Friday, May 09, 2014

Two very different PR, ER-EPR papers

I thought that the acronyms are sort of funny

Two contrasting papers on the Papadodimas-Raju theory of the black hole interior and the Maldacena-Susskind ER-EPR (Einstein-Rosen/Einstein-Podolsky-Rosen) correspondence have been posted to the hep-th arXiv today:
Daniel Harlow (Princeton): Aspects of the Papadodimas-Raju [PR] Proposal for the Black Hole Interior

Kristan Jensen, Andreas Karch, Brandon Robinson (Seattle+SUNY): The holographic dual of a Hawking pair has a wormhole
(TRF guest blogger) Andreas Karch et al. takes the positive attitude that the newest picture, ER-EPR, works, while Harlow sees a serious problem with another, related yet inequivalent, picture of the black hole interior, the Papadodimas-Raju (PR) theory.

PR and ER-EPR are not quite equivalent although the acronyms may be combined to a rather nice triangle. ;-) But I think that they are ultimately compatible. Both of them are right and complementary. PR tells us something about the freedom we have when we extrapolate dynamics of quantum gravity into the black hole interior; ER-EPR teaches us about the behavior of the Hilbert space in topologically nontrivial situations with an ER bridge (not really studied by PR at all, at least so far). Harlow's paper is against PR; Andreas Karch et al. argue in favor of ER-EPR maximally positively – so positively, in fact, that they (rightfully?) demand a part of the credit for the streamlined interpretation of the ER-EPR correspondence. But even though they focus on "different theories" of the black hole interior, you may see that PR and ER-EPR are "allies" because pretty much the same point that causes so much discomfort to Daniel Harlow is also what Andreas Karch et al. – which will be referred to as Jensen et al., to respect the alphabetical order – embrace so enthusiastically!

What is the point that may describe the "key" to both papers? I think that the point says that PR and ER-EPR may be interpreted as new dualities that allow several equally correct descriptions of the same degrees of freedom and their physical fate. Andreas Karch et al. are big friends of dualities so like your humble correspondent, they think it's cool to discover a new important duality and it's perfectly consistent.

On the other hand, Daniel Harlow seems to have a psychological problem with the very notion that there may be different ways to describe the same Hilbert space.

Let me clarify what I mean a little bit. I have to assume that the reader is mostly familiar with the basics of the ER-EPR correspondence and the PR picture of the black hole interior that have been discussed many times on this blog.

The new Jensen-Karch-Robinson paper builds on the Karch-Jensen followup to the ER-EPR correspondence in which Karch and Jensen argued that the world sheet of the QCD-like quark-antiquark string in the AdS/CFT correspondence has an Einstein-Rosen bridge on it, too. Because the quark and antiquark are entangled, the observations seem to confirm the picture that the entanglement is equivalent to a wormhole.

OK, the new Jensen-Karch-Robinson paper first "extends" the AdS/CFT picture to a Randall-Sundrum model. In the CFT language, this is represented by coupling the quantum field theory to gravity; in the AdS bulk description, the extension looks like a brane ending/cutting the bulk geometry. Things still work. The other point that they make is that the entangled Hawking particle pairs may be seen to be entangled; and may also be seen to be two sides of a wormhole, thus producing another confirmation of the ER-EPR paradigm.

If you're interested in their paper, you should read it and I won't try to reproduce everything they say. But I want to add a few more words of mine about a general philosophical theme that Jensen, Karch, and Robinson describe in their paper and that I fully agree with.

They want the ER-EPR link to be treated as another duality. Some degrees of freedom may be either thought of as living in two isolated places of the spacetime and being entangled; or they may be thought of as being geometrically connected by a wormhole. Moreover, the entanglement-like (EPR) description is more typical in the CFT side of the AdS/CFT correspondence while the wormhole-like (ER) description arises if you consider the bulk, too. In this sense, ER-EPR applied to states in the AdS space may be considered a special case case of the AdS/CFT correspondence, with the identification ER=AdS, EPR=CFT.

I've been defending the same interpretation of the ER-EPR correspondence as another duality for quite some time, namely from the birth of ER-EPR; see e.g. this October 2013 text, especially the section "Possibility to fix the gauge (light-cone gauge) in string theory & allowed topology change & ER-EPR". The point is that in a consistent theory of quantum gravity, the spacetime topology is not only dynamical but it even fails to be a "good quantum number". There is no operator that would count some invariants describing the topology (e.g. the number of Einstein-Rosen bridges in the spacetime). Instead, the same states in the Hilbert space may be built as excitations of a background with no bridges; one bridge; or many bridges. Some of these descriptions of the states may be "more weakly coupled" or "smoother" or "less excited", and therefore "less awkward", but in principle, all the descriptions – descriptions assigning the spacetime with any topology – are allowed.

