The Biggest Ideas in the Universe | 6. Spacetime

hello everyone and welcome to the biggest ideas in the universe I'm your host Sean Carroll and today's big idea is space time that should come as no surprise for those of you who are fans are the biggest ideas we did space and then we did time so now we're gonna do space time and you might want to know well why does this count as a separate big idea if we already did space and time of course secretly space time not so secretly is about relativity and in particular while we'll be concentrating on today is special relativity Einstein came up with two versions of relativity special in general will mention general relativity which is the theory of curved space time and gravity but today we're just gonna be focusing on what is space-time itself what why do you need an idea called space-time if you've already had space and you've already had time and I should say that as usual I'll be taking a slightly idiosyncratic point of view on the whole thing this is not a traditional course we're not going through in order all the big ideas like either a course or a history lesson what we're trying to do is to get at some aspects of these ideas in an interesting and new way so usually when you're introduced to space-time were to relativity there's a certain bottom-up approach in the sense that we start with ideas from Newtonian physics which includes absolute space absolute time and this is you know this is how we think this is how we intuitively think about the world that there's space and there's time and they're kind of very very different and then we show that through a bunch of arguments about symmetries and the speed of light and so forth you end up with space-time and special relativity so therefore since you might have been introduced to that way of doing it before I'm gonna be doing the other way I'm gonna is gonna be giving you the basic ideas of space-time from the top down and then we can show how those basic ideas recover our ordinary Newtonian intuitions and it's not that one way is better than the other I was actually just talking very recently with some colleagues of mine at Caltech about the best way to teach special relativity and there's definitely this difference between top-down and bottom-up approaches different people will learn better from different approaches so there's no right one but this is mine this is what I'm gonna do today so Newton would have told you that there was absolute space he might not have used these words maybe he deduced these words I'm not quite sure an absolute time by which we mean when you say something like what time is it right now at Alpha Centauri 4 light-years away in Newtonian physics there's an answer to that there's an answer to the question what time is it at every single event in the universe whereas if you want to say where something is you also get an answer to that but they don't mix together in any meaningful way so Einstein and in fact not even Einstein so much as his former teacher Minkowski or Minkowski if we're being German about it invented this four-dimensional space-time and of course it works equally well in higher numbers of dimensions as we talked about when we talked about space and why why do you have to do that why do you have to say forget about space and time separately think about four-dimensional space-time it's because the division of space-time into space and time separately still have not a bill the ability to talk and write at the same time you're allowed to do it right you're allowed to say okay here's what I'm gonna slice --is that I'm gonna cut through space-time to define one moment of time I'm gonna call that space but the point is the way you do that is not fixed by nature it is not something that is built in to the fabric of the universe it is a human convention it is something that you decide do you for your convenience you're gonna divide up space-time into space and time in a particular way this does not mean it's wrong to do it okay this idea of a human being coming along and deciding to simplify or conceptualize something in a certain way that is very convenient doesn't mean it's wrong it's a very convenient way of doing it I mean when we go through our lives we're not going to talk relativistically for reasons we will explain but the you shouldn't become too wedded to it especially if you're trying to understand fundamental physics so that's basically why we have to talk about space-time we'll get into more details obviously later but the other thing is how should we understand it and I have a very simple motto for how you should think about space-time space-time is kind of like space what I mean by that well there's a whole bunch of puzzles right and if you've ever been exposed to special relativity at all you I have all these paradoxes you're taught and you know hurts your brain to figure out how to answer them in my opinion essentially all these paradoxes go away if you really think about space-time as a universal thing in other words if you take this top-down approach start with space-time and then ask all your questions rather than trying to shoehorn relativistic situations into old fashioned Newtonian intuition we have intuition about space we need to extend that intuition about space to all of space-time and that will make our lives much easier so let me show you this in action consider the idea of the distance between two points now you might think that's a perfectly well-defined idea in space so let's do space and contrast it with space-time I mean I should say in case it's not perfectly obvious the definition of space-time is just the set of points in space and time so the set of events as we call them located with three numbers which are your locations in the three dimensions of space and a fourth number which is your moment of time so it's a four dimensional manifold we would say a four dimensional space in the notion of space that mathematicians use but the point is is more than that there's structure there's mathematical facts about space-time that come in to how it actually works out so that's what we're actually gonna be exploring in this talk so in space I'll draw space and you might have coordinates right so here's an x-coordinate here's a y-coordinate it doesn't matter how we label them or how we draw them or whatever and you might have points here's point a here's point B I want to get from point A to point B what is the distance between them well you could draw the straight line connecting them and calculate the distance that's something that you could do you could even use Pythagoras theorem if you wanted to however that's the distance you would that's that is only the distance you would actually Traverse while walking from point A to point B if you took of the straight line path no one is surprised to learn that if you take some other path then your distance