Why did people ever think to connect Black Holes and
thermodynamics?
In the early
'70s, people noticed a few similarities between the two.
One is
that both seem to possess an equilibrium-like state. I have a
box of gas. It can be described by a small handful of parameters
- say, pressure, volume and temperature.
Same thing with a
Black Hole. It might be described with just its mass, angular momentum
and charge. Further details don't matter to either system.
Nor does this
state tell me what happened beforehand. I walk into a room and
see a box of gas with stable values of pressure, volume and
temperature.
Did it just settle into that state, or did that
happen last week, or perhaps a million years ago? Can't tell.
The Black Hole is similar.
You can't tell
what type of matter fell in or when it collapsed.
Callender in his office at UCSD.
His book 'What Makes Time Special?'
won the Lakatos award in the philosophy of science in 2018.
Peggy Peattie for Quanta Magazine
The second
feature is that Hawking proved that the area of Black Holes is
always non-decreasing.
That reminds one of the thermodynamic
second law, that entropy always increases. So both systems seem
to be heading towards simply described states.
Now grab a
thermodynamics textbook, locate the laws, and see if you can
find true statements when you replace the thermodynamic terms
with Black Hole variables. In many cases you can, and the
analogy improves.
Hawking then
discovers Hawking radiation, which further improves the analogy.
At that point, most physicists start claiming the analogy is so
good that it's more than an analogy - it's an identity! That's a
super strong and surprising claim.
It says that
Black Hole laws,
most of which are features of the geometry of space-time, are
somehow identical to the physical principles underlying the
physics of steam engines.
Because the
identity plays a huge role in quantum gravity, I want to
reconsider this identity claim. Few in the foundations of
physics have done so.
So what's the statistical mechanics for Black Holes?
Well, that's a
good question.
Why does ordinary thermodynamics hold? Well, we
know that all these macroscopic thermodynamic systems are
composed of particles.
The laws of thermodynamics turn out to be
descriptions of the most statistically likely configurations to
happen from the microscopic point of view.
Why does
Black Hole thermodynamics hold?
Are the laws also the statistically
most likely way for Black Holes to behave?
Although there are
speculations in this direction, so far we don't have a solid
microscopic understanding of Black Hole physics.
Absent this,
the identity claim seems even more surprising.
What led you to start thinking about the analogy?
Many people are
worried about whether theoretical physics has become too
speculative.
There's a lot of commentary about whether
holography, the string landscape - all sorts of things - are
tethered enough to experiment. I have similar concerns.
So my
former Ph.D. student John Dougherty and I thought, where did it
all start?
To our mind a
lot of it starts with this claimed identity between Black Holes
and thermodynamics. When you look in the literature, you see
people say,
"The only evidence we have for quantum gravity, the
only solid hint, is Black Hole thermodynamics."
If that's the
main thing we're bouncing off for quantum gravity, then we ought
to examine it very carefully. If it turns out to be a poor clue,
maybe it would be better to spread our bets a little wider,
instead of going all in on this identity.
What problems do you see with treating a Black
Hole as a thermodynamic system?
I see basically
three.
The first problem is:
What is a
Black Hole?
People often think of
Black Holes as just
kind of a dark sphere, like in a Hollywood movie
or something; they're thinking of it like a star
that collapsed.
But a mathematical
Black Hole,
the basis of Black Hole thermodynamics, is not
the material from the star that's collapsed.
That's all gone into the singularity.
The Black Hole is what's left...
The Black Hole isn't a solid thing at the
center. The system is really the entire
space-time.
Yes, it's this
global notion for which Black Hole
thermodynamics was developed, in which case the
system really is the whole space-time.
Here is another
way to think about the worry. Suppose a star
collapses and forms an event horizon.
But now
another star falls past this event horizon and
it collapses, so it's inside the first. You
can't think that each one has its own little
horizon that is behaving thermodynamically. It's
only the one horizon.
Here's another.
The event horizon changes shape depending on
what's about to be thrown into it. It's
clairvoyant. Weird, but there is nothing spooky
here so long as we remember that the event
horizon is only defined globally.
It's not a
locally observable quantity...
The picture is
more counterintuitive than people usually think.
To me, if the system is global, then it's not at
all like thermodynamics.
The second
objection is:
Black hole thermodynamics is
really a pale shadow of thermodynamics.
