Wednesday, February 25, 2015

Black hole microstates from gluing an exterior with its delayed twin

...and a proof of state-dependence of interior field operators...

Kyriakos Papadodimas (CERN) and Suvrat Raju (Tata) have released a five-page paper that is full of hot ambitious ideas as well as cool, almost rock-solid arguments about the "holographic code" describing the black hole interior:
Local Operators in the Eternal Black Hole
They work with the eternal Schwarzschild black hole in the \(AdS_{d+1}\) space. They describe it using the tortoise coordinate, one that I and Andy Neitzke learned to love when we studied the quasinormal modes. This coordinate makes the \(rt\)-plane look "conformal" and some world sheet methods may therefore become applicable; I would like to comment on this point of mine in more detail later.

At any rate, the eternal \(AdS\) black hole may be holographically described using two conformal field theories, \(CFT\), and an eternal black hole state is a maximally entangled state\[

\ket{\Psi} = \frac{1}{\sqrt{Z(\beta)}} \sum_E \exp(-\beta E / 2) \ket{E,E}

\] The first thing they appreciate is that one may evolve this state in time, by a Hamiltonian (i.e. one may wait), to obtain many inequivalent states \(\ket{\Psi_T}\) that seem to have indistinguishable local physics, however:\[

\ket{\Psi_T} = e^{iH_L T} \ket\Psi = e^{i H_R T} \ket \Psi

\] One either asks the object to "wait" for time \(T\) in the left \(CFT\) only; or in the right \(CFT\) only. In both cases, one gets the same result but the result depends on \(T\) nontrivially.

Now, the eternal black hole should have \(e^S\) microstates and their interpretation for these microstates is simple: just take the "half-delayed" states \(\ket{\Psi_T}\) for all values of \(T\) in a huge interval \(0\leq T \leq \exp(S)\) or so. If you pick some convention for the phases etc. that allows you to define the original \(\ket\Psi\), you may pretty much label the microstate by a value of \(T\) between \(0\) and \(\exp(S)\) or so.

It may seem bizarre to parameterize a microstate by one real number \(t\in \RR\). I believe that there won't necessarily exist canonical values for the phases defining each state \(\ket E\) in the formula for \(\ket\Psi\) and each choice may be as good as any other. So there should be a definition of the microstates that is independent of these phases. I believe that the microstates may be mapped to all possible (antilinear?) \(\ZZ_2\) involutions of a certain kind on the Hilbert space.

Because this is an idea I am adding (even though they emphasize the need for the \(t\to -t\) transformation, so they probably realize all these things as long as the things are right), let me give a heuristic justification referring to some methods that are well-known to string theorists. Calculations on the mixed black hole background given by the density matrix \(\exp(-\beta H)\) use a "thermal circle", a Euclideanized time circle \(S^1\) of circumference \(\beta\). What happens if you replace the mixed state by a pure state \(\ket\Psi\)?

Well, I think that the Euclidean time relevant for the expectation values \(\bra\Psi \O\ket\Psi\) looks like a line interval \(I^1\) and the line interval may be written as \(I^1 = S^1 / \ZZ_2\). There should be many ways to pick the pure state \(\ket\Psi\). Correspondingly, there should be many ways how you can create the line interval, the \(\ZZ_2\) orbifold of the \(S^1\), from the circle \(S^1\). In string theory, such a \(\ZZ_2\) orbifold definition of "spaces with boundary" is known both on the world sheet and in the spacetime (especially in Hořava-Witten's heterotic M-theory).

Six weeks ago when I wrote about the monstrous beer conjecture, I suggested that the black hole microstates may be in one-to-one correspondence with gauge transformations behaving nontrivially on the horizon, i.e. with the volume of some gauge symmetries that may connect the two sides of the eternal black hole. It may look similar to what PR are saying now: they have some diffeomorphisms that behave nontrivially before the two sides of the spacetime are glued together.

However, now I think that the black hole microstates are actually not parameterized by the space of possible monodromies (around the thermal circle). Instead, they are parameterized by the set or space of possible \(\ZZ_2\) involutions of the gauge group. In the future, I want to spend more time on these comments of mine – and try to show how it works (and why one gets the right black hole entropy) in some favorite vacua of string/M-theory (including Matrix theory, perturbative string theory, and the monster-group pure \(AdS_3\) gravity theory).

