Saturday, April 17, 2021

Does the Universe have higher dimensions? Part 2

[This is a transcript of the video embedded below.]

In science fiction, hyper drives allow spaceships to travel faster than light by going through higher dimensions. And physicists have studied the question whether such extra dimensions exist for real in quite some detail. So, what have they found? Are extra dimensions possible? What do they have to do with string theory and black holes at the Large Hadron collider? And if extra dimensions are possible, can we use them for space travel? That’s what we will talk about today.

This video continues the one of last week, in which I talked about the history of extra dimensions. As I explained in the previous video, if one adds 7 dimensions of space to our normal three dimensions, then one can describe all of the fundamental forces of nature geometrically. And that sounds like a really promising idea for a unified theory of physics. Indeed, in the early 1980s, the string theorist Edward Witten thought it was intriguing that seven additional dimensions of space is also the maximum for supergravity.

However, that numerical coincidence turned out to not lead anywhere. This geometric construction of fundamental forces which is called Kaluza-Klein theory, suffers from several problems that no one has managed to solved.

One problem is that the radii of these extra dimensions are unstable. So they could grow or shrink away, and that’s not compatible with observation. Another problem is that some of the particles we know come in two different versions, a left handed and a right handed one. And these two version do not behave the same way. This is called chirality. That particles behave this way is an observational fact, but it does not fit with the Kaluza-Klein idea. Witten actually worried about this in his 1981 paper.

Enter string theory. In string theory, the fundamental entities are strings. That the strings are fundamental means they are not made of anything else. They just are. And everything else is made from these strings. Now you can ask how many dimensions does a string need to wiggle in to correctly describe the physics we observe?

The first answer that string theorists got was twenty six. That’s twenty five dimensions of space and one dimension of time. That’s a lot. Turns out though, if you add supersymmetry the number goes down to ten, so, nine dimension of space and one dimension of time. String theory just does not work properly in fewer dimensions of space.

This creates the same problem that people had with Kaluza-Klein theory a century ago: If these dimensions exist, where are they? And string theorists answered the question the same way: We can’t see them, because they are curled up to small radii.

In string theory, one curls up those extra dimensions to complicated geometrical shapes called “Calabi-Yau manifolds”, but the details aren’t all that important. The important thing is that because of this curling up, the strings have higher harmonics. This is the same thing which happens in Kaluza-Klein theory. And it means, if a string gets enough energy, it can oscillate with certain frequencies that have to match to the radius of these extra dimensions.

Therefore, it’s not true that string theory does not make predictions, though I frequently hear people claim that. String theory makes the prediction that these higher harmonics should exist. The problem is that you need really high energies to create them. That’s because we already know that these curled up dimensions have to be small. And small radii means high frequencies, and therefore high energies.

How high does the energy have to be to see these higher harmonics? Ah, here’s the thing. String theory does not tell you. We only know that these extra dimensions have to be so small we haven’t yet seen them. So, in principle, they could be just out of reach, and the next bigger particle collider could create these higher harmonics.

And this… is where the idea comes from that the Large Hadron Collider might create tiny black holes.

To understand how extra dimensions help with creating black holes, you first have to know that Newton’s one over R squared law is geometrical. The gravitational force of a point mass falls with one over R squared because the surface of the sphere grows with R squared, where R is the radius of the sphere. So, if you increase the distance to the mass, the force lines thin out as the surface of the sphere grows. But… here is the important point. Suppose you have additional dimensions of space. Say you don’t have three, but 3+n, where n is a positive integer. Then, the surface of the sphere increases with R to the (2+n).

Consequently, the gravitational force drops with one over R to the (2+n) as you move away from the mass. This means, if space has more than three dimensions, the force drops much faster with distance to the source than normally.

Of course Newtonian gravity was superseded by Einstein’s theory of General Relativity, but this general geometric consideration about how gravity weakens with distance to the source remains valid. So, in higher dimensions the gravitational force drops faster with distance to the source.

Keep in mind though that the extra dimensions we are concerned with are curled up, because otherwise we’d already have noticed them. This means, into the direction of these extra dimensions, the force lines can only spread out up to a distance that is comparable to the radius of the dimensions. After this, the only directions the force lines can continue to spread out into are the three large directions. This means that on distances much larger than the radius of the extra dimensions, this gives back the usual 1/R^2 law, which we observe.

Now about those black holes. If gravity works as usual in three dimensions of space, we cannot create black holes. That’s because gravity is just too weak. But consider you have these extra dimensions. Since the gravitational force falls much faster as you go away from the mass, it means that if you get closer to a mass, the force gets much stronger than it would in only 3 dimensions. That makes it much easier to create black holes. Indeed, if the extra dimensions are large enough, you could create black holes at the Large Hadron Collider.

At least in theory. In practice, the Large Hadron Collider did not produce black holes, which means that if the extra dimensions exist, they’re really small. How “small”? Depends on the number of extra dimensions, but roughly speaking below a micrometer.

If they existed, could we travel through them? The brief answer is no, and even if we could it would be pointless. The reason is that while the gravitational force can spread into all of the extra dimensions, matter, like the stuff we are made of, can’t go there. It is bound to a 3-dimensional slice, which string theorists call a “brane”, that’s b r a n e, not b r a i n, and it’s a generalization of membrane. So, basically, we’re stuck on this 3-dimensional brane, which is our universe. But even if that was not the case, what do you want in these extra dimensions anyway? There isn’t anything in there and you can’t travel any faster there than in our universe.

People often think that extra dimensions provide a type of shortcut, because of illustrations like this. The idea is that our universe is kind of like this sheet which is bent and then you can go into a direction perpendicular to it, to arrive at a seemingly distant point faster. The thing is though, you don’t need extra dimensions for that. What we call the “dimension” in general relativity would be represented in this image by the dimension of the surface, which doesn’t change. Indeed, these things are called wormholes and you can have them in ordinary general relativity with the odinary three dimensions of space.

This embedding space here does not actually exist in general relativity. This is also why people get confused about the question what the universe expands into. It doesn’t expand into anything, it just expands. By the way, fun fact, if you want to embed a general 4 dimensional space-time into a higher dimensional flat space you need 10 dimensions, which happens to be the same number of dimensions you need for string theory to make sense. Yet another one of these meaningless numerical coincidences, but I digress.

What does this mean for space travel? Well, it means that traveling through higher dimensions by using hyper drives is scientifically extremely implausible. Therefore, my ultimate ranking for the scientific plausibility of science fiction travel is:

3rd place: Hyper drives because it’s a nice idea, it just makes no scientific sense.

2nd place: Wormholes, because at least they exist mathematically, though no one has any idea how to create them.

And the winner is... Warp drives! Because not only does the mathematics work out, it’s in principle possible to create them, at least as long as you stay below the speed of light limit. How to travel faster than light, I am afraid we still don’t know. But maybe you are the one to figure it out.


