Einstein's General Theory of Relativity | Lecture 2

this program is brought to you by Stanford University please visit us at stanford.edu all right so let me let me just tell you what the story is with dark energy big rips whether dark energy will have a tendency to tear atoms apart and so forth we're not we're not in past classes we've studied a little bit about dark energy so for those who were there I'll just remind you for practical purposes let's forget the deep story but for practical purposes dark energy or cosmological constant or whatever you like to call it vacuum energy the observed accelerated expansion of the universe that is equivalent to a small repulsive force a component to gravity they say gravity is not repulsive well suspend judgment about that until we get to it in this class and imagine a small repulsive component to gravity which is not proportional to one over R squared like Newtonian gravity but in fact where the force increases with distance it increases with distance roughly the same way well how does a how about a spring supposing you have a particle on a spring connected to the wall or two particles connected by Springs how does the force depend on the seperation between them the force grows linearly with the separation between them of course until you break the spring but let's not break the spring the force grows linearly and it's proportional to the distance between the the particles at the end of the spring moss grows linearly there is some constant they're usually called the spring constant let's put it in and because it's attractive pulling it two things together there's a minus sign there imagine that we have such a spring okay now what and though with two particles attached to it what does the cosmological constant do what the cosmological constant the repulsive cosmological constant what it does a positive cosmological constant or a positive dark energy what it does is it adds a little bit of force of just this type proportional to distance but repulsive not attractive this is attractive that's the minus sign they're repulsive so it adds a little bit of repulsive force to the otherwise attractive force between the two particles on the end of the spring now I'm not really thinking about two particles on the end of the spring but it's a good model for us what I'm really thinking of for example is an atom an atom is a system which is composed of two or more particles held together by forces the forces are not entirely different than spring forces but what would happen if you added to this a little bit of much smaller than the spring constant the spring constant which holds atoms together is quite strong the dark energy forces are minut they're incredibly small until you get very very far away notice this force grows with distance forget this one we're going to add now a little bit of component the coefficient in front of it is called the cosmological constant or the dark energy also times R what would it do to the spring if there was a tiny tiny coefficient here and that coefficient is truly small for ordinary laboratory experiments all it would do would be to change the equilibrium position of the particles at the end of the spring a little bit it would effectively change the spring constant here make it a little bit smaller but what would happen if you made the spring constant a little bit weaker all that would happen is that the spring with the particles on it would be a little bit bigger that's all atoms would grow by an entirely negligible amount they wouldn't be torn apart they wouldn't be torn apart it would just change the equilibrium position of the electrons in the atom by a tiny completely insignificant amount and that would be the net effect on atoms it would be the net effect on almost anything that was otherwise bound together the reason is that this lambda here how small is it it's so small that this force doesn't become significant until you get out to the full measure of the entire global universe that's where it begins to get big but if you ask about it on the scale of the solar system it's negligible it's tiny by comparison with the other forces in the solar system gravity and other things and so what it would do is it would change the equilibrium position of the earth a little bit a tiny tiny bit or it would change the size of a galaxy a little bit and expand it out a little bit but it wouldn't overcome the forces which hold things together now there's a theory called the big rip it violates every principle of physics it does it violates every principle of physics why anybody would take it seriously I don't know but when serious theorists work out the equations and say there are limits between this and that such-and-such cannot be bigger than this so such as such cannot be smaller than that for deep fundamental principles somebody is going to come along and say yeah but what if those principles are wrong and such and such can be a little bit bigger or a little bit smaller than the bound that the theoretical principles tell you somebody will come along and say that they'll write a paper about it the paper will last firm till it gets to the New York Times and then it will slowly fall into the junk heap of bad physics ideas the big rip is one of those ideas but I'll tell you I'm not going to try to tell you right now are what is wrong with it will eventually come to it but it's basically the idea that this constant lambda here is time-dependent that it grows with time now there's no reason on earth to believe that not only is there no reason on earth to believe it there are very very strong reasons on earth to disbelieve it but you could say who cares what theorists say let's make a prediction of what would happen if lambda grew with time well if it grew with time eventually it would become stronger than the forces which bind together other objects at first it would become stronger than the forces which hold galaxies together and galaxies would fly apart then it would become stronger than the forces which hold the solar system apart solar system would fly apart then eventually it will become stronger than the forces which hold the spring together spring flies apart as I said that's called the big rip that this constant lambda the dark energy is not a constant at all but that it increases with time decreases with time is allowable increases with time violates some strong as I remember what you're saying lambda is a function of mass or whatever the Linda is just a numerical constant period well the point I'm getting at is what if there's no mass that man was still an event it's a force that would be there between any pair of particles if the particles are there if the particles weren't there there wouldn't be any particles to have a force on them so no according according the most current thinking it's called the cosmological constant for good reasons because all of the good theory says it ought to be a constant now could it not be a constant it doesn't violate any very very deep principles for it to decrease with time it violates big principles to increase with time so but we will come to this I just wanted to assure you to make you feel better about the future well the dark energy density in this room is tiny it's extremely small I'll tell you how small it is um take a cubic meter and how many how many protons worth a proton has a certain mass equals MC squared so there's a certain amount of energy in every proton ah roughly speaking I think the dark energy in this room per cubic meter uh would be I would have to work at it but it's roughly of order about a thousand protons a thousand protons in the cubic meter is a negligible amount of energy its gravitational effects on things are really negligible but if you had it smeared out through the entire universe that way then at sufficiently big distances away and then imagine now imagine now we live in a world which has an extra bit of energy that's causing repulsion but it's spread out throughout the entire universe in a smooth distribution then eventually at sufficiently large distances it would make an effect the repulsive effect on things which would become big but those distances are cosmological in size so it has an effect on the global universe but it doesn't have an effect on certainly not on laboratory physics it doesn't even have a significant effect on any kind of astronomical physics by astronomical I mean things smaller than the entire universe in fact for whatever reason this number happens to be such that it only that this force only becomes significant and comparable to other forces at something like the radius of the entire universe so it only becomes important under those circumstances people who want to tap the vacuum energy to get vacuum energy to do work and as a solution to the energy crisis have to deal with the fact that if they wanted to get a tank full of gasoline they would have to extract the energy out of a volume something like the orbit of the moon or something like that so there's not a lot of energy there yeah and no way to tap it but then that suggests that dark energy is really a relative and relative to what what does it mean let's let's let's I'm not sure what you mean by a relative energy do you mean well it gravitates it gravitates it has a it has the effect of a distribution of mass in the universe so it's another you mean our only differences of it important is that what you meant I'm not sure what relative a relative was being used on that's probably there's some particle associated with no no not now the dark energy