Popular science literature is replete with the necessity for superstring theories to live in at least 10 dimensions, requiring at least 6 of them to be compactified so that they are not observable. But can they predict that 4 dimensions are not compactified? Or conversely, for a superstring theory in e.g. 10 dimensions, are there arguments against compactifying 7 dimensions? or only 5? Arguments other than being ruled out by observation I mean.  My question has been marked as a duplicate of . Only the answer  is relevant to my question. It quotes a paper  by Karsh and Randall which seems to address my exact question:     We propose a new selection principle for distinguishing among possible vacua that we call the “relaxation principle.” The idea is that the universe will naturally select among possible vacua through its cosmological evolution, and the configuration with the biggest filling fraction is the likeliest. We apply this idea to the question of the number of dimensions of space. We show that under conventional (but higher-dimensional) FRW evolution, a universe filled with equal numbers of branes and antibranes will naturally come to be dominated by 3-branes and 7-branes. We show why this might explain the number of dimensions that are experienced in our visible universe.   But this is far too advanced for me as I can barely recognise buzzwords! And answer A10723 is of rather low quality, just a vague description of the goal of the paper which does not give any hindsight into the physics in there. In any case, this also mean that the other question, , mine is marked as a duplicate of, has even poorer answers than they look like at first glance. Following the  link on the page of Karsh and Randall paper turns out several citations which look to address my question and Q10075, but again at a level I can't grasp.
1376
I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a  on Wikipedia.  For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for &lt;3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two.  Am I being overly skeptical in thinking that while the force of gravity may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension?  Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self-aware structures form through completely different processes.  While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?
59768
I have often heard it said (by professional cosmologists) that if the universe is infinite, then there necessarily exist infinitely many copies of me repeated throughout.  The reasoning seems to be that any finite volume of space can contain at most finitely many fundamental particles existing in finitely many configurations.  From here it is implied that all possible (permissible) configurations must both exist and occur infinitely often.  This does not sound like a sound argument to me, so I have to believe that I have misunderstood the argument.    Is this unsettling implication true in an infinite universe, and if so how is it justified.
62043
My question has a silly formulation, but I want to know if there is some sensible physical question buried in it:   Suppose an exact copy of our Universe is made, but where spatial distances and sizes are twice as large relative to ours. Would this universe evolve and function just as ours?   Since mass is proportional to volume which is distance cubed, but strength of a rope is only distance squared, I think not everything would scale proportionally.   Does this mean that a universe with our physical laws that evolves like ours can only have one particular size? Or would the physical laws scale proportionally so that it evolves in the same way? If it has a particular size, what is it relative to? Another variation: suppose another exact copy of our Universe is made, but where everything happened twice as fast relative to ours. Would it evolve in the same way as ours?
39485
Here the lower box dimension and the upper box dimension are exactly what Falconer talks about in his book "Fractal Geometry". I already know that the upper box dimension is finitely stable since $N_\delta(A\cup B)\le N_\delta (A)+N_\delta(B)$, where $N_\delta(A)$ is defined as usual, and Falconer gives an example to show the box dimension may not be countably stable. (We know that, however, the Hausdorff dimension is countably stable).  But now I want to find the example to show the lower box dimension may not even be finitely stable.It seems that we can use von Koch curve to illustrate this, but what is the explicit explanation for this? Since the lower box dimension is monotonic, we need to find set A and B and show that   $\underline{\dim}_B(A\cup B)&gt;\max\{\underline{\dim}_B(A),\underline{\dim}_B(B)\}$.
98862
I'm totally stuck with the following question, I even don't know how to start:  For each natural number $n$ we consider a space $X_n$ that is obtained by removing $n$ distinct points from $\mathbb{R}^2$. We consider the 1-point compactification $X_n^+$ and we denote by $\infty_n\in X_n^+$ the point at infinity (so that $X_n^+=X_n\cup \{\infty_n\}$). Show that   $X_n^+$ can be embedded in $\mathbb{R}^3$. I do not have to write down explicit formulas for the embedding, but I have to explain my reasoning using pictures and mention what result(s) I use in order to reach my final conclusion If $X_n$ and $X_m$ are homeomorphic, then $n=m$.    Both questions I find very hard to answer. I really hope someone can help me!
97657
This question was triggered by a comment of Peter Shor's (he is a skeptic, it seems.)  I thought that the holographic principle and AdS/CFT dealt with that, and was enough for Hawking to give John Preskill a baseball encyclopedia; but Kip Thorne is holding out, as are others.  Why?
