Imagine that you're a grad student, postdoc, or a Milner prize winner who feels that his or her number of physics projects is limited now. I think that you should do a homework exercise and write a paper – as revolutionary a paper as possible – according to the following sketch.Off-topic, web:Some arXiv preprints may be converted from PDF to nice HTML with maths using arXiv Vanity. More info.

Analyze the quantization of QFTs and quantum gravity – or vacua of string theory – on the spacelike, hyperbolic slices in the Minkowski space\[

x_\mu x^\mu = R^2.

\] If you do it right, you should conclude with some insights about

- the black hole complementarity – the refusal of different slices to be independent – and therefore the information loss puzzle
- the horizon degrees of freedom and the Bekenstein-Hawking entropy

You should start by using the coordinate \(R^2 = x_\mu x^\mu\) as your time. On these equal-time slices with a fixed \(R\), you should write your canonical coordinates and momenta for a quantum field.

Try to define the Hamiltonian as the operator increasing the value of \(R\). Note that for different \(R\), these slices aren't quite identical – they have a different curvature. But there's a one-to-one map between the Hilbert spaces defined on these slices. The hyperbolic slices have the isometry \(SO(D-1,1)\) – basically the full Lorentz group without the Poincaré translations – and its boost-like generators behave as "momenta" on the hyperbolic slice.

In the context of the black hole information puzzle, one may define "strange slices" of the curved black hole spacetimes (and their Penrose diagrams) where some information seems to be doubled. Inside the black hole, these slices contain "all the information about the interior" i.e. the matter that has already fell to the black hole. On the other hand, the same space-like hypersurfaces also cross most of the Hawking radiation.

It's been a part of the lore for a few decades (which has been mostly proven qualitatively but it's still not understood too well "how it really works") that at these slices, quantum gravity should exhibit some kind of non-locality or black hole complementarity – the part of the slice that is inside the black hole shouldn't be quite independent from the slice that is outside. The exterior part of the slice should qualitatively behave just like the slices in the nearly flat space. One may naturally guess that the "strange phenomena" of quantum gravity only take place at the interior part of the slice or the relationship between the interior part and the exterior part.

A nice thing about the hyperbolic slices of the flat space that I started with is that they're a good model of the "long-term" evolution of some points inside the black hole, at a fixed radius of the orthogonal two sphere. Just like you need to "accelerate away" from the black hole if you want to stay a meter above the event horizon (and you may spend a very long time over there, the time is comparable to the Hawking evaporation time), you need to consider parts of a "curved space-like slice" if you wanted to stay inside the black hole, at a fixed distance from the horizon. Because this surface is spacelike – note that the time and space are interchanged inside the black hole, in some way – a massive object can't really stay at a fixed place. You need the equivalent of the superluminal motion.

**Some special phenomena at the hyperbolic slice of the flat Minkowski space should know a lot about the "strange phenomena" of the black hole interior – in the same way in which the Unruh radiation is a simplified toy model for the Hawking radiation.**

In particular, if there's some black hole complementarity – violation of the rule that spacelike-separated regions are completely independent and commuting with each other – this complementarity should be already visible, in some simplified way, on the hyperbolic slices of the flat space. Even though the slice may have an infinite proper volume, the amount of information it may store should be restricted in some way. The question is why and how. The curvature of this space-like hypersurface itself should already tell us that different parts of the hyperboloid are "less independent" from each other than they are believed to be at a flat slice.

This brings me to the second point. There exists a nice old argument of mine – also using hyperbolic slices – why the entropy in a region is bounded roughly by the Bekenstein-Hawking entropy \(S=A/4G\). How does it happen that the volume-proportional degrees of freedom are really absent and only the surface contributes?