is different right the distance you actually Traverse along a path is different than the shortest distance path distance between those two points this is so obvious that it's hard even to say people who have a little a domitor right where who are measuring how far they walk or I guess a Fitbit or something like that right they will get different answers walking between the same exact two points if they take different paths that's just again so obvious that we internalize it the distance depends on the path so our motto is space time is kind of like space and in exactly the same way the time elapsed between two events in space time is going to depend on the path so in space time the picture is gonna look very very similar except this is now all of space so it's X but also Y etc and this vertical Direction is what we label as time okay and then rather than a location in space you have an event a an event B so the event a is both a location space and a moment of time and you can travel between a and B that is not surprising and you could travel on a straight line which would be a constant velocity through space okay and then you could figure out well what is the space-time equivalent of the distance that we walked when we were just talking about space because now we're moving through both space and time right we're moving through X and we're moving through T in a sort of space and time separately way of talking about it well the actual the answer is that rather than bringing in a domitor along with you walking through space when you bring here as a clock you measure the amount of time elapsed along the path then you might say well why do I even need to do that I could just figure out what time I left and what time I arrived and that's the time that is elapsed but the whole point of space time is no that is not the time that is elapsed the time that your clock measures along a path is not the same as the coordinate time that you define throughout the universe exactly as over in our little picture of space the distance you travel is not equal to X or to Y when you're traveling along some diagonal path so when you travel along some different path through space-time the duration the amount of time elapsed depends on the path that you take this surprises us this is not what we're used to thinking about if you know if we synchronize our watches and what we're doing at pulling off a good spy caper we expect that if we meet up later even though we've taken different paths through space time our watches will still be synchronized if they're good watches as we'll see in a moment that's because we're all moving very slowly compared to the speed of light but this is the insight that makes space-time kind of like space the time elapsed along trajectories of objects moving slower than the speed of light people particles rocket ships whatever is analogous to the distance along a path okay so what this means is time is not absolute Newton was not correct about that time is something that different people measure one might almost say it is relative to the person moving and what their clock reads that's where the word relativity comes from that is one way in which it comes about the time coordinate which is something Universal we invented it we put it on space time we put coordinates on the universe but the point is it's not equal to the elapsed duration for any particular observer it might be by coincidence equal for some observers but it is not fundamentally the same kind of thing so that's interesting but of course we know that space time is not exactly the same as space right it's not quite the saying that there's there's clearly some differences you know one difference we've already talked about which is that if I have an object here it exists in both space right and left in this way and it is exist in time it is now and now and now but there's a difference in how it exists because there's a continuity through time right you mean the pen the pencil the Apple pencil still exists in more or less the same form from moment to moment in time but if you go in space it exists and exists exists then it stops existing very dramatically and no one is surprised by that it would be very dramatic indeed if the pen just disappeared so where is that coming from where is that difference coming from the answer is that there is a difference between space time and time namely well one way it shows up let's we'll get into the details later namely that for a straight line in space we all know that in space a straight line is the shortest distance path shortest issed that's not good shortest distance but in space time the straight-line is the longest duration right now I'm just asserting this you're gonna believe me because of the I'm the authority figure here we'll explain it a little bit better in just a second but when you go back to this picture and you say okay in the space-time picture when I was going on a straight line I said that will be a certain amount of time elapsed if I carry along my wristwatch on the curvy line and there'll be a different amount of time elapsed the rule is on the curvy line on the line where you are moving in some complicated way through both space and time that means you're not moving on a straight line you will always experience less time if you move more in space between the same amount of same two points you will move less in time so it's a bit of a trade-off there right I mean there's no such thing in space space if you move more and X move more and why it doesn't really matter but there's a trade-off in space time the more you move in space the less you move in time and we can make that a little bit more we can put that in a context maybe you're more familiar with the famous twin paradox I'm gonna put paradox in quotes because this is clearly not a paradox is just a feature of the universe that is surprising to you that's not that is not what qualifies as a paradox the twin paradox says okay here's our space time again so this is space this is time and you have a twin two twins that's what makes them twins they both start here one of them just stays home doesn't really move and evolves from A to B okay this is lazy twin they sit in their armchair this twin hops off in a rocket ship goes off very very fast here's the rocket ship and then they accelerate so they can go backwards they can reunite with their friend they left behind with their twin that they left behind not the best rocket ship I've ever drawn sorry about that okay these are two people who synchronize their watches they started cuz they were born at the same time at Point a they left one just when in a straight line in space-time one when in a curved line in space-time so the rule is this one will always have experienced less time than this then between who stayed behind so this is where it becomes purportedly a paradox although it's clearly not