The Biggest Ideas in the Universe | Q&A 4 - Space

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll and today we're doing the Q&A session associated with video number four which was space or at least idea number four the videos have been accumulating with the intro and the q and A's and everything and so I realized when we were talking about space when I did the video for that I realized that I didn't organize things in the ideal way this happened sometimes right casual informal this was as advertised so what I did was I started with the idea of dimensions of space three-dimensional space why is it like that could it have been different of their dynamical mechanisms that could change the number of dimensions and then I went on to Hamiltonian mechanics the position and momentum way of thinking about things yet another formulation of Newtonian classical physics and then asking the question why is position somehow special compared to momentum trying to answer it by saying because interactions in the real world are local in position but not in momentum so the second part the Hamiltonian momentum part is more important in some very real sense than the first part for one thing it's true like it's true that space is three-dimensional but all the different ideas that I talked about about why space is three-dimensional are all entirely speculative and they could all be completely wrong it's important to think about them it's important to appreciate that these ideas are out there and are contemplative all but I wouldn't put too much credence in any of them right now the state of the art is just not that good and second this difference between position and momentum is a really deep fact that people don't always appreciate and I think needs more attention as we go forward into trying to understand the fundamental nature of reality trying to understand the ultimate theory of everything quantum gravity and so forth even just understanding quantum mechanics this relationship about why position is singled out why we live in position space not momentum space is something that needs a lot more attention but by putting it second and by putting all this cool fancy speculative stuff first people spend a lot more time and attention on the number of dimensions of space question that's fine you know but I do want to sort of flip the switch for the Q&A video because I want to emphasize some things about Hamiltonians and momentum because that's really you know I think I think it is the more important part of this particular dialogue so let me talk a little bit about oops let me talk a little bit about phase space and hamiltonians well what I want to do here really is nothing that I didn't quickly mention in the first video but I want to emphasize that I want to go a little bit more detail so it really sinks in remember that the idea phase space was to plot something like X and P now X is position despite the fact that it begins with the letter P P is momentum and I'm suppressing dimensions here so if you live in a three-dimensional space there's three dimensions of X also three dimensions of P there will always be the same number of dimensions of X and P and then you have at every point you have the initial conditions for some physics problem you have the state of the system at one time if your Laplace is deamon the information you need to know besides the laws of physics is where are you in phase space it's from that that you can chug forward in time and figure out what will happen next in the Hamiltonian way of doing things what you do is you define a function H which is a function of X and P and it's basically the total energy so it's basically the kinetic energy plus potential energy of the system and the wonderful thing is from that from knowing what the Hamiltonian is just one function you can figure out all of the laws of motion of the system what do I mean by that why is that important or why do we care you know in Newtonian mechanics we wrote F equals MA and unlike before where I was just trying to keep things simple and approachable I'm drawing little vector signs over the force and the acceleration because they both live in three-dimensional space there's an acceleration and a force separately in each direction of space so even though this looks like just one equation there are three separate components to this equation okay it's a vector not what we call a scalar equation which we just have one component and there's also the relationship P equals M times V momentum is mass times velocity again three components for those equations of motion and the nice thing about the Hamiltonian is that all these six equations can be derived from this single function the Hamiltonian that's kind of an amazing thing that kind of compactification if you like from many equations into just one that kind of unification is what physicists like to see that's evidence that something cool is going on so what I wanted to do is sort of fill in some of the details I already mentioned this before I want to go and be a little bit more explicit just to drive at home so what I said last time was what you have is you know your position and your momentum so what you want to do is chug those forward in time you want to know what are their rates of change what are their derivatives in time and then you could just do calculus and figure out where the system goes it's going to go on some trajectory in phase space so you have X you have P what you want are DX DT this is the rate of change of position with time which most of us would ordinarily call the velocity V okay that's one thing you want and the other thing you want is the same thing but for Momentum's you want D P DT the rate of change of momentum and again in the Newtonian way of thinking about things we would say well position is mass times velocity the rate of change of velocity is acceleration mass isn't changing at all we're gonna simplify our lives thinking about it that way I once wrote a whole paper with my colleague Greg Anderson on particles whose mass changed with time which you could get in some cosmological scenarios you had there like we do all Newtonian mechanics because everyone always assumes that the mass is constant but anyway if P is ordinarily MV and