"From Newtonian Gravity to Einstein's Theory of General Relativity"

Case Western Reserve University's great thinker series proudly presents the origin science Scholars Program these lectures are presented by the Institute for the science of origins a partnership of Case Western Reserve University the Cleveland Museum of Natural History and ideastream with the assistance of Case Western Reserve University Segal lifelong learning program the College of Arts and Sciences and media vision it's my pleasure tonight to welcome a colleague and friend Andrew Tully did his undergraduate work at Oxford University and then his PhD at Cambridge University after that he crossed the Atlantic to this side and spent some time at Princeton University for postdoctoral work and then it was a distinguished research fellow at the Perimeter Institute in Canada from there he joined us here at Case Western Reserve University in the Department of Physics where he is an associate professor in the particle astrophysics and cosmology group and tonight he will tell us about from Newtonian gravity to Einstein's theory of general relativity please welcome dr. Andrew Tolly so yes so I shall be attempting to introduce to you I ensigns theory of general relativity for those of you that don't know this year is the hundredth year of this theory Einstein completed the theory in 1915 and all physicists would agree that this is still one of the greatest scientific achievements of the of the 20th century and I'm going to try and explain to you a little bit about this theory but we're going to start basic and we're going to start with Galileo and Newton and what was understood by Galilean and Newton and then well we'll slowly see how Einsteins developments and those of others through special relativity eventually led to what we now call the general relativity general relativity which is the current understanding of the theory of gravity so let us begin at the beginning so in fact many of the foundational principles of our current understanding of what gravity is can be traced back to Galileo and there are two key concepts and whilst these weren't called these were the names given by Galileo you can see these concepts in Gallo's work and what we now call the equivalence principle was the first one and the second one is the concept of relativity as according to Galileo so - the first concept that ilayya was concerned with how objects fall under gravity and as most of you probably know there was that there was a belief at the time that came from Aristotle that heavier objects would fall faster than light objects and Galileo gave a very interesting thought experiment to show why this this this couldn't be the case or at least he believed couldn't be the case and so he said well imagine two objects one light and one heavy so here's a big heavy object a big heavy ball in a light one and you touch them together by a string now if Aristotle is correct then the heavier object will fall faster under gravity than the light object and that will make the string between them taut but once the string is taut you can then think of that as a single object whose mass is the sum of the individual masses of the lights of the light ball in the heavy ball but then if Aristotle is correct then that combined object would fall even faster than any of the individual components which is obviously a contradiction so by that reasoning Galileo Galileo argued that Aristotle must be wrong and it must be the case that heavy objects fall just the say with the same acceleration under gravity as light objects and then that led to his famous Tower of Pisa thought experiment no one knows if he actually did it or not but he proposed a thought experiment where you stand at the top of the Tower of Pisa and you drop off two objects of different masses and if you neglect arid resistance then they will fall with the same acceleration towards the earth determined by gravity which in the earth is locally that acceleration is given by this number 9.81 m/s^2 we now know that's that really to be the right answer so that's the first thing and this will be this this will come back later is very important to Einstein's thinking about gravity but now let's come to the second one which is the concept of what we would now call relativity we find this already in Galileo's work on the dialogue concerning the two chief systems of the world and here's the text I'll read it for you but it's a little bit small but he starts off shut yourself up with some friend in the main cabin below decks and some large ship and have with you there some flies and butterflies and other small flying animals then he goes on to sort of talk about various experiments you can do with the Flies and butterflies this is a very poetic language relative to modern papers but this is how he described it and then we get to the crucial point when you have observed all these things carefully are you've done experiments with flies and butterflies and fishes though doubtless when the ship is standing still everything must happen in this way have the ship proceed with any speed you like so long as the motion is uniform so now the ship is moving and not fluctuating this way and that and you will discover not the least change in all the effects named nor could you tell from any of them whether the ship was moving or standing still so what Galileo is saying is if you're in a moving ship inside and you're not looking out the window if that ship is moving at constant velocity you can't tell you're in a moving ship everything locally to you looks identical to that what if the ship was at rest and that's really the concept what we now call the concept of relativity and to understand why that is true let's think about throwing a projectile initially at some constant velocity in the horizontal direction which then falls under the gravity in the horizontal direction if it has some initial velocity it just keeps that velocity and is an the velocity it has in the horizontal direction is not affected by gravity but in the vertical direction it'll fall under gravity and we'll create this parabola but the key thing is was gravity gravity affects how it falls vertically it doesn't affect