The Science of Everything Podcast - Episode 136: Introduction to General Relativity

Episode Date: May 11, 2023

An introduction to the conceptual and mathematical framework of Einstein's General Theory of Relativity. We begin by considering the key insight of gravity as a geometric phenomenon, and how the curvature of spacetime by matter explains the equality of inertial and gravitational mass. We then discuss the mathematics of general relativity, including geodesics, differential manifolds, covariant derivatives, the metric tensor, Christoffel symbols, the Riemann curvature tensor, the Ricci tensor, and the energy-momentum tensor. The episode concludes with a derivation and explanation of the significance of Einstein's Field Equations. Recommended pre-listening is Episodes 114 and 115: Special Relativity 1 and 2.  If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything  

You're listening to The Science of Everything podcast episode 136. Introduction to General Relativity. I'm your host, James Fodor. Now, this is an episode that has been requested for a very long time, and I have been meaning to do for a very long time, but it is a very complex and difficult topic, and I've had to do quite a lot of additional research, so it's been a while in the making, but here we are.

So in this episode, we're going to talk about the science and mathematics behind Einstein's theory of general relativity. In particular, we're going to talk about the notion of space time and how we can represent the curvature of space time using differential manifolds. We're going to talk about some of the mathematics behind covariant derivatives, Christophil symbols, the metric tensor and the Riemann curvature tensor and the Ritchie tensor, all of that building up to a discussion of Einstein's field equations in general relativity. Because of this background that needs to be given, we're not going to get much of a chance in this episode to talk about the solutions to Einstein's equations, experimental evidence,

and other scientific aspects of general relativity, that will be deferred to a future episode. So this is really an introduction and an overview to the mathematics and conceptual underpinnings of general relativity. Recommended pre-listening for this episode is episodes 114 and 115 on special relativity. This episode also is a bit mathematical. So if you have even a small background in calculus, that will be helpful, although I don't want to say that that's essential. Of course, the purpose of these episodes is conceptual, not in doing calculations, but I will be referring to concepts from calculus like derivatives. So if that's entirely foreign to you, it may be a little bit difficult to understand in parts. All right, so as I said, this is going to be an introduction

to the concepts and mathematics of general relativity. What we're going to do is, first I'm going to give an introduction to the general idea of general relativity and what the theory is about. And then we're going to step through the mathematics needed to really understand Einstein's field equations. It's relatively easy to sort of state in conceptual terms what Einstein's field equations say in terms of the relationship between space time and gravity, but understanding the actual mathematics behind it is quite a bit more involved. So that's what we're going to spend the majority of this episode doing. But before we get there, we need to know kind of what we're covering, and that will be the introduction part of the episode. So let's start at the beginning.

here. So let's talk about general relativity. General relativity is a scientific theory which generalizes the theory of special relativity, unsurprisingly, and in doing so it refines Newton's law of universal gravitation. General relativity provides a unified description of gravity as a geometric property of space-time, or four-dimensional space-time. The basic idea of general relativity is that the curvature of four-dimensional space-time is directly related to the energy and momentum content of matter and radiation in that region of space-time. Now, Einstein developed this theory from thinking about what's called the equivalence principle, which states that all objects accelerate at the same rate in a given gravitational field

independently of their mass or the composition. Now, we've talked about that before. In fact, that goes way back to the very first episode of this podcast explaining gravity. That's where I introduced the Newtonian theory of gravity. In some sense, this episode is a follow-on from that very first episode as well. well, because Newton provided a universal theory of gravitation in which each object was attracted to every other object that had a mass and would accelerate in accordance with the gravitational force applied to it. So, starting from that point of Newton's theory of universal gravitation,

Einstein was thinking about the way in which different objects move or accelerate in a gravitational field. And this is something we discussed in episode one. We know that every object accelerates at the same rate when it's placed in a gravitational field, independent of the mass or the composition of the object. Now, it doesn't appear that way because of wind resistance and other things like that, but if you ignore those, then you're considering only the force of gravity, then every object accelerates at the same rate.

So this idea is sometimes called the universality of freefall. Free fall, meaning an object is only exposed to the force of gravity, no other forces are present, all objects accelerate at the same rate. Now, this is actually quite strange. If you think about it, it doesn't really make very much sense. Why should all objects accelerate at the same rate when they're in a gravitational field? Because after all, this is not the case for other types of fields in physics, such as magnetic fields or electric fields.

If you place a given object in a magnetic field or an electric field, they don't accelerate at the same rate independently of their mass. The rate at which they accelerate is going to depend on things like their electric charge. What's unique here is that in the case of gravity, there's actually two senses in which mass appears, or two places that mass appears in the equations, two separate conceptual notions of mass. There's inertial mass and gravitational mass. Inocial mass describes how much an object resists acceleration when a force is applied to it. Gravitational mass describes how much an object exerts an attractive force on another object that it also has mass.

So an inertial mass is more of a resistive property of an object. Gravitational mass is an attractive property or pulling process. property of other massive objects. And the only reason that we have this universality of free fall, the only reason why all objects fall in a gravitational field at the same rate of acceleration, the only reason that happens is because inertial mass and gravitational mass are actually the same. Conceptually, they're different, right, but they have the same value. All experiments show, even extremely precise ones, show that their values are equal. So the inertial mass of an object is the same as the gravitational mass. And that's why, of course, normally

just talk about the mass. You don't distinguish them because, you know, they're equal to each other. But conceptually, they're actually different. And this is not the same for other forces, right? Because if you think about, let's say, think about an electrostatic force, you put a charged object in an electric field. So a force is exerted on that by the electric field. But every charged object doesn't accelerate at the same rate when placed in that electric field. Why? Because the amount of force that that object feels is dependent on its electric charge, but its resistance motion is determined by the inertial mass. That doesn't change, right? So the inertial mass and the electric charge are different, and so different objects will accelerate at different amounts depending on

their mass and how much, basically the ratio of their mass and their electric charge. And that's the case of all other forces as well, that basically there's a separate kind of charge or charge analog, which determines how much they're affected by the field, and then there's the inertial mass, which determines how much they are accelerated when they feel the force because of that field. What's unique about gravity is that those two things are the same. Inertial mass and gravitational mass. You can think of it as like a gravitational charge. It's how much the object feels the force of gravity.

