Off-topic, web: Some arXiv preprints may move converted from PDF to prissy HTML alongside maths using More info.Imagine that you're a grad student, postdoc, or a Milner prize winner who feels that his or her publish of physics projects is express now. I intend that y'all should produce a homework exercise together with write a newspaper – every bit revolutionary a newspaper every bit possible – according to the next sketch.
Analyze the quantization of QFTs together with quantum gravity – or vacua of string theory – on the spacelike, hyperbolic slices inward the Minkowski space\[
x_\mu x^\mu = R^2.
\] If y'all produce it right, y'all should conclude alongside some insights about
- the dark hole complementarity – the refusal of unlike slices to move independent – together with hence the data loss puzzle
- the horizon degrees of liberty together with the Bekenstein-Hawking entropy
You should laid about past times using the coordinate \(R^2 = x_\mu x^\mu\) every bit your time. On these equal-time slices alongside a fixed \(R\), y'all should write your canonical coordinates together with momenta for a quantum field.
Try to define the Hamiltonian every bit the operator increasing the value of \(R\). Note that for unlike \(R\), these slices aren't quite identical – they receive got a unlike curvature. But there's a one-to-one map betwixt the Hilbert spaces defined on these slices. The hyperbolic slices receive got the isometry \(SO(D-1,1)\) – basically the total Lorentz grouping without the Poincaré translations – together with its boost-like generators acquit every bit "momenta" on the hyperbolic slice.
In the context of the dark hole data puzzle, i may define "strange slices" of the curved dark hole spacetimes (and their Penrose diagrams) where some data seems to move doubled. Inside the dark hole, these slices comprise "all the data close the interior" i.e. the thing that has already roughshod to the dark hole. On the other hand, the same space-like hypersurfaces also cross most of the Hawking radiation.
It's been a business office of the lore for a few decades (which has been to a greater extent than oftentimes than non proven qualitatively but it's all the same non understood likewise good "how it actually works") that at these slices, quantum gravity should present some variety of non-locality or dark hole complementarity – the business office of the spell that is within the dark hole shouldn't move quite independent from the spell that is outside. The exterior business office of the spell should qualitatively acquit only similar the slices inward the nearly apartment space. One may naturally gauge that the "strange phenomena" of quantum gravity exclusively accept identify at the interior business office of the spell or the human relationship betwixt the interior business office together with the exterior part.
H5N1 prissy thing close the hyperbolic slices of the apartment infinite that I started alongside is that they're a practiced model of the "long-term" development of some points within the dark hole, at a fixed radius of the orthogonal ii sphere. Just similar y'all postulate to "accelerate away" from the dark hole if y'all desire to rest a meter to a higher identify the trial horizon (and y'all may pass a real long fourth dimension over there, the fourth dimension is comparable to the Hawking evaporation time), y'all postulate to consider parts of a "curved space-like slice" if y'all wanted to rest within the dark hole, at a fixed distance from the horizon. Because this surface is spacelike – banker's complaint that the fourth dimension together with infinite are interchanged within the dark hole, inward some means – a massive object can't actually rest at a fixed place. You postulate the equivalent of the superluminal motion.
Some special phenomena at the hyperbolic spell of the apartment Minkowski infinite should know a lot close the "strange phenomena" of the dark hole interior – inward the same means inward which the Unruh radiations is a simplified toy model for the Hawking radiation.
In particular, if there's some dark hole complementarity – violation of the dominion that spacelike-separated regions are completely independent together with commuting alongside each other – this complementarity should move already visible, inward some simplified way, on the hyperbolic slices of the apartment space. Even though the spell may receive got an infinite proper volume, the amount of data it may shop should move restricted inward some way. The interrogation is why together with how. The curved shape of this space-like hypersurface itself should already enjoin us that unlike parts of the hyperboloid are "less independent" from each other than they are believed to move at a apartment slice.
This brings me to the 2nd point. There exists a prissy quondam declaration of mine – also using hyperbolic slices – why the entropy inward a part is bounded roughly past times the Bekenstein-Hawking entropy \(S=A/4G\). How does it come about that the volume-proportional degrees of liberty are actually absent together with exclusively the surface contributes?