how it moves horizontally and so now let's imagine those two ships of Galileo here's one ship the ship is at rest and someone inside the ship just drops a ball to the earth and it will fall under gravity it'll accelerate down with a constant speed that's independent of its mass and then you have another ship which is a uniform motion at some constant velocity and imagine that someone on that ship then also drops the ball well from the point of view of the of the ship at rest the initial velocity of the ball will be a constant velocity because the person on the ship that drops the ball in itself is he's moving at constant velocity the velocity of the ship and so the ball will drop with a parabola but as the ball drops with the parabola the ship is also moving along and so then what you get is in the rest frame of someone who's sitting on the ship they don't see anything moving in the horizontal direction all they will see as the ball drop directly downwards as if the ship was at rest and that's the concept of relativity and so by reversing that argument in the rest frame what we call the rest frame of the observer who's inside the moving ship very well from their point of view the ship that used to be at rest is also in motion in the opposite direction and they say and the reverse of that argument is true so this is the concept of relativity that motion uniform motion is relative to the observer so we say that motion is relative and all observers who are in uniform motion are equivalent the fancy way of saying that in modern terms is that we call each observer who's moving at constant velocity an inertial frame or inertial frame of reference and we said that the laws of physics are the same in all inertial frames of reference so he summarized that a bit of poetic language again so the fish in the water will swim towards the front of the bowl with no more effort than towards the back and we'll go with equal ease to bait placed anywhere around the edges of the bowl and so on and so on so this is the concept of relativity according to Galileo now in the Galileo dives Newton was born a Newton incorporated into his famous laws of motion Galileo's concepts and in particular in Newton's first law which is which is written here in the Latin with Newton's first law says everybody persevere in its state of rest or in uniform motion in a straight line unless insofar as it is compelled to change that state by forces and press there on so in more ordinary terms he's saying if there is no force if an object is initially at rest it stays at rest if it's initially moving with constant velocity it stays a constant velocity but that has to be true because of Galileo's notion of relativity because if something is initially moving at constant velocity you can't tell if it's a rest because in somebody else's frame that object is at rest and if the if the object is at rest so you don't put forces on it is going to stay at rest so Galileans concept of relativity is built in to Newton's first law and what is often called the concept of inertia so inertia is essentially the name of this property that something will have some velocity some speed will continue with that velocity unless acted on by a force and so and furthermore in Newton's second law the famous one that determines how forces create acceleration in fact really it's the key that that forces great acceleration are not velocity so acceleration is the speeding up or slowing down of forces which is the same according to all Galilean frames of reference so all observers will see the same acceleration even if they don't agree with the velocity and furthermore Newton's second law tells us that the larger the mass of an object for the same force the smaller the acceleration and then tying that together with Galileo's argument that all bodies fall to earth regardless of the mass they'll fall to earth with the same acceleration under gravity then the only way that can be true is if the force of gravity itself is proportional to the mass of the object essentially so that the mass here cancels out in this formula so that you get that the acceleration is a constant independent of the mass and now we call this the equivalence principle in one language this is saying that the the effective mass seen in gravitational gravitationally the thing that determines the strength of gravity is the same as the inertial mass that comes in Newton's second law and again this will become important later now to complete Newton's story if you posit that the force is universal so let's think about an apple falling to the earth so we know that the force to explain Galileo is prediction that everything falls with the same acceleration then the force of gravity on the Apple from the earth must be proportional to the mass of the apple so it cancels out in Newton's formula but by the same token the force from the apple on the earth not something we tend to think about must be proportional to the mass of the earth but this is the same force there's only one gravitational force between the two and so you infer that the force of gravity must be proportional to both the mass of the apple and the mass of the earth and so in other words it's proportional to the product of the mass of the Apple and the mass of the earth and then there was only one missing piece which is how the force of gravity changed with distance in fact several scientists including Hooke and bully Aldous suggested that the way the force of gravity depends on distance is an inverse square law so that means as you double the distance you decrease the force by a factor of four a Newton showed that if that were true then you would precisely predict what were known to be the Kepler Kepler's orbits of planets around the Sun would be a direct consequence of that inverse square law and so we put put it all together and this gave him his final form for the force of gravity that the force of gravity is proportional between two objects of a given mass is proportional to the product of their masses an inverse square of the distance between them and there's some constant in there which we now call Newton's gravitational constant in in that formula and interestingly Newton himself was the first