a paradox it's just that there's no reason for these two people who went on different paths through space-time to experience the same amount of time time is not universal time is not absolute now I wanted to mention this in particular even though you've probably seen it before because it's a little bit misunderstood what is going on here so some people say I don't understand you're telling me that there's relativity there's symmetry between motion in different ways one twin stays stationary and from their point of view the other twin moves out and comes back but from the moving twins perspective from the perspective of the twin on the rocket their twin moved out and came back right so isn't there symmetry between them shouldn't the amount of time be the same by the principles of relativity hopefully if you've been following along you know that's not right when we talked about force we mentioned that there is a difference between inertial trajectories and non inertial trajectories so there is a difference between trajectories that are accelerated in some way and trajectories that are not accelerated in some way so there is no symmetry between these two twins because one twin is never accelerated and the other twin is that makes a difference in the physics of the situation whether or not you're accelerated matters however it would be wrong to say glibly that all of the difference in time between in the time elapsed for twin 1 and twin 2 is because of the acceleration what matters is the distance along the path the space-time distance the interval along the path you could imagine a new let me pick a different color to draw it in so let's imagine new trajectory those like this staying home then very quickly accelerating and accelerating back and then just staying home again ok that might not be the brightest color that I could have picked let me do it this way there we go okay here's a new trajectory the periods of acceleration on this trajectory are exactly the same you accelerate at the beginning when you start your journey you accelerate when you turn around and you accelerate at the end they just happen at different times okay but the amount of elapsed time along this green trajectory and this original white trajectory here are going to be completely different the length of these two pads in space-time is very different even Richard Fineman got this wrong when he explained it I'm sure if you had drawn these two pictures for him he would have understood it instantly but he did make the claim that was the acceleration that accounted for the difference in time it's not the acceleration it's the total amount of path distance that you travel you can't have a different path distance unless you accelerate at some point but the acceleration themselves is not the point you need to put it together in a particular way okay so that's the twin paradox now let's dig a little bit more deeply into why why is this difference between space and space-time coming into the fore the answer is it's the difference between Euclid Euclidian geometry or in fact Pythagoras you know Pythagoras's theorem the square of the hypotenuse is the sum of the squares the opposite two sides of a right triangle this is part of Euclidean geometry okay and you could lay down these rules for what you might think of as tabletop geometry the two-dimensional or even three-dimensional geometry of ordinary flat space okay what we're doing in space-time is replacing the geometry even though it's still flat in a way which we'll talk about in more detail later but it's a different kind of space a spacetime in fact and this is where Minkowski comes in it's amusing I think I've already said this maybe even in these videos but Einstein didn't like the idea that special relativity should be thought of in terms of four-dimensional space-time this idea was first proposed by Minkowski and Minkowski headband Einsteins teacher you know when he was in school but of course eventually he needed to Einstein updated his priors because he realizes a really helpful way of thinking about things Einstein thought it was just extra obtrusive Mattox sometimes things are extra of couse mathematics that don't really give any physical insight but Minkowski and geometry absolutely is necessary so let's be a little bit more detailed about this in space okay space here is Euclid again x and y okay and what you know is that if you have two points connected by a straight line then what you can do if there's a distance D between those two points you can figure out what D is in terms of these coordinates by saying well here's the change in X there's the change in Y and Pythagoras tells us that d squared is x squared plus y squared it's a right triangle okay that is sort of one of the foundation the pillars the building blocks of Euclidean geometry now over here in space-time professor Minkowski is going to give us a different formula he's going to modify Pythagoras's theorem so here is space XYZ etc here's time here are two events a and B and they're separated and what we want to do is figure out not the distance between them but the time elapsed okay so we do a very similar thing called the time elapsed tau because we've already used T for a coordinate sorry about that we already use D so we can't call it duration but here is X and here's T and Minkowski gives us a formula but the formula is slightly different than Pythagoras's the formula is tau squared equals T squared minus x squared and that minus sign makes all the difference in the world so this tau just be SuperDuper clear here this is the time elapsed measured by someone who will be moving along that trajectory between the two points this T here is the time coordinate and this is part of the essence of relativity to separate these two things in Newtonian physics time was absolute the time coordinate and the time lapse are the same what your clock reads no matter where you do where you go through the universe is going to be a universal time coordinate whereas relativity separates them and says the times more like space we can choose coordinates differently you know in space over here we drew an x-coordinate and a y-coordinate but we could have drawn different x coordinates and Y coordinates right we could have rotated our axes so that this was X prime this was Y prime and the point is just that the coordinates are not special the coordinates are not fundamental those are not what really matters they are human conveniences ways to label where you are in space but they are not built into the essence of space itself likewise T what you and I called the time coordinate of the universe for cosmologists we say oh yes this galaxy was formed when the universe was three billion years old or something like that that is a human convenience that is a coordinate that we put on the universe it is not necessarily the time elapsed as seen by an