Your Daily Equation #26: Einstein's General Theory of Relativity: The Essential Idea

everyone welcome to this next episode of your daily equation it may look a little bit different from the place where I've done the earlier episodes but actually I'm in exactly the same spot it's just that the rest of the room has gotten to be so incredibly messy with all sorts of stuff that I've had to shift my location in order that you don't have to look at the messy room that would otherwise be behind me all right so with that little detail out of the way today's episode I'm gonna start in on one of the really big ones the Big Ideas the big equations Einstein's general theory of relativity and just to give a little bit of context to this let me just note myself Oh different position I've got an angle myself differently sorry I think that's okay up on the screen good all right so we're talking about general relativity and to put this just in context of the other big vital essential ideas that really revolutionized our understanding of the physical universe starting in the 20th century well I'd like to organize those developments by writing down three axes and these axes you can think about say as the speed axis you can think about it as the length axis and third you can think about I cannot believe it Siri just heard me that is so irritating let's go away Siri Hey all right here back to where it was yeah I have to learn how to turn Siri off when I do these things anyway the third axis is the mass axis and the way to think about this little diagram is that when you are thinking about how the universe behaves in the realms of extremely high speed that takes you to Einstein's special theory of relativity which it just will have since that is the subject that I started with in this series of your daily equation when you go to extremes along the length axis and by extremes here I really mean extremes of very small not very big that takes you to quantum mechanics which in some sense really is the second major focus that I had in this your daily equation series and now we're on to the mass axis where when you look at how the universe behaves at extremely high masses that's where gravity matters that takes you to the general theory of relativity our focus today okay so that's how things fit in to that overarching organizational scheme for thinking about the dominant theories of the physical universe and so let's now get in to the subject of gravity the force of gravity and many people believed not far after say the late 1600s that the issue of gravity had been completely sorted out by Isaac Newton right because Newton gave us his famous universal law of gravity remember this is during the the Black Death way back in the late 1600s Newton retreats from Cambridge University goes to his family's home in the safety of the countryside there and in solitude through really the amazing power of his mental faculties and and creative ways of thinking about how the world works he comes up with this law universal law of gravity that if you have two masses let's say I have mass m1 and mass m2 that there is a universal force of attraction between them acting to pull them together and the formula for that is a constant Newton's gravitational constant M 1 M 2 divided by the square of their separation so if their distance R apart then you divide by R squared and the direction of the force is along the line connecting say their center center of masses and that seemed to be the be-all and end-all of the force of gravity in terms of describing it mathematically and indeed let me just just to get us all on the same page here is a little animation that shows Newton's law in action so you have a planet like the earth in orbit around a star like the Sun and using that little mathematical formula you can predict where the planet should be at any given moment and you look up into the night sky and the planets are just where the math says that they should be and we take it for granted now but Wow right think about think about the power of of this little mathematical equation to describe things that are happening out there in space right so so understandably rightly so there was a general consensus that the force of gravity was understood by Newton and his universal law of gravity but then of course other folks come into the story and the person of course that I have in mind here is Einstein and Einstein begins to think about the force of gravity and roughly 1907 or so and look he comes to the conclusion that sure Newton made great progress in understanding the force of gravity but the law that he gave us over here can't really be the full story right why can't it be the full story well you can immediately catch the gist of Einstein's reasoning by noting that in this formula that Newton gave us there's no time variable there's there's no temporal quality to that law why do we care about that well think about it if I was to change the value of the mass then according to this formula the force would immediately change so the force felt over here at mass M - given by this formula will immediately change if say I change the value of M 1 in this equation or if I change the separation if I move M one this way making our a little smaller or this way making our a little bit bigger this guy over here is going to immediately feel the effect of that change immediately instantaneously faster than the speed of light and Einstein says there can't be that kind of influence that exerts a change a force instantaneously that's the issue now small footnote some of you may come back at me and say what about quantum entanglement something that we discussed in an earlier episode when we were