the derivative of velocity is acceleration DP DT is MA which is just the force right so in other words the equations that we ordinarily get what I gave you what it is told you is true in Hamiltonian mechanics is that DX DT is P over m that is what you get from Hamiltonian mechanics and indeed looking at both sides of this equation P over m equals V is the same as P equals MV which we think is true and likewise what you get for the rate of change of momentum is the force and indeed F equals MA that's also true well I didn't quite tell you at least I don't remember telling you is how to actually get them from the Hamiltonian I said I said it in words but not in equations the answer is what you do is you take the slope of the Hamiltonian function in these two different directions okay in the X direction and the P direction in calculus you know we talked about taking the slope of a single curve that's the origin of calculus the beginning the first most obvious thing you do the derivative okay so now we're taking the derivative of a function of more than one variable it's not just X as a function of T that we're taking the derivative of its H as a function of X and P so what we do I'm gonna simplify my life here by erasing this what we do is what take what is called a partial derivative which means we hold everything constant except for one direction we can move in and we examine the slope in that direction so it's like you're standing on a hillside you can walk down the slope or you can walk up it or you can walk perpendicular so the slope is more or less zero right those are all different amounts of slope depending on what direction you go in this is the mathematize ation of that so the slope of the hamiltonian in the X Direction is called the partial derivative of H with respect to X funny dhdx and the partial derivative in the P direction is called funny D H DP so these funny DS which mean partial derivative I think we're invented by Laplace no no applause live knits the competitor to Newton to invent calculus they're they're sort of related to the Delta which is the actual Greek letter these funny DS that are written like this are called Dells and they're only used ever in calculus they're not letters in anybody's alphabet they're just special calculus symbols and what they mean is hold everything constant move only in one direction hold everything except one direction constant move in that direction calculate the slope this is known as the partial derivative okay and that's where we get these equations for the change of X and the change of P DX DT in Newtonian in Hamiltonian mechanics turns out to be the partial derivative of the Hamiltonian with respect to position with effective momentum so are these vector momentum so dxdt equals D H DP that's good what if I said what is the rate of change of momentum well you're probably going to guess given that there's some kind of symmetry between X and P that it's the partial derivative of H with respect to X with respect to position you are almost be correct it's minus that okay why is it minus that you know there's a deep Matt the answer that involves the symplectic structure on the manifold that we call phase space none of which we're gonna get to or at least unless this series of videos goes on for about 130 videos we're probably not going to get to the symplectic structure on phase space here but what we have well you'll just trust me is true is this equation relating momentum to the Hamiltonian and the derivatives of the Hamiltonian so there's us some kind of symmetries a little bit more complicated than just replacing X with P but the point is from this one function H you can just take its slope and figure out how fast position will change how fast momentum will change and that's all you need to move forward in time to integrate to get the solution that's how Hamiltonian mechanics work and what I really want to do here is dig a little bit deeper than that by doing an example you know like the spirit of these videos is not doing worked examples very much including all the math but you know sometimes that's kind of healthy thing to do so let's actually do it let's do an example of a Hamiltonian and how to get the equations of motion from it so what's the simplest example we can do the example would mean by an example well we have some function of X and P which is supposed to be the energy kinetic energy plus the potential energy we're going to pick the simplest function that we can pick right well what is the kinetic energy it's one-half MV squared and we said already that if you already plug in P equals mV that is 1 over 2m P squared that's the kinetic energy plug in P equals MV you could check that and then the potential energy you said is just gonna be some function of X but we don't want to be that abstract or that general we want to do a specific example so you want to actually pick some function let's just pick a nice symmetric looking function about 1/2 K x squared ok so just like there's a P squared there's an x squared and this looks very kind of even-steven in terms of what the position is doing and what the momentum is doing in this example in fact this example is super famous this is the simple harmonic oscillator why is it famous two reasons one because it's simple enough that you can solve all the equations exactly the other is things that are approximately simple harmonic oscillators appear all the time in the real world so let's imagine you have a wall here surface and then there's a box on the surface and it's frictionless surface and there's a spring connecting the holding box to the wall so there's some position at which we'll call x equals 0 where it's an equilibrium the springs neither pushing it or pulling it K can be thought of as the strength of the spring the spring constant and then when you move the box in one direction or another the spring will either push it back or pull it back to where it came from and if you push this back and forth this gives it has a potential energy V of X is in fact 1/2 K x squared so it looks like this it looks like a parabola as a function of X this is the simple harmonic oscillator that is its Hamiltonian so when you have questions like why is the Hamiltonian that why is it that