This "new kind of a duality" seems very profound to me and I think that Jensen, Karch, and Robinson have the same feelings. There is no inconsistency in this paradigm. Incidentally, the first sentences of their new paper say that "entanglement is [just] correlation but the entangled states can't be measured by themselves". It's exactly something that's been claimed in many older TRF blog posts.

Daniel Harlow's dissatisfaction

On the other hand, Daniel Harlow writes something about the \(1/N\) corrections in the PR picture – the general theory of the black hole interior that he reviews – and sort of agrees with PR that it works and has the potential to resolve all puzzles that were promoted in the context of the firewall paradox etc. However, the main point of Harlow's paper is some skepticism. The PR picture is "ambiguous", he thinks. Surprisingly, the people's disagreements about the "interpretation" of quantum mechanics began to play a prominent role in papers such as Harlow's. Until the middle 1990s, the epoch of the Duality Revolution, people would apparently agree what quantum mechanics was. So no stringy paper would ever talk about "interpretations of quantum mechanics". Only confused people were doing so. Suddenly, this business has penetrated to the quantum gravity research and it is turning out that at least someone must be understanding what quantum mechanics actually is.

The first skeptical sentence in the abstract of Harlow's paper says:
I argue however that the proposal has the uncomfortable property that states in the CFT Hilbert space do not have definite physical interpretations, unlike in ordinary quantum mechanics.
This sentiment of the author is elaborated upon in Section 3, "Do States Have Unique Interpretations?" on pages 15-18. Harlow is troubled by the fact that a black hole microstate \(\ket \chi\) may be written as\[

\ket\chi = A_\alpha \ket\psi = A_\beta \ket{\psi'}

\] for some equilibrium states \(\ket\psi, \ket{\psi'}\) and \(A_\alpha,A_\beta\in {\mathcal A}\) in various ways. Well, it's just a matter of trivial linear algebra that a vector in the Hilbert space may be obtained as an action by an operator on another state in infinitely many ways. It's true for every quantum mechanical system. What's causing discomfort to Daniel Harlow is clearly his assumption that the question "which equilibrium state or vacuum background is associated with \(\ket\chi\)" should be a well-posed question connected with a good quantum number, with some Hermitian operator(s). In other words, he assumes that it should be possible to measure "the equilibrium state \(\ket\psi\) upon which \(\ket\chi\) is built" by a measurement process.

He's wrong. He could figure out that he's wrong if he actually tried to defend his (indefensible) position, something he didn't bother to do – by describing what the hypothetical process that "measures the background or 'the' equilibrium state underlying a general state" actually is. He would find out that no such experimental algorithm exists and none may exist.

Even though we're talking about PR and not ER-EPR here, you see that it is pretty much the same question as the question discussed above, in the context of the Jensen et al. paper. Their – and my – answer is obviously that "the right equilibrium state" underlying a state \(\ket\chi\) isn't uniquely associated with \(\ket\chi\), and it's a good thing. Whenever we have a duality, and the ER-EPR correspondence is a duality, in fact, in some sense, a manifestation of the AdS/CFT holographic duality (at least some special cases of AdS/CFT and special cases of ER-EPR are the same thing), then we also have many ways to imagine "by what operators" we had to excite a ground state to get to the required state \(\ket\chi\), and so on. In a general field theory (it's enough to imagine a QFT on a curved spacetime background; or a QFT with moduli), there is no single "universally preferred ground state" that should serve as the basis to build the rest of the Hilbert space. A ground state – or, in this case, an equilibrium state – is non-unique. All states in the Hilbert space are actually equally real and equally allowed; that's what the linearity postulate of quantum mechanics demands. One description in the case of S-dualities may describe the same state as an excitation of a D-brane added upon a background; the other, S-dual description may see it as a complicated state of many fundamental strings, and so on. The ground state of one Hamiltonian may be obtained by a Bogoliubov transformation i.e. by an action of some creation operators on the ground state of another Hamiltonian (particle production). None of the descriptions is better than the other(s). They're exactly equivalent.