20305
Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's The Fabric of the Cosmos. I'm at chapter 7 and so far I understand most of general ideas he has talked about. Or at least, I think I understand them :-)  There is however one part, at the end of chapter 6 that I can't grasp.  It is about entropy and the state of the universe a few minutes after the big bang.  On page 171 he says:     Our most refined theories of the origin of the universe -our most refined cosmological theories- tell us that by the time the universe was a couple of minutes old, it was filled with a nearly uniform hot gas composed of roughly 75 percent hydrogen, 23 percent helium, and small amounts of deuterium and lithium. The essential point is that this gas filling the universe had extraordinarily low entropy.   And on page 173-174:     We have now come to the place where the buck finally stops. The ultimate source of order, of low entropy, must be the big bang itself. In its earliest moments, rather than being filled with gargantuan containers of entropy such as black holes, as we would expect from probabilistic considerations, for some reason the nascent universe was filled with a hot,   uniform, gaseous mixture of hydrogen and helium. Although this configuration has high entropy when densities are so low that we can ignore gravity, the situation is otherwise when gravity can't be ignored; then, such a uniform gas has extremely low entropy. In comparison with black holes, the diffuse, nearly uniform gas was in an extraordinarily low-entropy state.   In the first part of the chapter Brian Greene explains the concept of entropy with tossing the 693 pages of War and Peace in the air.  At first, the pages are ordered. The specific order they are in make sense and are required to recognize the pages as a readable book called War and Peace. This is low entropy. It is very highly ordered and there is no chaos.  Now, when you throw the pages in the air, let them fall and then pick them up one by one and put them on top of each other, the chances you get the exact same order as the initial state are extremely small. The chance you get another order (no matter what order, just not the one from the beginning) is extremely big. When the pages are in the wrong order, there is high entropy and a high amount of chaos. The pages are not ordered and when they are not ordered you would not notice the difference between one unordered state and another one.  However, should you swap two pages in the ordered, low entropy version, you would notice the difference.  So I understand low entropy as a highly ordered state with low chaos in which a reordering of the elements would be noticeable.  I hope I'm still correct here :-)  Now, what I don't understand is how a uniform mixture of hydrogen and helium can by highly ordered? I'd say you wouldn't notice it if some particles traded places. I'd say that a uniform mixture is actually in a state of high entropy because you wouldn't notice it if you swapped some hydrogen atoms.  Brian Greene explains this would indeed be the case when gravity plays no important role, but that things change when gravity does play a role; and in the universe right after the big bang, gravity plays a big role.  Is that because a reordering of the particles would change the effects of gravity? Or is there something else that I'm missing here?
22139
We know that space-time dimensions are 3+1 macroscopically, but what if 2+2?  Obviously it is tough to imagine two time dimensions, but mathematically we can always imagine as either having two parameters $t_1$ and $t_2$ or else in Lorentz matrix $$\eta_{00} = \eta_{11} = -1$$ and, $$\eta_{22} = \eta_{33} = 1.$$  Is there any physical reason for not taking this, like the norms become negative or something else?
7096
Ok so the universe is in constant expansion, that has been proven, right? And that means that it was smaller in the past.. But what's the smallest size we can be sure the universe has ever had?  I just want to know what's the oldest thing we are sure about.
47666
I answered a question here with a quote with the following quote from Cosmologist George Ellis.     “Similar claims [about a multiverse] have been made since antiquity by many cultures. What is new is the assertion that the   multiverse is a scientific theory, with all that implies about being   mathematically rigorous and experimentally testable. I am skeptical   about this claim. I do not believe the existence of those other   universes has been proved—or ever could be. Proponents of the   multiverse, as well as greatly enlarging our conception of physical   reality, are implicitly redefining what is meant by ‘science.’”—pg 39      “For a cosmologist, the basic problem with all multiverse proposals is the presence of a cosmic visual horizon. The horizon is   the limit to how far away we can see, because signals traveling toward   us at the speed of light (which is finite) have not had time since the   beginning of the universe to reach us from farther out. All the   parallel universes lie outside our horizon and remain beyond our   capacity to see, now or ever, no matter how technology evolves. In   fact, they are to far away to have had any influence on our universe   whatsoever. That is why none of the claims made by multiverse   enthusiasts can be directly substantiated.”—pg 40-41      “A remarkable fact about our universe is that physical constants have just the right values needed to allow for complex structures,   including living things. Steven Weinberg, Martin Rees, Leonard   Susskind and others contend that an exotic multiverse provides a tidy   explanation for this apparent coincidence: if all possible values   occur in a large enough collection of universes, then viable ones for   life will surely be found somewhere. This reasoning has been applied,   in particular, to explanation the density of the dark energy that is   speeding up the expansion of the universe today. I agree that the   multiverse is a possible valid explanation for the value of this   density; arguably, it is the only scientifically based option we have   right now. But we have no hope of testing it observationally.”—pg 42      “All in all, the case for the multiverse is inconclusive. The basic reason is the extreme flexibility of the proposal: it is more a   concept than well-defined theory. Most proposals involve a patchwork   of different ideas rather than a coherent whole. The basic mechanism   for eternal inflation does not itself cause physics to be different in   each domain in a multiverse; for that, it needs to be coupled to   another speculative theory. Although they can be fitted together,   there is nothing inevitable about it. … Nothing is wrong with   scientifically based philosophical speculation, which is what   multiverse proposals are. But we should name it for what it is.”—pg 43   Now with this in mind if these theories cannot be independently verified should their truth not lie in  method outside physics or what is also known as meta physics?