function of X and P the answer sadly is well that's the Hamiltonian that works to describe the physical behavior of this system right what you're really asking is why is the world like that and so sometimes that's the kind of question that has an answer sometimes it's not right now we're just gonna say the world is like that for this particular little world with a simple harmonic oscillator in it okay and we what we want to do is we want to actually find how these equations the rate of change of position is the partial of H with respect to P where change momentum is minus the partial of H with respect to X so how they work for this particular system the simple harmonic oscillator okay we're actually gonna do the math and so one little clue I'll give you one clue one math clue you know we're not about solving equations here in these videos but we want to do it now if you have a function y equals x squared and you want to take the derivative I'm going to take the slope the answer is dy/dx equals two times x more generally if you have X to some power when you take its derivative you bring that power down in front and you lower that power by subtracting 1 so 2 minus 1 is 1 X to the 1 is just X so the derivative y respect to X is 2x squared so what do you want to do we want to get DX DT we want to set that equal to the partial derivative of the Hamiltonian with respect to P so this means keep X constant if I look at the Hamiltonian if I keep X constant I just fix the one-half KX squared that doesn't do anything at all all it affects anything is the P part and I differentiate P in fact a differentiate piece so what I get is one over 2m times derivative P squared which is 2p and guess what that is just P over m that looks familiar doesn't it because that's what we said was the rate of change of position is P over m but we didn't assert it here we derived it we derived it from this very simple form of the Hamiltonian likewise if we said what is DP DT how quickly is the momentum changing over time that's minus the partial derivative of H with respect to X so that is - where's H over here here it is minus 1/2 K and then the derivative of x squared is 2x so it is minus K X what that means is exactly what our picture is showing here if we had this box displaced a little bit okay so that X is some positive number here's now x equals 1 let's say okay well this is saying that the force that is acting on our box is minus that displacement so the force is pulling the box back just like it should be doing so this is nice I mean I know it's math what some of you love math and this is all trivial and kind of boring some of you hate math and this is scary and intimidating no matter what the point is we can use these mathematical techniques to start from an extremely simple expression right here and derive all the equations of motion in a very very simple way in fact let me mention that what I said in in the previous videos was there's something called the Newtonian way of doing things it's a little bit confusing because sometimes the phrase Newtonian mechanics just means classical mechanics sometimes it means Newton's actual way of doing classical mechanics F equals MA right now I'm using it in the latter sense and we argued or I said that Newtonian mechanics is equivalent to something called the principle of least action it's a different way of thinking about classical mechanics and I also said that it's equivalent to this Hamiltonian way and this is in fact the demonstration what I just did with the math is a demonstration that you could derive Newtonian mechanics from the Hamiltonian it turns out also to be true that he could skip the Newtonian step and relate to the action to the Hamiltonian directly that's another way of doing it in fact Hamilton who came up with the Hamiltonian also came up with sort of the modern version of the principle of least action which is sometimes called Hamilton's principle but there's there's details and people argue about the nomenclature etc anyway all of this is to say this way of doing physics is very powerful certainly when you look at equations like this for the Hamiltonian PNX look very very symmetric there right but when you look at more complicated systems when you look at systems that are doing something other than being a simple harmonic oscillator you typically add more terms with X's in them you change the dependence on X you change the potential energy but you don't change the 1 over 2m times P squared the kinetic energy just is the kinetic energy usually there's always exceptions so that's where the asymmetry comes in at this level momentum and position look exactly the same it's only when you complicate the world and have many moving parts and many different potentials and many different forces that position and momentum start looking very very different ok that was all to make myself feel better because I wanted to give you more details about Hamiltonian mechanics let me actually answer some of the questions that were addressed after the video came out so questions one very simple one was about entangled particles I'm gonna write the question let's go say it in words because it takes too long so I wrote about entangled particles and I said I forgot which notation I used but if you have spin up and spin down I said look they could both be spin up or they could both be spin down or there could be some combination of both and a lot of people wrote in more than one person and said wait a minute I thought that when particles are entangled spins are always opposite so it's one is up one is down or one is down one is up the answer is these are two different possibilities that are both real possibilities the fact of entanglement has nothing to do with whether the spins are in the same direction or the opposite direction it's just about the fact that they are related to each other in some way so if your state was just this the spins are up both spins are up okay that's not entangled at all the one spin is up and that's completely independent of the fact that the other spin is also up right there just true to true facts these are entangled because once you measure one spin you have learned something about the other spin whereas this is not entangled in fact let