dark energy does not have the form of any energy that's associated with particles we'll come to it but I hope we'll come to it in the course of these lectures but no there's no particles in it now dark energy is to be contrasted with what's called dark matter dark matter is an entirely different thing a totally different thing and it really is made up out of particles or at least it's believed to really be made up out of particles but this is not where I was going tonight I just just wanted to for those who were involved in these exchanges of email I thought I would give you my view of it let's come back to the Newtonian laws of gravity we need to spend a little more time with Newton before we can move on to Einstein and when I say Newton of course many of the things I'm going to say Newton might not have really recognized but there are forms of or expressions of Newtonian physics first a little bit of mathematics and it's not really mathematics it's just all formalism ways of expressing the equations symbols all right first of all I told you last time we use the symbol del last times called del upside-down Delta and it's thought of as a kind of vector but it's not really a vector with definite magnitude and definite directions it stands just like every vector if I wrote any vector over here V we could say that's the same thing as giving its components the X component the Y component and Z component so one way of viewing a vector is just that it stands for three numbers the x y&z components of V could be a velocity for example okay the symbol del with an arrow above it stands also for three objects but those three objects are really derivatives derivative with respect to X derivative with respect to Y and derivative with respect to Z so whenever you see a del it always means you're differentiating something you're always doing the derivative of some kind a partial derivative or a set of partial derivatives of of something now for example how do you use it that's really called a differential operator it's not a set of numbers it's a set of operations but it also has three components so let's say let me the first example the operation del if it acts on a scalar now a scalar just means a function of position function of x y&z I'll stop writing X Y & Z soon enough and just call it X but take a function of XY and Z and apply to it the operation del what does it make it makes a vector Phi is not a vector but del times Phi is a vector and what are the components of the vector the components of the vector are just derivative of Phi with respect to X derivative of Phi with respect to Y and derivative of Phi with respect to Z so it creates a vector from a scalar that's one example of the use of del another example supposing it acts on a vector field it can also act on a vector field what does it mean let's take del dot a let a be a islets let's use V del dot V that's called the divergence of V we talked about it last time but just to give you a little more feel for how the symbols are used dot stands for that product if I have two vectors let's call one W for a moment let's call one of them W and let's call the other one V and I take the dot product of the two vectors what it stands for stands for the X component of W times the X component of V plus the Y component of W times the Y component of V Plus third term which is similar that's the meaning of W dot V the standard old-fashioned dot product between vectors which most of you have seen all right by the same rule we simply prefer we simply pretend that the symbol Dell stands for a vector it's like W but it's not a it's not a real vector it's just of operations and what this gives is the derivative of VX with respect to X plus the derivative of V Y with respect to y plus the derivative of V Z with respect to Z in other words it has the same form except replace W by this set of derivative operators and so it's a neat notation it's a notational device for doing certain kinds of derivatives partial derivatives on various kinds of things now if Phi is a scalar then del dot Phi is vector what if V is a vector what is Del dot V it's a scalar in the same way that the dot product between two vectors is a scalar the divergence of a vector is a vector these notations are pervasive throughout all the physics electricity magnetism gravity and without them we would we would not be able to wouldn't be able to function very well okay that that's just a preliminary warm-up and definition it's really just definition there's not only deep mathematics the deep mathematics is of course in the notion of a derivative or a partial derivative derivative along a direction now I mentioned of course that fie here is a function of position a function of position like that is called a scalar field it's a scalar and it depends on position depending on position like that makes it a field V the components of V may also be functions of position and in that case V is called a vector field all right so anything which depends on position is called a field and that's our basic notation alright let's come to the field to the gravitational field the gravitational field is a property at every point of space and let me define it for you if it first of all let's imagine that it's created by some masses so let's imagine some masses some mass points and I'll label them as usual i j k where i j and k run from 1 to whatever and let's call this over here the ithe mess now what we do to define the gravitational fields we invent another mass we imagine one more mass which we'll call the test mass the test mass is a little tiny mass that we move around in space and we examine the forces on it and in terms of the forces on it in different places in space we define a field and that field is a force field or strictly speaking a field of acceleration so here's what we do we take our little pest mass and we hold it we let it go and we see how much acceleration it accelerates with and in what direction acceleration is a vector it has a direction it has a magnitude and that acceleration the acceleration experienced let's draw a little test mass the test mass I'll just draw with a little cross in it there all right it may or may not be real but may just be in our imagination or it may be a very small little mass that we move around then explore the vicinity of the real masses and that acceleration is a vector it depends on position and I'm not going to write x y&z now I'll just write X X now standing for x y&z that acceleration and let me write it down for you let's call the vector from the ayþe mass to the test mass the vector pointing from the ithe master test mass let's call that R sub I last time I had defined the vector as the vector from the test mass to I now I'm defining it to go from I to the test mass those two vectors are equal and opposite to each other but by defining the vector R sub I as the vector from I to the test mass I will save myself a minus sign which direction ah sorry I think I want that I wanted to go the other way if I want to save myself the minus sign I want to make it go the other way yeah our eye goes that way from the test mass to the particle which is creating the gravitational field now which way is the force on this object due to the ice mass it's pointing toward the ayth mass right the gravitational force on the test mass is pulling toward the ayth mass and so it's along the vector R sub I not opposite to it and that will save me we're having to write a negative sign all right so now what do we know about this field of acceleration well it gets a contribution from every test from every real mass here and the contribution in particular from the ithe mass it's a sum a is the sum over all the masses some of all the real masses it depends on the gravitational coupling constant our Newton's constant it depends on the distance from the test mass to our that's R sub I from R sub I squared and what else and it depends on the mass of the I part achill but each particle has a mass the acceleration on this particle due to the eigth article is proportional to the mass it's proportional to G and it's divided by R sub I squared but everything that's written here none of the things that are written here are vectors G is a number N is a number and R R is the length of a vector it's square is just a number there's no vectors on the right-hand side so there's got to be something wrong with this equation what's wrong is I have to remind myself what direction the acceleration is and it's along the direction R sub I so the way to deal with that is to put another R vector here but then we have too many R's in the numerator we have to put another one in the denominator you have to make this cubed because the length of this is R another way to think about it is to leave it as R squared and think of this as a unit vector the symbol for a unit vector a vector of unit length is a vector with a little hat on top of it little hat on top of it means unit vector a vector of unit length all right so this vector here is a vector it tells you the direction of things it tells you the direction of the acceleration but the magnitude of it all comes from here and we add them all up the acceleration on the test mass is a sum of vectors all different their contributions this depends on position for example the further away we take it from the distribution of mass over here the smaller it is if we bring our test mass right up to one of the particles here is a great big force on it if we bring it on the other side the direction switches so this a here is a field it depends on