observer so what you see because of this difference here is that in space okay with this formula Pythagoras is formula the more you move in any direction yours adding more and more to the total amount of distance that you travel between these two points whereas in time there's a in space time there's a competition the more you move in X the less time you will actually experience that is why the twin that moves off and comes back ages less because they go on what is ultimately a shorter amount of interval in space-time because of this minus sign the more you move in space the less time you experience compared to someone who's not moving at all so that's cool that's fun and this this is you know fraught with meaning here because of course this is the foundation of talking about space and time in terms of geometry which will become crucial when we talk about general relativity and curved geometries but for right now it's it's kind of a interesting insight into why things are this way let me mention a little consequence of this insight that namely that some paths if there's this confident if there's this um give-and-take right there's this I don't want say conflict but it's the time and space had different signs when they come into this interval that we calculated they can competition is the word I'm looking for some pads will have zero time elapsed along them so you can compete so much that you don't experience time at all so here is again X T and if you go on what we've been would draw as a 45-degree angle actually let me not let me draw that as just a straight line okay and this I'll make the dashed lines so now you have a certain amount of interval X a certain amount of interval T and tau along this 45 degree line while tau squared equals T squared minus x squared so if x equals T tell the amount of time elapsed equals zero I'm you know biting my tongue here of course because there are things that actually do move this way in space-time we call them rays of light or photons or for that matter gravitons or other massless particles in the universe this kind of thing is exactly what light does and this is wide light beams or photons do not experience the passage of time from their local point of view from there if a photon could have a clock it wouldn't tick it would just remain frozen because there would be nothing happening but before we get into the details of that I have to sort of explain something that I've been that I sort of hid and maybe you noticed maybe you didn't depending on how closely you're paying attention when you say things like if x equals T okay there's an obvious problem with that relation X is measured in something like meters T is measured in something like seconds you can choose different units if you want but the point is no choice of Units make space and time be measured in the same units they're different things so if you're good at dimensional analysis not in the sense of the dimensions of space-time but in the sense of the units that we use to quantify physical quantities you would know that a equation like x equals T is just nonsense or even an equation like cao squared equals T squared minus x squared nonsense you can't add quantities that are defined using different units so what's going on why would I fool you in such a terrible way well the answer is there is a built in speed there is a speed we call it C and it is equal to 300 million meters per second or in much more convenient units one light year per year what happened there I don't know so this is the new feature that space-time has that space doesn't if you're going to combine space and time in some way you need a way to convert them back and forth to each other the way you do that how do you take something that is time and turn it into space well you multiply it by space divided by time right time times space divided by time equals space and space divided by time is a velocity and this see the speed of light 300 million meters per second or one light year per year that is it is 300 million light years per second right is 300,000 kilometers per second therefore 300 million meters per second yes C is the velocity that is the cool version factor between space and time so it happens to be a fact that light moves at the speed of light but from the relativity point of view that's not what matters about the speed of light did they mean the fact that light happens to move there is you know interesting and important for understanding light but we could easily imagine a world where relativity was still real and nothing moved at the speed of light it would just be a limit it would be this speed up to which you could never reach in this fake world where nothing was a massless object moving at the speed of light but there is still be the built-in speed limit the conversion factor that told you how to relate space and time that's what the speed of light really is we're gonna keep calling at the speed of light because that's what we call it but it is really the built-in speed limit that characterizes the conversion between space and time in Minkowski and geometry and there the rules are speed of light the speed of light is a limit it is the an absolute universal speed limit every one observes it if you try to accelerate up to the speed of light and then Beyond it turns out it would take an infinite amount of energy you're not gonna do that it's not going to happen particles that you know and love or objects that are made of many particles will never move faster than the speed of light and furthermore not only is it a speed limit but it is invariant oops in other words it looks the same to everyone this is the thing that gets a little bit tricky when you try to conceptualize it if you are on a train and you're moving at 30 miles an hour and you throw a ball forward at 10 miles an hour from the point of view of someone on the road watching the train go by the ball looks like it's going at 40 miles an hour 30 plus 10 okay the velocities just add whereas if you have a flashlight on the train and you point it in some direction the speed of light that you measure is exactly equal to the speed of light that was measured by someone on the street they do not add together there's a more complicated way in which you combine velocities and relativity so that the speed of light is the same to absolutely everyone which makes sense it's built into the fabric of space-time there's no preferred trajectories in space-time so if you're on a train you're on a rocket if you're on the ground as long as you're not accelerating to you spacetime looks the same and that means the speed of light looks the same and what that means is that there is a feature of space-time called light cones so if we draw this is my favorite part so if we draw our space-time diagram X and T and you have some point here okay and you measure X in lightyears and you measure T in years then