To localize the problem with Harlow's reasoning from a slightly different angle, let me copy some sentences from page 18:
We thus appear to have found a problem for the PR proposal; what is the bulk interpretation of the state \(V\ket\psi\)? Is the horizon excited or is it not?
The bug of Harlow's complaint is that a valid, predictive quantum theory isn't supposed to answer such unphysical questions. A valid, predictive quantum theory is supposed to (probabilistically) predict the results of measurements that may actually be done, and the quote above doesn't describe one!

In other words, and I have been explaining the very same point in dozens of blog posts about the foundations of quantum mechanics, a quantum mechanical theory (probabilistically) answers questions about observables, but it doesn't tell us what the questions should be. There may be different questions – about different observables building the Hilbert space by their action on different ground states or equilibrium states – and once we formulate the question well enough, we may find the corresponding operators in the quantum theory and the quantum mechanical apparatus allows us to calculate the probabilities of different answers according to the Born rule.

But there can't be a universal recipe by which a quantum mechanical theory would tell us which questions we should be asking! The existence of dualities – S-dualities, T-dualities, U-dualities, mirror symmetry, AdS/CFT correspondence, the identification of a 2nd quantized background with a coherent state of gravitons etc. on another background, but even the ER-EPR correspondence, or ambiguities by which we may assign an equilibrium state to a state \(\ket\chi\) in the PR proposal – tells us that there may be different sets of observables, or field operators on different backgrounds, that are equally potent tools to describe a given Hilbert space and its dynamics.

These dualities are not even some "revolutionary additions" made by the stringy physics. The most ordinary quantum-mechanical models have this feature, too. The very fact that there exist operators that don't commute with others guarantees that there is some freedom in the choice of the questions. In non-relativistic quantum mechanics, a wave function \(\ket\psi\) may be written as a function in the position representation or one in the momentum representation. They are Fourier transforms of one another. You may measure \(x\); but you may measure \(p\), too. You can't do both at the same moment. Quantum mechanics doesn't tell you that you should always measure \(x\). It doesn't tell you that you should always measure \(p\), either. There are no fundamentally privileged observables (Hermitian operators) acting on the Hilbert space! Again, it's a point explained in dozens of older TRF posts.

Dualities in string theory only differ from the \(x\)-\(p\) complementarity in the simplest models of quantum mechanics by having numerous, much richer systems of (competing) observables that are describing a much more complicated system. But it's always true that a state may be interpreted in many ways. Even in simple QM models, a state may be interpreted as a superposition of \(x\) eigenstates, or a superposition of \(p\) eigenstates. Or using one of infinitely many other decompositions. None of these decompositions is fundamentally better than the others! This is nothing else than saying that there are many questions that may be asked and the knowledge of the state vector provides us with the answer. In other words, Harlow's words, there are many "interpretations" of the state vector.

I believe that to disagree with this trivial point means to deny the basic postulates of quantum mechanics that have been known for almost 90 years.

In the case of the black hole interior, if an observer is able to parameterize the Hilbert space as a space of fields and objects living on a background geometry and if she can define and calculate the Heisenberg equations of motion for the operators, how they evolve towards the black hole interior, she may probabilistically predict her perceptions in the black hole interior. So yes, she may survive unless something "mundane" kills her. There is no firewall. But one may also get lost in the sea of some completely different operators whose values become singular and quantum mechanics will imply that they're singular, indeed. The absence of the firewall only means – and PR show – that the quantum gravity Hilbert space with all the known constraints allows the life and business-as-usual predictions to continue for an observer that has just entered the black hole interior (unless she wants to measure too complicated correlators with too many excitations, and so on). If she chooses to ask different questions than the questions about observables that are Heisenberg continuations of those from the black hole exterior, there will be different (and possibly singular) answers. If she chooses to ask no questions, there will be no answers. Her survival subjectively (or from the viewpoint of another infalling observer) means that she (or he) is able to continue to ask questions about local fields (including fields inside her body) and QM keeps on answering them in a way that was similar outside the black hole. That's it. We may also talk about her survival from the viewpoint of observers outside the black hole. The answers to all these questions is that nothing inside the event horizon is measurable and the ultimate state is a nearly thermal state of the Hawking radiation into which the infalling observer was transformed (apparently in a process including the death). The internal and external observers are allowed to ask different questions and they are actually doing so. It's no contradiction. A priori, all questions in a quantum mechanical framework are subjective. It's important to specify a question – an operator etc. – before quantum mechanics is supposed to produce the answers.