  1. I found this nifty animation of a Calabi-Yau manifold after it was mentioned here the other week.

    The space-travel theory I subscribe to is thus:

    ' is a fast ride
    Bring a laser gun and a ship with a warp drive
    That will go at the speed of light
    Times eight point five'

  2. In a susy symmetric rasberry multiverse indeed.

  3. Hello Sabine,
    many thanks for the very nice explanation of mini black holes

  4. in some way the wormhole may involve extra-dimensions. The easiest wormhole to understand is the Ellis-Thorne wormhole with the line element

    ds^2 = dt^2 – dr^2 – (r^2 + ℓ^2)dΩ^2.

    This means around the coordinate origin there is additional area to any sphere surrounding the origin. This is the same as the exercise of cutting a wedge out of a disk, making it conical, or near spherical, and then gluing that into another disk at a radial slit. This second disk is hyperbolic, or saddle shaped, and this results in a negative deficit angle, a surplus angle θ if you will and the curvature scalar R = θ/2πr^2 is negative.

    This runs into some question with the holographic principle. Of course, this is wrapped into some stringy ideas, but the basic idea with black holes can be understood without string theory. The above metric may be changed into

    ds^2 = (1 – 2m/r)dt^2 – (1 – 2m/r)^{-1}dr^2 – (r^2 + ℓ^2)dΩ^2.

    This is a wormhole with a black hole metric at a distance where r >> ℓ. However at r = 2m something funny happens, The horizon is reached, but the 2-sphere at this radius has area 4π(r^2 + ℓ^2). So is this area the holographic screen, or is the horizon?

    We might be tempted to say the Bekenstein bound is modified to,

    S = A/4ℓ_p^2 + 〈quantum corrections〉.

    However, these quantum corrections are likely to be near the Planck scale, maybe in fact higher string harmonics. The added ℓ^2 to the metric is likely to be on the order of a Planck/string length. Maybe with T-duality these higher harmonics might be some Nℓ_p in scale. Yet N cannot be that large in any practical sense. The classical wormhole then likely does not exist. Extra dimensions may play a role here, but they do not help with making a wormhole.

  5. But then, please don't send this theory to me... :)

    1. We want space travel though.
      Maybe all those TOE-physics fanboys can send Dr. Hossenfelder something useful.

  6. Hi Sabine,

    One reason it's hard to get an intuition for more than three spatial dimensions with some dimensions "curled up" is that you have to imagine two impossible things at the same time: 1. more than 3 spatial dimensions, and 2. dimensions that are "curled up".

    Is it possible to isolate just the "curled up" part? For example, by discussing a space with 3 dimensions (like ours), but with one of them curled up? Or even a 2 dimensional space (like Flatland), with one of them curled up?

    I suspect "curled up" is just an simplification of something that you can't really understand without math.

    1. Colin,

      Yes, that's why I keep showing that cylinder. It has one rolled up dimension, the other one is flat (and can be infinite). The rolling-up part isn't remotely as mysterious as many people seem to think. You just take each point in space and pin a compact object to it. Could be a sphere or a circle or a Calabi-Yau manifold or what have you.

    2. Thanks Sabine! To make sure I understand, the cylinder represents a two dimensional system, where one dimension is curled up and the the other isn't? If X is parallel to the tube, and Y is going around the tube, then can you go infinitely in the Y direction, and you would be going around and around?

      It seems like in order to curl, the Y direction is occupying a 3rd dimension.

    3. Colin,

      As you say, the cylinder is a 2 dimensional system. The embedding space (3rd dimension) is only there for illustration. It's not part of the cylinder.

    4. Thanks for your patience. It's still hard to get an intuition for. It would probably take numerous follow-up questions...

    5. This little Q & A is quite helpful, thanks.

    6. It might help to know that our standard spacial dimensions might also be curled, on a very large radius. The test would be (if immortal and possessing the required resources) to travel in a spaceship along our galactic north (determined by its rotation with the right-hand rule) and eventually find yourself re-entering our galaxy from the south. (And then to repeat the test in two mutually orthogonal directions.) This was a possibility I read about many years ago, but I am not sure current observations are still consistent with it.

      I think there is a slight analogy between the difficulty of imagining curled dimensions and the flat-earth versus spherical earth controversy, although the former is more complicated.

  7. I am thinking that as the old saying goes, there is more than one way to skin a cat.

    The particle accelerator mechanism may not be the only method of generating concentrated high energy. I wonder if the ability of coherent matter to accumulate energy in a condensate is another. I understand that a trillion trillion electrons can be converted into bosons and packed into a condensate waveform. That new waveform must have a huge energy potential. All those electrons carry an approximate accumulated energy potential summation of 10^29 Ev or 10^17GeV. That is around the grand unification energy level.

    Is this amount of energy enough to create a micro black hole?

  8. "[i]f you want to embed a general 4 dimensional space-time into a higher dimensional flat space you need 10 dimensions, which happens to be the same number of dimensions you need for string theory to make sense. Yet another one of these meaningless numerical coincidences...."

    Or maybe an argument for "special creation"? I've heard worse. For some as-yet unwritten tract in theology: "Hominids have ten digits on our two fingers, which has led us naturally to a base ten number system, which has made it inevitable that the above coincidence would seem resonant and confusing. All of which makes sense only if it is part of God's plan."

    1. Hi Christopher,
      The '10 fingers = base 10 = God's Plan' idea is an example of looking to make random coincidences and apparent correlations etc. fit into patterns (apophenia). Wikipedia has a few different manual counting systems that aren't base-10.
      So, 'special creation' most likely won't hold up if one considers all available information in different scenarios.

    2. Edit: Maybe it's not exactly apophenia, but the same idea of trying to make randomness fit a picture.

    3. C. Thompson, I agree. I was attempting humor. Always tricky on the web I guess.

    4. Got you. :)
      I found out some interesting counting methods, for what it was worth.

  9. The quantum fields of the standard model are sections of a fibre bundle and these can be thought of as higher dimensional spaces. Generally though, they are spoken of as internal spaces.

    Although they're fairly modern ideas, they go back to Cosserat brothers modelling of continua with additional symmetry groups attached to points. In a sense, the points have symmetry. They're called Cosserat media.

    This is a great way to think of quantum fields, they're basically waves in spacetime thought of as Cosserat media. This means even without going to quantum gravity we have atoms of spacetime that vibrate!

    And the quantisation of these fields is essentially excited states of these atoms - just like the atoms of Hydrogen and Helium.

  10. Hello Sabine,

    You like to write about higher dimensions quite often. You have also written about Planck units. That is all right.

    But why are you avoiding long-distance effects? What is the reason for that? Why does all the world, why does all the mainstream exclude spooky remote effects as an explanation for experiments done?


    1. Hi Stefan,

      What am I avoiding? I can't follow. I have a video coming up (in May) about what Einstein meant with "spooky action at a distance"!

