A new conservation law in GR discovered thanks to ER=EPR

An attack turned off the LHC: a terrorist (picture) made it necessary to stop the LHC. Due to a damage of the transformer, there won't be any beam up to next Friday. I still think that weasel words create many more problems than weasels.

I primarily view the ER-EPR correspondence (the equivalence of the non-traversable wormholes and the quantum entanglement) as an important conceptual finding that is directing people's research of the most esoteric, most quantum aspects of quantum gravity – the cutting-edge questions in this most fundamental part of theoretical physics. We learn about some new constraints in the rules that govern the Hilbert spaces in quantum gravity.

However, three Caltech authors, Remmen+Bao+Pollack, just showed a rather cool example of the wisdom that may also flow in the opposite direction:

Entanglement Conservation, ER=EPR, and a New Classical Area Theorem for Wormholes

Because the entanglement is the same thing as the wormhole and there exist some facts we may derive about the entanglement in general, there could be facts that we may derive about the pure simple classical Einstein's general theory of relativity, too.

This result is so pleasing that I immediately forgave these folks their collaboration with Sean Carroll in the past. Whoever co-authors a paper such as this one earns enough scientific capital to write three nutty papers e.g. about the Boltzmann Brains.




Juan Maldacena (with his nominal co-author Leonard Susskind) has already presented many such interpretations of various facts on both sides of the ER-EPR correspondence.




But Remmen et al. looked at one fact a bit more carefully – and they have actually derived a new, previously unknown theorem in classical GR using the wisdom of ER=EPR! What is it?

Well, it's simple. Under the unitary evolution that acts on the two subsystems separately (no measurements are allowed!), the entanglement entropy between two subsystems is conserved. Note that the entanglement entropy is nothing else than the von Neumann entropy of the reduced density matrix of the left subsystem:\[

S(L) = -{\rm Tr}_L (\rho_L \log \rho_L), \quad \rho_L = {\rm Tr}_R(\rho_{L+R})

\] A fun thing is that at least for a pure state of the composite system, \(\rho_{L+R}=\ket\psi\bra\psi\), this \(S(L)\) is the same as \(S(R)\) defined with the \(L\leftrightarrow R\) interchanged.

It's not hard to see that the separated unitary evolution of the two subsystems preserves the entanglement entropy. We just write the original entangled state \(\ket\psi\) as a sum of tensor products of vectors from the two Hilbert spaces. The entanglement entropy only depends on the coefficients of these terms but they're unchanged because the unitary evolution (of both parts) only "rotates" the ket vectors in the individual terms.

This decomposition also makes it obvious that it doesn't matter whether you compute the von Neumann entropy of the left or the right subsystem: the entanglement entropy depends on the (absolute values of the) coefficients in the terms where the left and right ket vectors enter "symmetrically"

Great. So the entanglement entropy is preserved under the unitary evolution. But the entanglement entropy has a clear geometric interpretation following from ER=EPR:\[

S = \frac{A}{4G}

\] where \(A\) is the area of a relevant surface inside the wormhole. Only when \(S\gg 1\), the problem in "quantum gravity" is well approximated by the classical GR. OK, this \(S\) is conserved under the separate evolution, so \(A\) must be conserved, too. It's more or less obvious if you just have a boring single wormhole that sits there.

But as the folks noticed, you may consider a complicated network of black holes and wormholes that overlap, interact, and connect two different regions in space. Imagine something similar to the usual pictures of the quantum foam. The time evolution will be complicated but there will still be a definition of some area \(A\) that is conserved in this complicated evolution.

The whole trick allowing them to find something non-obvious is that on the entanglement side of ER=EPR, the two subsystems are separated in a "clear way": we know what the two Hilbert spaces are. But on the geometric, GR-based ER side of the duality, the two regions aren't quite separated. They are connected into one spacetime geometry by the wormholes. This is why the conservation of some "area of the tunnels" isn't obvious. Similar tricks involving other dualities, e.g. the mirror symmetry, have allowed other people to solve some problems that seemed difficult – but that dramatically simplified in the dual description (e.g. counting curves on Calabi-Yau manifolds).

They define the relevant area, the "maximin" area (the term has previously appeared in the literature, e.g. in a paper by Wall, and the word is the "opposite" of "minimax"), and then they prove the theorem using tools of GR only (I hope). The definition of the "maximin" area appears already at the top of Page 5:

We define \(C[H, \Gamma]\) to be the codimension-two surface of minimal area homologous to \(H\) anchored to \(\partial H\) that lies on any complete achronal (i.e., spacelike or null) slice \(\Gamma\). Note that \(C[H, \Gamma]\) can refer to any minimal area surface that exists on \(\Gamma\). Next, the maximin surface \(C[H]\) is defined as any of the \(C[H,\Gamma]\) with the largest area when optimized over all achronal surfaces \(\Gamma\). When multiple such candidate maximin surfaces exist, we refine the definition of \(C[H]\) to mean any such surface that is a local maximum as a functional over achronal surfaces \(\Gamma\).

Achronal surfaces (or sets) are surfaces not containing any points \(q,r\) such that the latter is in the causal future of the former, which basically means "null or spacelike surfaces". OK, at any rate, there's some rule using some maximization and minimization and causal jargon in general relativity and when you go through the things, you may derive a theorem about an area that is conserved despite the complicated interactions and overlaps between many wormholes. Horizon mergers and the absorption of classical matter (obeying the null energy condition) is allowed.

Is there some deeper lesson to be learned? Does general relativity allow many such theorems that aren't obvious? Well, I would like to know.

Note that the direction of the insights is sort of opposite than one in the recent Strominger et al. BMS minirevolution. Strominger and pals want to find some previously unknown (although they were mostly known) laws in classical GR that allow them to solve the information loss puzzles etc. in new ways. I have had some doubts about that from the beginning especially because the "quantum side" of all these dualities is more reliable – you can be more certain about the laws that apparently follow from it. So it's just strange when we are expected to change our opinions about "how the quantum mechanical theory works" by some classical observations.

New insights found in GR thanks to the "dual quantum description" that may be used as a guide – results such as this new paper – seem more sensible to me.