me write another one here is one spin is up and the other spin is in a superposition of spin up and spin down so I wrote that badly sorry the whole point here is this is not entangled and this is not entangled either you don't learn about one spin by measuring the other one so you could have all sorts of different things it is true that this particular state where the spins are in opposite directions is a very popular one it appears a lot in a lot of real physical situations it's often the lowest energy state for these two spins to be in but it's not the only state we can be in all sorts of things that was one question here's another question how can a dimension be small so the reason why this is a question you know I said dimensions can be small but I also said that dimensions are not places to go like sometimes you know you watch Doctor Strange and you you know you see that there's a dark dimension with all the demons in it and you think of a dimension as a location out there in the universe and I try to emphasize that in physics a dimension is just a direction in which to go so people some people thought that there there seems to be some tension between that and they said you know how can a direction be small right I mean a direction exists at every point there's all the different directions you can go in so that's a very good question because it's I'm you're you've caught me this is using terminology in the slapdash ways that physicists often use it the example you know I gave was something like a cylinder very long skinny cylinder okay so here you have X in this direction and maybe Y is the direction that wraps around at any one point on the cylinder there are absolutely two directions you can move it right x and y this is two let's see it's not like X is better than Y or anything like that but at the same time if you go in the Y direction there's only so far you can go before you come back to where you left so we use the word dimension sometimes when we say small dimension we don't mean there's a direction you can move in that is small we mean there's a direction you can move in such that if you move in that direction you can only go so far okay that's what the phrase a small dimension of space-time means and the reason why that's an interesting thing is because if you imagine making that dimension smaller and smaller right you imagine making that circle shrinking down to smaller and smaller sizes it becomes indistinguishable from that dimension not being there at all so when you can only observe things at large distances the existence of a dimension of space that is much smaller than you can observe might as well not be there at all so there's a relationship in quantum mechanics and this is getting us too far afield but maybe in future videos this will be become clear small distances correspond to high energies and large masses of particles so big particle accelerators that can reach high energies are like microscopes that are looking at smaller and smaller distances but there is a limit right any one accelerator only goes up to a certain energy so above the energies that particle accelerators have ever probed if there is a dimension of space smaller than the distance corresponding to those energies they will be invisible to that is if the dimensions are at the Planck scale which is something like 10 to the minus 15 times the smallest distance we've ever probed we would not be able to see them at all roughly speaking that's what it means for a dimension or direction to be small okay another good question was string theory which we mentioned as a potential theory of gravity and other forces of nature how many dimensions and this is another great question perfectly a great question in the sense that is a perfectly valid question you're perfectly right to be confused depending on what you've heard about string theory before you might have heard that string theory has 26 dimensions you might have of space-time dimension the space-time in this case not just of space you might have heard that it has ten in way to further that it has eleven okay why do you have all these different numbers so the answer is they're just different versions of string theory or if you like there are different theories that can descend from string theory in some particular situation so the 26 dimensional number comes from what we call boson ik string theory you might have heard that in nature when we have fields or particles they come in two basic varieties bosons and fermions bosons can pile up on top of each other so like Higgs bosons photons and gravitons or bosons W and Z bosons fermions take up space so like electrons neutrinos quarks etc or fermions it's easier for various technical reasons for field theorists to work with bosons than with fermions so when string theorists first wrote down string theory they didn't put fermions in and what they found was their theory of boson ik strings only works in 26 dimensional space-time okay my only works what I mean is there's some symmetries that are really really crucial to the internal consistency of string theory and when you quantize the theory it turns out that those symmetries get broken unless you're in the right numbers of dimensions so this is called the critical dimension of string theory the the number of dimensions oops just dimension singular the number of dimensions in which string theory works quantum mechanically the symmetries are not broken but then when you add fermions it turns out that though again the weight that works the way that is consistent with all the symmetries etc you add an equal amounts of fermions and bosons and you get supersymmetry and this is where you get super strings super strings naturally live in ten dimensions that's the number of dimensions of space-time in which super strings can propagate and interact with each other consistently coherently not breaking any symmetries the eleven dimensions is kind of a sneaky thing so before there was super string theory there was super gravity super gravity is not a string theory it is a version of general relativity of Einstein's theory of gravity that has been made super symmetric okay so there's no strings involved it's field theory but it's super symmetric field theory