we didn't question his own theory it was a nice theory but he didn't he could see there were some problems one of his one of them was that the this notion of action at a distance regardless of how far of those objects were away from each other the force of gravity is always acting and so he said that one body may act upon another at a distance through a vacuum without the mediation of anything else buying through which their action and force might be conveyed from one another is to me so great an absurdity that I believe no man who has in philosophic matters the competent Faculty of thinking could ever fall into it but of course he did fall into it so that that's my summary of Newtonian gravity and the key points are Galilean relativity we'll see how that comes back later you nine Stein's thinking and Galileo's famous recognition that all objects will fall to the earth would same acceleration regardless of the mass at least if you neglect air resistance and and then Newton had it that gravity was a universal force he had one formula that always described the same force which was proportional to the product of masses of the two bodies in inversely proportional to the square of the distances but there is a there is a bizarre thing about this force as Newton recognized that its action at a distance so no matter how far away two objects are they will always feel each other's influence instantaneously Galilean relativity all this time how come Einstein gets all the credit for relativity that's a good question so I think people forgot at some level people forgot how much Galileo soar as well but I think it's also partly because now we own Stein's relativity is an important different has a different flavor to it and what we really remember is the significance of that discovery yeah but relativity really is Galileo's idea and you see clearly stating that concept so Hooke used to complain that Newton had stolen the idea from him is er that's right so when when hook when Utahns nted this at the Royal Society for the first time in his complete formula Hooke was there immediately accused him of plagiarism because he had himself thought of the inverse square law rose as well as in fact many other people had done but what Hooke record but hook did recognize what Newton got credit for and why he got credit is was the calculation of these Keplerian orbits which Hooke hadn't done at least with any the same level of accuracy and that's why we remember Newton for gravity even though of course he was not he did not invent gravity he did not even get all the deeds he was not even the first to think of all the details of the force but he was the first to convincingly demonstrate that this was the only formula that fit the astronomical data at that time thank you for joining us you've been watching dr. andrew Tully discussing Galilean relativity and Newton's gravity for more information on the origin science Scholars program please visit the Institute's website at origins dot case dot edu in the next part of the talk dr. Tolley discusses special relativity now back to the talk ok so the second part is we're going to fast forward in time to really the end of the 19th century and up up to up to then these notions Newton's laws of motion were not questioned and Galilean relativity was was not questioned to the late 19th century the 19th century physicist was concerned with the nature of heat and thermodynamics and also the crucial question of what are electricity and magnetism and what electricity and magnetism are was answered by James Clark Maxwell he though there were many parts of the theory that were built up over time but he was the first one to write down the complete description of electricity and magnetism in unified in fact to a single a single theory of electromagnetism and this was really was one of the towering achievements of 19th century physics and it just it fit all the observations of what what you needed to describe electric fields and magnetic fields and the motion of charged particles and so on in in his famous paper in 1865 he also predicted the notion of electromagnetic waves so these were waves in which the electric fields and the magnetic fields were oscillating in space and time a lot like water waves and he calculated the speed using his theory and he found that the speed was extremely close to what was understood to be the measured speed of light at that time and so of course it became very natural I think Faraday first suggested this or had suggested it before it came very natural to associate those elect those electromagnetic waves with light itself when you see here is saying the agreement of this result seems to show that light and magnetism are affectations of the same substance and again this is something we now know to be the correct answer and a few years later electromagnetic waves clear evidence that the light and radio waves and so on were electromagnetic waves was given by several people however the problem is and there wasn't clear at the time but we was clear in retrospect is that Maxwell's equations violated Galilean relativity so that means that they didn't have this property that they didn't the wouldn't couldn't be the same in all different Galilean inertial frames of reference so Maxwell if you're using Maxwell's equations in the ship at rest you'll get a different answer than if you used it in the ship moving at constant velocity and so there were two solutions to this and and the one that Maxwell chose was the first one is that we you should only apply his theory his equations in one particular preferred reference frame and that reference frame he supposed was the reference frame of some fluid that filled space and time I that was called the lumen if luminiferous ether there's another solution which of course is the right answer we'll get to is the Galilean relativity must be modified in some form but the first one is the one that Maxwell took and what what they imagined and and it's amazing in retrospect that this was the this was the general perspective but people imagined that there was space and time were really filled with some fluid that had to be stiffer than granite in order to determine the correct speed of