position and it's called the gravitational field all right there's a way to summarize this which is neat we talked about it last time I'm not telling you anything I didn't tell you about last time but the way to summarize it is in terms of the divergence of a let's look at the del dot a that's the divergence of a and let's look at let's just take a simple case the simple case let's suppose there's only one mass point if there's one mass point here then let's look at the gravitational field in the neighborhood of that mass point well it's pointing inward if I take my mass if I take my test mass and move it around in the vicinity of the the heavy mass here I'll discover a gravitational field which varies from point to point always pointing toward that mass right so the gravitational field the vector field points inward toward that point the point in the inward pointing this of it is called the divergence of a field the divergence of the acceleration field here is due to masses and it's proportional to the mass at every point there's a divergence which is proportional to the amount of mass at that point to express it mathematically we have to invent the concept called mass density now we all know what we mean by mass density it's not a new concept for most of you mass density simply means our take a volume of space ask how much mass we're now imagining that either that the mass is continuously distributed not in the form of point particles or that there are so many point particles tiny tiny point particles that we might as well effectively think of it as being distributed then the amount of mass in a unit volume or the amount of mass per unit volume is called Rho equals mass per volume we take a small matter we take a small volume count the ball the mass in it Delta M over Delta V is called is called the the mass density the mass per unit volume and of course it varies in general from place to place here there's no mass per unit volume here there's mass per unit volume so the mass per unit volume is itself a field it's the density field on the left hand side here we have the divergence of a the divergence of a a is a vector field but it's divergence is a scalar the relationship between masses and gravitational field which is basically the relationship that that Newton wrote down here it is can be re-expressed in the form that del dot a is equal to minus 4pi i'll play in a moment well I'll remind you where the 4pi comes from why it's there times the mass density oops times Newton's constant the strength of gravity is always proportional to Newton's constant so the effect of a mass is always proportional to Newton's constant it's proportional to the amount of mass that's the Rho and this is the basic equation that you can think of as a field equation it relates two fields the field of acceleration to the field of mass density and the constant G is just a constant in the relationship this is called Gauss's law why is there a minus sign here incidentally well the minus sign simply takes into account that if you have a mass point there the acceleration field is pointing inward so you don't really have a divergence you have a convergence of the gravitational field but a convergence is just a negative divergence so that's why the negative sign is there right this is called Gauss's law but there's another thing which is named after Gauss and it's Gauss's theorem this is a law of nature this is equivalent to Newton's relationship between the acceleration field and the mass densities it's something that you get from experiment or if not well from experiment or observation but there's a mathematical theorem which is Gauss's theorem which also has to do with divergences of vectors so let me remind you about Gauss's theorem it is so central to everything in certainly in gravity theory that I feel justified in spending another 15 minutes on the ten minutes on it um Gauss's theorem says that if you have a if you have a field which has a divergence that divergence is a scalar and you take some region of space any region of space that has a boundary those it could be a sphere doesn't have to be a sphere it could be a sphere with two ears sticking out of it ah it should have a topology of a sphere in other words it shouldn't be a donut but it should not have holes in it or anything like that even then it's okay but let's just take it to be a chunk of space like that then what Gauss's law says is that if you integrate the divergence over the volume the x dy DZ inside the region and how do you do that what does that mean that means you break up the region into little into little cubes three-dimensional cubes you break it up into twenty three-dimensional cubes in each cube you take the divergence of a you multiply it by the volume of the little cube and you add them all up okay that's integrating the divergence of a over the interior region here what it says or what Gauss's theorem says is that that integral is equal to a surface integral a sum over the surface so let's draw the surface here let's draw a piece of the surface and also now break up the surface the bounding surface also in the little squares little cells alright and in each cell the field a has a component perpendicular to the cell and they also have components in the directions parallel to the surface but a has a direction perpendicular to the surface let's call that a perpendicular or a sticking the component of a sticking out of the surface it says that Gauss's theorem says that the integral of the divergence of a over the interior is equal to the integral over the boundary that's usually called be Sigma 4 Sigma stands for little surface area stands for little surface here the integral D Sigma of the component of a sticking out of the surface in other words it's just adding up all the components of a sticking out of the surface the I told you what the significance of this would be for the flow of water if divergence of a stood for the divergence of the velocity field of water then it would just be the amount of water being pumped into a region here is equal to the amount going out through the boundary but this is Gauss's theorem let's just isolate Gauss's theorem now the integral over a volume of the divergence of a is equal to the integral of the over the surface of the component of a perpendicular now that's Gauss's theorem now let's put them together let's put them together to see how they work together let's imagine first that we have a distribution of mass which is spherically symmetric that means it doesn't have any dependence on angle in space that the distribution of mass has the symmetry of a sphere symmetry of rotational invariance another word there's another word for having symmetry of a sphere anybody know what it is it's ice a tropic isotropic means the same in every direction about a center about a central point here okay all right so it could be a ball of material but it doesn't have to be a solid ball of material but it should be shaped like a sphere okay so we have some mass in here and I want to know what its gravitational field is what I do is I surround it nice and spherically symmetrical e concentrically with a sphere I apologize for the fact that my drawings are two-dimensional if I had some way to make them three-dimensional I would but I can't so they're two dimensional versions of surrounding a spherically symmetric distribution of material by a shell by an imaginary mathematical shell can the red be seen all right so the red shell around here now let's see what this says all right first of all we integrate up del dot a but what is Del dot a from Gauss's law not from Gauss's theorem but from Gauss's law we know that del dot a is four pi minus four pi times the mass density times G so let's work that out so we're going to get we're going to get minus 4 pi we're going to get G and then we're going to get the integral of the mass density over the interior of the sphere the minus 4 pi g can come on the outside of the integral since only a constant and the integral itself just gives us this integral is just what you see here now what's the integral of Rho over of a volume that's the mass integrating up the mass density just gives you the mass enclosed so the mass enclosed within the red sphere okay so let's rewrite that that's minus 4 pi G times the mass within the sphere if the sphere if the red sphere is bigger than let's call this a planet let's call it the planet if the red sphere is bigger than the planet then it's a entire mass of the planet in there minus 4 pi mg or minus 4 pi G M is the left hand side what about the right hand side of the equation the right hand side of the equation says take the perpendicular component of the acceleration of the gravitational field now if the mass distribution is isotropic the gravitational field will also be gravity be isotropic in other words the component the perpendicular component of the field would be the same everywhere it will be pointing out with the same component everywhere is on this sphere so that means that the perpendicular component of a doesn't vary from point to point on the sphere and that means that this integral here is just the perpendicular component whatever it happens to be times the integral of these Sigma what is the integral of the Sigma mean it's just the area of the sphere it's just adding up the little area cells of the sphere and that's the area of the sphere which is 4 PI R squared R here not being the radius of a planet but being the distance from the centre of the planet to the red sphere ok so now you see why the 4pi