  11. A Probably Silly Question: Why does the number of these extra dimensions shrink when Super Symmetry is added?

  12. I have developed a pre-spacetime, pre-quantum theory, valid at the Planck scale. This is a theory in eight non-commuting octonionic dimensions, which because of the homomorphism SL(2,O) ~ SO(9,1) is equivalent to ten dimensional Minkowski space-time. There is an additional time direction, known as Connes time, in which evolution takes place. Classical systems live in four space-time dimensions. Quantum systems penetrate all the eight octonionic dimensions. The thickness of the extra dimensions is of the order of Compton wavelength of the penetrating quantum systems. To overcome the problems of string theory, one gives up the rules of quantum field theory at Planck scale, and replaces them by Stephen Adler's matrix-valued [Lagrangian] trace dynamics. This octonionic theory does well as regards unification of gravity and the standard model, and as a prediction we derive the correct value 1/ of the low energy fine structure constant. Interestingly, the extra dimensions help understand how non-local quantum correlations arise without violating special relativity. Also there is an important confirmation of the Kaluza-Klein idea as to the number of dimensions needed to describe various interactions: four for gravity, additional two for weak interactions, and additional four for strong interactions. In our work this arises precisely because of the following homomorphisms between division algebras and Minkowski-spacetimes in various dimensions:
    SL(2,C) ~ SO(3,1)
    SL(2, H) ~ SO(5,1)
    SL(2, O) ~ SO(9,1)
    The number of space dimensions goes from 3 to 5 to 9 as we add weak interaction and the strong force, to gravity. "

    1. Tejinder Singh, superluminal quantum correlations are perfectly explained in the classical way, additional spatial dimensions are not needed here (see details in my blog). The usual representation of the rotation group in hyperbolic space is impossible, so a modification of the SRT geometry or at least a new interpretation of SRT is needed for compatibility with the rotation groups in Euclidean or complex space. How exactly the extra dimensions are not supported by the physical facts observed to date, Dr. Sabine Hossenfelder perfectly explained in her video.

    2. Tejinder Singh wrote: "[...] as a prediction we derive the correct value 1/ of the low energy fine structure constant"

      That's an interesting number! Do you have an analytic formula, or is it the result of numerical calculations? How accurate is it? How does it depend on the number of dimensions? Is there a preprint?

    3. Hello Werner. There is an analytic formula. The low energy fine structure constant is given by

      (9/1024) exp [2/9 - \sqrt{1/6}] ~ 0.0072971 = 1/137.04006

      It is accurate upto two parts in 10^7, and the accuracy improves greatly if higher dimensional corrections are taken into account. These are discussed in my preprint titled `The characteristic equation of the exceptional Jordan algebra: its eigenvalues, and their possible connection with the mass-ratios of quarks and leptons ' searchable online. Thanks for your interest. Best...Tejinder

    4. Thanks Igor. In my papers I explain how the said problems with extra dimensions [`not supported by physical facts observed to date'] are overcome in my work. One must give up on the rules of quantum field theory at the Planck scale, and replace them by Adler's trace dynamics. When that is done, the said problems with extra dimensions no longer arise.

    5. Tejinder, thanks for the intriguing preprint. Unfortunately, due to my lacking mathematical expertise, I find it hard to relate it to the physics as I have understood it so far. It will take me some time to dig into the literature you quoted.

      By the way, it is misleading to count also the leading zeros in quoting the accuracy; I'd say the accuracy is five significant figures (30 ppm). It's also unclear how the higher dimensional corrections are calculated, but perhaps I'll understand it after more careful study. :-)

    6. Michael Rios wrote a paper last decade on the role of SO(9,1) with exceptional groups. SO(10) = B4 has the exact sequence

      B4 ---> F4 ---> OP^2

      the projective Fano plane. SO(10) is in the kernel of F4, and modulo SO(10) is a cohomology.

    7. @Tejinder Singh:

      Very interesting. I'll look up your papers.

      I've just learnt that the standard model gauge group can be expressed via octonions over at the n-category cafe blog. This was discovered by work by Todorov and Dubois-Violette.

    8. @Tejinder Singh:

      How does your additional time variable, the Connes time arise? Is it anything to do with Connes theorisation of the origin of time, his notion of thermal time arising from Tomita-Takesaki theory?

      It's interesting since I had a brief discussion on this blog about theories in physics with additional time directions. I stated that I knew of none and that I was dubious about their physical relevance given causality. Then I discovered hyperbolic meta-materials which can be thought of having two time directions and the two time theory of 12d supergravity and now your own theory!

    9. There is an attempt to construct a theory of quantum mechanics as an emergent theory from a deterministic and local fundamental dynamics. In such an attempt, time is a two dimensional parameter, not reducible to one time parameter. From one side, there is a dimension time that describes the hypothesized fundamental deterministic dynamics. The role is similar to T. Singh "connes's time", but the nature of totally arbitrary. The second dimension of time refers to any time parameter used to describe classical or quantum dynamical systems as they emerge. This second dimension of time comes with the additional property of emergence too, since all time parameters are associated with emergent dynamical systems. By construction, emergent quantum mechanics is time parameter invariant respect to both notions of time. There certain similarities between this approach to the foundations and some points of the theory from T. Singh.

      In classical dynamical systems, see for instance the book of V. Arnold on mathematical methods on classical dynamics, there is the slow time parameter and the fast time parameter, associated with a slow dynamics and fast dynamics. G. 't Hooft is trying to see the emergence of quantum systems from this point of view, which is somehow similar to the point discussed above, but different.

      I think none of these ideas of higher dimensional time refers to a geometric high dimension of time, like 9+2 theories.

    10. Thanks Mozibur, for your interest. You can find more about `octonions and the standard model' through the recordings of the ongoing Perimeter conference of the same name. It includes talks by Todorov, Dubois-Violette, and also Cohl Furey, who also has a wonderful set of video recordings online, on `Division algebras and the standard model'.