light propagating through it so the there was a there's fluid called the ether in which the light was a wave inside this ether and it had to be stiffer than gravity gravity gravity but it had to be perfectly inviscid which basically means that it nothing interacts with it so as the earth goes round the Sun is going through the ether but nothing happens to it so this is this is a sort of crazy concept how can something be so stiff but have no viscosity anyway that's what that's what he proposed and in fact many other people proposed that at the same time and so people went to look for the motion of the earth relieve that because by that point because no one thought the earth was a rest people knew the earth was moving and so if it's moving it must be moving through the ether and so we should be able to detect the actual motion through the ether by the modification of the propagation of light and of course there were many experiments to look for this and this is of course the most famous one right here at case which is the Michelson Morley experiment which was a particular type of optical experiment and interferometer that was intended to figure the speed of the earth through the ether by the fact that if the earth was moving then the experiment was moving then if if the light went down this way versus this way in one part that would have to go longer than the other and you would see then by combining the two together and looking for interference you would be able to figure out the velocity of the earth and famously it gave a no result they couldn't find any motion of the earth through the ether and this was a significant problem and many physicists attacked this too trying to understand and so it was first George Fitzgerald who proposed a rather crazy idea that when objects move through the ether they somehow become contracted so their size becomes changed as they move through the ether and this is the this this he argued could be used to explain that we contracted in precisely a way to cancel out the effects you would see in this experiment so he wouldn't see any interference pattern and this was developed into a complete theory by Lorentz and well really finally by Poincare you put it all together and the idea and the idea that Lorentz had was that as you move through the ether you have a different notion of space there different notion of time so now something that's moving through the ether will have a a different we'll see space differently will see time differently they'll measure different time on a clock in such a way to cancel out any effects in these experiments that were looking for the ether so this is slightly bizarre thing so things change as they move through an ether so that you can't ever find that you have anything that was what was that was what they were positing an Einstein just turned it round he didn't really give any new mathematics Lorentz and Franco he already had the mathematics he simply said well if you're creating these crazy formula to try and the notion that things change in their length and so on and time is different to say that why you to explain why we never see our motion through the ether maybe the ether just isn't there and of course that that is the right answer and in fact what God did affect what Einstein did is we reintroduce Galileo's concept of relativity that all inertial reference terms are equivalent so that as we move at constant velocity then the laws of physics should still be true and so Maxwell's equations should apply in the ship at rest and should apply in the ship and moving at constant velocity but of course he's still needed to explain these experiments and so like Lorenson Fitzgerald and Poincare a and so on he still had to imagine that space and time measured by observers who were moving at different velocities would have to be seen as as different things and so we we returned to Galileo's experiment and here we think about the two ships again and and some objects moving described by some position in the horizontal direction and the vertical and some time T and now let's imagine a moving ship in the in the reference frame of the moving observer the same space and time will get modified through this complicated formula with square roots here and so on which depends on the speed of light now don't worry too much about the precise formula but it's clear that this formula is much more complicated than Galileo's formula and it describes how an observer moving at some constant velocity will measure differently space XY and time T relative to an observer at rest these are what we now call Lorentz transformations by the same token so he through these transformations he was able to show that Galileo's notion of relativity is still true in other words if you go to the reference frame of someone who's locked up inside the hull of the ship then they will not know that they're moving at constant velocity and they will see what's going on in the other ship as as as as they will see but basically different notions of space and time relative to that and and all this happens in a rather magical way so that for the two observers so observer one you can think of the ship at rest and observer two is the ship moving at constant velocity they measure space and time in a different way such that they always agree on the speed of light and that's the property of the lens transformations in fact what Einstein did was to derive those transformations based on assuming that all observers no matter what their velocity is will see light to be traveling at the same speed so this was quite a radical concept although it was the only thing that was consistent with Maxwell's equations and so here they here to emphasize this point here the picture is if someone is on it on the moving ship and they shine a torch then light goes out of the speed of light according to Galileo from the reference frame of the ship at rest then that light should be going at faster than the speed of light because you'd think it would be the speed of light plus the speed of the ship so he thing could be going faster but according to Einstein the speed of light from this torch is also the speed of light in vacuum the same speed even even even