was in Gauss's law why Gauss put it to his law it was in order to be able to cancel a lot out at this point it was arbitrary you could have defined things differently now we solve this for the gravitational field and what does it tell us it tells us that the gravitational field of a mass is minus mg over R squared something we already knew that's Newton's law that the acceleration due to a massive object is mad the massive out the mass of the object times G divided by R squared but it gives us a nice we talked about this last time I'm just going back over it it also tells us that it doesn't depend on the mass having all been concentrated at a point all it depended on was the mass was spherically symmetrical e distributed okay so that's that's I did this again to point out to you that del dot a equals minus four pi Rho G is equivalent to a being the sum over all the mass points of Newton's characteristic expression for for gravitational field so that's cool huh it's a nice way to express things this summarizes the very very complicated equation on top of it it doesn't look so complicated but if you wrote it out for lots of mass points and so forth it would become complicated whereas here's a way to express it where it's rather simple yeah um while we're at it somebody asked me last time about the gravitational field within the circuit within the earth what is it like within the earth here we did the gravitational field outside the earth we can now carry out a calculation of what's going on inside the earth if we like it's kind of interesting all we have to do now is take our red surface same calculation but now take the red surface inside the earth but now we have to be careful because the amount of mass within the red circle the red sphere is not the total mass of the planet anymore it's only the mass within the red sphere so let's do the calculation over again the right-hand side stays the same the right-hand side is still 4 pi in other words the the perpendicular component of the field times 4 pi times R squared where this in R is now the radius of the inner sphere here what about the mass within the red sphere how much mass is there within the red sphere well for that we have to make some kind of assumption about how the mass is distributed throughout the red sphere let's make the assumption which is not so bad for the earth it's not a bad approximation for the earth that the mass density within the earth is uniform the mass density is uniform inside the earth and then when you get to the surface of the earth it drops abruptly to zero that's not exactly true it's denser near the centre then than near the outside but let's take that as a model a uniform mass per unit volume within within the earth then how much mass is inside this red region here it's not the total mass of the earth now it's different so let's calculate it it's the volume of the sphere times the mass density this is equal to minus the mass within the sphere and the mass within the sphere is the volume of the sphere 4/3 PI R cubed that's the volume of the sphere times Rho 4/3 PI R cubed times Rho I think I got it right Oh R on the side G good thank you all right we can cancel out some things we can cancel out before PI's and now we can divide by R squared if I divide by R squared what I get is a perpendicular is equal to minus 1/3 Row 1 G 1/3 Rho times G times R what does it say this says that the acceleration of a test mass of course we would have to drill a hole through the earth if we wanted to drop a particle we can't drop particles if they're stuck in the rock but if we had some sort of particle well either we drill a hole through the earth and drop particles down or we drop particles which don't interact with the with the material of the earth neutrinos don't interact much with the material of the earth we drop a particle what does it do well it experiences an acceleration which is toward the center that's the minus sign here the perpendicular component of a is negative that means it's toward the center it's got some constants in it Rho G and 3 those are just numbers but it grows linearly with the distance from the centre of the earth so it's an acceleration which means a force if I were to multiply this acceleration by the mass of the test particle on both sides then I would have the mass the test particle the mass on the test particle is toward the center and it's proportional to the distance away from the center there's a name for such a system it's a harmonic oscillator a force which is proportional to distance in other words a particle displaced from the center will be pulled back to the center exactly as if it were on a spring with a spring constant that depends on the mass density depends on the gravitational constant and it depends on three so if we dropped a particle if we drill the hole through the earth and drop the particle it would just oscillate up and down up and down like a harmonic oscillator I forget the the time constant I think it takes about ten minutes for it to go through the earth and come back or something of that order magnitude so and this is not important to the general theory of relativity it's just another example of how to use Gauss's law and Gauss's theorem to solve what would be a very hard problem if we had to add up the contribution of every particle in the earth I mean just think about it how hard that would be yet this that this method allows us to to do it easily okay any questions the distribution of the matter outside of the shell it's not just negligible zero that's a good point okay right so let's let's talk about this yeah that's right that was one of the consequences and one of the curious consequences I'm sure it gave Newton some pause when he really realized that but that's right it is only the material inside the sphere which contributes to the gravitational field the material on the outside all cancels out the gravity from here and the gravity from here and the gravity from here they all cancel out Newton knew it and he proved it without calculus which is quite a the reason he proved all these things without calculus is because none of his uh none of his audience knew any calculus and so he had to do everything geometrically but we see it from Gauss's law Gauss's law tells us among other things that the gravitational field of a spherically symmetric mass only depends on the mass within the radius R and doesn't depend at all on the mass outside the radius R Bar a consequence of this I think we talked about it last time but might as well just say it again is that if the mass was entirely distributed on a shell imagine the mass was on a shell thin shell then inside the thin shell there would be no gravitational field at all particles would move around as if there was no gravity at all on the other hand particles outside the shell would see the gravitational field of the shell exactly the same as if it were a point at the center so that's a rather remarkable fact and it's special to the one over R squared law notice that tells you assuming Gauss's law and Gauss's theorem you get the one over R squared law I'm having a hard time avoiding those two figures at the surface of the earth I get a four that I hate it oh did I miss I may have mr. four would see where we're uh which we're so I go back I mean to go back of stuff here now the expression there on your right the lower there you're equating that to the expression on the board to your left at the surface of the earth where we left this in this while those two gravitational acceleration should be the same on the surface what should do the same with water a bar on the right board should be the same as a perpendicular on the Left board at the surface of Europe Oh at the surface of the good did I make a mistake I lose a factor for somewheres I may have left lost effect you miss is it a factor of four that's missing for pie from which one oh you're absolutely right they should agree at the surface of the earth so someplace I lost the four and a pie but this one this one of course is correct that's one person one person does somebody see where I mean zero with the mass you can even for classes they're on both sides of the equation is to be functional for mass well what do you have role when you have mass so let me go back and go back and do it i ah okay all I hear is a buzz Oh okay when I roll in one is best yeah yeah yeah oh okay no it was right it was right yeah yeah yeah yeah yeah yeah oh really that's right that's right no I all right to those who didn't follow the last discussions together that was right are one more concept which is important before we get on to relativity in the equivalence principle where we go the equivalence principle in particular is the idea of gravitational potential just a one one point the gravitational field is a field of acceleration it seems like no it jumps at the shower it jumps from jumps at the shell as you go from outside the shelter inside the shell of your riveted discontinuation the feel exhausted well if the shell is a finite if the shell is a finite thickness of course then it's smooth but if the shell was literally an infinitely thin shell then it jumps right the gravitational field jumps the equation you had here said that the gravity force or the acceleration varies linearly from the center of the earth yard and yet until we get to the upper we get to the boundary of the earth who also said see if we consider a series of shells the force on an object is all the shells outside could be neglected in all the shells inside right but the mass of a subscale