And inertial mass and gravitational charge, think of that in quotes. Those are the same, right? And we just call that the mass. And therefore all massive objects fall at the same rate or accelerate at the same rate in a gravitational field. So it's different from all the other forces. Because of this universality of free fall, because there's no difference as far as we can tell,

inertial and gravitational mass, it means that there's no way to distinguish, at least in small regions, between inertial motion and motion under the influence of a gravitational force. There's no difference between being attracted by a gravitational field and being in a rocket which is accelerating. Again, this is not true for other types of forces, like electric forces, magnetic forces, and so forth. The underlying reason for this is because of the equality of inertial and gravitational mass. So all of this that we've been talking about is called the equivalence principle. And this is what distinguishes gravity from all of the other fundamental forces or types of interaction. So if you take an accelerometer, which is a device that measures

acceleration, and you pull on it with a rope, then the accelerometer will measure a force. If you pull on it with a magnet, the accelerometer will measure a force. If you put it in a rocket and blast it off, it will measure an acceleration. And it's a force. So all other types of fundamental forces, if you apply them to an accelerometer, it will measure an acceleration. And therefore, or if you imagine being inside a box with the accelerometer, you would be able to tell that that force was being exerted on you. So you can tell the difference between the force applying to you and not applying to you. Now this is different to gravity. If you put an accelerometer in a uniform gravitational field, it won't measure an acceleration because it's in freefall.

That's the same as astronauts in orbit of the Earth are weightless. They're in freefall. They're in a gravitational field where they're being pulled to Earth, but they don't feel that acceleration because they're not pressing up against anything. can't tell if you're actually in this situation. You can't tell if you're in a gravitational field unless there's sort of something else to compare it to. The reason for this is because inertial mass is the same as gravitational mass, so the resistance to motion is the same as the amount that the motion is affecting you, and so that in the sense cancel each other out, so to speak, and you can't actually tell. Okay, so that's all the background to what Einstein was thinking

about, and this was sort of a mystery for quite a long time, and Einstein sort of came up with a solution for this as to why is it that the inertial mass and the gravitational mass are the same? Is that just some sort of grand coincidence or what's going on here? Einstein's key insight was that actually gravity isn't really a force. Rather, free fall is simply the natural motion of objects. Or in other words, free fall is inertial motion. This was Einstein's key insight with respect to general relativity. It's hard to over-emphasize how dramatic an idea this is,

because from Newton onwards, everyone thought of gravity as a force. Gravity applies a force to something like any other types of force. But Einstein said, no. gravity isn't a force. Any object that is only under the quote-unquote force of gravity, so therefore any object that's in freefall is actually just experiencing inertial motion. Inertial motion being it's just moving in a trajectory under its existing velocity in accordance with momentum. You might say, well, how on earth could that be true? It certainly seems like when you drop something off an airplane, it is pulled to the ground, you know,

it starts accelerating towards the ground. Well, that's the whole insight of general relativity. Instead of explaining gravity as a force, Einstein said, no, no, no, objects in freefall do not experience any force. They're actually just experiencing inertial motion. The reason why it looks like they're experiencing a force is because they're undergoing inertial motion in curved space time. So whereas Newton thought about gravity as an extra force that acts on things, Einstein thought of gravity as basically just part of the shape, well, literally, actually, part of the shape of space time. Gravity is not a force that kind of acts on things. Rather, when objects are moving under the influence of gravity, that's actually just their natural motion,

which to us looks like a force because we can't distinguish the two. We can't distinguish between whether a force is acting on something or an object is just in free fall because of this equality between inertial and gravitational mass. So Einstein formulated this idea that gravity, in general relativity now, is not a force, it's simply objects moving along their shortest path, but in curved space time. The idea is that massive objects cause local curvature in space time, which deflect objects off a path that they would otherwise take.

So if space time is completely flat, the object will just, you know, do what Newton thought it did and continue traveling at the same velocity in a straight line, right? That's in flat space. But in curved space, the idea is, well, if space itself is curved, then the object is going to travel the shortest path between points that it can, but in curved space. And so that actually turns out to be a curved path. and that can give the appearance of an attractive force. But the idea is that, remember, that we detect a force based on acceleration. So if something's accelerating away from its previous motion, then we say, oh, there's a force acting on that. And I'd say, said, well, actually, in the case of gravity, that's not the case.

There is no force there. The reason it's kind of curving or moving off of its previous path is actually because it's traveling on the sort of straightest path that it can, but in curved space. Another way to put this is that we think the object is moving on a curved path in flat space. That's sort of how we perceive it. But actually what's happening is that the object is moving on a straight path in curved space. So it looks to us like it's on a curved path in flat space, but it's actually a straight path in curved space. One way you can visualize this is if you imagine a rubber membrane, like a two-dimensional membrane,

that's kind of stretchy so you can deform it. And you place some billiard balls on it. the billiard balls sort of sink into it and bend it so that there's a depression. And the heavier the billiard ball, they're more they kind of depress and push down on the rubber membrane. Now, imagine we're rolling marbles across this membrane. If you roll the marbles across a part of the membrane that's flat and doesn't have any billiard balls on it, then the, you know, the marbles would just roll straight across and as if they were just rolling across the table. You know, nothing fancy happening there.