though this guy's at rest and this ship is moving and but to achieve that he had to imagine as a rensin and Fitzgerald had done that space and time measured by different moving observers are different and it was his professor Hermann Minkowski that tied this together and recognized that really what was going on is that space and time were not separate things but should be wedded together in terms of a four-dimensional space-time geometry and he delivered a nice address in which he says the views of space and time which I wish to lay before you were sprung from the soil of experimental physics and therein lies their strength their radical henceforth spaced by itself and time by itself are doomed to fade away into mere shadows and only a kind of union of the two will preserve in an independent reality and he was exactly right and to explain that I need to tell you a little bit about geometry and remind you for what we mean when we talk about two-dimensional geometry that can be summarized by Pythagoras theorem if we're in two dimensions we want to measure the distance between here and here and we do so by doing a triangle like that 90 degrees then the distance from this point at this point is given by it's that distance squared is given by the sum of the squares on the other side and that's Pythagoras this famous theorem and that theorem that theorem basically encodes what we mean by two-dimensional geometry and then later that was generalized to the notion of three dimensions we all know we're in three dimensions of space this way this way and this way and to compute a distance in three dimensional space then you take the distance if you have a cube for example you want to know the distance between here and here you you you take the square of the sum of each of these three sides and then that is equal to the square of this distance so this is essentially just Pythagoras theorem in three dimensions so what do you do when you get to four dimensions well you just do it again except now there's a twist and the twist is that you put in a minus sign here and it's that minus sign that distinguishes between space and time but nevertheless this is a four dimensional geometry in which to compute the distances between two points in space and time you just use pythagoras and you add up the four things except you don't add them all up you minus the amount in time multiplied to get the unit's right by the the speed of light and here's a copy of some notes from Einstein I think around 1913 in which you show him clearly having taken in this concept and when you got four coordinates here X Y Z ICT this this is the time direction with a square root of minus one in here that was minkowski's notation and you see him clearly using that concept now a key feature about this is that space-time geometries are very different than space geometry so whilst we're saying that we have a four-dimensional space a four-dimensional space-time is something quite different it has extra structure because of that minus sign that's the only difference there's the only difference we know of between space and time that minus sign there it has a big influence because it means that sometimes the distance between things in space-time is positive its distance squared is positive sometimes it's negative and sometimes it's zero and the cases where it's zero are precisely for things moving at the speed of light and so here is a picture imagine a point and you imagine you send out a light beam that could go in any direction this is a picture of that light beam moving forwards in time and here's a picture of it going backwards in time if you could imagine that or you really mean the light coming in to converging to that point and that creates something like although a light cone the cone of light propagating from out from a point and along with any point on that cone the distance between here and here in space-time is zero this light cone distinguishes so in relativity it's also true that nothing can travel faster than the speed of light essentially because of this formula and so the light cone separates those points that you can reach from this point by traveling at slower than the speed of light and similarly backwards in time those points that can reach here by traveling slow in the speed of light and so those are what we call the that's that's really the future in other words those points that we can influence through interactions and so here we have the future and here we have the past that's the region of space and time that can influence us and then there's a whole other region that's neither the future nor the past which we cannot interact with because it would require something traveling faster than the speed of light to reach at that point this kind of geometry is much more complicated than what you would get if you had a four dimensional space geometry but it was really this geometry even though in cost keys recognition was the relativity were just transform it the relativity was telling you that we really lived in a space-time geometry and space and time are wedded together in this way and we should really from now on not compute the distance of things and the time of things but we compute the distance in space time according to this formula so in summary the key recognition Einstein is that the eath of the luminiferous ether that people imagine that light was propagating in did not exist and then Maxwell's equations really are true in all frames of reference all the inertial frames of reference or all reference frames in which are moving at constant velocity but measured but to explain the null results of the Michelson Morley experiment and other experiments measurements of space and time had to be modified in a way that their relative to the observer so not everyone agrees on how long something takes in time not everyone agrees how long something is in space but everybody agrees on the length in space-time according to that particular formula and all this works in a magical way such that the speed of light is the same in all frames of reference and so this is really the key recognition from special relativity is that we really do live in a four dimensional geometry don't you have