whatever is not linear that you headed oh no no no that's right that's right we cancelled out yeah okay so let's let's go back now let's go back let's go back we had on the left hand side we have on the left hand side of the equation the gravitational field times the area R squared that was one side of the equation the other side of the equation had the mass which you correctly say was proportional to R cubed divided by R squared and we get a proportional to R that's what we did the R squared the the area Grose's R squared the mass Gross's R cubed and therefore the gravitational field is like 1 over R place to write that equation again requirement so when is your instance this one okay it's Gauss's law again we take a sphere which is smaller than the size of the earth okay one side of the Gauss's law gives us the total mass within that sphere is that create clear who asked the question yeah is that clear all right so one side of the equation is the mass within that sphere what is that that's 4/3 PI R cubed times the density well we also have to throw in a G the G ya 4/3 PI R cubed times Rho is the mass within that sphere and we multiply it by G that's one side of the Gauss's equation here the other side of the Gauss's equation is the integral of the component of a sticking out from the surface that's equal to the surface area 4 PI R squared times a perpendicular and there's a minus sign there so we have on the left side something like R cubed on the right hand side R squared I can cancel the 4 pi and then divide by r but cancel a 4pi I get G over 3 R cubed Rho is equal to minus R squared a perpendicular and now divide by R squared dividing by R squared leaves one power of r there so it tells me that the gravitational field is linear just like a spring okay on the surface of the earth you can equate that to mg divided by R squared yeah mg is 4/3 m is 4/3 PI R cubed smooth think of 4 PI W 0 this one is missing a 4 pi missing the 4 pi multiplying G to do is intimacy it varies you should have to be the forcing function wrong think of rows double the mass of the earth is 4/3 PI R cubed 4/3 PI R cube the row right okay so on the one side you have 4/3 PI R cubed you have a 4 PI and the other side you don't take a little bet oh ok I thought I did it carefully but maybe I made a mistake let's like let's think about how many think I'm right all right let's see let's do it during the break will doing the break I'll go through it with you and it is not entirely possible that I make numerical errors oh how you get to care is by doing the integral yes that's where he messes before yeah yeah so when when you cook we'll work it out in detail during the break okay I was going to talk about gravitational potential but I think I'll do that next week and I want to get on to the equivalence principle a little bit of some new physics this of course was all old physics and it's taken us another hour to go through Newtonian mechanics repetition of what we did last time but let's let's move on now to the equivalence principle now the equivalence principle of course is the principle that says that a field of acceleration is in a certain acceleration and gravity are the same thing but to make it specific the elevator analogy is a good analogy now for reasons that are a little strange I'm going to have my elevator accelerate horizontally it's just because I want to use the coordinate and call it X instead of Y you'll see why we'll get to it all right so we have an elevator somebody stands in the elevator I want to emphasize that this has nothing to do with the with this room it could be in any direction that the accelerator is accelerating it could be accelerating by means of a cable or whatever Rockets I don't care whatever are pulling it and it's being pulled with an acceleration let's call that acceleration little G what about an object that's dropped by the passenger within the elevator all right that object according to Newton's laws in a frame of reference outside the elevator in an inertial frame of reference watching this that object standstill but what does it do relative to the what does it do relative to the person in the elevator well the elevator person gets accelerated that way with an acceleration G but he doesn't say is accelerated he just says I'm standing still and he sees the little thing that he dropped accelerate toward the floor and of course accelerate toward the floor with exactly the acceleration G but not toward the right he sees it accelerate toward the left so he says there's a field of acceleration inside the elevator there's an a field inside the accelerator pointing downward everything he drops gets accelerated downward exact downward meaning to the left exactly as if there was a gravitational field a field of acceleration pointing pointing to the left how big is it well it's just exactly equal in magnitude to the acceleration that the elevator is moving with little G and sorry says there's an acceleration field of strength G pointing toward the floor of the elevator okay looks like gravity sounds like gravity smells like gravity are the conclusion that Einstein came to and this you might say to you so why did it take on Stein to say this almost any small child who's been in an elevator would have recognized it that gravity the effects of gravity the effects of gravity on everything that this person does juggles balls does whatever he wants to do is exactly identical to what he would experience if we were in elevator being accelerated gravity and acceleration are the same thing according to Einstein in other words the physical effects of being in an accelerated elevator are identical to the effects of gravity now as I said that seems almost trivial I'll tell you what let's take a break and then come back to some some discussion of its non-trivial 'ti why Einstein was called Einstein instead of Joe palooka all right so let's talk a little bit about how Einstein use this idea of gravity being the same as acceleration he didn't just say that but he used it and let's go through one application of it and then start to discuss some ideas about the general theory of relativity so now let me put now let me return to oh wow Oh before yeah before I do that before I do that I read there was some reason why I put the elevator horizontal it had to do with a graph that I want to draw let's talk a little mathematically about what it means to be in an accelerated frame of reference the accelerated frame of reference was the was the frame of reference as seen by the passenger in the elevator his frame were his or her frame of reference okay so X is horizontal that's why I had the elevator going horizontally because I wanted to use X and I want to plot X along the horizontal axis and along the vertical axis I'm going to plot time T this is all of space time X well I haven't plotted the other directions y&z but time and X and that's the world inside inside and outside now let's first talk about what it means for a frame of reference to be uniform velocity before we get to acceleration let's just suppose the elevator we're moving with uniform velocity not accelerated let's call the bottom of the accelerator floor let's call that X prime equals zero in other words we have two frames of reference me standing on the outside and I have a coordinate which I measure by meter sticks and I might measure it from right over here 1 meter 2 meters 3 meters four meters and so forth alright so that's my frame of reference and I call my coordinate X and here are the ends of a meter sticks neither sticks and the first made of stick this is X is x equals 0 x equals 1 x equals 2 and so forth alright now what about an observer somebody in the elevator who's moving with uniform velocity he has his coordinates he stacks up his meter sticks in here his meter sticks I'm not the same as my meter sticks my meter sticks is standing still in my frame his are moving in my frame and he will call the bottom the floor of the elevator he will call that X prime equals zero prime means in the frame of reference of the passenger primed for passenger ok so there's also an X prime coordinate let's take X prime equals zero that's the floor of the elevator the floor of the elevator is moving and so the floor of the elevator looks like that as time goes on the floor of the elevator moves further and further to the right not only does the floor of the elevator but x equals 1 follows along and x equals 2 follow along those were intended to be parallel I'm not sure they are so the moving frame of reference it measures distances relative to the floor of the elevator over here whereas the stationary frame of reference measures distances relative to x equals 0 over here now what's the relationship between X and X prime well let's take an any old point over here this is a point in space and time it's a point of space at some time and let's ask what its x value is its x value is the distance to here in other words it's the coordinate that the stationary observer sees just call it X what about X prime X prime is the distance to the floor of the elevator the elevator is moving so X prime is this distance what's the difference between them the difference between them is just the distance that the elevator has moved velocity of the elevator times time that's how far the elevator has moved in an amount of time T it's moved with its velocity it's moving with a uniform velocity V times T so what is X prime here's X prime over here X prime is equal to X minus VT that's the relationship between the prime coordinate and the