They'll just go in a straight line. But now imagine that we roll the marbles past the billiard ball, not straight into the billi ball, but like near it. And so the marble passes across the part of the rubber membrane that's deformed that's that has a sort of bends down because of the mass of the billiard ball. And so if you think about what would happen if you do that, what happens to those marbles as they roll close to the billiard ball and therefore are influenced by the local curvature of the rubber membrane? What will happen is that the path of the marbles is bent. They will be deflected around the billiard balls and will kind of curve as they move away from them. Now,

That's not because a force is acting on the marble. It is if you actually did an experiment like this, right? There's a force acting on the marble. But if we imagine the analog for true space time, it's not because there's a force acting on the marble. What's actually happening is that the marble is moving in a straight line, but on curved space. And so it's just that what straight looks like locally changes because the space itself is curved. Another way to understand this is if you look at a map of the world, like a 2D projection,

there is no way to accurately project all of the earth's surface on a two-dimensional map without distortion of shape, relative shape, and or size, because the earth is the surface of the sphere, so it doesn't project to a flat surface. And so what happens is if you draw a straight line on a map between two points that are far enough apart on the earth, you know, it will look like you've drawn a straight line. But then if you were actually to travel between those points, let's say you take, say my hometown is Melbourne, Australia. Let's say, my hometown is Melbourne, say I was to get in a plane and fly to London. You imagine getting out a globe and pointing to Melbourne and then tracing a path as straight as you can, like the shortest line possible from Melbourne

to London, just across the globe as if you're flying there. So you imagine that path. That path is called a geodesic. We'll come back to this. This geodesic path. And then you imagine drawing that path out on your flat map. So you go from your globe to your flat map and you draw out the path. The path will look curved on that flat map. It will look like you're taking the long way around. Like, why don't you go the straight way just between, you know, draw a straight line with a ruler on the flat map, right? Now, the reason you don't do that is because that's that flat surface that you're depicting there, that's not a fully accurate depiction of distances and shapes of the true three-dimensional surface of the globe. And so if you draw a geodesic on it, if you draw the actual line that's the shortest distance, it looks curved.

The path is curved, but it's still the shortest path. It looks like on the flat surface that it's taking a detour, but it's actually not. It only looks that way because of the inaccuracy of the projection. That's kind of like what's happening to us in our universe, according to Einstein, that what looks like the deviation in motion of an object that's kind of taking the long way around, it looks like it's not going straight. Actually, it is, I mean, it's curving, it's moving through curved space, but it's actually taking the shortest path possible.

It's just that it's taking the shortest path possible in curved space. And the shortest path possible in curved space or on a curved surface like the earth actually looks like it's kind of a detour if you project it onto a flat surface. And so that's why it looks like there's a force that acts on the objects that move them away from traveling on a straight line. In fact, they are traveling straight in the sense of the shortest distance between two points. It's a bit tricky to talk about straight because in flat space, straight line is the shortest distance between two points and is literally like straight.

It doesn't curve. Whereas in curve space, a straight line, as in like without any curves, is not the shortest distance between two points. That's actually what's called a geodesic, which is the shortest distance, the shortest path between two points on a curved surface. And so according to Einstein, all objects in the universe travel along geodesics when the only force that acts on them is the force of gravity, or in general relativity, actually, no force is acting on them. They're just traveling along geodesics and traveling on the shortest path possible between two points in curved space time. And that's what the analogue of the rubber membrane and the billiard balls is supposed to represent.

The billiables represent massive objects, like stars, for example, or even galaxies at a larger scale. The marbles represent any other type of object. Theoretically, the marbles deform the membrane as well. It's just they're so much smaller than the billiard balls. You can kind of ignore that. And so the marbles travel along the curved membrane, but they're deflected because of the curvature of the membrane. And then it looks like from a sort of a two-dimensional point of view, or a flat point of view, it looks like their path has been bent, but really their path is straight on a bent surface.

They are travelling the straightest line they can, but it's just that the surface itself that they're travelling along has been curved. And that's like how spacetime is curved, according to general relativity, and every object in the universe is travelling on a geodesic, every object that's in free fall is travelling along a geodesic, which itself is generally curved because mass curves and bends space time. The key point to understand here about why it is that in general relativity we can understand gravity as not being a force is that when you have two massive objects, instead of thinking about it as if they exert a force on one another, which is how we think about it in Newtonian physics, where one exerts a force on the other, which is proportional to the product

of the mass, that doesn't happen in general relativity. Instead, what happens is that the paths that they travel through space time curve through curved space. I mean, they're actually moving sort of straight, but through curved space. And because space is curved, the paths of these two objects move closer to each other. So massive objects look like they're exerting a force on each other, and thereby cause each other to accelerate towards each other. But what actually happens is not that. It's actually, it looks like that because we're inside the curved space.

But what's really happening is that the objects are continuing on their merry way, but space has become curved because of their mass, and therefore they move closer to each other in curved space, just like balls rolling over a curved surface, like an inward curve surface will move closer to each other because of the curve that kind of causes the past that they're traveling to move closer to each other. One way you can think about this is if you had people standing on opposite sides of the earth, standing at the equator and then walking up towards the North Pole, initially it would start off such that they're separated by a very

wide distance. But as they get closer to the North Pole, they get closer and closer together until they meet up at the North Pole. Now, that's not because they sort of walked towards each other, so to speak. It's not because a force pulled them towards each other, certainly. Really what happens as they were traveling on their separate geodesics. It's just that the curve of the earth in this case meant that their GAD6 met up at the North Pole. And so it looks like they were sort of pulled towards each other, but really they were just traveling on a sort of their own GED6 and they met up at the North Pole because of the curvature of the Earth. So in general relativity, that's what happens when any massive objects are pulled towards each other or appear to be pulled

towards each other. What actually happens is that the objects are just moving on their merry way along their GED6, but space curves because of their mass, and so it looks like they're pulled closer to each other. And this is how Einstein reconciles the idea of, or explains the idea of gravitational and inertial mass being equal, because according to this, they're part and parcel of the same thing. inertial mass is resistance to changes in the velocity of an object. And gravitational mass is just the extent to which the object bends space time, right? And the bending of space time is directly what causes the object to move in a different direction to would have otherwise.

