Solving the paradox of cold Black holes with high entropy

Black holes can described by just two external parameters: mass and rotation. [1] They are also generally considered “cool” with their “temperatures” close to absolute zero. But they are also the most entropic objects in the universe. How can such simple cold objects harbor so much entropy? This has always boggled me. Below is my speculation on why this must be true.

The holographic principle, a groundbreaking yet currently unverified concept, posits that a black hole’s information is encoded on its event horizon. Black holes entropy also grows with its area rather than volume, which is another thing that has always surprised me. This principle is a result of mathematical and theoretical work from leading thinkers such as Bekenstein and Hawking. However, there’s no empirical evidence for this, as these very same theories suggest black holes will evaporate only over deep astronomical time. A sun sized black hole for example may “evaporate” in 1064 years. This number is hard to fathom. So let’s use an analogy. Let’s take the current age of the universe, estimated at 13.7 billion years, and compress it down to a second. If each second tick in this clock is 13 billion years, and using that clock we measure13.7 billion years, we’d still not be close to this number. If we created another clock that ticked one second every time our first clock showed 13.7 billion years, we’d still not be close. We would have to repeat this process 3 times to even approach the number!

The same challenge applies to the famous Hawking radiation. While compelling from a theoretical perspective, it lacks empirical evidence. This is because the ‘temperature’ of most stellar-sized black holes would be very close to 0K, much below the temperature of the cosmic microwave background left behind from the big bang. Supermassive black holes should be ‘cooler‘ still. We’d be waiting for a very long time to empirically observe it as shown in the illustration above!

Typically, cooler objects have lower entropy than hotter ones.[2] For example, solid ice has less entropy than liquid water because the molecules have fewer degrees of motion. This suggests that objects like neutron stars, which are so dense that their density approaches that of atomic nuclei, must also have low entropy, especially when they cool down in deep astronomical time. But, intriguingly, research indicates that when adjusting for temperature differences, the entropy of neutron stars appears to converge towards that of black holes, a finding that challenges our conventional understanding of these dense objects. On the one hand, this shouldn’t be surprising because neutron stars are the closest objects to black holes. They have escape velocities approaching the speed of light, and adding ‘only a little more mass‘ can cause them to collapse into a black hole. But on the other hand, this should be surprising because we expect them to have low entropy as cool, solid, super-dense objects with very few degrees of motion allowable (minimal number of microstates) for any given particle in them.

How do we reconcile these disparate interpretations? My conjecture is that we can do so intuitively by considering the length of time they exist and the span of space they interact with. In other words, by thinking of black holes as phenomena rather than objects. Consider a star. It has very high entropy as a hot ball of gas. But the energy it sends out into the universe is highly localized, high-frequency radiation which can do quite a bit of work before turning into ‘waste heat’. Overall, this should cause the matter surrounding the star to get more disorganized over time. This also allows open-loop systems like life on earth, which locally decreases entropy (and local information processing) while broadly increasing entropy elsewhere in the universe.

The star is only able to do this when the matter around it is less energetic than the star itself. To prove this to yourself, consider the early universe after the big bang when the universe was hotter than the Sun. Paradoxically, it should also be more ordered than now because entropy can only increase with time. Now imagine somehow creating the sun at this time. It would be physically impossible because gas wouldn’t be cool enough to gravitationally bind together nor fusion any more viable than “outside”, but consider it a thought experiment. It would actually be cooler than the universe around it! So, it wouldn’t increase disorder/entropy but serve as a sink of entropy. But this will only be true until the universe around it cools down sufficiently. Once that happens, the star goes back to increasing entropy. So while it locally seems like it decreases entropy, it’s only when you look at it at as an object in space time rather than an ongoing phenomenon.

Now let’s apply the same logic to black holes and neutron stars. Most objects in the universe would suffer heat death far more quickly than these objects. They are ‘more stable’ in that way. If we fast forward enough, most starts will die, quasars will run out of accretion disks and everything turns dark. Then black holes, once the darkest objects in the cosmos, ironically must emerge as the most luminous entities. This paradigm shift occurs due to Hawking radiation, which should become dominant when the “temperature” of black hole as measured by hawking radiation exceeds the “temperature” of cosmic microwave background left behind from the big bang and any other background left behind from now long gone luminous objects. In this era, the radiation from evaporating black holes effectively makes them the most significant sources of “light”, a stark contrast to the universe we currently observe. This is because most stars, including long-lived red dwarfs, would have died eons ago. The expanding universe and other phenomena, such as the transformation and eventual dissolution of galaxies and the life cycles of various types of stars, would have contributed to the increase in entropy. Planets would have gone much sooner. Now what increases disorder in this universe where there seems to be no order at all? Well, our trusty luminous black holes. And they should continue doing so for the rest of the foreseeable age of the universe until everything is torn apart. Ergo, they must be the most entropic, as only they are able to increase the entropy of something that seems like it can’t get any more disorganized.

[1] As suggested by the ‘no hair theorem’, externally we can only observe their mass, rotation, and potentially electric charge. In practice, black holes are likely to be electrically neutral overall due to the nature of matter falling into them. That leaves only mass and rotation.

[2] Though this relationship can vary based on microstate configurations and other thermodynamic factors, I’m opting for simplicity here.

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