Harlow also has trouble with the state-dependence of the operators in the PR approach. It is really the very same problem as one above but I don't want to reformulate all these things in yet another language, especially because I have explained why the state-dependence is OK, consistent, and unavoidable in the quantum physics of black holes in August 2013.

I would say that Daniel Harlow who has written a dozen of papers related to quantum gravity isn't a string theorist and this vice seems to be correlated with the confusions about quantum mechanics. It doesn't mean that string theorists never write papers building on confused and deeply deluded "interpretations" of quantum mechanics which are drowning in the popular misleading buzzwords (the word "measurement" or a variation of it appears 45 times in Harlow's paper! All Harlow's comments about "the measurement's being given by a unitary non-Schrödinger transformation" are wrong; the measurement is no real process and if there were a nontrivial one, it would never be unitary). After all, Lenny Susskind, a co-father and giant of string theory, has done it several times, too. But I still think that the degree of acceptance of quantum mechanics – even in contexts where quantum mechanics look counter-intuitive to others – is much higher among the string theorists than it is among other physicists.

This correlation may arguably be interpreted in the opposite way, too. A big portion of the difficulties that some physicists face when they are trying to embrace string theory is their inability to fully accept something much simpler and more fundamental, namely quantum mechanics. String theory – and its picture of quantum gravity – is really nothing else than taking the lessons of quantum mechanics seriously even when we are talking about the spacetime geometry.

And that's the memo.


  1. When I have the lots of money I will visit if I can find a good reason to go.

    I the meantime I will walk to dog and look for that young beaver I saw yesterday. I'll send a picture if you want.

  2. The moon dust is electrically repulsive and get into everything and messes IP the bearings. Too bad.

  3. The rusb have a new quantum tunneling machine that can send you right inside the BH for about 10 sex.

    Not sure how I come out though??

    They are going to test it with a politician xxx er I mean a monkey next week.

    It uses a lensed quantum sync tech. They all tunnel at the same time during a large energy release......

  4. They are rehearsing for the moon takeover by taking over the Arctic and the N. Pole...(actually we Canadians are more annoyed with the USA's not agreeing to Canada's sovereignty over the NW Passage. Maybe Russia will support us if we support their Lermontov shelf claims.)

  5. Walking as an activity has declined in North America unfortunately. I used to hike trails in our beautiful National Parks amongst jewels of alpine lakes and scree...still do on occasion. Urban walking usually involves dogs (or in our case, gray wolf cross), but that is OK. Brisk walking is perhaps the best route to sustained health.

  6. kashyap vasavadaMay 9, 2014, 6:10:00 PM

    Interesting suggestion Lubos: "If we had enough helium-3, it could stimulate the research of the
    thermonuclear reactors. And the reactors burning helium-3 could turn out
    to be simpler to construct than those that are being built today."Are people doing serious theoretical work on this? My understanding is that the current problems with thermonuclear reactors has not much to do with availability of tritium and deuterium or that tritium is unstable.

  7. Sounds like a job for robots.

  8. LOL, Gordon. At some moment, I would be surprised that North Americans have forgotten e.g. mushroom hunting. Now you seem to remind me that it may be a corollary of their having forgotten walking. ;-)

  9. ER/EPR or ER=EPR?

  10. LOL, the literature insists on =, which I find strange in a name of anything, and prefer a simple hyphen although the slash could also be OK.

  11. Well, I followed the AdS/CFT notation :-)

  12. Tatchell is clearly a crackpot, but I don't really trust the link you gave to Science Daily, it puts a lot of emphasis on CO2 influence on climate and I tend to be skeptical about that, the plankton part of the story is likely true though, my guess would be because of iron fertilisation of the oceans (but maybe something with CO2 'sounded better', you know)

  13. There are no international laws regulating the annexation of territories on the Moon" There is - the Outer Space Treaty of 1967

  14. how big of a coilgun do we need to launch rocks of rare earths so they can land in the black sea?

  15. A big rubber bans should do as there is no air to slow it.

    It can be mounted flat horizontal on the moon. Just want until it is angled right and shoot. Actually this means it the gun is on the boundary between the back and front of the moon. You can have as many guns as you like side by side.

    The better design is to park it first in the Lagrange point. I've got my flag to claim one of them already. When needed on earth we just give it a small kick for transport down. We use a 8 layer foam shield to fly it down with GPS right to your parking lot.

  16. I'd be a little bit surprised to learn that a significant number of Americans ever knew mushroom hunting. I recall absolutely nothing about it from childhood except, from school, that it's something Russians do. I know that doesn't make much sense, considering the fact that we eat them. I mean, how could we ever have adopted the custom of eating them, if no one had the custom of "hunting" for them?