So there's a direct connection between the two types of mass. Through the geometry of space time, you don't need to postulate this sort of two separate things which are sort of for no clear reason are equal to each other. And that's how in general relativity we get gravitational attraction without any actual force. It's all geometry. It's geometry plus the effect of matter on bending that underlying geometry. So it's quite a remarkable and elegant theory in that way. And that leads us to Einstein's field equations of general relativity.

These equations are what describes this relationship quantitatively. So the idea in the words of John Wheeler, space time tells matter how to move, and matter tells space time how to curve. So all of the matter and energy in the universe contributes to the curving and bending of space time. Space time as bent and curved by the existence of matter within it, then determines the trajectories that matter moves through space time. Just like in our analogy, the billiard balls causes the membrane to bend or deform, and then that bent or deformed membrane changes the trajectory of the marbles that we roll across it. So that's kind of analogous to the situation here.

Of course, bear in mind that in our rubber membrane analogy, the reason why billiard balls are deforming the membrane is because they're being pulled down by gravity. But in the case of general relativity, it's not like there's some supergravity that exists outside of space time and that's pulling things into space time. You shouldn't think of it like that. The idea is simply that matter and energy has an intrinsic capacity of bending space time. It's not because it's being pulled by something outside of space time. It just does that. It just bends space-time locally.

So that's the idea of general relativity, and Einstein's field equations describe this mathematically. Basically, the equation is sort of fairly simple. On the left-hand side are terms that describe the geometry of space-time, and on the right-hand side of the equation, are terms that describe the matter and energy content of the universe. So in a sense, that is the key concepts of general relativity. It's all about Einstein's field equations,

and the idea that the geometry of the universe, the curvature of the universe, is determined by the mass energy content of the universe, and in effect, they're actually directly proportional to each other. Einstein's field equations say that the curvature of the universe is proportional to the mass energy content of the universe. That's loosely what they say. Of course, in order to understand that, we need to understand how do you quantify the entire distribution of matter and energy in the whole universe, and how do you quantify the geometry of the whole universe? How do you do that mathematically? and how do you put them into an equation? Well, it turns out that in order to do that properly and mathematically rigorously, you need to do a lot of work.

You need to have first a way of describing curved space time. Then you need a way of describing smooth motion in that curved space time, and that requires us to be able to deploy the tools of calculus, because we need to be able to reproduce the results of Newtonian physics, which involves calculus, so we need to be able to describe velocities and accelerations and things like that. In addition, we need a method for computing distances in curved space time. Distances are kind of easy and flat space time, but they become more difficult in curved space time, so we need to have a method for doing that, and that's something called the metric, which we'll talk about.

Then we need a way to formally describe the curvature of space time. It's easy enough to talk about it in vague terms and make analogies like to a rubber sheet or to the curvature of the Earth as a sphere. But obviously we need a rigorous way of mathematicizing that for the universe as a whole. We then need a way to describe the matter and energy content of space time as a whole. And finally, we need an equation that relates them to each other. I've kind of already mentioned that. The equation is pretty simple. It just says, again, a bit loosely, but it says that the curvature is proportional to the matter and energy content.

There's a constant proportionality in there. But we also need some justification for that equation, and there's a few finer points there as to where it comes from and exactly how it's constructed. So these are all the mathematical ingredients that we need. We need to define space itself. We need to explain how motion works in that space. we need to explain how curvature works, how distances work, we need to explain matter and energy and how to quantify that, and then we need to relate them in an equation. So that's what we're going to do. That's what we're going to talk about for the rest of this episode. Obviously, I'm only

going to give you a general introduction to the mathematical concepts behind what we're doing here. We're not going to go through calculations and things like that. That's too difficult for an audio podcast, and that's not really the point of the show. But hopefully it will give you some idea about what's happening and a bit of a deeper insight into Einstein's field equations. And also, So it might be useful as a starting off point for further study of your own into general relative, if that's something that you're interested in. All right, so let's go through the list that I mentioned of sort of key ingredients that we need in order to mathematicize this at the moment sort of vague notion of the curvature of space time

is equal to or proportional to the matter and energy content of space time. Well, the first thing that we need is a way to describe curved space time. Now, I've been talking about space time without really explaining what I mean there. I'm leaning a bit on the previous episode we did on special relativity. So maybe have a look at that if you're a bit unfamiliar with this idea. But I'll just reintroduce it briefly. The idea here is that we think about space as having three dimensions, and then we think about time as being something separate to that.

But in special relativity, and especially in general relativity, we kind of get rid of that idea for the most part, and we just think of them as being part and parcel of the same thing. So we don't have three dimensions of space and one of time. We have four dimensions of space time. and they're all part of something called the space-time manifold. I think the easiest way to think about this is that a manifold is a sheet. Now, that's not quite right, because a manifold doesn't have to be two-dimensional.

