In the early 1970s, people studying general relativity, our modern theory of gravity, noticed rough similarities between the properties of black holes and the laws of thermodynamics. Stephen Hawking proved that the area of a black hole’s event horizon—the surface that marks its boundary—cannot decrease. That sounded suspiciously like the second law of thermodynamics, which says entropy—a measure of disorder—cannot decrease.
Yet at the time, Hawking and others emphasized that the laws of black holes only looked like thermodynamics on paper; they did not actually relate to thermodynamic concepts like temperature or entropy.
Then in quick succession, a pair of brilliant results—one by Hawking himself—suggested that the equations governing black holes were in fact actual expressions of the thermodynamic laws applied to black holes. In 1972, Jacob Bekenstein argued that a black hole’s surface area was proportional to its entropy, and thus the second law similarity was a true identity. And in 1974, Hawking found that black holes appear to emit radiation—what we now call Hawking radiation—and this radiation would have exactly the same “temperature” in the thermodynamic analogy.
This connection gave physicists a tantalizing window into what many consider the biggest problem in theoretical physics—how to combine quantum mechanics, our theory of the very small, with general relativity. After all, thermodynamics comes from statistical mechanics, which describes the behavior of all the unseen atoms in a system. If a black hole is obeying thermodynamic laws, we can presume that a statistical description of all its fundamental, indivisible parts can be made. But in the case of a black hole, those parts aren’t atoms. They must be a kind of basic unit of gravity that makes up the fabric of space and time.
Modern researchers insist that any candidate for a theory of quantum gravity must explain how the laws of black hole thermodynamics arise from microscopic gravity, and in particular, why the entropy-to-area connection happens. And few question the truth of the connection between black hole thermodynamics and ordinary thermodynamics.
But what if the connection between the two really is little more than a rough analogy, with little physical reality? What would that mean for the past decades of work in string theory, loop quantum gravity, and beyond? Craig Callender, a philosopher of science at the University of California, San Diego, argues that the notorious laws of black hole thermodynamics may be nothing more than a useful analogy stretched too far. The interview has been condensed and edited for clarity.
Why did people ever think to connect black holes and thermodynamics?
Callender: 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.
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 toward 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.
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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.
Craig Callender explains why the connection between black holes and thermodynamics is little more than an analogy.
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.
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.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
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