The Hawking Paradox: Resolved
Hawking radiation, the information paradox, and the no-hair theorem
In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition. The work appears to resolve a paradox that Stephen Hawking first described five decades ago.
In a series of breakthrough papers, theoretical physicists have come tantalizingly close to resolving the black hole information paradox that has entranced them for nearly 50 years. Information, they now say with confidence, does escape a black hole. If you jump into one, you will not be gone for good. Particle by particle, the information needed to reconstitute your body will reemerge. Most physicists have long assumed it would; that was the upshot of string theory, their leading candidate for a unified theory of nature. But the new calculations, though inspired by string theory, stand on their own, with nary a string in sight. Information gets out through the workings of gravity itself — just ordinary gravity with a single layer of quantum effects. All’s good with this but what even is the Hawking Paradox and why was it seemingly unsolvable till now?
Well, Hawking’s paradox rises from the ashes of a failed attempt to unify quantum mechanics — the discipline of incredibly small things — and general relativity — the discipline of massively large things. These two theories, although remarkably accurate in their own domains, unfortunately, do not play well with each other.
In the 1970s Stephen Hawking found that an isolated black hole would emit radiation at a temperature controlled by its mass, charge, and angular momentum. Hawking also argued that the details of the radiation would be independent of the initial state of the black hole. If so, this would allow physical information to permanently disappear in a black hole, allowing many physical states to evolve into the same state. However, this violates a core precept of both classical and quantum physics — that, in principle, the state of a system at one point in time should determine its value at any other point. In other words, it violates determinism. Specifically, in quantum mechanics, the state of a system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future. Though this seems to be an issue with Hawking’s paradox.
As you probably have already made out, Hawking’s paradox concerns itself with black holes. Black holes are bodies with incredibly high density —so high that the center converges to a singularity.
Intuitively, we understand that greater density would mean a more profound curvature of spacetime. Now black holes are objects that have crossed a very crucial threshold of density — the curvature of spacetime is so profound that not even light escapes it. Any information can only travel inwards once it crosses the event horizon. And because nothing escapes a black hole, we have no way of knowing what is beyond the event horizon — information is lost.
The conservation of information is quantum unitarity, the law that the quantum mechanical wavefunction always evolves coherently, no pure state ever turns into a mixed state. These are clean google terms. The classical analog is Liouville’s theorem.
What we understand is that information can be created but never destroyed. This doesn’t play well with black holes. So, we’re met with a paradox.
In fact, general relativity extends itself to establish a constraint to the information we can know about any black hole: mass, charge, angular momentum, and radius. This is pretty much everything fundamental we can know about a black hole. Other quantities like entropy, the outer stationary limit surface, and the innermost stable circular orbit are quantities that are derived from this. However, the information in a black hole is lost forever. This didn’t sit right for many scientists. What does it mean for information to be destroyed? Is information conserved? If it is, how does that work with black holes?
The No-Hair Theorem
In general relativity, strong gravity can warp the geometry of spacetime so much that black holes are formed. The interior of a black hole, where curvature becomes infinite, is an extremely complex configuration that defies our current theories. But according to general relativity, these complexities might be hidden to observers outside the black hole’s horizon. The no-hair theorem states that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum. All other information (for which “hair” is a metaphor) about the matter that formed a black hole or is falling into it “disappears” behind the black-hole event horizon and is therefore permanently inaccessible to external observers.
As such we come to the conclusion that every isolated unstable black hole decays rapidly to a stable black hole; and (absent quantum fluctuations) stable black holes can be completely described in a Cartesian coordinate system at any moment in time by these eleven numbers:
- Mass (M. Energy, if you want to call it that)
- Linear momentum (three components: Px, Py, Pz)
- Angular momentum (three components: Jx, Jy, Jz)
- Position (x, y, z)
- Electric charge (Q)
These numbers represent the conserved attributes of an object which can be determined from a distance by examining its gravitational and electromagnetic fields. All other variations in the black hole will either escape to infinity or be swallowed up by the black hole. Cleverly enough, by changing the reference frame, one can set the linear momentum and position to zero and orient the spin angular momentum along the positive z-axis. This eliminates eight of the eleven numbers, leaving three which are independent of the reference frame: mass, the magnitude of angular momentum, and electric charge. Thus any black hole that has been isolated for a significant period of time can be described by the Kerr–Newman metric in an appropriately chosen reference frame. The LIGO results provide some experimental evidence consistent with the uniqueness of the no-hair theorem. So we’ve good reason to believe that the no-hair theorem works as intended.