supersymmetry I should say I really should tell you what that means supersymmetry is a symmetry between bosons and fermions it is not a symmetry that exists in experiments as far as we have yet been able to tell but it might be hiding from us it might be there we just haven't yet observed it it's very very popular among many theoretical physicists so super gravity is a version of gravity that is super symmetric for the same kinds of reasons as we talked about super gravity naturally lives in 11 dimensions and this is fun actually there's a fun thing to talk about here because it's a little bit of interesting history of physics in the 1980s 1990s there were already people working on super gravity but it super gravity doesn't have all of the nice features that string theory has it's there are certain features of string theory that were hard to prove would also hold just in super gravity so there was a plucky minority of people who worked on super gravity just out of stubbornness and they liked it they they weren't really looked upon very highly by the super string people but then it turns out that super gravity which naturally lives in 11 dimensions contains even though it doesn't contain the strings two-dimensional brains so remember I think I mentioned something like this this is a brain a brain is a two-dimensional sheet instead of a one-dimensional string if you like a string is a one brain a two-dimensional sheet is a two brain a three-dimensional jelly is a three brain there are eight dimensional brains and the whole bit you can imagine anything like this that's a it's a back construction from the word membrane because back you know hundreds of years ago no one invented a word for an eight dimensional object who knows why so it very very naturally supergravity contains these two-dimensional brains but they weren't fundamental they were just you know they seem to be features of the theory but it was hard to know what status they had it so forth then the 1990s joke polchinski especially but a bunch of other people realized that string theory also contains a bunch of higher dimensional objects a brain like objects of all sorts of different numbers of dimensions so the idea that string was strengths areas not just a theory of strings became very popular and then the idea of super gravity with its two brains was sort of reexamined and it was ed Witten who actually figured out ed Witten you know the most famous string theorist alive he realized the following thing imagine you have a two-dimensional brain but you also have one compact dimension of space okay so this is let's say you have eleven dimensions of space-time that's ten dimensions of space let's imagine you have nine dimensions of space that are big and then you have one dimension of space this one that is small so this is where this is a particular configuration in which super gravity could live right and then you can take a brain and you can wrap it around so in fact I realize that this is not the best drawing to make let me unwrap this because I need to draw more than one big dimension I drew one big you mention this way and he'd draw two of them so what I want to draw is something like this so what this is supposed to mean is imagine you're identifying the top and bottom here so this is actually a circle and these are two big dimensions and that's the one small dimension so this is three dimensions that I've drawn here right two big one small I'm suppressing the rest of the dimensions but imagine this is part of an 11 dimensional space-time okay you get that so if I if I zoomed in here what this means is if I'm in this surface if I walk upward and I get to the top I suddenly boom appear back in the bottom it's just like a one dimensional torus which we call a circle okay so this is two big dimensions in one circle and imagine this is part of super gravity so I've suppressed the other what is it if I have nine dimensions of space total No ten dimensions of space total nine of them are big one of them is small good that's what I drew but now I have two-dimensional brains living in this so there's two possibilities one is the two-dimensional brains can sort of stretch across the big dimensions but there's another possibility that they could just like be embedded in the small dimensions so imagine the two-dimensional brain did something like this right so what this is saying is there's two dimensions but one of them is wrapped around the small dimension of space and the other one dimension is stretched into the big dimensions of space then to the point of view of someone who only lives in the big dimensions some low-energy observer that two-dimensional brain looks one-dimensional that is to say it looks like a string so what women realized was you could start with 11-dimensional supergravity compactify it along one dimension and the two-dimensional brains of supergravity would become one-dimensional strings so he realized there's a relationship between string theory and super gravity and the relationship is they're both two different aspects of the same underlying theory which he named M theory that's what M theory is it's the unknown we still understand it 'mother theory magical mystery theory behind it all and the 10 dimensional superstring theories there are five of them if you want to count are all different limits different approximations and different circumstances of M theory and 11 dimensional super gravity is another limit another approximation of M theory so that's the relationship between the 11 dimensions and the 10 dimensions the 26 dimensions just a you know is a good first try okay the 10 10 dimensions and 11 dimensions are real honest but different limits of the underlying M theory was probably more than you wanted to know or more than I wanted to say but this stuff is so cool I just can't resist okay continuing on are there alternatives to compactification what I mean by this is I said are there I said there are ways to hide but extra dimension by making it small are there ways that we couldn't see an extra dimension even if it weren't small even if it were big well the answer is yes otherwise I wouldn't be talking about this and the answer comes exactly from this idea of brains so imagine that you are here's