unprimed coordinate and this you better get used to this idea of relating frames of reference like this ah that's the relationship between the moving frame and the stationary frame now there's one other thing in Newtonian physics who are not yet doing relativity in Newtonian physics all observers measure the same time time is the same for everybody and so we can write another equation that T prime is the same as T this is Newtonian physics this is not yet relativity this is not yet Einstein it gets modified in Einstein but this is the relationship between the coordinates measured by the moving observer and the coordinates measured by the stationary observer it's a coordinate transformation X prime is X minus VT T prime is equal to T now what let's suppose now that there was an object which is standing still in the outside frame of reference this guy drops his ball here the ball stands let's suppose it's tan let's say it stands still in my frame of reference just for fun how does it move in his frame of reference ah if it's standing still in my frame of reference what is it doing in his frame of reference I'm in particular interested in the velocity and the velocity all right so if a ball is standing still in my frame of reference it means X is equal to a constant it's just standing there still the elevators going I the person in the elevator is looking at the ball sees it moving alright if X is a constant then X prime is of course just minus VT but what I want to do is differentiate this equation with respect to time what does it say it says that X prime that I'm using dot to indicate time derivative standard notation we've used it before a dot indicates a derivative with respect to time that's equal so X dot but if X is constant if it's standing still in my frame of reference that's zero what's the derivative of minus VT with respect to T just minus V so this was a long-winded way to say that the velocity of the ball in the scene from inside the elevator is minus V of course it is the object was standing still in my frame of reference the observer is moving past it with velocity V and of course sees it moving backward relative to him with velocity minus V that's all this says what does it say about the acceleration as seen in the moving frame of reference let's differentiate it one more time differentiating it one more time that gives us the acceleration in the prime frame of reference what's the derivative with of V with respect to T if V is constant zero so if it's not accelerating in my frame of reference it's not accelerating in his frame of reference very simple okay trivial in fact but now let's suppose that the accelerate the elevator is not moving with uniform velocity but that it's moving with uniform acceleration then the distance that the elevator moves is not VT what is it supposing it's moving with uniform acceleration a acceleration equals little G the acceleration due to gravity then how does the floor of the elevator move relative to me if it's accelerating like that well it doesn't move on a straight line it moves on a curved line and moves on a parabola if it's uniformly accelerated relative to me it's moving on a parabola and that parabola is x equals one-half GT squared is this known to everybody this is a standard formula for uniform acceleration yell out if you don't know it okay this is this is the elementary calculation of a thing moving with uniform acceleration so the moving observers the floor moves to the right one-half GT squared along with a floor all the other meter sticks the end of all the meter sticks do the same thing they accelerate off on a curved path in space-time an acceleration means a curved path in space-time so let's let's see what the relation between X Prime and X is again same thing here's X Prime here's X what's the difference between them the difference between them is x minus one-half GT squared this distance is one-half GT squared the distance from here to here X Prime it's ex minus GT squared this is the new transformation for transforming to an accelerated frame of reference okay let's say again that the passenger drops a coin or whatever it is and that the coin happens to be standing still he throws it in just such a way that it's standing still in my frame of reference that again means X is a and what does it say about ex-prime it says that X Prime well let's get out out what does it say about the velocity if X is standing still and we differentiate this we get the velocity in the prime frame of reference zero from differentiating X standing still in my frame of reference but what happens if we differentiate minus one-half GT squared minus GT but I'm interested in the acceleration that the observer in here sees so I differentiate it again X prime double dot the acceleration seen from inside here what do I get when I differentiate it once more does minus G this is a mathematical statement of the simple trivial observation that acceleration works the same way as gravity the accelerated elevator gives rise to an acceleration for anything that you drop which is just minus G just like an ordinary gravitational field of the earth so anything which is dropped in this accelerator outside in this elevator accelerates toward the floor not toward not in the direction of the elevators acceleration but in the backward direction that's the minus sign here and the accelerates backward with the acceleration minus G so that's a mathematical statement but what I want to bring out is that it really has to do with using a set of curved coordinates it has to do or that acceleration is really the choice of an a curvy linear set of coordinates to describe space-time that's the real lesson here acceleration an accelerated reference frame is described by a coordinate transformation to a set of curved coordinates all right so we need to keep that in mind the connection between gravity curvature were beginning to develop but having done that let's now come back to some simple physics one of the things that I'm Stein became instantly famous for we can work it out qualitatively it's the bending of light okay so we first his first step was to say I really mean it when I say that the laws of physics in a gravitational field are the same as the laws of physics in an accelerated reference frame and in particular not just the laws of Newtonian mechanics but the laws of everything nobody at that time had the faintest idea how gravity effects electric and magnetic fields they didn't know in fact I don't even know that the question was asked at that time in a serious way and so it wasn't known for that reason it wasn't known how gravity affects the motion of light rays well Einstein said is I now know how gravity should affect the motion of light rays so here's his elevator now I'm drawing the elevator vertical vertical the elevator will accelerate upward and let's suppose a light beam is shined from one side to the other first suppose the elevator is not accelerating let's suppose the accelerator is stand of the elevator is standing still and you shine a light beam what does it do it goes in a straight line and it goes right across the elevator it hits this point over here now what happens if the elevator is accelerated upward well let's send the light beam just arbitrarily let's send the light beam at exactly the instant when the elevator begins to accelerate it's not yet moving it's just beginning to accelerate so the light beam starts moving off horizontally but now the accelerate can say accelerate the elevator is accelerating upward and so where does the light beam hit the other side of the elevator down here somewheres and it travels in a curved path in fact that curved path will be a parabola again it'll travel on a parabolic orbit and that parabolic orbit you can work out easily just from the for the motion of the elevator and what do you find you find that the light ray has a downward component of acceleration which is equal to G the same acceleration if the elevator is L is accelerated upward with acceleration G then the light ray will have a downward component of acceleration which will also be G so by that little argument nothing more than that Einstein deduced the fact that light Falls in a gravitational field as I said this was unknown nobody even suspected it that the existence of heavy masses which create gravitational fields would cause light to do anything other than travel in a straight line in fact light doesn't travel in a straight line when it passes through the earth passes across the earth why because the gravitational field of the earth can be thought of as an upward acceleration and so light will fall ah let's go through question yeah suppose you shine a light in the direction of the elevation yeah so that it's a little more complicated it's still it still it still has an effect and it affects the energy effective speed no because that depends on right the acceleration in the vertical direction will correspond to change in energy of the you know it won't affect this because one more question yeah the the edited experiment did verify general relativity yeah if you take plain old Newtonian mechanics and assume that a photon is a particle of mass you're he over c-squared hmm and just use Newton's gravity equation will it property let's do it let's do it look so that's good effect if that's general relativity predicts exactly the same thing by a factor of two uh-huh but let's let's say let's see if we can roughly work out the bending of light by the Sun okay the question is this if a light ray passes the Sun and clearly