You remember I mentioned the rubber sheet analogy, so that would be a manifold in that context, but that's two-dimensional. That's a two-dimensional manifold. In general relativity, we're interested in four-dimensional manifolds, because there are four-dimensions, three of space and one of time. And you can't really visualize a four-dimensional manifold, because we're kind of used to three spatial dimensions, and we can't really imagine more than that. So I encourage you to think of a manifold in going forward

as if it's a sheet that we can then bend and kind of warp and curve in different ways. But that will only be a two-dimensional analog. You just have to sort of vaguely imagine or keep in the back of your mind that really this actually should be four-dimensional, not just two-dimensional. So a manifold is a, technically it's a topological space,

but I won't try to define that rigorously. So think of it as it's a sheet that locally resembles Euclidean space near each point. Euclidean space is, just flat space. So basically the idea is I want to be able to describe curved geometries, but I want to ensure that locally, like if I get near enough to each point, it looks flat. Now, that might sound a bit trivial. Like, well, doesn't everything kind of look flat if you zoom in enough? Technically not. There are certain, if you know what a fractal is, that's an example

of something that no matter how much you zoom into it, it never looks flat or smooth at least. But what we're interested in here is a topological space, so think of it like a sheet, but in more dimensions, which can be bent and warped and curved globally, but if you zoom in close and close enough, it looks flat. Now, the earth is a great example of this. The surface of the earth is obviously, it's obviously a sphere, right, so it curves around, and so it's very much not flat. But locally, you know, as human-sized people walking around Earth, it looks flat locally, right? So this is a good example of a manifold. The reason we need it to be locally flat is so we can apply some of the tools of mathematics and calculus and things like that. We have a nice ordering and arrangement of points

in space. All right, so far so good. So there's a concept of a manifold from geometry. We can kind of, topology specifically, we can kind of grab that. But what are we going to do with our manifold? A manifold so far is basically just a, think of it again as a sheet, which locally is kind of flat. But so far we don't have any way of describing motion along or through that sheet. Because remember, what we ultimately want to be able to describe is movement through this map. movement through space time, right? That's the whole point. We need dynamics. We need to be able to describe objects and particles moving through it. So we need to introduce some more formalisms here. We need to make more assumptions with our mathematics. And what we're going to introduce is

the ability to do calculus on this manifold. And this makes it what's called a differentiable manifold. That just means it's a manifold where you can do calculus, where you can take derivatives and things like that. For those of you who do not know what a derivative is, a derivative is essentially a way of describing rates of change. So velocity is a good example of a derivative. If you imagine describing the position of, let's say, your car at different times in the day, you might describe it as displacement probably is a better way to describe it. You might describe it as located at zero. Say your house is zero, right? And say your work is 100 away from that. Sometimes in the day, your car might be at 0.0, sometimes it's at 100, sometimes it's at 50. It changes over time, right?

If you take the derivative of that sort of series of displacement values, what that's telling you is the rate of change of the position of your car. Or in other words, the speed of your car. How fast is it moving? Sometimes the derivative will be zero because it's not moving anywhere, and then sometimes the derivative is positive. So derivatives are just rates of change. And if that's still too abstract, just think of it as a velocity. Derivists don't have to be about position as such, but that's sort of useful for us, because most of the calculus we're interested in relates to movement of particles or objects through spacetime, and so that's about velocity anyway. So we need to introduce additional formalism that allows us to do calculus and talk about rates of change and

velocities and accelerations and forces and all of that cool stuff. We need to be able to do that in this curved manifold. Now, you might just sort of think, okay, well, you know, we know that this manifold is locally Euclidean, which means it's locally flat, so why can't we just do calculus like we do in an ordinary flat space? Like how Newton did it, right? He assumed everything was flat. So what's the problem here? Why can't we just, you know, take a derivatives of our functions and describe how particles move over time? Well, you can sort of do that, but there's a catch. And the catch is now the manifold is curved. We're not just dealing with nice, easy, flat Euclidean space. We're dealing with curved space time. And that makes things a bit

more complicated. To understand why, I think the easiest way to understand this is, again, to think of the example of the curved surface of the Earth. So think of a sphere. Imagine that you are standing at the North Pole, and you've got a spear. A spear is pointing due south. Now, for those of you who are paying attention, if I'm standing at the North Pole, any direction that I face is due south, so it kind of doesn't matter what way you're facing, but anyway, as long as you're not facing up, like, to the sky. So it's facing due south, right? Your spear is pointing due south. Okay. Now, let's imagine that then you start walking, and you keep that spear pointed like in front of you as you're walking. You don't rotate it from side to side.

You just keep walking. And let's imagine that you're feeling very energetic today, and you walk all the way to the equator. Okay, well, very well done. You've walked from the North Pole to the equator. Let's imagine zooming out now and looking at the direction that your spear is pointing in, as you've walked from the North Pole right down to the equator. Imagine looking in it from like a great distance. So you'll sort of have to imagine the sphere is very long to be able to see it nonetheless. Think of yourself at the North Pole. If you imagine sort of looking at this from some ways away, the spear is going to be pointing kind of along the surface of the Earth, but sort of nearly perpendicular to like the top of the Earth. So it's going to be pointing like kind of nearly

parallel to the line of the equator. It's going to be pointing, you know, like off to the side from the view of the North Pole, so to speak. But but then think about walking downwards towards the equator. As you continue to walk along, you're walking along a curved surface. And so you're moving from being at the North Pole to moving down to the equator. As you do that, your spear is gradually rotating. It's not rotating locally with respect to the ground. You're still holding it out in front of you parallel to the ground. It's just that because the earth itself is curved, over long distances, your spear is actually going to be pointing in a different direction. When you get to the equator, it's actually going to be

