Strange Mathematical Term Changes Our View of Black Holes

Calmet and his graduate student Folkert Kuipers were examining quantum effects near the event horizons of black holes, something that is fiendishly hard to pin down. To tackle this, the researchers employed a technique to simplify their calculations. As they were working, a strange term appeared in the mathematics of their solution. After months of confusion, they realized what this newly discovered term meant: It was an expression of the pressure produced by a black hole. Nobody had known this was possible before, and it changes the way scientists think about black holes and their relationships with the rest of the universe.

Perhaps one of the @physicists can sort out my confusion on a key point. I don’t understand the presumption that Black Holes don’t produce pressure. I thought Hawking’s argument made clear that they did.

This is my naive understanding. Pairs of particles are popping into existence and the annihilating each other all the time. At the event horizon of a black hole, one particle can fall into the black hole, while the other speeds away. So they can’t annihilate each other. Ergo Hawking radiation.

But that’s also a clear argument for pressure too. The radiation exerts pressure. Of course it is tiny in comparison to gravity, but that’s the case here too.

So what is new here exactly? It can’t be the idea that black holes exert pressure, is it?

Perhaps they found more pressure than exerted by Hawking radiation alone? Or perhaps it is by a different mechanism?


In regular thermodynamics, there is what is called the 1st law of thermodynamics. There are many different versions of the 1st law, but the one that is most relevant is: (change of energy) = Temperature*(change of entropy) - Pressure*(change of volume)

When people worked on black holes in the later 20th century, they found an equation that looks reasonably close to the 1st law of thermodynamics. They found that:
(change in energy of BH) = Temperature*(change of entropy of the black hole).

\Delta E = T \Delta S

This is called the 1st law of black hole thermodynamics. This equation looks almost like the 1st law of thermodynamics, but just missing the Pressure*(change of volume) term.

P \Delta V

This work states that, if certain quantum assumptions are true about black holes, a correction to the 1st law of black hole thermodynamics can be derived that gives the missing Pressure*(change of volume) term.

\Delta E = T \Delta S - P \Delta V


Showing my ignorance here… Would the pressure exerted by black holes explain, to any extent, the acceleration of the universe’s expansion?


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Unfortunately the pressure, if it exists, is too small to do that. Technically, it’s also not the right ‘kind’ of pressure.


Thanks @PdotdQ. I added in some equations to your post to clarify to onlookers. Please do correct the if I got something wrong.

So, what I am understanding from you, is that they found out for black holes that, taking QM into account, that this holds true:

\Delta E = T \Delta S - P \Delta V

So the P \Delta V term is due to QM. Notably, it should be an exceedingly small outward pressure and the \Delta V is exceedingly small too. Right? If that is the case, there would be almost no hope in detecting it, unless perhaps it arises merger dynamics somehow. Though I am doubtful it would be large with respect to Gravity.

So here is my question then, assuming what I am saying here is about correct.

Is that new term, the P \Delta V, due to this?

Or is it the pressure deriving from some other interaction between QM and black holes?

Thanks for writing this out. I would say that everything in that term, including the T \Delta S part, is due (in some way) to the quantum nature of black holes. Even if the original T \Delta S can be derived without quantum, Hawking needed to use quantum theory to figure out the exact equation for the T \Delta S.

The original Hawking calculation is not complete, and corrections can be made to it. That’s what this paper is trying to do; it’s one of the papers trying to compute corrections to Hawking’s original calculation (this is somewhat of a cottage industry, there are many groups doing this, often with wildly different techniques and starting assumptions). So, in a way, everything is due to quantum, not just the pressure term.

Correct, it should be exceedingly small.

I would say no. First off, that is an explanation of the Hawking radiation, which is related to, but not the same as the 1st law of black hole thermodynamics. Secondly, that explanation is not correct. I think that explanation first came from Hawking when he tried explaining Hawking radiation to a general public. There are parts of that explanation that is reminiscent to his actual derivation, but in the end it is not what’s actually going on. With all due respect to the late Dr. Hawking, I think that explanation causes more confusion than understanding, and should be avoided.

The actual reason for Hawking radiation has to do with what happens when you apply transformations (i.e., changing reference frames) on quantum fields. It turns out that vacuum is not transformation invariant: if you change your frame of reference, what you observe as vacuum changes. Some frame of references will see the vacuum as something that is filled with particles, ergo Hawking radiation.


Okay but Hawking radiation is still radiation. Radiation exerts pressure. So it’s been known for a long time that black holes exert pressure. Is that the pressure of the new term? You are saying “no.”

Correct. There is a difference between the pressure exerted by a black hole itself, and the pressure exerted by radiation that the black hole emits.

Imagine having a piece of coal that is glowing hot (emitting radiation). There is a difference between:

  1. The coal has pressure (due to how the carbon atoms arranged in the coal behave when I press against it)
  2. The radiation that the coal emits has pressure

This is completely analogous. Now, it may be that these two pressures turn out to be mathematically related to one another, but a priori we can’t say that they are the same pressures.



So we know the pressure is tiny compared to the gravity involved. But that isn’t saying much, because gravity is immense near a black hole!

How much pounds per square inch would this be? If it arose from Hawking radiation, it would be tiny on a human scale. But it doesn’t. So maybe it’s more?

Hmm, I don’t know the numbers, but that’s an interesting question to ask.

I would also warn the following: yes, the authors of the paper are interpreting this extra term as the “pressure” of the black hole. But I am not 100% convinced yet that this is not just a term in a mathematical equation that looks like the pressure term in the thermodynamic 1st law, but has nothing to do with the real pressure of the system (or at least, not pressure in the same sense that we are used to).

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