starting from this point light will move out at 45-degree angles into the future whether it goes left or right I'm only drawing one dimension of space but of course really there's more dimensions of space so what I can do is sort of complete this into a cone describing the interior of the cone as all the points you can reach going slower than the speed of light okay so we do so this is light moving at 45 degrees this line that I drew is a massive particle massive particles move slower than the speed of light and this whole light cone that I've drawn is basically you start at the event you imagine doesn't have to be real but you imagine sending out rays of light in every direction they move through space-time and every real particle has to go at that speed or slower so the number of points the set of all points you can reach like starting at one event and going into the future is the future and it takes the form of this cone that is the light cone there's also a past light cone you can imagine all the beams of light that came to you from the past and again there's more dimensions of space so we fill in the diagram and this is the set of points which you would call your past so this is where the rubber hits the road in terms of the difference between space-time and Newtonian absolute space and absolute for Newton there was the future there was the past and then there was now there was simultaneity so I'll draw this smaller diagram Newton you can still draw space-time diagrams in Newtonian physics space and then if you're a point right here this is what the diagram looks like future past now okay whereas the Minkowski your Einsteinian diagram has this cone structure where there's the future there's the past and then there's a set of points which we simply call space like a set of space like separated events in space-time and you want to say well are those points you know as an event right here is that happening at the same time as the event there you're just not allowed to ask that question that's just not well formed thing to say in relativity and and you'll emphasize that a little bit more there's no so there's no absolute notion of simultaneity but there is instead what it's replaced by is the structure of dividing all of space-time into three regions from any one point that the point has a future like oh and a past light cone and a set of space like separated points where you can't get to without going faster than the speed of light in some sense that's what replaces the notion of now but it's obviously a lot bigger than the notion of now in Newtonian mechanics so you might you might ask a perfectly good thing to ask when faced with this kind of picture is why don't we notice why isn't that completely obvious to us why do we think that there is something called now right why why did Newton who is a pretty smart cookie not draw these light cones I mean by the way actually let me draw up something first to be a little bit more yeah the reason why this was my favorite part is because it's the light cones that are really the true structure in space-time so I don't wanna this is such a pretty picture I don't want to redraw I'll just redraw it again I don't want to erase it I'll just redraw it again we've been drawing space-time diagrams like this there's space there's time and then you say okay here's a point a an event and there's light cones going into the future and coming in from the past okay and every point has a light cone here's B okay and the light cones by the way continue on eyes don't draw them but they don't end they go on infinitely far into the past and future and from point B there's also light cones going out let again it's the same idea it's just a set of points that you can reach going at or slower than the speed of light okay and you can extend these into the future so when you have two points you can talk about a region up here that is in the future of both of them a region down here that is in the past of both of them and everything else is neither in the past or future of both of them simultaneously but the point is these light cones are the real structure built into space-time if you in my dream world of perfect physics which is not even a dream world I want to become true but if let's put it this way if the right way to teach physics was just to start with the absolute truth and laid on everyone rather than to build it up gradually right by going from Newtonian mechanics to relativity to quantum mechanics if you just started with the truth at the top and then explained how it actually matched on to our experience you would never draw the time coordinate and the spacecraft coordinate because those things are not built into the fabric of space-time those are human choices the things that are built in are the light comes the light cones are the real thing that defines how space-time goes sadly that's never gonna happen and probably it shouldn't happen it would be like to to divorce from our everyday experience to make much pedagogical sense but that is how space-time actually works okay that's the that's the other thing when to say but now we can read ress this question I was just about to say why don't we notice why don't we think that way why is it hard for us to imagine that that's how space-time really is well the answer is when I drew this picture here I was careful to show to indicate this was X measured in lightyears this was t measured in years a year is not so long the experience years but a light year is really a lot of far distance for us right most of us will not travel a light year in our lives the nearest star is 4 light years away so why don't we notice well it's because when we actually in our lives would draw space-time diagrams it were we to do that and we measured space and time in more reasonable human units let's say X was measured in meters T was measured in seconds and then you have an event you can still draw the light cones right but the light cones are not at 45 degrees if this is you know 1 meter 2 meters 3 meters and this is 0 1 second 2 seconds 3 seconds 1 you remember the speed of light is 300 million meters per second so one second later starting from here I should be at x equals 300 million I can't even possibly draw that right my my dashed line would be very very close to just a perfect horizontal line it's not exactly there but that's almost what it is if I'm trying to draw my light cone okay in both directions it's almost just a horizontal line this is what light is doing and that's my future light cone this is my past light cone which is doing almost exactly the same thing right I'm Way exaggerating the difference here just so you can see it in truth it would be infinitesimal will be less than a pixel you wouldn't be able to perceive it at all so this is the light cone in human units meters and seconds coming from this point and look it looks almost like