pointing down, right? Imagine that you've actually got now. that your globe in front of you. And you can actually do this, right? Imagine, you know, you place your index finger pointing sideways at the North Pole and then just slide your finger along the surface of that globe until, until you're now, your finger is pointing down. It'll be on the equator. The point of all that is that the direction that your finger is pointing, or that the spear is pointing, if you're imagining the person walking, it changes, right? It starts off pointing kind of side to side from the perspective of the North Pole until it's pointing up and down from the perspective of the North Pole. It's pointing like from north to south

because you're now at the equator and the arrow is pointing downwards. Hopefully you can visualize that. The idea is simply that if you go from being at the North Pole to pointing a spear out in front of you to then walking all the way to the equator, now the spear is pointing in a different direction. It doesn't look like it's pointing in a different direction from the perspective of the person holding this spear because to them it just looks like they've gone from holding it out in front of them to holding it in front of them after a long walk. Locally it doesn't look like there's any change, but actually from the global point of view, there has been, because they've moved on a curved surface. What's the relevance of that, you might ask? Well, the problem with that is that

totally screws up calculus, because the whole issue with calculus is being able to describe precisely in some kind of overall coordinate system how the position of an object is changing. But we can't do that here, because in what looked locally like we were just sort of moving in a straight line, actually it turns out we were curving and something was bending, right, that the spear is now pointing into a different direction. We have to find a way of describing the motion of, you know, that guy with his spear. We have to find a way of describing his motion along that path in a way that can subtract out the effect of the curvature. Because we don't want to describe how the space itself is curving.

We want to describe how the person is moving in curved space. And those are different things. See, in flat space, space is flat. So space doesn't curve, it doesn't move, doesn't change. So we don't have to worry about this normally. But now that we're working in curved space, there's two different things going on. When I move from one position to another on, let's say, the two-dimensional surface of the Earth, there's two different things that have changed.

One is that I've changed position and maybe I've changed direction I'm facing. You know, I can move around and so forth. But the other thing is that the space itself has changed because it's curved, right? It's different to what it was before. So we have to kind of factor those two out from each other or separate them out from each other. There are ways of doing this mathematically. I won't try to describe sort of the derivation of this because it's a bit too complicated for audio only. The point, though, is that we have to do more.

mathematical work and introduce something called the covariant derivative. If you are familiar with regular calculus, you know what the derivative is, right? You know, D-Y-D-X, you take the derivative and you compute stuff. The covariant derivative is like that, but with extra steps, because it factors in incorporates the fact that the space itself has changed as you've moved from one position into another. Space itself has curved, so you have to factor this in when an object moves from one position to another. The object might have moved or changed or rotated, but it's also possible that it's just space that's curved and you need to separate out those two effects. Now, we can figure out ways of doing this mathematically. What it does is it complicates the math

quite a lot, and you'll see how much it complicates it when we get to the end with Einstein's field equations. This might seem like it's a detail that, oh, well, do we need to know about this? But you kind of do, because these covariant derivatives is really what makes Einstein's field equation so difficult. If you try to solve Einstein's field equations in flat space, it's very easy because everything's flat and covariant derivatives become irrelevant. Covariant derivatives just turn into plain old normal derivatives in flat space, in Euclidean space, and so you don't have to worry about all these issues. But in curved space time, you do have to worry about them. And in particular, in order to actually compute this covariant derivative, this basically think

of it as like curvature adjusted derivative, you have to introduce this mathematical construct called a Christophel symbol. Christophil symbols are just numbers, or they're actually functions of the coordinate system, but you can just think of them as a whole bunch of numbers. And they tell us how much of the unwanted component of a vector change we need to subtract off. Basically, when I'm moving from one place in my curved space time to another, how much of the change in the vector do I need to get rid of in order to just factor out the curvature of the space

and just be left with the motion of the vector itself? How much of the unwanted bit do I have to remove? That's what the Christophel symbols tell me. me. Christophal symbols are denoted with a capital lambda symbol. It looks like a capital f, except the lower horizontal line is missing. I'm just mentioning that if you've sort of seen any of these equations before. Christophel symbols basically tell us how to adjust a normal derivative and make it a covariant derivative. And so we can do calculus properly in curved space time. So that's really good. We can describe the motions and trajectories and accelerations

of objects now. But you might be asking, well, hang on, how do we get these Christophil symbols? was you told me that they're these numbers that make corrections for the curvature of space time, but where do they come from? Well, they can be calculated by taking partial derivatives of the metric tensor. Now, you might be wondering, what the heck is the metric tensor? I haven't explained that yet. Well, yes, because that's coming next, right? This is why general relativity is difficult, because you have to introduce a lot of concepts and kind of explain them one at a time, so it can become a bit tricky. The metric tensor is a mathematical object that describes distances in curved space time. I'll say a bit more about that in a moment, but the metric tensor

is crucial. It's at the heart of general relativity. And so it's a very important concept to be aware of. And what we'll see later is that in fact, pretty much all of this stuff on the left-hand side of Einstein's field equations, remember, that's the equations that relate the curvature of space-time with the matter and energy content of spacetime. And on the left-hand side of Einstein's equations is the curvature geometry stuff, and on the right-hand side is the mass energy content stuff. It turns out that everything on the left-hand side pretty much, the geometry curvature part, everything there can be expressed in terms of the metric tensor or its derivatives. Now, that turned out to be very complicated mathematically, but it's all reducible down to one

single object, basically. It's the metric tensor. But the point that's relevant here is that Christophil symbols tell us how to fix derivatives to get covariate derivatives so that we can do calculus and talk about velocities and things in curved space time. And that's what they're important for for the moment. Okay, so let's come back to our list that we were talking about in terms of trying to construct the mathematics needed for general relativity. We've introduced the notion of how we're going to describe curved space time. We're going to describe it as a manifold. Again, think of that as like a sheet, but with more dimensions. Then we've introduced a way of describing motion in curved space time, making it a differentiable manifold, and introducing this notion of covariant derivatives,

which allow us to describe velocities and accelerations in a way that adjusts or correct. for the curvature of space time, and those use Christophel symbols. The next thing on our list is we need a way of describing distances in curved space time. Now, distances are easy and flat space time. I mean, you can just grab a tape measure, so to speak, or something that is equivalent to a tape measure and measure the distance, right? But, as you might have guessed, it's more complicated in curved space time. One way to visualize this is to think about those two pathways that I drew on Earth.