At any rate, Papadodimas and Raju are careful about the description of the gauge transformations that do change the state or don't change the state, and some deficit coming from "extra gauge transformations" that may be inserted at the horizon (or the locus where the two CFTs are glued together) is responsible for the diversity of the black hole microstates.

Note that four weeks ago, when we talked about a new paper by Polchinski et al., the idea that some gauge-dependent building blocks should be used to construct the field operators in the bulk was exploited, too. Papadodimas and Raju are playing with a similar idea – but in a very different way, technically speaking.

They explicitly construct the interior field operators that are state-dependent, and argue that state-independent operators for the black hole interior can't exist because one would either get some paradoxical predictions from them, e.g. the negative expectation value of \(N\), the number operator for a harmonic oscillator, or a firewall that appears after a long time and ruins the classical expectation that the eternal black hole is really eternal and not changing (even after these exponentially long times).

The number of people in the world who are working on these fundamental aspects of the spacetime is very small. It should be higher. I think that the good graduate students etc. should understand that the state-dependence is consistent and mostly proven to be necessary and they should try to write down explicit formulae for many things that have been discussed by qualitative, verbal language only so far. It may be much more possible to calculate the precise final answers to all deep questions than you may think.

In particular, if microstates of an eternal black hole (and, more generally, two arbitrary regions of spacetime glued along the horizon) are some \(\ZZ_2\) involutions inside \(CFT_L \times CFT_R\), as I was arguing, it should be possible to define and solve this problem in all string/M-theory vacua and explain the universal formula for the black hole entropy and resolve all remaining puzzles linked to the observation in the black hole interior.

Another paper: fine-tuning in some SUSY models may be redefined to be small

Nanopoulos and Du, Li, and Raza talk about the supernatural MSSM. They argue that the right measure of electroweak fine-tuning should be redefined so that it is very modest if all the mass parameters may be linked to a single SUSY-breaking parameter. That's the case of the no-scale supergravity and SUSY models with the Guidice-Masiero mechanism. These models have some extra advantages – but also disadvantages. Concerning the latter, stau seems to be the lightest superpartner. That's bad as an explanation of dark matter because stau isn't dark. To solve this dark matter problem, they introduce an axino and the stau decays to the axino after a millisecond up to a minute.


  1. Seems to me that 2-D CFTs' infinitesimal conformal transformations form infinite-dimensional Witt algebra. Only chiral fields are invariant with respect to the full infinitesimal conformal group. So, Luboš,

    1) How did they destroy spacetime chirality?
    2) Test for spacetime chirality, DOI: 10.5281/zenodo.15107, 10.5281/zenodo.15439

    3) I do not doubt the road apples. I want to see the pony.

  2. I didn't know one can still find that stuff nowadays, but boy, it was fun when I was young.

  3. I see Roger Piece Jr is under investigation now.

  4. Dear Lubos, I don't really understand what is going on here, but to my knowledge local operators can't be defined in quantum gravity. Is this wrong ?

  5. Donbass is too close to Dumbass for me to overlook.

    The Arab arms connection to Ukraine has the smell of the gutless Obama Administration on it.

    Make sure we take every opportunity to quote the dirtbag Pachauri's statement of Religious Beliefs.


  6. Hi Uncle Al, I see you're on about this Eötvös experiment again. Seems you have some kind of strange notion that the vacuum might turn out to violate parity and that we live in chiral world...

    Well good news, because we do, and this has already been an experimentally established fact for more than 50 years!

    The only global involution under which our vacuum is invariant is CPT (product of charge, parity and time reversal symmetries - I'm not talking about SUSY here..)

    They did the experiment - here's your pony!

  7. Sorry meant to read Roger Pielke Jr.

  8. kashyap vasavadaFeb 26, 2015, 2:24:00 AM

    I have a simple question. From all this complicated stuff, can one have any idea how Gravity behaves inside a black hole? In high school classical gravity physics, it would go linearly inside a sphere of uniform density. It cannot possibly be true in a black hole. Does string theory still give singularity at the center?

  9. Probably Luboš will elaborate on this but I'll point out his posts on state-dependence a-la Papadodimas-Raju and its connection with absence of background-independent local operators,

  10. Dear John, they can be and they must be able to be defined approximately, as in effective field theory because we know experimentally that the effective field theory is a valid approximation.

    The right statement - weaker than your bold claim - is that there's not necessarily any exact quantum description of the quantum gravity physical system that would use these local field operators and nothing else.