      And you are spot on about the Connes time - it the same as the one proposed by him based on the Tomita-Takesaki theory. For the last couple of years, I also struggled with this two times thing. Then I asked: *which* is the time that *flows*? And i believe that is Connes time, and that Connes time is the same as Newton's absolute time, which is the same as cosmological time [measured say by CMB temperature]. If we label Connes-Newton time by T, and the clock time [what we measure via motion of the clock hands] in Newtonian mechanics by t, then we know that T=t. But only T flows, t does not. Motion takes place in time T, and we use t to measure flow of T, but t by itself does not flow. At a given T, a body is as location (t, x) in spacetime. In Newton mechanics, relativity is Galilean relativity, and we have t'=t, x'=x-vt. In special relativity, (t,x) and (t', x') are related by Lorentz transformations, but neither t nor t' flow. I think it is a mistake that in special relativity and general relativity we have forgotten the absolute Connes-Newton time T. It was never lost. When we say time in relativity is not absolute and simultaneity is relative, what we mean is that t and t' are not equal. But neither t nor t' flow. What flows is T, even in relativity. By flow of T we mean that with increasing T, the entropy of the universe increases. Entropy csn never increase with t or t' - these are reversible mechanistic time labels of a space-time manifold. In relativity too, a body is at (t, x) or (t', x') at a given T. We can ask: what about causality? If in a frame of reference (t, x) an event at time t_1 influences another event at time t_2, isn't t_2 in the causal future of t_1, and doesn't it mean that time t is flowing from t_1 to t_2? No, it does not mean that. A moment's thought will show that `influence' implies increase of entropy! Else we can never operationally deduce that an event at t_1 has *influenced* an event at t_2. So what is flowing is the time T, from T(t_1) to T(t_2), and we interpret this as flow of t fom t_1 to t_2. Having a time-like separation between the first event and second event is a necessary condition for the first to influence the second, but not sufficient. We also need the entropy to increase. Similar reasoning holds in general relativity. And there is another subtlety in GR: the black hole area theorem. When we say that in any process involving black holes the total area of the black holes always increases, we are led to ask, but increases in which time? I believe the relevant time here is again the Connes-Newton time. Best .... Tejinder

    11. Thanks for your comments Topolino. I completely agree with your last point: having Connes time does not mean we now have a 9+2 theory. We continue to have a 9+1 Minkowski spacetime. As Connes time evolves, the position of a `particle' in this (9+1) spacetime changes from one to another.

      And yes there is some similarity with 't Hooft's work, I agree. The underlying dynamics in the octonionic theory is deterministic [but non-unitary] and from here emerge quantum theory as well as its indeterminism. The indeterminism is a consequence of having coarse-grained over the fast dynamical variables... i.e. not observing the dynamics at Planck tine resolution.

    12. @Tejinder Singh:

      Thank you for your informative comment. Personally, I've always been interested in a quantum thermodynamic origin of time flow. In fact, Rudolf Haag noted in his book, Local Quantum Physics:

      >But can one isolate events with absolute precision? Can one describe the universe as a set of events increasing in number as time goes on? This would neccessite the introduction of irreversability at a fundamental level and a revision of the concepts of time and space. In quantum mechanics this is circumvented by limiting the class of events considered to the interaction of the 'microscopic system' with a macroscopic measuring device.

      The first results that I know of that reintroduced time in an irreversible fashion in spacetime was causal set theory. It's great to see more work along these lines as I'd increasingly grown sceptical of a physics that ignores the irreversability of time and its 'flowing' nature.

      I came across Cohl Furey's videos on Youtube a while ago, and I agree that they were marvellously done. Very clear and to the point.

      Your own paper was very clear though I think it will take some time for the ideas to sink in.

      Thanks also for pointing out the workshop on the Octonions and the Standard Model at the Perimeter Institute. The lectures look very interesting. I'm particularly intrigued by the magic pyramid of super gravity theories classified by the division algebras R, C, H & O as I've been getting to grips with supergravity recently.

      Best Regards,
      Mozibur Ullah

  13. Dr. Hossenfelder,
    Another great video!
    Interesting that even with multiple dimensions you still can get the 1/r2 applied to gravity. I saw a video where they basically said that the 1/r2 applied to detected LIGO gravitational waves was showing we had only three dimensions - not so fast I guess.
    I do have a question on gravity. If gravity moves at the speed of light and experiences aberration delay aren't all the planets in our solar system being pulled to the old position of the sun in which case we will end up with an unstable orbit (tangential accelerations no longer pointing towards the centre of mass?).
    With respect to warp drive, I thought you can theoretically warp space faster than light (galaxies are accelerating away from each other far off faster than light)? Slower than light is warping space useful as opposed to say accelerating yourself using pre-positioned and staged fusion devices?

  14. What would be a universe with three space dimensions and two, three or more time dimensions be like?

  15. When "curled up dimensions" considered, highly possibly it's all about the uncertainties we don't understand at Planck level.

  16. I am intrigued by the string theory of micro black holes that develop in structures called bubbles of nothing within anti-de sitter (AdS) space. This phenomenon seems to be related to Bose condensates which may explain a unique type of erosion which causes matter to disappear. As an identifier mechanism, I beleive that Bose condensation can consolidate into a lattice structure called a supersolid. The supersolid lattice takes on unmistakable distinctive polygonal shapes: hexagon, triangle, and sphere that identifies the action of Bose condensation present at high density. These shapes appear in the erosive patterns that Bose condensation(BCE) can leave in its wake.

    Acoustically driven x-ray emission and matter collapse in lead

    Pertinent to this introduction, one paper listed above that appears to be connected to the appearance of BCE is really mind blowing. A paper authored by a Mexican physics team that describes the bombardment of a lead foil (sandwiched in plastic) with sound shockwaves (coming from a simple yet powerful system to create the cavitation shockwaves) and they observed, not without a fair share of surprise, X ray emissions and also a polygonal shaped hole in the foils where the lead simply “disappeared” and was nowhere to be found (take in account it was encased in plastic, and the small debris they could find inside of it did not account for even 1% of the missing material). They propose that the matter “collapsed” and produce some conjectures about how this could be explained invoking some electron involvement, from an accepted theoretical frame of reference, but obviously fall short of explaining and express their intention to keep working on this.

    I am really impressed and excited by this result and beleive that the indicators point to the action of a quasiparticle based condensate formation when phonons become entangled with electrons. Extra dimensions must be involved in the movement of lead to some other place under a state of coherent superposition. There is a class of experiments that show possible action within extra dimensions possibly within the body of the lead and/or condensate that might advance the study of both Witten/Sen singularities and extra dimensions.

  17. I was hoping that the solution to obtaining the type of energy needed to make an Alcubierre drive function would involve something along the lines of dark matter, dark energy, or some sort of gravity but since nobody has mentioned them in any explanations I've come across, I assume not.

    1. I was cheering for WIMPs that you could convert to photons by the Primakoff effect. Then you could use staged fusion devices (like Orion but with the fusion devices pre-positioned on your path) to get to 0.2c and then you turn on your Primakoff effect generator to push on all the WIMPs you are passing by for relativistic speeds. There would be your 'propellantless' flight.
      Unfortunately no WIMPs. And Axions if they exist are pretty light, I don't know if 0.2c is fast enough to use them.

    2. Hi Craig,
      I hadn't come across the Primakoff effect before, that's something for me to look into.
      Here's hoping something useful might turn up for sub-C flight.

  18. Many of you,

    we have the following elementary particles:
    Electrons, neutrinos, protons and photons, as well as the anti-particles.
    Then there are the generations, whose representatives are all unstable.
    These have charge, mass, spin.