Einstein has already proved that light falls in a gravitational field so he wanted to estimate the first thing he did before he had the full apparatus of the general theory of relativity he had as much as I've explained up till now and he said I will try to estimate how much the light bends in other words what angle does the does the light ray get the accelerator what's the right word deflected yeah by what angle does the light ray get deflected all right we can make an estimate now at this point he was reduced to estimates because he didn't know the details of how the spherical earth with its varying gravitational field would really affect things but in particular he was interested in the light ray which just skimmed the surface of the sun room got as close to the sun what why because the effect would be maximal the closer the light ray got to the Sun of course he wasn't interested in light rays that went through the Sun light rays don't go through the Sun a light ray that just barely skimmed the Sun how much it got deflected okay let's see if we can figure it out first of all in passing and passing through the when it passes close to the Sun when it passes close to the Sun it has an acceleration toward the Sun which is just the acceleration field of the Sun so when it's in the vicinity of the Sun let's break it up when it's when it's out beyond the distance comparable to the radius of the Sun it's too far from the Sun to feel anything whether it's when it's within some region comparable to twice the radius of the Sun here it feels the gravitational pull of the Sun all right what's the acceleration that it feels downward well for that we just have to work out the acceleration due to gravity when we have to work out is how much acceleration is there at the surface of the Sun I think we already had a formula for that right the acceleration due is M mass of the Sun G over R squared that's the go ahead of that right mg over R squared downward right everything gets accelerated downward light as well as everything and all with the same acceleration so mg divided by R squared is the acceleration the vertical component of the acceleration right how much velocity does the how much vertical component of the velocity now vertical doesn't mean toward the Sun it means vertical on the blackboard here how much vertical component the velocity does it have after it passes the Sun well to calculate the change in velocity you multiply the acceleration by time acceleration times time is the change in velocity so you take this acceleration and multiply it by the amount of time let's call it delta T that it takes for the light beam to cross across the radius of the Sun now we can estimate that because we know how fast light moves it moves with the speed of light so how long does it take it takes a time delta T which is approximately twice the Wraiths this distance here divided by the velocity that's velocity times time equals distance so time is distance that it travels divided by the velocity but what velocity the velocity of light the velocity of light going across here so that's 2r divided by C so let's plug that in that gives us another two the two is not important R divided by C and what is that that's twice mg over R C right that's the downward component of velocity here but that's not the angle the angle is basically the ratio of if we have a small angle relative to the horizontal is the horizontal small angle relative to the horizontal that's the velocity this is the direction of the velocity after it passes the Sun then the downward component here the the angle is the ratio of the downward component to the horizontal component the downward component divided by the horizontal component is essentially the angle at least if the angle is small so the angle what's called theta is equal to the ratio of the downward component here it is 2 mg over R see that's the download' component of the velocity divide that by the horizontal component of the velocity and that gives you the angle another horizontal component of the velocity is 1 over C and here's the answer to mg over RC squared that's a very small angle typically mainly because C is a big number and G is a small number so G is a very small number C is a very big number C squared is even bigger and so that's typically a small angle when the when the light ray goes across the Sun so that was the crude estimate that Einstein made I can't remember what the what the correct answer is from the general theory of relativity it's within a factor of two of this I think maybe it's just exactly either twice as a half of it I can't remember and so every light beam that passes by the edge of the Sun gets deflected by that much that's that was Einstein's prediction it was verified by looking at stars passing across the Sun during an eclipse somewhat we don't need to go through that most of you know the story but the main point here is that I'm Stein used the idea that gravity and acceleration are the same something that was known to everybody but he used it in the context which nobody had ever thought of namely he applied it to the motion of light and to electromagnetic phenomena in general okay so that's a now yes gravity is like acceleration but that idea is pretty much limited to a uniform gravitational field uniform vertical gravitational field it doesn't really make sense when you try to think about it globally in terms of the Earth's gravitational field when you stop thinking of a Flat Earth approximation so here's the earth The effect of the Earth's gravity on anything over here out here Einstein said is exactly the same as if you were in an elevator over here accelerating upward with with a certain acceleration well first of all you can the acceleration has to be different in different places different direction in different places different magnitude in different places and so there's no overall sense in which you can take the gravitational field of the earth and replace it by a single accelerated frame of reference the acceleration has to be this way over here this way over here this way over here this way over here so there's certainly no sense in which the full gravitational field of the earth can be replaced by saying you're in an accelerated reference frame what you can say is for a small amount of time and over a small amount of different distance distances which are too small to detect the fact that the gravitational field has different directions in different places in other words on a small elevator for a small amount of time the effect of the Earth's gravitational field is the same as the effect of an accelerated elevator but only for a small amount of time why because unless you allow the acceleration to change with time but for a small amount of time you can pretend that small amount of time and a small amount of space not enough space to feel the change in the direction of gravity you can say that gravity has the same effect as acceleration but if you really think about it I mean you have to think much more deeply about what the or the gravitational field of the whole round earth is now this is related to something else if you're freely falling near the surface of the earth you don't feel the effect of gravity you feel you can tell the difference between free fall and not free fall but there's no difference between falling at the surface of the earth and being at the middle of space with no gravitating objects at all what about what about somebody falling over here can somebody throw only falling away just in freefall can they or can they not tell the difference between falling and being in an elevator so to speak or being uniformly accelerated what's the right word and a person over here tell it they're falling as opposed to being in outer space can they tell the difference between gravity and acceleration and the answer is yes they can tell the difference because the gravitational field of the earth is not uniform it points in different directions and it varies with distance so that means an observer is an observer a cubic observer a cubic observer has a different gravitational field on them in different places the bottom of his feet is being pulled harder than the top of his head so it's being stretched that way are the gravitational field is sort of pinched in toward the center here and there's some sense of being squeezed this way you know it does a call those are tidal forces we talked about tidal forces last time tidal forces are real you can feel them but in order to feel them you have to be large enough to sense the variation in the gravitational field so the right statement is if you're small enough if you're small enough and for a small enough amount of time then the gravitational field you don't feel it in falling and it can be completely replaced by being in an accelerated frame of reference gravity and acceleration are the same but if you're big enough to feel the curvature of the Earth's surface and if you're big enough and you wait long enough enough time and you can feel the variation of the radial component of the gravitational field then you can definitely tell the difference between being in outer space and being in freefall freefall isn't so free you get squished one way stretch them other way these are tidal forces the effect of tidal forces are an obstruction to eliminating the gravitational field by replacing it by an accelerated frame of reference all right you can't really replace a gravitational field by an accelerated frame of reference except locally and over small times so the equivalence