some big dimensions okay and let's three big dimensions of space and let's wrap around these three big dimensions of space let's wrap around a brain okay a two-dimensional brain just to make our lives easy so this is a brain and this is space that's easy enough to do but the important thing is people realized very early on in the theory of d-branes that from boat in ski and others but there could be particles and fields stuck to the brains okay that it's there's every by the way every joke relating B or a and E to be our AI n has been made they've been Papers written about brain surgery and all that stuff okay so forgive me for that I'm not try to make them here anyway you can imagine people people made of particles who essentially are only living on the brains right cannot get out into what is called the bulk the bulk is the rest of the universe so you might think that we don't have to compactify at all what if there's a three-dimensional brain embedded in nine dimensional space of one of those superstring theories or ten dimensional space of super gravity well it doesn't quite work because there is one force of nature that cannot be confined to the brain not at least in ordinary circumstances which is gravity gravity always reaches out into the bulk because gravity is a feature of space-time itself it's not something you can confine to a brain so there are ways that the extra dimension could be bigger than you think but you have to sort of fool around a little bit one way is that we live on a brain and the extra dimensions are bigger than you know an atom but small enough that we don't notice gravity leaking out experimentally these days that's about 1/10 of a millimeter so if the whole size of the extra dimension is smaller than 1/10 of a millimeter and we are stuck on a brain inside that extra dimension that is compatible with experiments as we know it today the other way to do it is if the geometry of space in the bulk is so highly warped that effectively gravity doesn't get out there at all if you change the geometry of space from being flat to being bent then it's harder and harder it can be harder and harder for gravity to get out this was a theory invented by Lisa Randall and Raman Sundaram warped compactification x' and Lisa wrote a whole book about it warped passages again no evidence experimentally that any of this is true there's upper limits on how much it could be true but the point is you can imagine different scenarios and you know in the 1990s this was full employment for theoretical physicists to imagine different ways you could try to hide the extra dimensions these all were wonderful real theories that made predictions and we want to look for the predictions that we haven't seen any of their predictions come true yet it might happen tomorrow it might happen in between when I record this and when the video appears but right now it hasn't happened yet so again all this talk about extra dimensions so far is entirely hypothetical ok getting a little bit more deep if you want it is already pretty deep right but there there was a question about locality and holography which I'm gonna not say a lot about because we don't quite yet have the tools on our plate to think about the holographic principle but I respect this question because it's really completely on point when it comes to locality the point is there are things called black holes here's a black hole and you know black hole will get into it but a black hole has the feature that if you're a person outside if you fall in you can't come back out it's a one-way street ok but Stephen Hawking showed there are particles that will be released Hawking radiation and what that means is that eventually the black hole will evaporate away will lose all of its mass and it will disappear and there's something called the black hole information loss puzzle relating the amount of information quantum mechanically that you could put into the black hole and get it out by the Hawking radiation the reason why I'm mentioning this is because we're not I'm not going to nearly enough details to understand it but the insistence that the information that goes in must also eventually come out is just Laplace's daemon doing the insistent insisting right it's the quantum mechanical version of Laplace's daemon it's physicists being absolutely convinced that there is a conservation of information over time that the information that exists in the universe at any one moment of time is the same at any other moment that's what convinces them the information must eventually come out of the black hole but there's a lot of you know thought experiments associated with this and the whole thing and what people eventually realized Hawking radiation also implies that there's entropy that the black hole has but the entropy turns out to be proportional to the area of the black hole's event horizon a black hole as a sphere or a spherical topology anyway and it has an area for these event horizon and you can calculate the entropy and what that seems to imply is that the amount of information contained in a black hole is distributed on the surface area of the black hole now I'm there's no reason you should believe me when I say that I haven't filled in the details there's a relationship between information and entropy and it's a relationship in the case of black holes between entropy and area whereas if you just had space if you have space and locality if you really believe in locality as I've been pushing here locality might make you think well locality would make you think that at every location in space there's a bit of information or these some bits of information some amount of information and that information is separate from the information that is at some different location in other words the amount of information should scale as the volume of space there are subtleties here because of quantum mechanics and entanglement and so forth but the number of particles that I could put in the box should be proportional to the volume of the box over here with a black hole it seems to be proportional to the area which is a different number so Gerard a tuft Nobel prize-winning physicist then leonard susskind another famous physicist who appeared as a guest on the wine