focusing our attention on quantum mechanics you will recall that when I discussed the spooky action of Einstein we noted that there's no information that's traveling from one entangled particle to another there is an instantaneous according to a given reference frame correlation between the properties of the two distant particles this one's up and the other ones down but there's no signal no information that you can extract from that because the sequence of results that the two distant locations are random and randomness doesn't contain information so that's the end of the footnote but bear in mind there really is a sharp distinction between the gravitational version of the instantaneous change in a force versus the quantum mechanical correlation from entangled particles all right let me put that to the side so Einstein realizes there's like a real issue here and just to bring that issue home let me show you a little example here so imagine that you've got the planets in orbit around the Sun and imagine that somehow I'm able to reach in and I pluck the son out of space what will happen according to Newton well Newton's law says the force drops to zero if the mass at the center goes away so the planets as you see are immediately instantaneously released from their orbit so the planets instantaneously feel the absence of the Sun a change in their motion that is exerted instantaneously from the changing mass at the location of the Sun to the location of the planets no good according to Einstein so Einstein says look perhaps if I understood better what Newton had in mind regarding the mechanism by which gravity exert its influence from one place to another Ison says maybe I would be able to calculate the speed of that influence and maybe with you know hindsight or or better understanding a couple hundred years later maybe on his time says to himself I will be able to show that in Newton's theory the force of gravity is not instantaneous so Einstein goes to check in on this and and he realizes as many scholars had already realized that Newton himself is kind of embarrassed by his own universal law of gravity because Newton himself realized that he had never specified the mechanism by which gravity exerts influence he said look if you have the Sun and you have the earth and they're separated by a distance there is a force of gravity between them and it gives us a formula for it but it doesn't tell us how gravity actually exerts that influence and therefore there is no mechanism that Einstein could analyze to truly figure out the speed with which that mechanism for transmitting gravity operates and therefore he was stuck so Einstein sets himself the goal of truly figuring out the mechanism of how gravitational influences are exerted from place to place and he starts at about 1907 and finally by 1915 he writes down the final answer in the form of the equations of the general theory of relativity and I'm gonna now describe the basic idea which I think many of you are familiar with of what Einstein found and then I'll briefly outline the steps by which Einstein came to this realization and I'll finish up with the mathematical equation that summarizes the insights that Einstein came to all right so for the general idea Einstein says look if say you have the Sun and the earth right and the Sun is exerting an influence on the earth what could be the source of that influence well the puzzle is there's nothing but empty space between the Sun and the earth so Einstein ever the capable genius to look at the most obvious answer if there's only empty space then it must be space itself space itself that communicates the influence of gravity now how can space do that how can space exert any kind of influence at all Einstein ultimately comes the realization that space and time can warp and curve and through the curved shape that can influence the motion of objects right and so so the way to think about it is imagine that space this is not a perfect analogy but imagine space is sort of like a rubber sheet or a piece of spandex and when there's nothing in the environment the rubber sheet is flat but if you take a bowling ball say and you put it in the middle of the rubber sheet the rubber sheet will be curved and then if you set marbles rolling around on the rubber sheet or on the spandex the marbles were now going to curve trajectory because they're rolling in the curved environment that the presence of the of the bowling ball or the shotput creates in fact you can actually do this I did a little home experiment with my kids you can see the full video online if you like this from a few few years ago but there you see it we have a piece of spandex in our living room and we have marbles that are rolling around and that gives you a sense how the planets are nudged into orbit by virtue of the curved space-time environment through which they travel a curved environment that the presence of a massive object like the Sun can create let me show you a more precise well not more precise but a more relevant version of this warp it so you can see it at work in space so here you go so this is grew this grid represents 3d space it's a little hard to picture it fully so I'm going to go to a a two dimensional version of this picture that shows all the essential ideas notice that space is flat when there's nothing there but if I bring in the Sun the fabric warps similarly if I look in the vicinity of the earth the earth 2 also warps the environment and now focus your attention on the moon because this is the point the moon according to Einstein is kept in orbit because it's rolling along a valley in the curved environment that the earth creates that is the