to say the earth is moving with respect to something that's right so the motion is relative that's right so that's that's what the the ship experiment is done moving so the you you're not moving if you're moving a constant velocity relative to someone but if you're accelerating then you are moving because everyone will agree that you're accelerating so the motion is relative only for what I call inertial observers those moving at constant velocity yeah you can always change to the frame of reference in which your address yeah so what does it mean that the earth is moving around the Sun and not vice versa so the question is what does it mean that the earth is moving around the Sun and not vice versa and in some sense it's true that I can go to the frame of reference of the earth and I see the Sun moving around me or vice versa I can go to the frame of reference of the Sun and move around and the earth is moving around me and neither are wrong but both of both of those frames are not what we call inertia frames so they special at the concept of special relativity actually doesn't apply in those particular frames because there really there is some acceleration going on in that there's gravity involved and that really then we have to go to general relativity which is the next section so it in reality the earth and them both the earth and the Sun are really removed moving around the center of mass of that solar system with the center of mass is in turn moving around the galaxies and so on and so on so motion is always relative but but there is a difference between constant velocity motion and accelerated motion so the the big finding was that the speed of light is constant and everything else had a change to accommodate that and that's all from Michelson Morley experiment here or other people also that was the most fervent ative experiment showing that there were many experiments done around the same time to be different types of experiments looking for the motion through the aether but that was the most definitive experiment that really gave the conclusive evidence that the speed of light was the same in all frames of reference yeah and that there were certainly influential in Lawrence's thinking and Fitzgerald's thinking it's we don't know how influential it wasn't in terms of Einstein's think where there's some experimenters said they found something goes faster than light a few years ago or oh yeah the question was was this whether some experiments measuring fast and speed of light or I guess to translate maybe measuring the motion through the ether and I'm sure there were B but those effects always went away on further detailed measurements yeah so I'm sure that I haven't looked at the history about too much but I'm usually that's the case oh recently oh no no no to date there is no violation of special relativity that we know of yeah yeah so that's right oh there was there was a there was a recent statement about neutrinos possibly fasting traveling faster the speed of light but that that went away on that turned out to be an experimental era we hope you've been enjoying the origin science Scholars Program with dr. Andrew Tully dr. Tolley was awarded a faculty early career award by the National Science Foundation to fund his work on the nature of gravity in the second part of our talk we learned about special relativity Einstein's theory describing the behavior of fast moving objects in our final segment dr. Tolley will discuss general relativity Einstein's theory of gravity now back to our talk ok so I moved to my third part which is general relativity and this is this was Einstein took about 10 years to reach this from special relativity which at some level it gives you an idea of the conceptual leap that went going from special relativity to the general authority but let's essentially let's let's start up and let's think about repeating Galileo's experiment Einstein did do it this L of elevators but let's think about doing ships so let's imagine you put a ship up there in space and then you drop it down it's under freefall under gravity and this is where Galileo's concept of the equivalence principle comes in we know that everything falling under gravity if we neglect air resistance will fall to earth with the same acceleration so if an observer is inside the the ship and they drop the ball the ship is falling down with given acceleration and the ball is falling down given acceleration and so of course the ball does not fall down with inside the ship relative to the ship the ball of it on the right-hand side remains remains a fixed position whereas on the for the ship which is on the sea you drop a ball and of course it just falls down with fixed acceleration so so now what we're doing is we're thinking about going to the reference frame the accelerated reference frame therefore what we now call the free-falling frame the ship that's falling downwards and in the free-falling frame you drop a ball and it doesn't fall downwards it just stays there so in some sense there is no gravity inside this frame that follows because because of galilei's equivalence principle so it locally you just don't feel any effect of gravity because nothing moves down of course from the outside observers point of view it's because both the ship and the ball are accelerating downwards but if again if you're inside the ship I'm allowed to work in the frame of reference of of that observer you don't see any effect of gravity and of course this is an experiment we can actually do now and so at least the concept if you're in an accelerated frame of reference I you're in freefall towards the Earth you just let yourself go under gravity that's locally equivalent to being at rest without any gravitational field the concept of freefall or reversing that if you're at rest in a gravitational field so you're sitting on in an elevator that's on the earth that's locally equivalent to being having an elevator which is accelerating upwards as the elevator accelerates upwards you'll effectively feel a force that you'll attribute to gravity even though it's really just acceleration so it was this recognition that there was something equivalent between acceleration and gravity and