principle is a limited idea that makes sense only in this limited sense okay there's an obstruction to getting rid of the gravitational field and replacing it by freefall by um by a field of acceleration and that obstruction is tidal forces there's no way that you can construct an accelerated frame of reference which will completely get rid of the gravitation the way that we would if the gravitation were uniform gravitational form you go to an accelerated frame of reference falling down with uniform acceleration and gravity is gone in that frame of reference no way to do it for the real Earth the real earth is a gravitational field which is created by mass and mass corresponds the divergence of gravitational field it's the divergence of the gravitational field which gets in the way of trying to replace it by a uniform acceleration so uniform acceleration is or how to say it eliminating gravity by replacing it by a accelerated frame of reference there's an obstruction and the obstruction is the masses in the universe create a divergence of the gravitational field I'm beginning to get tired so I think I should probably quit any minute now we need to you need to review some special theory of relativity perhaps I'll take another ten minutes and just remind you of the things we learned about special relativity in previous courses and which I want you to know let's see yeah but the inside the elevator is where there's special relativity yeah we of course are studying it in the approximation the moment of the Tony in physics yeah I don't think I want to do special relativity tonight I think I won't do that instead I want to talk about geometry a little bit geometry and curvature curvature is a concept we're going to need and I think we can just discuss it without discussing physics for a little for a little bit how do I describe the geometry of this blackboard the geometry of this blackboard is flat space just flat two-dimensional space how do I describe its geometry the way a mathematician would describe its geometry is by specifying the distance between any pair of neighboring points if you know the distance between any pair of neighboring points you can rebuild the entire geometry from that knowing the distance between any pair of neighboring points nearby points is enough to rebuild the geometry of this blackboard and to discover that it's flat let's take a pair of points let's characterize them by coordinates x and y or just let's call them X sub I I runs from 1 to 2 and here's another point at position X plus DX and y plus dy two points separated by a differential distance DX and dy what's the difference what's the distance between them the distance between them is just DX square plus dy squared or that's the square of the distance between them the square of the distance between them that's called the s squared is the X square plus dy squared that's just Pythagoras theorem the X dy the S the S squared is the x squared plus dy squared now that is something which is true if you use for your coordinates nice rectangular Cartesian coordinates what would happen if I replace the Cartesian coordinates on the blackboard by some crazy set of coordinates coordinates for which the axes were curved and some highly complicated kind of way there are lots of different coordinate systems that you can use to describe the blackboard they don't have to all be drawn as a rectangular grid we could use polar coordinates polar coordinates look like this all kinds of other coordinates and in general the lines of constant coordinate lines of constant X and the lines of constant Y would be curves on the blackboard that would be a general description of the blackboard not tied the nice rectangular mesh like this all right so let's suppose now this could be x equals 1 x equals 2 x equals 3 is y and his x equals 0 is y equals 1 y equals 2 y equals 3 I have a grid if I tell you the value of x and y you can tell me exactly where the point is on the blackboard but the grid is no longer a rectangular grid the distance between two points is no longer given by DX square plus dy squared there's a general formula for any kind of curved coordinates like this it looks like this it's a d s squared has some coefficients whatever it is it will involve these little differential displacements and it will involve them quadratically to the second powers we're going to go over this again I want to lay it out for you now we're going to even prove it but it's going to be some quadratic form this is called the quadratic form the x squared the x times dy and the y squared it'll have all of these things in it and furthermore these G's will depend on where you are there'll be numbers there'll be numbers but they'll really be fields they'll depend on x and y x x I'm really getting tired so I should probably quit the next five minutes I don't know how many of you have seen this for that sort of formula how many have not seen it how many of not all right so we're going to work it out next time and I'll show you why this is true but before I do let's just go through what some of these things mean what does it mean to have a G one two here in the original formula you had nothing multiplying the x times the Y there's the X square is the Y squared I say more generally if you think about general sets of coordinates with curvilinear coordinates then there'll be terms with the x times the Y you know what those mean when there's a term DX dy it means the coordinate axes are not perpendicular that's what that means what about G 1 1 and G 2 2 they'll also have to do with the relative spread if we were to make coordinates like this which were much denser in one direction than the other direction then the G 1 1 G 2 2 would be different than each other if the coordinates curve and things change from place to place then these G's depend on position but the general formula for general curved coordinates for the distance between two points look something like this and it contains three functions of position G 1 1 G 1 2 and G 2 2 those functions are called the metric and we're going to go through again I'm going to show you why that's true but we're still just describing the blackboard we're describing the blackboard and the blackboard could always be described by these coordinates here when it's true that you can describe a geometry in this simple way like this the geometry is called flat even if you decide to describe it in a more complicated way if you can find new coordinates that simplify it in this form it's called flat this is the hallmark of a flat geometry Pythagoras theorem if you like there are geometries which are not flat there are geometries that no matter how you choose the coordinates you cannot reduce it to this form those geometries are called curved nor example of curved geometry the surface of a sphere the surface of a sphere you cannot choose coordinates on the surface of the sphere which makes the distance between points the x squared plus dy squared it's always going to be more complicated than that no matter how you choose the coordinates what about the surface of a cylinder is the surface of a cylinder curved no it's not it's flat because you can always choose exactly the same coordinates that you would choose if you unrolled it you can't unroll a sphere and spread it out flat on a surface but you can spread out a cylinder and make it flat on the surface cylinder is flat the sphere is not how about let's see what other kind of a saddle surface a saddle surface shaped like a saddle is also not flat can't be flattened out ah a cone what about a cone hmm yeah a cone is flat Everywhere's except at the tip the cone is flat everywhere is accepted tip because you can always take a cone here's a column right there's a cone at the tip something funny is going on but anyway is else you can unwrap it and spread it out on the table so you can use coordinates on a cone except for the tip which look like this cones are not flat but only at the tip cylinders are flat planes are flat spheres are not flat how do you tell what's whether a surface or given metric like this describe something flat or not well the answer is curvature which I haven't defined we're going to define it the answer is curvature but curvature is the obstruction the flattening out the coordinates tidal forces are the obstruction they get getting rid of the gravitational field by changing coordinates to accelerated reference frames though that connection is not accidental curvature is the obstruction to flattening a surface out or to thinking of it as flat tidal forces are the obstruction to removing a gravitational field by coordinate transformation to an accelerated frame of reference that's the connection we want to build up the curvature of space and time is essentially the same thing as the effect of a real gravitational field the real effects of gravitational fields are tidal forces and that's what we're going to find that curvature of space-time is the same as tidal forces ok let's quit now before I fall over and exhaustion the preceding program is copyrighted by Stanford University please visit us at stanford.edu

Channel: Stanford

Views: 661,870

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Keywords: science, physics, theory, of, relativity, force, gravity, dark, energy, time, lambda, particle, repulsion, spring, constant, attraction, big, rip, density, universe, vacuum, test, mass, vector, component, del, derivative, gravitational, field, accelerati

Id: s8UrYIZhm60

Length: 107min 37sec (6457 seconds)

Published: Thu Jan 22 2009