Take a sphere in the flat 3D space, a slice of the Minkowski space. You may consider a flat slice describing the phenomena inside this sphere. But you may also slice the 4D spacetime by the hyperbolic slices \(H^3\) with a fixed curvature – similar to the shape of the mass shell in the momentum space. Because most of the hyperbolic slice is "nearly null", it has a very small proper volume. In fact, when the region of the hyperboloid is much larger than the curvature radius, the proper volume scales like\[

V = A \cdot R_{\rm curvature}

\] i.e. the product of the proper surface and the curvature radius of the hyperboloid. The very analogous scaling exists in the anti de Sitter space and is a way to heuristically explain why holography isn't so shocking in the anti de Sitter space – the volume of the slices scales with the surface, anyway. Our story is just a counterpart of that argument, a counterpart using a purely space-like slice. The usual low-energy description with quantum gravity neglected only starts to break down once \(R_{\rm curvature}\) approaches the Planck length. In this limit, when you assume the simplest density of entropy per unit proper volume – one bit or nat per proper Planck volume – you will get the total entropy that is of order \(S=A/4G\).

So there should exist some "manifestation of holography" that also applies to the hyperbolic slices of the flat Minkowski space discussed from the beginning of this blog post. The degrees of freedom on such slices should behave holographically – and be effectively stored at the boundaries of the hyperboloid. Why is it so that the big volume of these hyperbolic slices doesn't contribute independent degrees of freedom? The reason must be analogous to the derivation of the thermal Unruh radiation – which is the radiation seen by observers moving along

*time-like*hyperbolic trajectories in the flat spacetime – except that some of the logic must be reversed.

The Unruh thermal radiation may be derived from the relationship between the two different vacua – ground states of the "two different Hamiltonians" (the regular time translations in the Minkowski space; and the time translations of the Rindler space which are boosts of the Minkowski space). Similarly, on the hyperbolic slices, you should define the "regular momentum" as well as the "Rindler-like momentum" corresponding to the boosts of the Minkowski space.

At later times, general solutions inside the future light cone could be written in terms of some eigenstates of both types of the momentum operators. There should be something like a Bogoliubov transformation. And that counterpart of the Bogoliubov transformation should relate the descriptions based on the two slicings – flat slices and hyperbolic slices – in a way that is analogous to the Unruh story in the Rindler space. One should see some many-to-one maps between the natural Hilbert spaces and derive some seed of the black hole complementarity which would be the master toy model that is used in all black holes.

**I believe that the hyperbolic space-like slices contain many fewer degrees of freedom than expected from locality because the condition of a restricted momentum (which is really a boost generator in the Minkowski space) constrains the size of the physical Hilbert space immensely.**

If the story above doesn't look complete to you, it's because it isn't really complete at this moment. I have more – and some equations – than what I wrote above but what I have isn't complete, either. Maybe there's a chance that someone finds the full story, finds all the "analogous new things" to the Unruhology that may be derived on the hyperbolic space-like hypersurfaces as slices of the Minkowski space. I am talking about some elaboration of a formalism that basically assumes the effective field theory. At the end, the effective field theory can't be a complete description that tells you all the details about the microstates in quantum gravity – you need the full theory of quantum gravity, a.k.a. string/M-theory, for that. But the qualitative features of that full story must have some

*interpretation*in terms of the local effective field theory. The local effective field theory must vaguely understand the character of its own breakdown.

It would be fun if some or numerous grad students, postdocs, and Milner Prize winners tried to write a term paper clarifying how far they can get in analyzing all these situations and ideas. ;-) At the end, I believe that the building blocks may be combined so rationally that one may construct a

*proof*of the black hole complementarity and other novel phenomena in quantum gravity. Strict and naive locality we're trained to believe from the Minkowski space must be "restricted" in some way in quantum gravity at the end but the reasons and basic character of this restriction may be understandable in terms of clever enough slices and geometry applied to the effective field theories, I think.

Incidentally, Michael Dine whom I know well from Santa Cruz etc., also as a once-time anti-anthropic co-author, got the latest Sakurai Prize. Well-deserved, congratulations, Mike.

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