a surface it doesn't look like a huge three-dimensional region of space or four-dimensional region of space it just looks like a moment it looks almost like a sheet of simultaneity and this is what we in our everyday life call the nap if you extend the now from here meters away you're fine you're moving and your experience in life is so slow compared to the speed of light that for all intents and purposes there is something called what's happening simultaneously to you a few meters away it's only if you're moving close to the speed of light or if you're talking about things that happen light-years away or something like that that the structure of relativity really becomes obvious and important and inevitable okay that was one point to make here's another point to make spacetime is a wonderful example of unification we went from two different notions space and time and we combined them into something called space-time and this is something that happens over and over again in physics in the history of physics where you have things that looked like that they were separate but you can really find a common origin for them both you unify them it happens so often it's so wonderful when it does happen that sometimes we talk about you know grand unification in particle physics or the theory of everything but you know we don't know ahead of time what the ultimate theory is going to be like I would I would think that probably the ultimate theory will have even more unification than we have now but remember we have to actually look at what nature does not what we want it to do however space-time is a wonderful example of unification so it's not just that space and time themselves are unified but a whole bunch of other things get unified at the same time so let's as it were so let's I'm gonna you know be bad be typical be standard and draw my space-time diagrams with X's and T's because that's what people do and that helps us understand so let's consider some moving particle I don't know why my path gets little bumps in it like that but okay so here's a moving particle and it has a velocity right but you got to think carefully about this in fact what I'm gonna do is I'm gonna draw a separate diagram oops this is a spacetime diagram as we've been drawing I'm gonna draw a separate diagram for space okay so this is space-time let's just draw our more traditional space diagram x and y and we might draw here's know light cones like thinking my curve can go up and down at all of it once if it's going through x and y it's not a spacetime diagram I can have a velocity right here's a trying my best to draw a little arrow here velocity equals it's a vector it's the derivative as you know of the position X with respect to time and so velocity is a vector what does that mean we put a little arrow over it and it has components there's a velocity in the X direction there's a velocity in the Y Direction is a velocity in the Z direction if we're in three-dimensional space and okay it makes sense what is the velocity is the rate at which we are moving through space as a function of time what about space time you know we're moving through both space and time and the idea of a rate at which we move through space time is a little bit different because time is passing inevitably right like in space maybe your trajectory is just you're just sitting there you're not moving at all in space-time you inevitably move in some way even if it's just forward in time so in some sense the rate at which you move through space time is always one it is always one second per second but that's just a limited sense because of course we can still compare the clock reading that we have when we carry along with us to this time coordinate and that will change as we move and so forth so what we do is we define something called the four velocity this is the three dimensional velocity so this is the MU where the MU is a Greek letter it's a it's an index the MU has components the T for time V X V Y Z it's supposed to be a capital D I know this is capital B for space time little the I know you can tell the difference but they're different trust me and what they are every single one is just the derivative with respect to the time elapsed for you the proper time is it's called a relativity the time on your watch of the coordinate X mu which is X here is X Y Z and this is X zero is called time so in other words this is a little bit more than you need to know for this level of understanding but everything we did in Google three dimensional geometry has an analogue in four-dimensional space-time geometry we put time as the zeroth dimension why do we do that probably because we very often will imagine more than three dimensions of space or less than three dimensions of space but we almost always have one dimension of time so if you made time the fourth dimension and counter your dimensions zero one two three four four time then when someone wants to come along and do 10 dimensional string theory they're gonna have to boot time to be the tenth dimension or something like that it's more convenient just to keep time as the zeroth dimension and have as many number of spatial dimensions as you want okay fine that all fits together very nicely but the reason why I'm saying this is because we have momentum and momentum in good old Newtonian mechanics it was a vector it was the mass times the velocity we have a four momentum in space-time and it is called P mu and it is M times V mu and it has it has components there's a momentum P zero in the time direction px py PZ and the miracle the reason why I'm going through all this effort is p0 is the energy and then px py PZ is just a three dimensional momentum so the point of this is that not only our space and time unified together but energy and momentum are you together energy is the time version of momentum now proving this to you I'm just stating it but this is one of the beautiful aspects of relativity that these things that were separate in Newtonian mechanics become unified in relativity and that's just a very beautiful aspect of it okay and let me let me wind down a little bit with matching this on two other ways that you might have been already exposed to of thinking about relativity so if you're ordinarily taught by a regular person relativity you will be taught a lot about length contraction length contraction and time dilation or the other is the other thing the time dilation we secretly already talked about right we talked about the different amounts of time passed by a twin and who stayed behind in twin who went away length contraction is the other thing you're going to think about in the conventional ways of doing relativity and the reason why I don't like this sort of way of introducing relativity as much is because we talk about time dilation length contraction like it or not you're still stuck in the Newtonian viewpoint