    We have a handful of good experiments, such as:
    - Double-slit experiment
    - Pair annihilation and generation
    - Quantum entanglement experiments
    - Lifetime of muons

    What the elementary particles do not have is exact location and exact momentum. Nevertheless, a mathematics is used, which lives from infinitesimal changes.
    Why are you doing this? Why are you using this mathematics? Are you thinking that it is suitable?!

    The double-slit-experiment shows clearly,
    that micro-objects are neither wave nor particle.
    Nevertheless light and electrons are always shown as wave
    or particles.
    Why are you doing this?

    In the pair annihilation, objects with mass (electron and positron) are switched
    into objects without mass (light).
    Nevertheless, you use a mathematics which was developed for the position determination of planets.
    A planet changes only its position. Its mass never changes.
    Why are you using this mathematics for it then?

    With the experiments to quantum entanglement the one end (e.g. the one photon) knows
    with at least 1000 times speed of light, what happens at the other end, has happened.
    How will you explain these superluminal velocities?

    There are muons, which originate in 20km height. Although they have only approx. 1µs life span,
    many of them make it to earth. 3*10^8m/s * 10^-6s = 300m
    How can the muon measure the atmosphere to 300m, while we measure it to 20km?

    You guys really think you can answer even one question with more dimensions in place and/or time?


    1. Yes, Stefan! Please see my post above. Best - Tejinder

  19. I imagine an experiment in which I am sitting at a desk and the experimenter places a pen in front of me and offers me one million dollars to pick it up. I believe the offer to be genuine and that there are no catches. Yet I don't pick up the pen and forfeit the million. The experimenter offers me the million if I can simply reach out and touch the pen. Again, I don't. Then the experimenter offers me the million if I can avoid picking up the cup of coffee next to me for about a minute. I immediately pick up the coffee.

    So why? Because the experimenter has just watched a video of what I do in the next hour which has been sent back in time. I can't avoid doing what is in the video.

    Now I wonder why my brain has acted in a way so contrary the way it normally operates, just because some information has been sent in time. Physically, how does that work?

    Again, it seems implausible (not impossible of course) that this scenario can occur. So my guess is that it is impossible for information to be sent back in time, which means that it would be impossible for anything to travel faster than light.

    If I am wrong and if it is ever achieved then it would be fascinating to see this experiment carried out. Either I would be able to do one of these simple actions and pick up the million, in which case we would have evidence of multiple parallel universes (otherwise where did the video come from?) or else we will have evidence for super determinism.

    Sadly even if faster than light travel is ever possible it will almost certainly not occur in my life time.

  20. Sure lol why not I’ll do it

  21. A theory has some chance of being correct if it starts throwing light on difficult problems whose resolution was never the original motivation for developing the theory.

    What exactly happens during the double slit interference experiment with electrons?

    Quantum theory without classical time - the octonionic theory referred to above, was developed to achieve a formulation of quantum theory which does not depend on classical time. To arrive at that goal, we are compelled to bring in extra dimensions - a total of eight non-commuting octonionic dimensions. Classical systems live in a sub-space, our four dimensional classical space-time. However, quantum systems live in all eight dimensions. The extra dimensions are not curled up - rather the extent of the extra dimensions is the extent [Compton wavelength] to which a given quantum system penetrates the extra dimensions. Classical systems are classical precisely because their penetration depth into the extra dimensions is much smaller than Planck length.

    We have previously suggested that quantum non-local correlations manage to avoid violating special relativity because two space-like separated 4-d spacetime points are causally connected through the extra dimensions. The points maybe billions of light years apart in three-space, and yet only a Compton wave-length apart through the extra dimensions - this is certainly allowed by theory.

    In this light, consider the classic double-slit interference experiment with electrons. In three space, it is true that an electron goes though only one of the two slits. The slits are disconnected in three-space, and it is impossible for the electron to go through both slits in three-space. But in the extra dimensions the slits are connected, and only a Compton wavelength apart. Through these extra dimensions, an electron goes through both the slits, resulting in the observed interference pattern. A classical object such as a ball has an extremely small Compton wave-length, less than a Planck length: to see it going through both slits, we will have to observe at Planck scale resolution.

    1. Dear Tejinder,

      Let me disagree a bit your explanations of quantum interference and non-local quantum correlations.

      The Compton length scale cannot be the scale suggested for the explanation of Young-Feynman interference phenomena. For the electron, the Compton wavelength is of order 10^-12 m. However, experiments show interference even if the slits are much more separated  (500 nm) at least. The relevant dimension for the double slit experiment is the domain of the wave function, or something related with it.  In this sense, there is also an explanation of interference phenomena from the point of view of emergent quantum mechanics as discussed above.
      Furthermore, the explanation provided by Hooft is also more amenable, where the interference phenomena is determined by the wave function.
      Similar remarks apply to the apparent faster than light quantum correlations. It is not the Compton wavelength what counts, but the support of the wavefunction.

      To assume a special status of the Connes-Newton time is against the main principles of general relativity. Is this not a problem? At the end, one of the first Einstein achievements was to liberate physics from Newton's absolute time as an unnecessary assumption. An even bigger intellectual achievement is to liberate physics of the special status of the concept of coordinate systems  should have an attached  physical meaning: general covariance is the name, and the equivalence principle its support. If one wishes to keep this point of view, then Connes-Newton time cannot be anything of special significance. Hence, why should it be so relevant for a theory? That is, is your theory general covariant?

      Please, take this coments with an honest pretension of clarifying further these difficult questions. Even if my comments are in order, it could be that your ideas have a deep and penetrating insignt in Nature.

      Best regards.

    2. Hi Tejinder,

      applying knowledge from one area to another is certainly often helpful.

      Double slit experiment
      Although only implied, your explanation for the double slit experiment does not work in at least 3 points:
      1. Sabine begins her video showing all three dimensions with her hands and a graph in the background.
      Dimensions are always perpendicular to each other. This is also true for your extra dimensions.
      Suppose you have two points with the coordinates (0,0,0) and (1,0,0).
      If you start from (0,0,0) and move in y-direction you will never get to (1,0,0) but move away from both points.
      New dimensions do not form a short circuit. (Sabine also mentioned this.)

      2. For a short circuit you would have to bend the x-dimension.
      But there is no reason for that.

      3. You always mention the Compton wavelength as penetration depth in your extra dimensions.
      The Compton wavelength is a constant.
      But the interference pattern is variable and depends for example
      on the momentum (de Broglie wavelength) of the electron.

      I am afraid you are betraying yourself.
      I am sorry.


    3. @Topolino

      Thank you for your valuable insights.

      1. I agree [and also with Stefan on this point] that it is the support of the wave function, not Compton wavelength. My apologies for the mistake. I inadvertently carried over Compton wavelength from another context in the proposed theory - this length gives the ball-park estimate as to whether a system in the theory be treated as being quantum or classical.