I was
surprised to see the analogy wasn't as thorough
as I expected it to be.
If you grab a
thermodynamics textbook and start replacing
claims with their Black Hole counterparts, you
will not find the analogy goes that deep.
Video:
Craig Callender explains why the connection
between
Black Holes and
thermodynamics
is little more than an analogy.
Peggy Peattie for Quanta Magazine
For instance,
the
zeroth law of thermodynamics sets up the whole theory and a
notion of equilibrium - the basic idea that the features of the
system aren't changing.
And it says that if one system is in
equilibrium with another - A with B, and B with C - then A must
be in equilibrium with C. The foundation of thermodynamics is
this equilibrium relation, which sets up the meaning of
temperature.
The zeroth law
for Black Holes is that the surface gravity of a Black Hole,
which measures the gravitational acceleration, is a constant on
the horizon. So that assumes temperature being constant is the
zeroth law.
That's not really right. Here we see a pale shadow
of the original zeroth law.
The counterpart
of equilibrium is supposed to be "stationary," a technical term
that basically says the Black Hole is spinning at a constant
rate.
But there's no sense in which one
Black Hole can be
"stationary with" another Black Hole. You can take any
thermodynamic object and cut it in half and say one half is in
equilibrium with the other half. But you can't take a Black Hole
and cut it in half.
You can't say that this half is stationary
with the other half.
Here's another
way in which the analogy falls flat. Black hole entropy is given
by the Black Hole area. Well, area is length squared, volume is
length cubed.
So,
What do we make of all those thermodynamic
relations that include volume, like Boyle's law?
Is volume,
which is length times area, really length times entropy?
That
would ruin the analogy.
So we have to say that volume is not the
counterpart of volume, which is surprising.
The most famous connection between Black Holes and
thermodynamics comes from the notion of entropy.
For normal
stuff, we think of entropy as a measure of the disorder of the
underlying atoms.
But in the 1970s, Jacob Bekenstein said that
the surface area of a black hole's event horizon is equivalent
to entropy.
What's the basis of this?
This is my
third concern.
Bekenstein says,
if I throw something into a
Black Hole, the entropy vanishes.
But this can't happen, he
thinks, according to the laws of thermodynamics, for entropy
must always increase...
So some sort of compensation must
be paid when you throw things into a Black Hole.
Callender keeps in
his office
a wooden model of a perpetual motion machine.
The
original design dates back
to the 13th century.
It does not
work.
Peggy Peattie for Quanta Magazine
Bekenstein
notices a solution.
When I throw something into the
Black Hole,
the mass goes up, and so does the area. If I identify the area
of the Black Hole as the entropy, then I've found my
compensation.
There is a nice deal between the two - one goes
down while the other one goes up - and it saves the second law.
When I saw that
I thought, aha, he's thinking that not knowing about the system
anymore means its entropy value has changed. I immediately saw
that this is pretty objectionable, because it identifies entropy
with uncertainty and our knowledge.
There's a long
debate in the foundations of statistical mechanics about whether
entropy is a subjective notion or an objective notion. I'm
firmly on the side of thinking it's an objective notion.
I think
trees unobserved in a forest go to equilibrium regardless of
what anyone knows about them or not, that the way heat flows has
nothing to do with knowledge, and so on.
Chuck a steam
engine behind the event horizon.
We can't know anything about it
apart from its mass, but I claim it can still do as much work as
before. If you don't believe me, we can test this by having a
physicist jump into the Black Hole and follow the steam engine!
There is only need for compensation if you think that what you
can no longer know about ceases to exist.
Do you think it's possible to patch up Black Hole
thermodynamics, or is it all hopeless?
My mind is
open, but I have to admit that I'm deeply skeptical about it.
My suspicion is
that Black Hole "thermodynamics" is really an interesting set of
relationships about information from the point of view of the
exterior of the Black Hole. It's all about forgetting
information.
Because
thermodynamics is more than information theory, I don't think
there's a deep thermodynamic principle operating through the
universe that causes Black Holes to behave the way they do, and
I worry that physics is all in on it being a great hint for
quantum gravity when it might not be.
Playing the
role of the Socratic gadfly in the foundations of physics is
sometimes important.
In this case, looking back invites a bit of
skepticism that may be useful going forward.