scape podcast proposed this idea of the holographic principle that this relationship between entropy and area was real it was physical it was it was meaningful that there was a real sense which we can think of black holes literally as being defined by information living on their surface area so that seems to be at odds with the idea of locality and all I'm gonna say here is yes it is yes it is and this was this cause has cause continues to cause great consternation there's another version of holography called the ATS CFT correspondence invented by juan malta sane and where things are much more specific and it's perfectly clear that locality is violated in some sense that's a situation where you have one theory one theory of physics okay one Hamiltonian if you want which can be thought of as two different space times two different field theories living on two different universes and those space times have different numbers of dimensions so there's no one-to-one map between locations in one's face time and locations in another so locality has to be violated some way and the upshot of this for just answering this question is when gravity becomes important all these are gravity right a DSC of T black holes the holographic principle this only comes in when gravity becomes important when you have non-gravitational quantum field theory this idea of locality and all the different locations in space being separately defined makes perfect sense the number of things that can happen scales with the volume not with the area gravity changes things and there's actually other reasons to think that once you have quantum gravity locality can't be sacrosanct anyway so rather than saying locality is absolutely the most important thing how can we reconcile these weird features of it I think that we should bite the bullet and say look once gravity is in the game locality is just an approximation locality manifests itself it emerges it's an emergent phenomenon in certain situations where gravity is not that strong and where gravity is strong like in a black hole or in these anti-de sitter spacetime ABS universes locality is not going to be exactly right it's still approximately right in the same in the same set of limiting circumstances so that's the thing we need to understand not how to save locality perfectly but why locality is even approximately true that's why I'm emphasizing these questions about momentum and hamiltonians because we take locality for granted and it's important in our brains to undo that and to remember that didn't have to be that way and to start asking ourselves why it is that way rather than some other way ok so I think I got to go you know even we have we have some of us even in quarantine we have things we need to do so there's a few other questions that were asked that I'm just gonna very very very quickly and in fact I'm gonna love them into one big question how do extra dimensions X DS affect the world we observe this is a great question because I said that while they're so small we can't observe the dimensions directly but there can be indirect effects it turns out the answer is in a lot of ways the parameters of physics so what I mean by that is the mass of the Higgs boson the mass of the electron the charge of the Higgs boson and the electron the number of dimensions of space obviously the number of families of particles you know if you know a little bit about particle physics there's something called the electron and it has a friend of the electron neutrino but then it has a heavier cousin the muon and mule has its own neutrino and there's another heavier cousin the tau has its own neutrino so there's three generations of these particles and in ordinary particle physics that's just a fact when you have extra dimensions you can explain that fact the number of generations of particles can be tied to the ways in which the extra dimensions are curled up and in fact all of these parameters the masses and charges and so forth depend on the size and geometry of the extra dimensions in principle this is the origin of the landscape of universes in the string theory multiverse if you have lots of extra dimensions you can curl them up in lots of different ways and the parameters of physics all look different in those different ways so even if the extra dimensions are invisible they are still very important to us and you know you can go on to ask well do they have anything to do with things like dark matter or dark energy maybe probably not but it's possible people that absolutely explored these things again this is the full employment thing like even when we don't know the final answers we can have very very well posed questions when you have a new idea like extra dimensions it opens up the space of possibilities it makes it very very broad so old questions not even that old but questions like what is the dark matter when you say well I might have extra dimensions could that have anything to do with dark matter the answer should always be maybe until you investigate it very very carefully and that's what we do right now there is no known direct relationship between extra dimensions and dark matter or even dark energy that there's clearly a potential relationship between dark energy and extra dimensions it's too much to go into right now but it could it could possibly exist because the dark energy could be one of the parameters of low energy physics right the energy density of the vacuum clearly can depend on the extra dimensions but we don't have any explanation people tried I've tried other people have tried we've written papers but we don't have any compelling explanation for why extra dimensions would make the dark matter or dark energy have exactly the properties they do so it's still an open question that's why this is a fun thing to do even though we're doing the biggest ideas and pretty basic ideas like the existence of space it's not that hard to go from the existence of space to what are the properties of space to could these help explain some of the most pressing mysteries that we have in physics today

Channel: Sean Carroll

Views: 74,957

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Id: tBWNRo7rmuQ

Length: 48min 4sec (2884 seconds)

Published: Sun Apr 19 2020