also the idea that you can somehow get rid of gravity by just changing to an accelerated eye a free-falling frame of reference there was influential in Einstein's thinking in the development of general relativity and so let me let me say that more concretely so at every point in space and time if you're in a gravitational field we can always go to the frame of reference of someone who is free-falling ok so just just let them fall under gravity and not stay where they are and and locally for each of those those people it's as if they don't feel any effect of gravity they feel themselves just to be at rest so if you're inside an elevator so you can't see out or you're inside a ship so you can't see out then you don't know that you're in a gravitational field however so to see where the notion of curved space-time comes from we're saying that locally we don't feel any effect of gravity but each of those frames of reference each of those locally free-falling frames of reference have to be patched together in some particular way and the way they get patched together is not entirely perfect so they don't exactly align up with each other so here's my sort of is a patchwork quilt to sort of give that idea so at each point in space each one of these patches represents a free-falling frame of reference where there is no gravity so where the gravity really is is in how these different patches patch together and then and this leads to the notion of a curved space-time a curved space-time is when this patch doesn't quite fit up with an X patch and doesn't quite fit up with an X patch and so on and so as you move along this space-time the you get you get rotations and movements and and so on and this is how you can understand this is how Einstein understood how planets move around the Sun why they move on elliptical orbits and why light is bent by the Sun because really the light is always travelling in a straight line in a local patch in a local free falling frame of reference is you see it's moving in a straight line here in a straight line here in a straight line here in a straight line here but where the curve comes from is the fact that each of those local patches don't quite match up together now in reality it's something much more smoother than this but this this gives the the idea so as as the earth goes around the Sun it's actually moving along as great line but we don't see it as a straight line because the space around the Sun is patched together in in a in a curved way that creates the curvature but locally the earth and also light bending around the Sun is always moving in a straight line and so if the geometry the gives rise to gravity in this picture so here's another example of a curved space-time so if you go to the Peter Lewis building you know each individual you make up this curvy curvy building here but each individual brick is basically an ordinary brick and what what makes the curvature is how you fit them together so you think of each individual brick is one of these free falling frames of reference in which an observer in that frame of reference sees no gravity but where the gravity really is is in how they get patched together and how you create that curvature of space-time and so one of the predictions of general relativity was that because of this space-time curvature light would a massive object like the Sun would Bend the passage of light around and in fact also Newton Newton's theory would predict that but Einstein's theory predicted that the angle that he would bend as the light went around the Sun would be twice that of Newton and that was a one of the first distinct predictions of Einstein's theory relative to Newton's theory and people went out to look for it specifically as Sir Arthur Eddington went out and looked at the solar crips of 1919 and looked at how the position of the Stars got moved during the Eclipse to see precisely the the the change in position because of the light being bent around the Sun when the Sun was within the path and the he gave the first evidence that this really did occur from that and this immediately catapulted Einstein to international frame and now this effect which we now call gravitational lensing so the fact that objects massive objects curved space-time and therefore can act like a lens in a sense that light pasmo passes through curves is now something we observe everyday both from stars and galaxies and even clusters of galaxies and the whole the cosmological scale so so this is the key realization of Einstein is the equivalence principle led him by a long chain of reasoning to the notion that space-time is really a curved geometry and a curve but you build up a curved geometry by patching together locally flat geometry so each of like those bricks in the Peter Lewis building so it locally it looks flat and you compute the distance between two points locally but how you compute the distance over here is different than here is different than hearing it's different here and that's how you encode the geometry of space-time and we do this through something called the metric which is a ten component object which describes basically how you change your rule how you change Pythagoras's theorem to describe the curved space-time geometry but all it all it is is an instruction rule to say if I'm at this point in space-time how am I supposed to compute a distance in space-time given a distance in space and given a distance in time and I do it one way here later in a different way over here they do it a different way over here and it's that rule for how it changes that describes the curvature of space-time and that in turn then describes gravity itself how objects will move through the curves space times this is a is a very sophisticated concept but it's now well tested that this is the this is the current best picture for what gravity is so you might conclude that from the equivalence principle that you can't really have a distinguish between being in a gravitational field an accelerated frame of reference that that's how I told it to you so far well that's sort of true at zeroth order but it's not really true if you're a bit more careful because you can feel the effect of gravity through tidal