mechanism by which gravity operates and if you pull back you see that the earth is kept in orbit around the Sun for exactly the same reason it is rolling around a valley in the warped environment that the Sun creates that's the basic idea now look there are a bunch of subtleties in here maybe I'll uh maybe I'll quickly address them right now you could say to me a hey look with the example of the spandex which is the at-home version of the Sun warping the fabric around it if I put up of a bowling ball or a shot put on a rubber sheet or a piece of spandex the reason why it warps the spandex is because the earth is pulling the object downward but wait I thought we were trying to explain gravity so our little example now seems to be using gravity to explain gravity what are we doing well you're absolutely right this metaphor this analogy really needs to be thought of in the following way it's not that we are saying that Earth's gravity is causing the environment to or rather Einstein is telling us that a massive energetic object merely by virtue of its presence in space warps the environment around it and by warping the environment I mean warping the full environment around it of course I have difficulty showing that fully but actually let me just give you this this little visual here that you know it gets partway toward it now you see that the full 3d environment say is being warped by the Sun it's harder to picture that one and the 2d version is pretty good to keep in mind but the 3d one is really what's happening we're not looking at a slice of space we're looking at the entire environment being influenced by the presence of a massive body within it all right that that is the basic idea and now I wasn't spend I want to spend just a couple minutes on how it is that Einstein came to this idea and it's really it's really a two-step process so step one Einstein realizes that there is a deep and unexpected connection between accelerated motion acceleration and gravity and then he realizes that there is another unexpected and beautiful relationship between acceleration and curvature curvy space times curvature and the final step then of course will be he realizes that there is a connection therefore between gravity and curvature so this link right over here is forged if you will through acceleration being the common quality that leads you both to an understanding of gravity and understanding of curvature therefore a link between gravity and curvature okay so let me just quickly explain those links the first of which happens in well it was always there but Einstein realized it in 1907 1907 Einstein is still in the Patent Office in Bern Switzerland he had the great success in 1905 with a special theory of relativity but he still is working in the Patent Office and he has one afternoon what he calls the happiest thought of his entire life what is that happiest thought the Hat happiest thought is he imagines a painter who is painting the exterior of a building on a hind ladder he matches that painter falling off the ladder falling off the roof and going into freefall he doesn't take this thought all the way to the impact to the ground the impact is not his happiest thought the happiest thought happens during the journey why he realizes Einstein realizes that the painter during this descent will not feel his or her they will not feel their own weight what do you mean by that well I like to frame it this way imagine that the painter is standing on a scale that's velcroed to their shoes and they're standing on the scale on the ladder kind of a hard image but imagine that they're now falling as the painter Falls the scale Falls at the same rate as the painter therefore they fall together which means the painters feet don't exert a push on the scale they can't because the scale is moving away at exactly the same rate as the feet are moving downward too so looking down at the reading on the scale that painter will see that the reading drops to zero painter feels weightless painter does not feel their own weight now give you a little example of that that again this is sort of a annapolis so general relativity but it do do it at home physics right this is a dy I version of the general theory of relativity so how can you establish without falling off the roof of a house in a more safe manner how can you establish that freefall this kind of accelerated downward motion accelerated downward motion can in some sense cancel out the force of gravity well I did an example of that on The Late Show with Stephen Colbert some years ago and they did a nice job filming it so let me let me show you the basic idea so imagine you have a bottle filled with water and it's got some holes in it the water sprays out of the holes of the bottle of course why does it do that because gravity is pulling on the water and that pull forces the water out of the holes in the bottle but if you let the bottle go into freefall like the painter the water will no longer feel its own weight without feeling that force of gravity nothing will pull the water out of the hole so the water should stop spraying out of the holes and check this out really does work all right here we go during the descent look in slow-mo there is no water spraying out of the holes during that accelerated motion that descent so this is what we mean here about this relationship between acceleration and gravity this is a version where the accelerated downward motion faster and faster as the bottle of water or the painter Falls the force of gravity is canceled if you will by that downward motion you might say well wouldn't you cancelled why is the bottle falling why is the painter falling that's gravity