  11. Dear Kashyap, the laws of physics are always the same, and by the equivalence principle, the black hole interior - a piece of that region - must behave just like any other region of nearly empty space, so we know what must happen there, at least to low-energy probes. It's the same thing that happens outside or anywhere.

    Yes, the same black holes must have the singularity according to string theory, the singularity predicted by classical GR. It has one in the sense that the curvature invariants etc. exceed all reasonable bounds in some region - so that it's guaranteed that the classical GR has to cease to be a good description.

    It doesn't mean tha things are "strictly infinite" over there. Indeed, quantum gravity - string theory - often regulates things to be finite. But it may be infinite and more importantly, the conclusions about the "inevitable death" (of a macroscopic object) in the region (near the singularity) that one derives from classical GR are correct.

  12. I loved the last paragraph (the only one I can intuitively appreciate)! ;-)
    I probably loved it extra much because it made me think that, at this the highest and hardest level of fundamental theoretical physics, the virtuosi of (something like} ~'twisted hyper-dimensional vector spaces with strings thrown in and tying it all together'~ might actually have a serious reason to adopt and apply my apparent psycho-philosophical dual to your "conservation of redundancy of representation", namely my Tolerance Principle[d attitude]. %-}

  13. Ha ha glad you liked it, but you shouldn't take it too seriously, it's a just a vague heuristic that "seems" to pop up in certain situations in Algebra and Model theory and nothing rigorous!

    I have no idea if something like that applies here to possible representations of a CFT and I'm certainly not qualified to speculate on advanced topics in ST/QFT...

    Just an interest layman!

    Cheers, Liam

  14. Imho, If our material universe has at least one symmetric anti-material mirror copy universe,
    ( Max Tegmark: “Is there a copy of you reading this article” ) then parity violation seems to be a must in each universe. Both Higgs fields should have a preferred opposite spiral rotation effect.
    The Higgs field oscillations should perform left- tor right handed spirals.
    See: the tetrahedral chiral vacuum lattice.

  15. Good question about the singularity Kashyap, and, as usual, great answer and link by Lubos.

    I must admit that I have been brainwashed after reading quite a few (usually considered) authoritative books, popular articles and blog posts.

    Brainwashed to assume that any singularity and related infinity of certain physical parameters, like mass density, curvature, etc. implies a crisis and incompleteness of underlying physical theory.

    Then I noticed that Lubos in his articles on arXiv often refers to black holes as massive elementary particles, a kind of a big mass nonextremal electrons (if charged).

    Then it clicked that in QFT we have point particles and corresponding infinite mass density at a point, along with infinite value of Coulomb potential, yet nobody seems too worried about that.

  16. Tony, good observations. Just to avoid confusions, when I was kid, it was also natural for me to believe that "every quantity" has to have a well-defined finite value in a consistent theory.

    The problem with that expectation is that not every quantity has to be confirmed as the "right, well-defined observable" which may be used to formulate the exact laws of physics. Most observables from approximate theories aren't quite well-defined, and there's nothing wrong with that.

    They may be calculated to have naturally infinite values, and that doesn't imply any unavoidable inconsistency because there may exist other observables whose values and evolution is well-defined - often observables that are rather unknown (or look contrived) in the approximate theories.

    And if one is talking about "non-dynamical" parameters of backgrounds, they may be infinite in perfectly consistent theories, too. The singularity at the fixed point of the C^2 / Z_2 orbifold is perfectly singular. It doesn't have to be resolved. But strings and branes propagating on this background give perfectly good finite predictions for everything that may actually be measured.

    And the third general point is that even if quantities are regulated, like the maximum curvature near the black hole singularity in some cases, it doesn't change anything about the singularity's being the end of the world.

    At least for those 3 classes of issues, the idea that a "good theory eliminates all singularities and all their previous major consequences" is invalid.

  17. Yang and Lee are not relevant to gravitation. All manifestations of trace chiral vacuum background selective to matter are curve fit: Parity violations, symmetry breakings, chiral
    anomalies, baryogenesis, biological homochirality, Chern-Simons repair of
    Einstein-Hilbert action.

  18. Italian spelling note: Giudice-Masiero mechanism.
    Thank you.

  19. This is what power and money can do to your post. No matter how corrupt or bad you are, as long as you have power and authority, nothing can
    stop you. Mass Effect