  17. Cool, I didn't know.

    I still think that no one will be afraid of ignoring the treaty.

  18. I guess that you mostly eat champignons and they are mostly grown in "gardens" of a sort, so it's not hunting, it's just harbor. Hunting is when the things grow in the wild, and in that case, the mushrooms are more diverse.

  19. Good article, a lot to try and understand.
    Time, as a function of distance, from the center of a Black Hole?
    Is there a similarity to gravity, inside a solid (spinning) sphere?

  20. Lubos, can you maybe write a blog post in response to this video? It's the cult of personality Elon Musk talking about science vs engineering, taking the view that engineering drives science.

  21. He's a great technological leader but I don't think that this video is so deep that it would produce a meaningful blog post.

    Science is a separate business from engineering so engineering isn't really a player in it at all. What we could debate is which scientists are important and how they depend on engineering. Obviously, engineering is important for experimental science, mostly, and the boundary between engineers and experimental physicists is really blurred.

    On the other hand, science is in no way just experiments - or clever experiments, and most of progress in high-energy physics of the latest decades has been in the theory - despite the running of the LHC. I am saying trivialities that Musk seems to softly deny them but he does so softly which means that nothing can be said to be "strictly wrong". It's just some self-evident bias, his overlooking the bulk of science and its independence of the practical applications.

  22. Hi Lubos,

    Just to clarify something; do you disagree with the Maldacena-Susskind interpretation of the EP=EPR?

    As I understand it at least I think it is clear that the original intention of Maldacena and Susskind was not to formulate EP=EPR as a correspondence. What they are saying is that the entangled pair is really connected by an ER bridge in the same physical space-time. Even if we have entangled particles they will be connected by an ER bridge although a sort of Planckian one (whatever that means).

    Unless I got it wrong.

  23. I am not aware of anything in the Maldacena-Susskind paper or your comment that I disagree with.

    It is not clear to me what's the difference or tension you are trying to point to - or fabricate. A duality means that both sides of the duality, both descriptions are equally real.

  24. I would vote for a BH interior of Higgs particles compresses by Casimir pressure, together with adouble boundary Fermion repelling horizon

  25. Example, JKR paper states at page 3:

    “Similarly, one presumably should conclude that we can think of the entangled black hole pair as a classical wormhole geometry in the large area limit or as an entangled state of the two sets of microscopic degrees of freedom at any scale. Just as for the emergent black hole geometry, these should be complementary descriptions that are not simultaneously tractable. This is the description suggested by our holographic analysis.”

    I think Maldacena and Susskind would disagree with this interpretation. The two descriptions are not complementary. They are exactly the same and simultaneously tractable i.e. entanglement means ER bridges and ER bridges mean entanglement at any limit.

    Unless I have not understood the Maldacena/Susskind paper correctly.

  26. I have no idea how two inequivalently sounding descriptions of a state vector may fail to be complementary.

  27. Nice article!

    The penetration of the quantum interpretation fog into the quantum gravity business brings me into a dimemna: (conistant) quantum gravity I love and favorite the tag on physics site, whereas quantum interpretation I deem useless and therefore ignore the tag ...

    From the nice explanations, the duality character of ER/EPR is intuitively very clear. Are there people working on nailing it really down (by the concrete duality transformation etc ...) too?

    Right from the start, I had a potentially dumb question concerning the dynamical interpretation of a wormhole on a worldsheet and put it down on PhysicsOverflow

    The site could do with some more activity BTW ... :-/

  28. Eelco HoogendoornMay 10, 2014, 12:18:00 PM

    The moon may make sense for general mining purposes; especially if you want to get real about building spacecraft in bulk, since you arnt trapped in some relatively huge gravity well. Also, the minable volume of the moon is much greater, since the earth is largely molten lava; and a lot of its elements are concentrated in the core, which we arnt getting to any time soon. The moon gives us more equal access to whatever early supernovae bestowed upon our solar system. But yeah, those are all rather long term factors which I do not at all expect to become relevant in my lifetime. Indeed, solar panels on the moon are pants on head retarded.

  29. Hi Dilaton! A lot of what Lumo is divulging about his physics intuitions and interpretations and attitudes (given that he does so in fairly ordinary language, mind you!) feels intuitively right to me, too! ;D
    However, I might as well also say that I have a sense I have almost reached a state of satisfied saturation as far as my ability to intuitively absorb Lumo's *thus communicated* insights (specifically re dualities or complementary mathematical descriptions of What Is going on).