Hairy Black Holes?
We say black holes are not “hairy” because we don’t have any extra information that can tell us about what happens inside a black hole. Something like this:
What we’re saying here is a little profound. We’re saying that, according to general relativity, it doesn’t matter how a black hole was formed or what objects fell into it in order to give it the mass it has. If two black holes have the same mass, charge, momentum, and the other bits of information we’re allowed to know, then they’re identical to each other — doesn’t matter how they came into existence.
In other words, if two black holes have the same mass, they’re indistinguishable from each other. We can’t ever find out their origin. But there’s a bigger problem here.
Enter: Hawking radiation
If we account for certain aspects of relativity and quantum mechanics, it quickly becomes apparent that black holes must emit some thermal radiation. These particles are generated a little bit outside the event horizon. Meaning this doesn’t violate our principle from earlier — particles don’t actually come from within. Now the mechanism involved here stems from the principle of antimatter-matter annihilation. At any given instant, pairs of matter and antimatter particles are popping into and out of existence. They’re incredibly unstable and annihilate each other almost instantaneously. At the edge of the event horizon, the same phenomenon occurs. Instead this time, one of the particles in the antimatter-matter pair falls “into” the black hole. In other words, it gets sucked in. It’s well known that these particles come in pairs. So for one of them to fall into a black hole, would mean that the other one has nothing to annihilate with. Each antimatter particle annihilates with its corresponding matter particle. And since now matter cannot be created, the black hole must lose the mass equivalent to that of the particle it’s eaten. On unimaginable time scales, this results in black hole evaporation — the black hole shrinks due to the radiation which it emits. This is called Hawking radiation:
Hawking radiation is the thermal radiation predicted to be spontaneously emitted by black holes. It arises from the steady conversion of quantum vacuum fluctuations into pairs of particles, one of which escaping at infinity while the other is trapped inside the black hole horizon.
Now the issue here is that Hawking radiation is independent of what goes into the black hole. In other words, for black holes that are identical, the radiation emitted will be the same. This tells us virtually nothing about the matter trapped beyond the event horizon. So although Hawking radiation is successful at explaining how information is not lost, it struggles to explain why information cannot be retrieved.
But things don’t just stop there. Since black holes lose mass through radiation, they must die eventually — go out in an underwhelming cosmic end. Their death would mean the permanent loss of all the information that went in. Because at some point, the Hawking radiation would stop accounting for the total infalling mass. This doesn’t work well with quantum mechanics.
Now the following description here is an oversimplification of where we stand today but I believe it captures the essence of it all. Everything I’ve said so far implies that information can be completely lost with the death of a black hole. Most of modern physics disagrees with this. And this is the paradox. For an incredibly long time, it was suspected that to resolve this paradox, we’d need a new branch of physics. There wasn’t a satisfactory answer in either general relativity or quantum mechanics. Recently, however, we seem to have discovered a potential solution.
Two caveats: first, it was long thought that the solution would lie in some combination of general relativity and some theory of quantum gravity. This quantum gravity theory, if correct, would not only fix the Hawking paradox but would also allow us to unify physics. We’d have a theory of everything. Secondly, this relatively popular bit of research is just one possible explanation for the Hawking paradox.
With that said, this new explanation essentially studies the gravitational field generated by a black hole incredibly far away from it. In our fundamental theory of gravity, its strength never really becomes zero. It decreases by a factor of 1/r² but there’s always a non-zero value:
More interestingly, the explanation we’re talking about studies gravitons — theoretical particles postulated to be the bosonic force carriers of gravity. These gravitons are thought to be the particles for a g-field just as how photons are particles for an electromagnetic field. As well as this, the explanation considers the black hole’s wave function evolution rather than treating it as a large general relativistic mass. Again an incredibly large oversimplification but the explanation actually calculated the gravitational field strength at any point away from a black hole actually depends on the quantum state in which the black hole is in. This is directly related to the information which we wouldn’t otherwise have access to. Based on our gravitational field measurements, we can verify the accuracy of this explanation and potentially have a solution to the information paradox. In other words, the black hole has hair — quantum hair but hair nonetheless. Now of course, there’s a lot more to this paper that I haven’t gone into and a lot more’s still there that I don’t even understand yet but hopefully this article has given you an insight into the Hawking paradox. Anyway, with that, I’m going to end this here. And as always, thank you for reading!