forces and here you see that's basically because in a real gravitational field things move say if you're in the earth things to move towards the earth and so these two objects if in their inner elevator falling towards the earth they're going to get slightly closer together because they're moving towards this point and so and more if say an astronaut is falling towards a heavy object like a black hole or something they'll become stretched by the the gravitational field and so it's those tidal forces the stretching type of forces that you can actually use to measure the local curvature of space-time and in some sense this is the real this is the real part of gravity these tide tidal forces rather than not just this sort of constant acceleration down so generality leads to many weird and wonderful predictions such as black holes here is here is a geometry of black holes a black hole is a space-time which is curved so much there is a region of that space-time where light simply can't get out because the gravity is so strong and there's a very clearly defined distance from the center of the black hole which we call the event horizon that describes that and around the round that round the black hole the space-time is curved in this complicated way given by in this case this is the particular metric and here's here's a picture of a black hole the diagram representing adduct the black hole his space going along to the right and time going up and here you think about matter you there's some star that collapses to form a black hole here's the collapsing matter the no black hole at this point and then as you go forwards in time eventually at some point the black hole forms and you have this radius which is called the event horizon out of which light can't escape and the reason I can't escape is what's happening is that the here the space-time curvature is represented by the bending of these different light cones so this is the local local light cone you can think of as the local free-falling frame of reference that describes the well which way light will move so this is the future this is the past future past and the light cones get bent by the gravitational field such that at the event horizon you see they're tipped in such a way that the future any-any light sent out at the boundary of the event horizon will only ever go inside the black hole and never outside and so it's that bending of those light cones that is what is responsible for the fact that black holes are black in the sense that they have this event horizon and one of the other great predictions of general authority which came pretty early on and was noted by several people Freeman lore met Robertson Walker is that because of because of equations of general Authority there's a notion of curved space-time the the universe should be expanding or should be expanding or contracting one or the other but it's essentially should becoming either more curved or less curved with time and the right one is that it's actually becoming less curved with time and you can think and you can describe this geometry in very simple terms as a simple modification of Pythagoras theorem in which you each time T you compute the distance and you multiplied by some number and that number changes with time so as you go into the future now you use a bigger number and so that's effective as saying that space is bigger but really what it is is like this I say at this time this distance is say 3 centimeters and then at a later time I say it's 4 centimeters and then a later time it's 5 centimeters and so on and what's really happening is that that this region here is becoming more and more curved and I should change how I compute my distances and that's how we understand that the universe indeed is is expanding and this was famously yeah the famous observations of Hubble which showed that galaxies were moving away from us because the light from them was redshifted confirmed that expansion which had effectively been predicted theoretically already at that point and so here here are the final equations and hopefully I'm not giving you the full details but here is the sort of transition from Newton's equations to Einstein's equations in Newtonian theory the force gravity really is a force which is given by the product of masses a Newton constant divided by the distance squared in Einstein's equations that you have to solve to determine these metrics they amount to essentially there are 10 equations for ten unknowns this metric which measures how Pythagoras theorem gets modified by the amounts are simply saying that this curve the local curvature of space and time is proportional to the density of energy and momentum and so on so so in summary the equivalence principle tells us that locally the effective gravity can be removed just by changing frames of reference changing to a free-falling frames a frame of reference but where gravity remains is in the notion of how those frames of reference patched together in other words how you build up a particular curved geometry from patches which are locally flat geometries and space-time curvature gives rise to tidal forces and Einstein's theory tells us that the amount that space and time is curved is proportional to the local density energy momentum density and stress and pressure and we'll hear more about the predictions of general relativity next week from Professor Dr I'm going to end this lecture is part of the origins science Scholars program of the Institute for the science of origins a partnership of Case Western Reserve University the Cleveland Museum of Natural History and ideastream it has been brought to you with the assistance of Case Western Reserve University's College of Arts and Sciences Segal lifelong learning program and Media vision for more information on the origins science Scholars Program including a full video archive please visit the institute's website at origins dot case dot edu you

Channel: Case Western Reserve University

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Keywords: Case Western Reserve University (College/University), cwru, case, origins, science, tulley

Id: Y45lf2xMzjA

Length: 52min 0sec (3120 seconds)

Published: Thu Dec 10 2015