29204
I can think of three general ways of explaining why the universe contains more matter than antimatter:  (1) Near the Planck time, the universe had zero baryon asymmetry, but at some later time, determined by some GUT energy scale, the  were satisfied, and the baryon asymmetry became nonzero.  (2) Nonconservation of baryon number occurs only at Planckian energies. Near the Planck time, the baryon asymmetry evolved from zero to some nonzero value.  (3) The Sakharov conditions have never been satisfied. The baryon asymmetry has always been nonzero, and has simply scaled as expected. (Apparently one  $\Delta n=n_B-n_\bar{B}\propto s$, where $n$ is number density and $s$ is entropy density).  It seems like most theorists are interested in #1, but is there any reason why 2 and 3 aren't possible?  2 seems pretty reasonable, since for the reasons given in , we have good reasons to think that baryon number is not conserved under Planckian conditions.  3 also seems reasonable to me, since even if baryon number is nonconserved at Planckian energies, that's only one of the three Sakharov conditions. I don't see any obvious fine-tuning objections to #3, since the scaling of baryon asymmetry with cosmological expansion isn't particularly drastic (not an exponential decay or anything). Is there something unphysical about maintaining $\Delta n\propto s$ all the way back to the Planckian era?   Some people might object to #3 on aesthetic grounds, since we "expect" the initial conditions of the universe to be symmetric, but that seems weak to me. After all, we don't object aesthetically to the fact that homogeneity was an imperfect symmetry of the early universe, and we even accept that the early universe was in a thermodynamically unlikely state.
105416
A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character.  To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS/CFT and String Theory the following sentence often appears: "The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$".  The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov/Nambu-Goto action) to a low-energy effective description. Of course, I might be asking a question for which the only concise answer is: "go read GSW". Here's hoping otherwise ;)
70324
Wasn't the density of the universe at the moment after the Big Bang so great as to create a black hole? If the answer is that the universe/space-time can expand anyway what does it imply about what our universe looks like from the outside?
15479
It seems there are a lot of respected physicists appearing on pop-sci programs (discovery channel, science channel, etc.) these days spreading the gospel of   Three examples, quickly: 1) Many programs feature Michio Kaku saying that he is on a quest to find an equation, "perhaps just one inch long," which will "describe the whole universe."  2) Max Tegmark has come out with  in which he expresses the gut feeling that "nothing is off-limits" to science.  The subtitle of this book is My Quest for the Ultimate Nature of Reality.  3) In the series  there is talk about a search for the "God equation."  (A good counter-example would be Feynman, but his self-described "non-axiomatic" or "Babylonian" approach does not seem popular with physicists today.)  Is there any sense among physicists that it might be impossible to articulate the "ultimate nature of reality" in equations and formal logic?  It seems to me that physicists are following in the footsteps of the 19th century mathematicians (led by Hilbert) who were on a similar quest which was put to rest by Gödel's incompleteness theorems in 1931.  Is there any appreciation for how the Incompleteness Theorems might apply to physics?  Has any progress been made on  for the 20th century?  Shouldn't this be addressed before getting all worked up about a "God equation?"
96383
In  (to ) it has been noted:     One more point to make about the paradoxical decomposition to more parts than elements [...]   currently we do not know of any model of +$\lnot$ where such decomposition does not exist.    Namely, as far as we know, in all models where choice fails there is some set which can be partitioned into more parts than elements.   (Just for clarification: From the context of the above references the comparison "more parts than" is obviously meant in terms of the  of the suitable  being strictly larger than the cardinality of the suitable initial set itself.)  Now, I find the negation or denial of such "paradoxial partitioning" interesting; i.e. the statement (proposition, "$\lnot$PP"):  "Each partition of each given set has cardinality less than, or at most equal to, the cardinality of the given set."  And I like to further explore how this suggested "proposition $\lnot$PP" (being considered along with ZF, of course) relates to "the standard set theory including Axiom of Choice", . Therefore  My questions:      Are there any models of ZFC known (or could there be any such models, in principle) in which there is some set which can be partitioned into more parts than elements ?  And:  Are there any models of ZF known (or could there be any such models, in principle) in which there is no set which can be partitioned into more parts than elements, and which is not also a model of ZFC ?  (And just for reference: Is there a conventional or concise way of expressing the suggested "proposition $\lnot$PP" in terms of standard notation, such as used in the sources linked above? Has it perhaps been discussed already, by some other name? ...)
60494
The AdS/CFT correspondence states that string theory in an asymptotically anti-De Sitter spacetime can be exactly described as a CFT on the boundary of this spacetime.   Is the converse true? Does any CFT in a suitable number of spacetime dimensions have an AdS/CFT dual? If no, can we characterize the CFTs which have such a dual?
44215
As a start I assumed  which is the reciprocal of  as the frequency upper limit, but not so sure. On the other hand, is there a lower limit? Can we consider the reciprocal of our universe as a theoretical limit which is not yet limited, or just assume 0? I am just curious to know if you have any ideas with thanks.