      2. Connes-Newton time: Connes time T arises naturally as a consequence of the so-called Tomita-Takesaki theory in the context of Connes' non-commutative geometry. It then provides an ideal scalar measure of time flow in that epoch of the universe when classical space-time does not exist. That then raises the question: do we have two times after the space-time manifold emerges? So I am saying of these two times, only the Connes time flows, whereas the time t of the spacetime manifold does not flow: it defines the coordinate charts of the manifold, covariantly. And yes general covariance is there in the (t,x) system on the manifold. Say the action of the theory is S = \int dT L(q, q-dot) where dot denotes time-derivative wrt T. In the emergent theory the action becomes S = \int dT \int d^4x L(q(x)). The Lagrangian is now the same as that of general relativity, including matter. This Lagrangian is generally covariant, in the usual way, under transformations of x. The T is sitting outside the GR integral, as if a fifth coordinate, but it is not a part of the general covariance analysis. We vary the GR action to get Einstein equations and interpret them as evolution equations in a covariant time t. As if this time is flowing. This interpretation, i.e. that t flows, is where I have concerns. Perhaps we should interpret solutions of Einstein equations as giving us the metric field g({\bf x}, t) for all t and all {\bf x}. And say we keep track of Connes time T by the CMB temperature. Then we can say that at a particular value of T [flowing time] the metric g at the location (x,t) has such and such value. This gets rid of having to deal with two flowing times, and gives t the same (covariant) non-flow status that {\bf x} has. In one of the comments above, I elaborated a bit wrt special relativity and the relation between T and t. Thus at present I am more comfortable with retaining an absolute flowing time T in GR as well, and identifying this with Newton's absolute time T, and separating this entirely from the spacetime manifold (x,t). I think doing so helps greatly with relativistic QM as well. One can treat t now as an operator, on the same status as operator {\bf x}, and let T play the role of a scalar time evolution parameter [Stueckelberg formulation].

      Am happy to continue this discussion on Connes-Newton time. Thanks..Tejinder

    4. @Topolino

      “The relevant dimension for the double slit experiment is the domain of the wave function, or something related with it.”

      Back in 1996 I had what I thought was a stellar idea, based on a symmetry transformation, that over the following several years I elaborated on to explain multiple features of atomic physics, that I didn’t think were adequately addressed in the Standard Model. That further elaboration was the supposition that de Broglie, or matter waves, consisted of a pair of real physical variables. The properties of the field, embodying these variables, would then account for the suppression of synchroton radiation from orbiting electrons. In later years I further realized that this putative field might explain the double slit experiment as this postulated field had effectively zero mass (per wave cycle), and thus (I assumed) was not restricted in range. Additionally, the nature of this field would induce intense warping of the local metric and figured that would provide the ‘machinery’ to effect the interference process at the slits.

      I was giddy with excitement as only an amateur can be about such informal ideas. At the time I was blissfully unaware that General Relativity forbids negative energy densities in our Universe. Upon discovering that, and knowing this concept entailed negative energy densities, it was clear that the hypothesis wasn’t viable; at least within the framework of General Relativity. That was unfortunate as I had later extended the idea to explain the Dark Sector in Cosmology. In fact, I thought I had figured out what the ‘machinery’ was that underlay MONDian behavior, which even allowed that phenomenological theory to be extended into realms which it previously hadn’t embraced, as I expounded on over at Stacy McGaugh’s tritonstation blog, in the thread “Bias all the way down”.

      As stated over at that “Bias all the way down” thread I still plan to post a paper, summarizing these ideas, at viXra. And, as indicated upthread, I’m rather keen on conducting experiments with type-1 superconductors employing liquid helium. As insane as it sounds one idea that crossed my mind was to encase a niobium ‘bullet’ within a tough insulating jacket, and after lowering it to its critical temperature shoot it out of a ‘cannon’ (rifle or maybe air gun), and monitor for accelerations signals along the flight axis of the bullet. I can see the headlines: “Amateur mad scientist blows hole in wall of research facility narrowly missing lab director with exotic bullet. Swat team called in”.

    5. Dear Tejinder,

      Thank you for your kind reply. As you are happy with continuing the discussion concerning Connes-Newton time, let me ask you some issues, to clarify the concept.
      1. What is "Tomita-Takesaki theory in the context of Connes' non-commutative geometry" ?I know very few. I read that in that context the modular automorphism of a certain von-Neumann algebra is invariante respect to the outer group of the von-Neumann algebra, providing the definition of Conne's times (that is, the additive reals). On the other hand, Newton's time can be generally formalized by means of Einstein-Cartan's theory. Can you explain why they are the same concept? Is there a logical lap? In principle, they are different notions. I would like to ask you how to take that they are identified... are they the same?
      2. You are identifying Conne's time with a cosmological time. This is rather odd to me because of the following. The cosmological time that you use is a concept that appears in a particular class of solutions of GR. It is definitely a coordinate meaning attached, while GR is general covariant theory. It is therefore odd that you attach an interpretation of something resembling independent of coordinates to something that has a coordinate meaning.
      3. Probably the most important point, for me, is the following. You suggest that Conne's time plays also a role at the quantum level, at the quantum experiments level. If I am correct, what you suggest is that it is through the flow of Conne's time that real situations emerge. But then, it seems to me that such a concept leads to an unobservable character of the Conne's time. An observer at motion at very fast speeds, close to the speeds of light, will observe a different rate of flow than a student in a Lab doing an interference experiment, isn't it? The speed of the flow will be different, in apparent contradiction with any absolute interpretation of T. That is, they cannot observe Conne's time in the experiment, if it is absolute. Thus such a time that flow in a quantum experiment cannot be identified with Conne's time. It seems to me better to forget conne's time as a key ingredient in your theory and make it fully covariant, at least, T-parameter invariant.

      4. You wrote the following:
      "Say the action of the theory is S = \int dT L(q, q-dot) where dot denotes time-derivative wrt T. In the emergent theory the action becomes S = \int dT \int d^4x L(q(x)). The Lagrangian is now the same as that of general relativity, including matter."
      In the second action, let us call it, S_2, in order that the average operaion \int  dT is not fully trivial, the lagrangian L(q(x)) cannot be general relativistic. It needs to depend upon T.  Also, I do not understand how you pass from the first action to the second action. It seems to me that the fields $q$ are not the same than in the first. Are they?

      Let me ask again on the issue of the relation between non-uintarity and collapse models. What is the status of collapse models with respect to experiment? I remember now that Adler's theory also derived in a close relation with a collapse model and that it is not in as bad a position with respect to experiment as the GRW model. What about Penrose-Diosi theory? I said before they were almost falsified recently. This is not true, as you probably know: only a certain version of the model has been (probably) falsified.

      Best regards.

    6. Dear David Schroeder,

      Thank you very much for the comments. I cannot say much,
      except two things:
      1. Negative densities can be found easily in quantum mechanical systems.

