Thinking

# Black Holes and Quantum Entanglement

Note to regular readers–apologies for not writing all month. I have been really busy with my research in the mathematical domain. My ongoing work is on the question of the persistence of quantum entanglement around rotating black holes. This is interesting because, first of all, no one understands by what underlying mechanism entanglement works. I outlined it my post on the nature of reality, but let me give a shorter explanation here.

Entanglement for soccer moms

Suppose you have two fair coins. Imagine that every time one comes up heads the other comes up tails, i.e., they are perfectly correlated–even though they still have probability 1/2 of coming up heads individually! This is basically the case of maximal entanglement. Of course, we don’t observe this with coins but that is because of decoherence so that the probability of this happening with coins is vanishingly small. What is crazy is that this actually happens with quantum phenomena like spin, as has been verified experimentally innumerable times. No one knows by what mechanism such coordination takes place so this is a very mysterious phenomena. One would like to understand it better.

Rotating black holes

Theoretically, it’s clear that entanglement persists at arbitrarily large distances in flat spacetime. Might this be true for curved spacetime? This is quite relevant since we quite obviously live in the domain of general relativity (GR). In fact, our GPS devices would be a few hundred yards off if they did not make GR corrections to Newtonian mechanics. Essentially, one wants to know if this works the same way in spacetimes that are exact solutions to Einstein’s equations of general relativity. Mathematically, rotating black holes are just an interesting example of such spacetimes with just enough symmetry to allow for analytical solutions (Crucially, the Dirac equation for spin-1/2 particles separates into purely radial and axial equations which can then be solved explicitly.) [Nerd alert: This has to do with the existence of Killing-Yano tensors, which not only guarantee the separation of variables, they also ensure complete integrability–which means that the number of constants of motion that exist equal the dimension of spacetime. For a freely falling particle these are the rest mass, energy, angular momentum and the surprising fourth first integral called Carter’s constant which comes from the Killing-Yano tensor as well.]

Now, one would like to investigate whether entanglement persists in the extremely curved vicinity of a rotating black hole, maybe with one particle inside the event horizon? The point being that the resolution of each particle’s spin is then independent of the curvature of spacetime (gravity). Or, more interestingly, that it gets entangled with the black hole itself.

Since the spin of a particle couples to the curvature of spacetime, spin-spin entanglement spills over into entanglement of spin and momenta which are both described by the spinor representing the particles. (Entanglement is expressed by both particles having the same wave function which is just a spinor in differential geometry.) A rotating black hole has a very interesting feature. The event horizon is the boundary of the black hole–from which even light, and therefore nothing else (current results about superluminal neutrinos aside) can escape. There is another horizon outside it called a Killing horizon. Between these horizons, in what is called the ergoregion, you have to rotate with the black hole; it takes infinite energy not to. I suspect that this spilling of spin entanglement into spin/momenta entanglement reaches a limit as one hits the Killing horizon and enters the ergoregion. However, this is an open question.

The vicious interior

The interior of a rotating black hole is considered unphysical. Mathematical physicists literally call it vicious, which is a technical term for a region where time travel is possible. In fact, the situation is much worse. One can go from any event–a point in spacetime (t,x,y,z)–to any other event in the interior by going enough number of times around the ring singularity (it is quite literally a time machine). However, the case with one observer inside and one outside is still of purely mathematical interest.

As fascinating is the (mathematical) existence of wormholes. In a maximal extension, one wants to account for the entire history of all photons (light rays or null geodesics). Now Kerr spacetime (an isolated rotating black hole) has a maximal extension with an infinite tower of spacetimes smoothly connected by wormholes.

Information loss

The topic under discussion is of course related to the question of whether information is lost inside black holes. Do we lose the information contained in the internal degrees of freedom of particles that disappear inside a black hole? We have good reason to believe that information contained in any physical system is conserved. Hawking and Thorne had a bet with Preskill and Don Page on this. Hawking conceded the bet in 2004, prematurely in my opinion.

So far

What I understand so far is that one can correct for the curvature of spacetime and recover the entanglement in this regime. It is mathematically nontrivial–a hard and messy exercise in differential geometry, but so far it seems doable. Things are moving quickly and the hope is that we will have an explicit demonstration soon and move on to investigating the ergosphere.

I hope you found this as fascinating as it seems from the trenches.

Back on Earth

In Bahrain, there were sham elections, protests and further brutal crackdown by the US-Saudi backed al Khalifa regime. Saleh is miraculously back in action in Yemen further complicating the situation which has begun to look more and more like a civil war. Anwar al Awlaki, the American born al Qaeda ideologue was killed by a US predator drone strike in Yemen recently. Liberals who care about his citizenship have raised questions about his extra judicial killing even as they were barely done celebrating the extra judicial killing of bin Laden. On the tenth anniversary of 9/11, Americans erupted in predictable triumphalism.

The Syrian National Council was established by dissidents in Syria while it’s key neighbours Turkey and Iran have started pushing against the Assad regime. The US has been reported to prepare for a post-Assad Syria and has decided to let it’s client state, Turkey, manage the transition–as the policy tensor has long expected. The Assad regime just retook Rastan, a strategic town between Hama and Homs. Things are still very fluid. There was even movement in Saudi Arabia where King Abdullah granted women the right to vote men into powerless positions on a toothless body.

The Palestinian authority is in intense negotiations with the Americans and Europeans over getting recognized as a state by the UN. Americans want them to live with the status quo and the Europeans want the Vatican option (observer status without recognition as a sovereign state). The Palestinians are inspired by the Arab spring, which is quite interesting and tells us something about the changing nature of the international discourse. I hope to write about that soon.

Others who have been inspired by the Arab spring are radicals who are occupying Wall St. The protest seem limited but they are spreading across North America. It is probably going to be a dud but it has the potential to be the biggest game changer of them all.

Next up: a review of A Brief History of Neoliberalism by David Harvey. I hope to post it sometime later this week. Stay tuned.

[Update: I apologize for the delay. It turns out that Harvey’s central thesis is more or less based on Duménil and Lévy’s classic Capital resurgent. In fact, I have seen numerous references to this book in the work of Giovanni Arrighi, Noam Chomsky and Kevin Phillips et cetera. I cannot do justice to the review without reading it. The meat of the claim is that the ruling elites went on the warpath in the 1970s and 80s in a bid to restore class power, whence the shift to finance, neoliberalism and the Reagan revolution. As is clear from my writing here, my understanding is closer to Robert Reich’s thesis. Note that we are all in agreement that we live in a regime that is functionally an oligarchy, “rule of owners”. The disagreement is more nuanced. I have a problem with assigning intent to imaginary “ruling groups” when purely structural/institutional/sytemic explanations suffice. But I will try to keep my mind open as I read Duménil and Lévy. Let’s see if they can convince me.]