but I'm saying not from our experience watching the painter fall not from our experience watching the bottle of water fall I'm saying that if you put yourselves in the shoes of the painter or you put yourselves in the shoes of the bottle of water whatever that means then from that perspective the free-falling perspective from your perspective in that accelerated trajectory you don't feel the force of gravity that's what I mean now the important point is that there's also a reverse to this situation accelerated motion can not only cancel out gravity but accelerated motion can mock up it can sort of fake a version of gravity and it's a perfect fake again what I mean by that well imagine that you are floating in outer space so you really are completely weightless right and then imagine that someone causes you to accelerate right they tie a rope to you and they accelerate you say well let's say they accelerate it like this they accelerate you upward right and imagine that they do that by putting a platform under your feet so you're standing on this platform in empty space feeling weightless now they attach a rope or crane whatever to a hook on the platform on which you're standing and that crane that hook that rope pulls you upward as you're accelerating upward the board under your feet you're gonna feel it pressing against your feet and if you close your eyes and if the acceleration is correct you'll feel like you are in a gravitational field because how does a gravitational field say on planet Earth feel how do you feel it you feel it by virtue of the floor pushing up against your feet and if that platform accelerates upward you will feel that pressing against your feet in the same way if the acceleration is correct so that's a version where accelerated motion creates a force that feels just like the force of gravity you've experienced this and an airplane at this beginning to taxi and it's about to take off as it accelerates you feel pressed back in your seat that feeling of being pressed back you close your eyes and it can sort of feel like you're lying down the force of the seat on your back is almost like the force you'd feel if you're just a lying say on your back on a couch so that's the link between accelerated motion and gravity now for part two of this so that's 90 nose so for part two we need the connection between acceleration and curvature and this there are many ways I mean I it's not the history is fascinating and again it's mentioned before because I kind of loved the piece we have this this stage piece like Falls you can check it out where we go through the whole history of these ideas in a stage presentation but there are actually a number of people who contributed to thinking about gravity in terms of curves released Einstein's recognition of this and there's one particularly beautiful way of thinking about it that I like it's called the ehrenfest paradox it's not actually a paradox at all paradoxes are usually when we don't understand things at first and there is a seeming paradox but ultimately we sort it all out but sometimes the word paradox isn't removed from the description and let me give you this this example that gives us a link between acceleration and curvature how does it go remember accelerated motion means a change in velocity velocity is something that has a speed and a direction so there's a special kind of accelerated motion where the speed the magnitude doesn't change but the direction does and what I have in mind here is circular motion circular motion is a kind of acceleration and I now like to show you is that circular motion that accelerated motion naturally gives us the recognition that curvature must come into play and the example I'm going to show you is is a familiar ride you may have been on it you know at an amusement park or carnival it's often called the tornado ride I described this in the elegant universe but I'll show you a visual in just a moment you know it's a ride you you stand on the circular platform that spins around and you actually feel your body pressed against a circular cage that is moving it's attached to this circular platform and that outward force that you feel I'm making me strong enough that sometimes they actually drop the bottom of the ride out with you're standing on so you just hovering there and sometimes in midair but your body is pressed by the circular motion against the cage and there's enough friction hopefully that you don't slip away and fall alright that's the setup here's here's the issue all right so here is this circular ride imagine that you measure the circumference of this ride from the outside not on the right itself so you lay out these rulers and whatever you find I think in this case there are twenty-four rulers twenty-four feet you can also measure the radius and you can get a number for that as well and indeed if you look at the relationship between the circumference and the radius you will find that C equals two pi R just as we all learned in junior high school but now imagine measuring this from the perspective of someone on the ride itself the accelerated observer well when they measured the radius they're going to get the exact same answer because that's moving perpendicular to the motion no Lorentz contraction but if you measure the circumference look what happens the rulers are all instantaneously moving in the direction of the motion so that you're all shrunken contracted therefore it takes more of those rulers to go all the way around in this particular case I just imagine that it's 48 of those rulers 48 rulers for the circumference equals 48 radius is unchanged again that's moving perpendicular to the instantaneous direction