      2. I would like to suggest that, if you really think that your idea is important and you want a truly professional feed-back, send the manuscript to a professional journal. You do not need to pay for this and the worst thing is that the manuscript is rejected without a comment, after some weeks (in math, the situation can be much worse.... with manuscript sent back after years without a comment, but in physics, the journals are much more agile). In the middle case, your manuscript will come rejected, but with comments of someone that has read the manuscript. This can be very useful for you, because then you can either reflect from the comments and throw the manuscript to the bin, or you can correct the manuscript and send it to another journal. 
      Sending the manuscript to a public arxiv could be as sending to a resonant box... that may confuse you even more.

      Best regards.

    7. @David Schroeder
      Hi David, your 'nobium bullet' sounds like a great scenario for a story.
      'Nobium Bullet' is also a name for a music group or album. :)

    8. @Topolino, Thank you for your kind words and very thoughtful suggestion. I very much like that idea. Currently the concepts in the paper are not buttressed by a mathematical analysis that would evaluate the model against real world data. I'm going to do my best to rectify that deficiency before submitting the paper to a professional journal. It would be wonderful to have feedback from referees that are professional physicists. With or without the math they undoubtedly would be able to spot flaws in the model that I completely overlooked.

      There was a reference to a paper that quantitatively assessed the limits for negative energy densities in nature that I came across a few days ago, but didn't read. That was on my reading bucket list but somehow lost track of the reference. It should be easy to find.

      @C Thompson, That is amazing that a musical group would come up with such an unusual combination of words. I have dabbled in science fiction, and actually incorporated some of the physics ideas that relate to the proposed bullet experiment in a sci-fi story set 300 years in the future.

    9. Hi David, I meant, that to would be a great name for a band or a track or album. If you publish any writing online, I'd like to read it if you don't mind.

    10. Oh I see, I didn't realize that's what you meant, partly, I think, as I was in a morning fog having gone to bed late. I do have one story that's at a link on my personal website. But it's against the rules to post links to personal websites, so I'll have to somehow obtain your email or vice versa. Alternatively, I could upload the story to some website for viewing. I'll have to look into that.

      As I was doing my daily bike ride today I was thinking about the evolution of extra dimensional ideas and could see it was quite a logical progression, as described in many books I’ve read. But the original classical 5D theory of Kaluza using the metric machinery of GR is quite complicated as I just discovered by browsing the Wiki page on Kaluza-Klein theory. If I put my mind to it, I might be able to understand this at a level where I could see if a crude idea I came up with (“symmetry transformation” in comment at 7:50 AM, April 23, 2021) might manifest with some particular configuration of curled up dimensions.

    11. @C Thompson,

      I found the perfect website to upload my short story called “WritersCafe”. It’s 53 pages long (double-spaced). I’ve got considerable editing to do, but as soon as that is finished and I upload the story to the site, I’ll provide the URL.

  22. @ Teijinder Singh,

    I clicked on your homepage and was delighted to see a smorgasbord of fascinating topics that I can scarcely wait to dig into. As a non-scientist, retired engineering technician, I've always had a deep interest in the fundamental sciences. I was positively captivated by higher dimensional theories when I first read about them in layman-level, popularization books in the 80's. Then, as the 80's morphed into the 90's, an explosion of new ideas emerged within the professional scientific establishment. Some didn’t stand the test of time, but others like Alcubierre’s warp drive generated immense interest in the popular press and critical assessment by specialists in General Relativity. Problems with Alcubierre’s metric were very quickly identified and proposed solutions to them were proffered up by other scientists, along with new variants of the original metric. That process is still ongoing today.

    I can say, without exaggeration, that I was electrified by all these developments, especially as someone fascinated by space travel, and the possibility of humanity someday reaching the stars. To add icing on the cake, nearly simultaneous with these novel ideas were reports coming from various quarters of anomalous phenomena associated with condensed matter systems. This seems to have had its beginning with calibration checks by Janet Tate, et. Al, on exquisitely fabricated quartz spheres coated with a thin layer of niobium, that were to be used in the Gravity Probe B project designed to detect frame dragging induced by the Earth’s rotation. This anomaly inspired other scientists such as Martin Tajmar and C. J. de Matos to hypothesize that the apparent increase in the mass of Cooper-pairs discovered by Tate’s group was actually the result of the hypothesized quanta of the gravitational field, the graviton, gaining mass within the rotating superconductor. This mass increase, they speculated, created a hugely enhanced (by some 30 magnitudes over standard theory) gravitomagnetic field that ‘dragged’ on the Cooper-pairs giving the illusion of a mass increase of those pairs.

    But also in the 90’s seemingly fringe experiments were coming to the attention of the public by controversial figures, also claiming anomalies with superconductors. For example, there was the brouhaha around Eugene Podkletnov and his experiments that was written up by Charles Platt in a 1998 article in Wired magazine. But in the same time frame amateur experimenters like Frederick N. Rounds were reporting odd behavior in the readily available high temperature superconductors like YBCO. Combined with the results of hundreds of experimental runs by Tajmar, et. Al, at the Austrian Research Center, between 2003 and 2006, this made me wonder if some genuine new physics was showing up in these experiments.

    Since YBCO hobbyist chips were readily available I figured I could play to. Using my electronics and metalworking skills, along with experience in machine shops, as an engineering technician, I fabricated my own specialized circuitry, homemade cryostats, etc. I tried zapping YBCO chips with discharges between 500-600 volts, monitoring for signals with a 1 milli-g resolution ADXL203 accelerometer chip. I ran into problems making electrical contact with the badly warped commercial YBCO chips, and eventually gave up, without finding anything interesting. I then purchased a length of niobium-titanium wire, probably salvaged from an MRI machine, drilling and tapping holes on either end of a 1 inch section to accept 4-40 screws. That solved my electrical contact issue. Since this alloy superconducts only at liquid helium temps. I will need to contact a university or lab that works with liquid helium to continue the experiments.

  23. Dr. Hossenfelder,

    "Flatland" written by Edwin Abbott Abbott and published in 1884 used a two-dimensional world/civilization in order to discuss the issue Mr. Abbott wanted to address. Not a physics book at all. However, it does allow for some imaginative questioning, how would the flatlanders recognized/see and/or understand three-dimensional beings and/or objects? How would any type of communication occur between the 2D and 3D worlds? By extension these questions would apply to 3D and 4D worlds/universes. Just to make a point, no math involved with these questions and understanding relationships between dimensions.

    Just one more item for me on this topic, the physics of our universe, no matter how many dimensions it may have, is at least 13.8 billion years old. The physics we know about today is less that 400 years old, and modern physics is only about 100 years old. It seems to me that we should be a little more retrospective as we try to move forward and understand the universe we live in. Example, where in the physics of the universe that we know of right now does it say that unification and/or a "god equation" is needed or required? Take it one step further, how do we know that our knowledge right now is complete thus allowing for correct unification? It just seems to me that we need to understand what we know before moving forward with trying to understand what we don't know about 13.8 billion year old physics.