Take a sphere inward the apartment 3D space, a spell of the Minkowski space. You may consider a apartment spell describing the phenomena within this sphere. But y'all may also spell the 4D spacetime past times the hyperbolic slices \(H^3\) alongside a fixed curved shape – similar to the shape of the majority vanquish inward the momentum space. Because most of the hyperbolic spell is "nearly null", it has a real modest proper volume. In fact, when the part of the hyperboloid is much larger than the curved shape radius, the proper book scales like\[
V = H5N1 \cdot R_{\rm curvature}
\] i.e. the production of the proper surface together with the curved shape radius of the hyperboloid. The real analogous scaling exists inward the anti de Sitter infinite together with is a means to heuristically explicate why holography isn't together with so shocking inward the anti de Sitter infinite – the book of the slices scales alongside the surface, anyway. Our storey is only a counterpart of that argument, a counterpart using a purely space-like slice. The commons low-energy description alongside quantum gravity neglected exclusively starts to pause downward in i trial \(R_{\rm curvature}\) approaches the Planck length. In this limit, when y'all assume the simplest density of entropy per unit of measurement proper book – i fleck or nat per proper Planck book – y'all volition larn the total entropy that is of lodge \(S=A/4G\).
So in that location should be some "manifestation of holography" that also applies to the hyperbolic slices of the apartment Minkowski infinite discussed from the outset of this weblog post. The degrees of liberty on such slices should acquit holographically – together with move effectively stored at the boundaries of the hyperboloid. Why is it together with so that the large book of these hyperbolic slices doesn't contribute independent degrees of freedom? The argue must move analogous to the derivation of the thermal Unruh radiations – which is the radiations seen past times observers moving along time-like hyperbolic trajectories inward the apartment spacetime – except that some of the logic must move reversed.
The Unruh thermal radiations may move derived from the human relationship betwixt the ii unlike vacua – solid soil states of the "two unlike Hamiltonians" (the regular fourth dimension translations inward the Minkowski space; together with the fourth dimension translations of the Rindler infinite which are boosts of the Minkowski space). Similarly, on the hyperbolic slices, y'all should define the "regular momentum" every bit good every bit the "Rindler-like momentum" corresponding to the boosts of the Minkowski space. At afterwards times, full general solutions within the hereafter low-cal cone could move written inward damage of some eigenstates of both types of the momentum operators. There should move something similar a Bogoliubov transformation. And that counterpart of the Bogoliubov transformation should relate the descriptions based on the ii slicings – apartment slices together with hyperbolic slices – inward a means that is analogous to the Unruh storey inward the Rindler space. One should run across some many-to-one maps betwixt the natural Hilbert spaces together with derive some seed of the dark hole complementarity which would move the main toy model that is used inward all dark holes.
I believe that the hyperbolic space-like slices comprise many fewer degrees of liberty than expected from locality because the status of a restricted momentum (which is actually a boost generator inward the Minkowski space) constrains the size of the physical Hilbert infinite immensely.
If the storey to a higher identify doesn't aspect consummate to you, it's because it isn't actually consummate at this moment. I receive got to a greater extent than – together with some equations – than what I wrote to a higher identify but what I receive got isn't complete, either. Maybe there's a peril that somebody finds the total story, finds all the "analogous novel things" to the Unruhology that may move derived on the hyperbolic space-like hypersurfaces every bit slices of the Minkowski space. I am talking close some elaboration of a formalism that basically assumes the effective patch theory. At the end, the effective patch theory can't move a consummate description that tells y'all all the details close the microstates inward quantum gravity – y'all postulate the total theory of quantum gravity, a.k.a. string/M-theory, for that. But the qualitative features of that total storey must receive got some interpretation inward damage of the local effective patch theory. The local effective patch theory must vaguely sympathize the grapheme of its ain breakdown.
It would move fun if some or numerous grad students, postdocs, together with Milner Prize winners tried to write a term newspaper clarifying how far they tin give notice brand it analyzing all these situations together with ideas. ;-) At the end, I believe that the edifice blocks may move combined together with so rationally that i may build a proof of the dark hole complementarity together with other novel phenomena inward quantum gravity. Strict together with naive locality we're trained to believe from the Minkowski infinite must move "restricted" inward some means inward quantum gravity at the halt but the reasons together with basic grapheme of this restriction may move understandable inward damage of clever plenty slices together with geometry applied to the effective patch theories, I think.
Incidentally, Michael Dine whom I know good from Santa Cruz etc., also every bit a once-time anti-anthropic co-author, got the latest Sakurai Prize. Well-deserved, congratulations, Mike.