of the motion which is all in the circumferential direction right radius is going this way circumference is going that way so there's no change in the measurement of the radius which means C will no longer equal to PI R you say itself what how can see not equal to PI R what does that mean well when you learned that C equals 2 PI R you were talking about circles that were drawn on a flat surface it must therefore be the case that from the perspective of the person on the ride laying out those little rules and feeling that gravitational force right there accelerating that feel that force pulling them outward from their perspective it must be that circle is not flat must be curved it must be the case you know sort of a poetic image of this of you will over here kind of dolly esque picture those circles are warped they're curved clearly C will not equal to PI R for those particular warped shapes so that's kind of an artistic version of it but the conclusion is that the accelerated motion of the ride which we know gives a connection to gravity also gives a connection to curvature so then that is the linkage that we were looking at the accelerated motion from the circle gives rise to the feeling of a gravitational like force that accelerated motion gives rise to measurements from the perspective of the person experiencing that acceleration that do not satisfy the usual rules of flat Euclidean so-called geometry and therefore we learned that there is a connection between gravity and curvature and now I can bring back the image that we had before with a little more insight from that description so again here is flat 3d space when there's no matter go to the 2-dimensional version just so we can picture it bring in a massive body like the Sun and now that gravity gives rise to this curvature and again the moon why does it move around the moon in some senses is being nudged around by the curvature in the environment or said another way the moon is seeking out the shortest possible trajectory we called geodesics we'll come to this and that short as possible trajectory in that curved environment yields the curved paths that we would call a planet going in to orbit that's the basic chain of reasoning that leads on Stein to this picture all right so then what is the equation um just gonna write down the equation and subsequently subsequent episodes I'm gonna just in this episode be satisfied to just give you the basic idea and show you the equation I will unpack the equation later on but what is the equation well Einstein in November of 1915 at Elektra the Prussian Academy of Science writes down the final equation which is our mu nu minus 1/2 G mu nu R equals 8 pi G over C to the fourth times T mu nu what in the world does that all mean well this part over here is the mathematical mathematical the mathematical still early for me the mathematical way of talking about curve that sure ok and this fella over here is the way you talk about energy and mass also momentum but we call it mass energy one since we learn in special relativity that mass and energy are two sides of the same coin you recognize that mass is not the only source I mean that clumpy object like the earth is not the only source for gravity energy more generally is a source for gravity and that is captured by that expression over here T mu nu I'll describe this not today but in a subsequent episode and that is Einsteins equation for the general theory of relativity not to really understand this equation you need to understand all of these gadgets that we have here the Ricci tensor the scalar curvature you need to understand the Riemann curvature tensor to understand those this is the metric on space-time you need to understand that and I really do mean space-time in fact we're talking about the gravitational pull of a planet like the earth or the Sun the imagery that I showed you with the Warped environment you know it helps your mental thinking about things but in the usual way the we set up our coordinates it's actually the warping of time not really the warping of space that's the dominant influence in causing an object to fall whether I drop an object here or whether it's the moon perpetually falling toward the earth as it moves in the tangential direction thereby keeping itself in orbit so time is is really quite important to this you can't just think in spatial terms at all but to understand all those mathematical details we have to unpack the the mathematics if you will of differential geometry I will do a little bit of that in subsequent episodes but I hope this gives you a feel for the basic insight of the general theory of relativity why it is that Einstein came to this realization that gravity necessarily involved a curvature of space don't keep that tornado ride in mind again no analogies are perfect but it does help you catch the essential links between say accelerated motion and gravity the water drop the painter between accelerated motion and curvature the tornado ride and then it's the genius of Einstein it puts it all together as we will see and unpack in subsequent episodes okay that's all I wanted to do today that is your daily equation until we meet next time looking forward to that until then take care

Channel: World Science Festival

Views: 88,134

Rating: 4.9459105 out of 5

Keywords: Brian Greene, Albert Einstein, General relativity, special relativity, quantum physics, quantum theory, Quantum mechanics, #YourDailyEquation, #DailyEquation, math, daily math equation, Mathematics, number theory, daily series, math series, New York City, World, Science, Festival, 2020

Id: 2cO8Qq-fthQ

Length: 34min 40sec (2080 seconds)

Published: Tue May 12 2020