  24. One interesting point that isn't emphasised enough in string theory is that the Calabi-Yau manifolds they compactify the theories on are actually vacuum solutions to 6d GR with vanishing cosmological force. I find that much more comprehensible physically speaking than just the bare name, Calabi-Yau manifold.

    I'm saying force rather than constant as it seems a bit odd to name something by it's coupling constant. It would be like call gravity, the gravitational constant or electricity, the electric constant.

    1. It's called the cosmological constant because it's constant.

  25. Hello Sabine, I have a question, I may be naive, I am not a specialist; My question is -Can a closed temporal magnitude linked to the three spatial dimensions plus time be used to describe many closed spatial quantities? ; I mean, If I add another time dimension; but closed, could it eliminate the excess of spatial dimensions? , varying that time closed interval could accommodate any closed magnitude? Thank

    1. I don't know what this means, sorry.

    2. Your question is a bit odd. The closest interpretation and possible answer I can think are that de Sitter spacetime in 4-dimensions has topology M^3×ℝ, for M^3 = S^3, a three sphere for positive curvature or M^3 = ℝ^3 Euclidean space if this is flat. These spaces foliate spacetime along a real number line. Anti-de Sitter spacetime is M^3×S^1, where the S^1 is time. Time is on a circle. There are closed timelike curves! Normally we consider this AdS spacetime in a region where i^+ and i^- meet in a circle, so there is only one closed timelike curve that we “shove off” to conformal boundary. This is a conformal patch.

  26. Isn't Campbell theorem telling s that a 4D spacetime/manifold can be embedded in a 5D flat or Einsteinian universe? Why do we need 10 D for this?

    1. I'm talking about global embeddings.

    2. Embedding of manifolds is strange. In general, by the Whitehead theorem a manifold of N dimensions is only guaranteed to embed in a manifold of 2N dimensions. A manifold of N dimensions only embeds in one of N+1 dimensions if there are some conditions met, such as orientability.

      A torus is in one embedding a donut, really dough-naught, and we can map that to a square area. These then tile up in a general embedding of R^N inside N^{N+1}. The curvature of the donut has both positive and negative parts. The inner part is a saddle shape with negative curvature and the outer is elliptical with positive curvature. The average curvature is zero, just as a square tile is. These positive and negative curvatures are manifestations of embedding, sometimes called extrinsic curvature given by Gauss' second fundamental form or the Gauss-Codazzi equations.

      Wilhem de Sitter found ways 4-dim spacetime could embed in 5 dimensiions. This led to the de Sitter and anti-de Sitter spacetimes. These can be visualized as a cone with an outer continuous hyperboloid, the de Sitter space, and two hyperbolic regions in the + and - cones, which are the anti-de Sitter spacetimes.

      The role of AdS_5 is found in the Maldecena result on AdS_5xS^5 and the symmetries of AdS_5 are equivalent to conformal fields CFT_4 on the conformal boundary of AdS_5.

  27. Ah. I think Lawrence rather meant the Whitney Theorem?

  28. Still one more question. Whitney's theorem requires generically 2N => 8 dimensions for global embedding of a 4D spacetime. Sabine mentioned 10 D. I am still missing two... Any additional hint? Thanks

    1. Well space-time isn't Euclidean. The 10 is from Stephani's GR textbook. It isn't further explained in the book, but I think it can't be more than 10 just because the metric has only 10 independent components in 4 dimensions. Having said that, as Lawrence points out, it is certainly the case that specific space-times can be embedded into lower-dimensional spaces. Now, how this fits together with Whitney's theorem, I don't know.

    2. Hi again,

      I just coincidentally saw that Stephani's textbook actually pre-dates the theorem. So quite possibly he didn't know of it. Ie, I guess that 8 is actually the correct number. (Though of course if you can embed it into 8 you can also embed it into 10...)

    3. For spacetime that is orientable our spacetime can be embedded in 5 dimensions. This is what de Sitter worked, where he looked at a metric

      s^2 = t^2 + u^2 - x^2 - y^2 - z^2,

      a flat 5-dimensional spacetime. For the signature of u positive the constraint equation on this, setting s = constant gives a hyperboloid exterior to a light cone, or de Sitter space. For the signature negative it gives two hyperboloids in the light cone. This is anti-de Sitter space.

      The 10 dimensions come from some technical world with the bosonic string in 26 dimensions. It the fields are put in a supersymmetric doublet this constrains the theory further into 10 or 11 dimensions. John Baez, an exponent of the idea of E8 or octonions in physics, wrote a paper first decade of the century on how 8 dimensions of octonions is related to 10 or 11 dimensions.

    4. Sorry, I got confused about what we are talking about. The thing is, I actually decided to *not* include the comment about the embedding space in this video. I only just saw that I accidentally left it in the transcript. The comment will appear in the video coming up on Saturday, alas without the remark about the numerical coincidence. I'll add a note in the info below the video that actually the number should be 8, not 10.

      As to the 10 dimensions of string theory, as Lawrence says, different argument entirely.

  29. Yes I know about the string the 10 (Supersymmetry and / or conformality)/ 26 (bosonic / closed strings). I was really interested in the coincidence (as well as the KK extension to SM that would also be 10 (per part 1), or 8 if the extra dimensions could be shared or 7 if local embeddings at each point was sufficient, ...).

    Ok I appreciate the answers.



    PS BTW I found a copy of Stephani's book :)

  30. It is for sure complicated …
    You may also like: AN EMBEDDING FOR GENERAL RELATIVITY AND ITS IMPLICATIONS FOR NEW PHYSICS - Embedding of GR solutions in a 5D Einsteinian spacetime (or flat per So solutions of GR also seem to support just N+1 dimensions. I assume it is a local embedding but to be double checked.
    Reading Stefani's book, I had to go to the original paper: It seems that 10D is the maximum extra dimension to embed with the metric considerations proposed by Stephani. These are different constraints than Whitney. And BTW it seems also a local embedding not global (above 1)D, I guess many solutions exist). 8D is solution of the Stephany problem for 4D spacetime where Killing vectors exist. But again it is for the problem formulated by Stephany; but 8D match Whitney (Except that it is local and so the generic lower limit is 5D for GR solutions).
    Amazingly, yes global embedding and Whitney was not known and so the estimate in the original paper is: at most 87 D (: ) ) for global embeddings …
    Anyways; 10D vs. 8D comes from different constraints on the embedding spacetime. So 8D is sufficient globally but to uniquely define the metric and have isometry as proposed by Stefani, it’s 10D or less.
    The maximum/at most and or less qualifications are still puzzling me.

    Thanks for teh chat It was fun and interesting

    1. Hi shm,

      Yes, thanks for pointing out. I never really looked into this, now I am wiser :)


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