Remember when I was taking my flight from Melbourne to London, if I measured the length of that pathway on my globe, you know, drawing the path on my globe, which curves around from one to the other, if I measured the distance of that pathway, that's going to be very different to if I draw a straight line on my flat map with a ruler and then measure the length of that.

Those paths are not going to have the same length, because basically one is incorporating the curvature of space and the other one isn't. And of course, there are many other ways that spacetime can be curved. I've been using the example of a sphere because that's sort of easy to understand. But in fact, there are many, many other complicated types of geometries it can have as well. So we need to have a way of adjusting distance measurements

and making sure we compute them properly to incorporate the fact that spacetime itself is curving. And that is what this metric tensor does. The metric tensor is pretty much the central object in general relativity when it comes to the curvature of space time. It describes the local geometry of space time. So it describes how it's curved. It describes the shape of space time. Differential manifold don't have to have a metric tensor. Like you can do the maths of differential manifolds, but without introducing this idea of a metric tensor. A metric tensor is additional mathematical structure that we're just going to suppose we're going to demand or define that this structure exists on our differentiable manifolds so that we can define distances

and angles. And that's useful because we think that, well, in order for this theory that we're constructing to be useful of the real world, to be able to do physics, we're going to need to be able to define distances. So we need to have a metric tensor that allows us to do that. The metric tensor is commonly written as a four by four matrix. The reason it has, it's four by four, is because there are four space-time dimensions, one of time and three of space. And it's a four-by-four matrix, essentially because it describes the amount, very loosely, you can think of it as describing the amount that each dimension kind of bends in or affects or interacts with all of the other dimensions.

So that's why it's four-by-four, each dimension in terms of each other dimension. The metric tensor is symmetric, which means that it has ten independent components. This is important because the metric tensor fully describes the geometry or the curvature and the shape of space time. So theoretically, the whole universe. Now, you might be thinking, how can you possibly describe the shape of four-dimensional space time,

the whole universe, with just ten numbers in our metric tensor? That's a very good question. There's a few ways to answer this depending on the way you look at it. One way to look at it is, well, you can't really, and so anything that we do is really just going to be a very crude approximation

of the general overall shape. We're going to abstract away from all of the specific little details of this galaxy, here in this galaxy there. We're looking at like overall shape. Another way to think about this is theoretically, and this is sort of the deeper insight, I think,

theoretically those 10 numbers would be enough to describe the overall metric tensor for the universe, assuming general relativity is true. But the thing is, the metric tensor is not just a 4x4 matrix of numbers. Each of the components of the tensor, in general, is going to be a function of the parameters that specify

where we are in space time. So there's different coordinate systems that we can use to describe our position in the manifold. An example of a coordinate system that should be familiar to everyone is your X and Y coordinates, your Cartesian coordinates of the simple two-dimensional graph, right? Y is how far up you are, and X is how far side to side. So those are two coordinates, two numbers that you need. And you can imagine having a matrix in which you write, in this case it would be a two-by-two matrix, right, because there are just two dimensions, in which each of the four elements of that matrix was a function of those coordinates. So each element in the metric tensor is not going to be literally a number, it's actually going to be a function of space time coordinates. And so theoretically,

that could be a very, very complicated function. It could be a function that has lots of little blips and blobs and complicated shapes, and you can imagine it doing all sorts of weird and funky things. In practice, of course, it's going to be very difficult, basically impossible to know what that true function of the true metric tensor should be for the real universe, because it's going to be incredibly complicated. I mean, theoretically, any massive object, will have an effect on that. And so each little proton is going to cause the tiniest little bump in it. And there's no way you can actually describe that accurately. So we approximate, we compromise, right? And we'll talk about some actual real solutions to Einstein's equations that are actually sort of feasible and can be computed.

We'll talk about that in the next episode. We won't have time to get into that here. For our purposes, you can think of the metric tensor as a bunch of numbers, although really they're their functions of the coordinate system. But if it's easier to think of them as numbers, then just think of them as numbers, especially because in simple coordinate systems or in simple geometries like flat space, for example, they are just constant numbers. So if space time was completely flat, if it was as Newton thought of it, then the metric tensor would just be a diagonal matrix of one. So that means that you have ones down the diagonal and everything else is a zero. That means it's nice and flat and no dimensions affect or like curve into or interact with the others and everything's

simple. And in a space like that, you kind of don't even need to worry about the metric tensor because it doesn't really tell you anything. When the metric tenor, The tensor becomes relevant is when space starts curving and then things start to become complicated, and now some of these elements start to become non-zero, or not different to one. Okay, so we introduce this idea of the metric tensor. The metric tensor is this 4x4 matrix, the elements of which are functions of the coordinates of space time, so in theory they can be very complicated, but in practice we usually assume they're fairly simple, so that we can actually solve the thing.

