black hole.  But for a solar mass black hole the Hawking temperature is very tiny (10-7 degrees above absolute zero), and there is only enough energy to radiate massless particles (photons and gravitons), in radio wave frequencies.  Unfortunately, the number of Hawking photons emitted is so small that detecting them would be quite hopeless, even if we had a measuring device right next to the black hole.

As the black hole emits radiation, it loses energy and very slowly shrinks.  So Hawking’s theorem doesn’t apply; the area decreases.  However, it turns out that the radiation which comes out has even more entropy than the black hole did.  No matter what you do, the total amount of entropy always goes up.  So it seems like black holes obey the Second Law even when you take into account quantum mechanics.

Now we come to the surprise twist.  Event horizons seem to have entropy, yes, but no one knows what the entropy is counting!  In ordinary matter systems, the matter counts the number of ways for the matter atoms to be arranged.  But black holes aren’t made of matter, they are made of spacetime.  By analogy, the entropy of the black hole probably tells us the number of possible configurations of whatever ingredients spacetime is made out of!

The size of the black hole entropy suggests that these ingredients are probably about 10-35 meters in size (Planck length).  This is about 20 orders of magnitude smaller than a proton or neutron.  So it is not surprising that space and time seem continuous to us at human scales.  But when you get small enough, they may really be made out of little chunks that can’t be subdivided!  But there’s no way to do experiments to measure distances that small, so we can’t know for sure what’s going on.  But we’re grateful for every tantalizing clues we can get.

God could have chosen to tell us what are the laws of the universe.  But he didn’t.  Instead He allows us to gradually find out what they are.  There’s a certain thrill to knowing how things work, but there’s also a sense of sublime wonder when you come to the limits of our understanding.  I love thinking about quantum gravity because one continually comes up against these deep mysteries.  Glory be to God!