  30. Thanks Lubos! I thought it was interesting that Musk seemingly limits science to high energy physics when he says no scientific discoveries are being made without colliders. Then he implies that Maxwell (along with Faraday and Galileo, who I am more sympathetic with) was driven by engineering to make scientific discoveries.

  31. Thanks, Gunther!

    Well, quite on the contrary for Galileo. He did some engineering but he was driven by pure science that he founded - one could even say by his desire to "clarify theology".

    He did some work on developing telescopes because he needed it for something in theory, not the other way around. He did lots of pure theory, too.

    Faraday and Maxwell were rather analogous but in electromagnetism. Obviously, people wouldn't learn about electromagnetism without some experiments. But we don't use the word "engineering" for all experiments. Moreover, there is a lot of important theoretical work going on. I don't know what Musk is talking about.

  32. Lubos do you see a difference between the following two statements?

    JKR write:

    “We have demonstrated that the holographic dual of entangled pairs in mSYM on curved backgrounds and/or coupled to dynamical gravity is given by a classical string worldsheet whose worldvolume exhibits an ER bridge.”

    I would replace it with:

    Entangled pairs in mSYM on curved backgrounds and/or coupled to dynamical gravity are connected by an ER bridge (some sort of ER bridge, of Planckian nature I suppose).

    Am I wrong?

  33. Dear Giotis, your version is clearly misleading because mSYM is a non-gravitational theory so it can't have any Einstein-Rosen bridges - an Einstein-Rosen bridge is a connected pair of black holes, a solution of Einstein's equations, and mSYM is simply not a theory with Einstein's euations, topology change, and dynamical metric tensor at all!

    Well, I could say, more carefully, that the normal mSYM-like description or formulation of the physical system doesn't use anything like dynamic geometry, variable topology, and Einstein's equations. Do you really disagree?

  34. Hi Lubos,

    No I don’t, but I’m not talking about a pure field theory like mSYM without gravity but I’m referring the specific construction of the paper i.e. entangled pairs in mSYM on curved backgrounds and/or coupled to dynamical gravity.

    Basically I’m referring to a Hawking pair (quark anti-quark) in a theory with dynamical gravity. Check chapter V of the JKR paper.

  35. OK, Giotis, if you talk about a gravitational theory, there are wormholes in it. But you are still misunderstanding one side of ER=EPR.

    ER=EPR doesn't say and never said that only ER is OK and EPR is wrong. It says that you may describe the same state either as two a priori isolated black holes - or any other objects - whose microscopic degrees happen to be entangled (EPR); or as a state living on a topologically different geometry, one with an ER-bridge.

  36. " ... Lenny Susskind, a co-father and giant of string theory ..." Who might be on the list of the top 20 giants of string theory?

  37. “String theory - and its picture of quantum gravity – is really nothing else than taking the lessons of quantum mechanics seriously even when we are talking about spacetime geometry.” Does string theory imply that the equivalence principle is wrong near the Planck scale? Quantum field theory might imply that the universe has at least two vacua: a standard quantum vacuum and another quantum vacuum with a greatly larger average value for the Higgs field. The unstable vacuum would be likely to be the one involved in the big bang and there would probably be physical upper bounds for the energy density. In other words: (Inert mass) * Acceleration = (Intensity of the gravitational field) * (Gravitation mass) * (Scale factor related to the Planck length). In terms of the GR field equations, the simplest idea might be to use

    T(mu,nu)/(1 – (T(mu,nu)/T(max))^2)^1/2 as the replacement for T(mu,nu).

  38. I first came across this system when walking from benesov train station to konopiste. It's fabulous. Simple, straightforward and effective.

  39. Hey, you Czech's tops in hiking technology, you PATsy? Wait a minute. My smartphone has how many GPS locator capacities, which most Yankees have now. Problem solved, right?

    What if I'm out of battery power? Then your many trails are a wonderful backup! But battery power is improving fast and in a hiking group, it will be redundant.

    Ooh, and then there's that growing space junk problem, that (ala "Gravity) could take down the whole satellite system. (Oops.) Not so good.

    Oh well - a few kinks for the 'superior' trail guiding system to work out. So good thing that when I get lost in Czechia country there's a cool old tech backup route finding system - no power or fragile satellite technology to depend on. You win, Dr. Lubos