that says that there's something called length and there's something called time separately you're not really diving in to the waters of special relativity where all there is is space-time and the speed of light you're still dividing up space-time into space and time by talking about length and by talking about time separately and that's exactly what gets you into trouble so nevertheless you can talk about them here's how we would draw the phenomenon of length contraction so length contraction is if you told in when you are exposed to special relativity things moving close to the speed of light contract they get shorter they seem smaller what does that mean what is that about so I would not like you to think of it that way I would like you to think of the fact that things exist in space time not just in space so if we can idealize a little rod like the pencil if we can idealize it as one-dimensional in space okay so we ignore the thickness of it we just focus on the one dimension of space then in space-time it is two-dimensional so a one-dimensional a one-dimensional rod in space is two-dimensional in space-time so on my picture how would I draw this rod well it has a beginning point and an end point and then they persist through time right both forward and backward for that matter and so let me see if I can do this correctly I tried to set this up and not perfectly correctly but I'm trying to draw in fill in in between this is supposed to be the extent of the little rod okay and you ask yourself well what is the length of the rod and what you mean is usually you would sort of draw some line in space-time and you could measure its length that's something that exists and call that L but notice that you made a choice right you made a choice of which way which line along which to draw from the left side the beginning of the rod to the right side to the end of it I could have drawn the line this way called that now or let's call it L prime just to be special okay and I would have gotten a different answer right because we know the the formula l squared equals x squared minus T squared and so if I drew it with some extent in time in other words if I measured the beginning of the rod and the end of rod at two different moments of time then I would have gotten a different length for it and in fact you know this line that we drew right here looks longer on the way that I drew it but we know that it actually has an extension time so it's actually shorter in space-time interval and therefore it is in fact contracted that is the origin of length contraction okay length contraction comes from the fact that the way that we drew it in the rest frame of the object is the longest possible length that we could have associated with this the what I would like is to live in a world where we don't associate lengths with things at all we can talk about the length of something in its rest frame that's okay we're allowed to talk about it but if you want to think about it if you really want to get the intuition right it's better to think about the whole two-dimensional extent and in the same vein you know there's something that we often hear talked about in discussions of special relativity which are frames of reference okay so I will connect my way of doing it with the traditional way of doing it and then tell you why I don't like it I don't love don't love it it's fine but I don't love it so here's what I mean your spacetime diagram X T we can think of these X and T coordinates as being associated with a particular observer what I mean by that is let's imagine here we have a person and they are starting here and they just stay stationary they don't do anything in other words they just stay at x equals zero okay and what that person does imaginarily not really but we assume that they could do this is to send out little rocket ships infinitely fast okay breaking the speed of light and they move at you know perpendicularly so they send out little rocket ships that go this way so what they do is essentially construct a coordinate system all throughout space time because these rocket ships carry clocks with them and so they can measure not just the distance perpendicular to how this person is moving but also the time you could also measure the time going up okay and so this is how you would construct a coordinate system all throughout space time based on what is seen by this observer and just by the way I'm saying it is obviously not something that would ever actually happen that would never actually be done but you it's something you can imagine doing but then you could say oops I don't want to do it that way then let's say we do the same thing but from the perspective of a different observer so I'll write my original X&Y coordinates here but let's imagine that my observer is moving at some velocity okay and they define x equals constant to be where they are okay and T which we'll call t coops I said that was why that was not right so this is supposed to be T the same time coordinate but our new observer locates their center of time at themselves so that's T and you can say well if they were to send out rocket ships infinitely fast where would they go and their reference frame and the answer is if you if you know that the speed of light defines a light cone at 45 degrees so this is C speed of light then the rocket ships would do something like this so they would not and you might think that's a little bit weird because this what I'm doing here is very very analogous to rotating coordinates in good old Euclidean geometry right we have x and y and we can just rotate coordinates to x prime y prime but usually that's fine but usually what we want to do is keep them perpendicular to each other okay and you might say well here the yellow lines I drew this does not look perpendicular that doesn't look perpendicular at all that's at some angle but remember the minus sign that is in Minkowski in geometry it turns out that these lines really are perpendicular to each other in the Minkowski and sense and so this is the new X prime coordinate and this is the new T prime coordinate and they're at this angle than what remains fixed is the speed of light the light cones remain fixed because the light cones are what is built into space-time in this way of thinking about things so what happens because of this is very interesting because I can I'm gonna use more and more colors this is awesome so let's imagine an event here and an event here B okay so you notice that to the first observer in that is say in the coordinates of the first observer which is just the white labeled X and T a happens before B there's space like separated so in some sense you shouldn't compare them timewise but in this coordinate system in this frame of reference a happens before B to the second observer in other words in the yellow coordinates B is before and because B is under that little first line that we drew okay the yellow line so what happens before what is not the same two different observers in different reference frames if those two things are so far away from each other that there's space like separated so something can happen right now right here in your room and you say well there's another event at Alpha Centauri 4 light-years away what is happening at the same time the answer is there's no such thing as at the same time there's at the same time in a reference frame in a coordinate system that is setup relative to an observer but that's your choice that is not built in to the framework of the universe now the reason I don't like talking this way is because what I would rather do instead I would rather just think locally in other words rather than being an observer who stretches out a coordinate system all across the universe I'd rather just say what is it that one observer sees so there's this temptation relativity to talk about you know how fast is a clock ticking from your point of view when you're far away but it's out there moving close to the speed of light or when it's near a black hole or something like that I wouldn't just uh nask that question I want to just say you can ask questions about what you see coming from that clock what the reading is that you get from light rays going from the clock to you or you could ask about what a person standing next to the clock sees and the answers are different okay because you're they're doing different things in space-time but the short answer to the second question what is the person standing next to the clock see they will always see a clock moving at one second per second that's why because not only are they moving differently or they different different gravitational field but the clock is also and they're being affected in exactly the same way and the whole point of relativity is to them they're not being affected at all everything is perfectly normal from their point of view so rather than saying clocks move faster when they are stationary or when they are far away from gravity or something like that or saying the clocks slow down when gravity is strong or they're moving close to the speed of light I like to say the clocks move at one second per second all the time but what you can do is compare clocks when they leave you and then they come back and they're back at the same point in space one of the reasons why I'd like to emphasize this perspective is because it extends very naturally to general relativity well we've been talking about so far is just special relativity we're not gonna do generality completely right now that's for a future video but the punchline the general point oops of general relativity is that space-time still exists still crucial but now space-time is curved it has a geometry and that's not such a surprising thing maybe you know if we go way way back to the beginning here and we talked about where to go yeah we talked about Euclid versus Minkowski we are talking about geometry this is what is going on space-time has a geometry there's a way of talking about it at the simple level of Euclid versus Minkowski the only difference is a minus sign okay there's all plus signs in Pythagoras's theorem there's a minus sign that appears in the medical skiing way of doing things general relativity is going to guess what general is this it's going to say that this the difference in geometries could be much more dramatic and because of that this trick over here of having an observer and extending uniquely a reference frame all throughout the universe is not going to work they will in generally general not be possible to do that so in special relativity different observers might attach different notions of simultaneity very far away in general typically you can't even do that in generality there's no way to attach a notion of simultaneity generally there can be special cases but in general in the most generic case you just not gonna be able to do that at all rather you should think locally you should ask what individual observers actually experience and let me just mention that let me just emphasize one one feature here all of this general relativity and special relativity or these are part of classical mechanics so special relativity yeah I was not very beautiful special relativity is the theory of a fixed background space-time and the particulars of it is the speed of light is invariant that latter fact speed of light being invariant is still true in general relativity but special relativity is an arena on top of which other theories happen electromagnetism quantum electromagnetism etc there are relativistic theories theories that play in the playground called special relativity whereas general relativity is a theory by itself that is all playground because it says that space-time is not fixed space-time is dynamical it responds to matter in energy and we observe we feel we interpret the curvature of space-time as gravity that's what gravity is according to special to general tivity is the curvature of space-time so despite special and relative in general being the terms here general relativity is more general than special relativity because it allows for not just flat space-time geometry but for arbitrary space-time geometry but at the same time general relativity is a specific theory Einstein made a specific choice for the equations of motion of the gravitational field of the curvature of space-time and those define what we mean by general relativity so both of these are by themselves classical theories in the sense in the following sense they're not Newtonian they're not absolute space and absolute time but they can fit perfectly naturally into the Newtonian laplacian paradigm in both special relativity or general relativity you give me some positions you give me some momenta I have equations that will let me chunk forward in time and figure out where the particles or where the fields are going to be next it's quantum mechanics where things really become different so some people put the difference between classical physics and the rest of the world at both relativity and quantum mechanics they group together relatively in quantum mechanics as post-classical I think that's not the best way to do it these are words they're arbitrary we can make them up however we want but I think that it's better it's more productive to associate quantumness with the departure from classical physics relativity is great spacetime is great it's still the classical world so maybe next time we could think a little bit more about the quantum world

Channel: Sean Carroll

Views: 328,864

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Id: Cg2tOUTE2F4

Length: 63min 21sec (3801 seconds)

Published: Tue Apr 28 2020

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[u/bigaus25]

Hopefully I can finish this without cumming