And this metric tensor tells us the shape, the overall shape of space and time. Now, there's one very important aspect of the metric tensor, which I haven't mentioned, and it actually applies to other mathematical objects that we've been talking about as well, such as covariant derivatives and velocity vectors, which I haven't really gone into detail, but we can talk about velocity vectors as describing the velocity of a particle located at some position in space time. Now, all of these concepts, you know, the covariant derivatives and velocities and the metric tensor itself, all of these things need to be true of the universe independently of the way we describe the universe. Now, what I mean by this is that

there are different ways of describing the same thing mathematically. A good way to explain this is to think about the position around the circumference of a circle. Now, you can describe that using X, Y coordinates. You can say on this many X units and this many Y units, and this is a position on the circumference. But a different way of describing that would be to use what are called radial coordinates or polar coordinates. So in polar coordinates, you don't specify an X and a Y. You don't specify an up and a down. You specify a radius, which is a distance away from the center.

And then you specify an angle, which is how far you've rotated. around. Now, maybe you'll have to think about this, but any position on two-dimensional space can be described either using polar coordinates or using Cartesian x-y coordinates. You can do it either way. It doesn't matter. Up or down or basically distance away from the middle and then angle around. It gets you the same answer in the sense that you can always describe the same positions. It's not like one can describe some, but it misses out on some others. They're both just as good. They both get the job done. The difference between them is just that the actual numbers that we get are going to be different, of course, because in one we're dealing with radius and angle,

the other we're dealing with X and Y. So the important point here is to understand that the coordinate system, the formalism that I use to describe something, is different from the thing itself. The position of something in space time, or its velocity for that matter, is not affected by the coordinate system I used to describe it. And so when we're constructing our formalism for velocity and distances and everything in space time, you better be sure that this formalism gives the same answer regardless of the coordinate system that I use, if I choose to use. Cartesian coordinates, X, Y, or Polo-coordinates, or some weird coordinate system. I mean, there's really an infinite number of coordinate systems you could use.

You could come up with whatever you like, really. You could say the number on this diagonal and that diagonal, like whatever you want. The point is, the answer should not be affected by the coordinate system that you choose. Because if it does, then that's, well, which one's right, right? You know, we'd have to go and experiment and find out. So we want to ensure that the formalism kind of builds it in automatically, such that regardless of the coordinate system you use, the answer is the same. And getting the same answer regardless of the coordinate system is called invariant,

right? So it's invariant to the coordinate system. It stays the same regardless of the coordinate system you use. It gives the same answers. Some of the details of the calculations may look different, of course, but the answer will be the same. It's just like if you measure velocity in meters per second or you measure it in feet per minute, right? Those units are different, but the answer should be the same if I ask you, how long is it going to take to get there? If you convert the units, the calculation may be different, but the end answer should be the same. In order to ensure that that is true for our mathematical theory that we're constructing, we use objects that are called tensors. Now, I've just mentioned one of them, the metric tensor.

In fact, most of the objects that I've been talking about are tensors, so pervariant derivatives, those are tensor objects, and a velocity vector is also a tensor. Confusingly, Christophel symbols are actually not tensors, but I won't get into explaining why that is. The point, though, is that many of the objects that we've been talking about and appear in general relativity, especially relativity as well, are tensors because they're invariant to the coordinate system use. It doesn't matter whether you use your polar coordinates or your Cartesian or something else. They're always the same, because they're an object that exists out there independently of people and independently of our descriptions of them. Now, we need to make sure that the mathematics respects that, and so there are certain properties that we demand. When I say demand, it means that we sort of define them and ensure that they're true. We demand these properties whole. of the mathematics that we're doing. And that actually allows you to derive certain relationships between, say, the metric tensor and the covariant derivative and other of these mathematical objects that I've been describing.

I'm not going to get into the details of that here, but I just want you to understand what the notion of a tensor is as something that's invariant to coordinate system and invariant under certain types of transformations, like if you go from describing it in one coordinate system to another, all of the underlying physics should be the same. Surface-level calculations look different because you're using different coordinates,

but at the end you get the same answer. And that's actually, that comes us back to the metric, because that's what the metric actually gives us. It gives us space time intervals between events. Or we can use it to determine the space time intervals between events, which are sort of like the distance, but distance in space and time, considering them both together. And that's important because we're going to be able to need

to incorporate that into a full theory of motion through space. We need to be able to find distances, so we need the metric tensor to do that. So the metric tensor defines kind of shape and distance in curved space. And it's a tensor, which means it's invariant of coordinate systems and our descriptions of it. And so that means that there's certain extra mathematical properties that have to hold true as well. All right. So we've talked about describing space time as a manifold. We've talked about describing velocity in terms of these special covariant derivatives,

which requires to make these adjustments to normal derivatives using these Christophil symbols. We've talked about the metric tensor and how it allows us to describe distances and sort of the shape of space time using coordinates that are arbitrary, but we'll give the same answer because of the properties of the metric tensor as long as we obey the rules. we're going to talk about curvature. So we've said that the fundamental idea of Einstein's theory is that the curvature of space and time is proportional to the matter and energy content of space and time. So we need a way of quantifying curvature and putting it into an equation. To do that, we use something called the Remun Curvature Tensor. That's another mouthful. So Riemann is just the guy

who sort of came up with this in differential topology. So that's just a name. And curvature tensor, well, we know what curvature is. So we're talking about curvature. And Tensor, it's another one of these objects that's invariant of the coordinates. So that's all that that means. This is the most common way, the Ruman curvature tensor is the most common way to express curvature in Riemannian manifolds. So the Riemann curvature tensor basically describes curvature as the amount by which a vector put at a point somewhere in the manifold. So I stick a vector at a point in the manifold, and then I rotate it around a very small square loop. Let's think of sticking a pin in my rubber manifold. All right.

stick a pin in it that kind of points in a particular direction along the manifold, not up and down, but horizontally sort of across the manifold. Now what happens is imagine moving that pin sideways, up, and then backs the other way, and then back down. So like left one centimeter, up one centimeter, right one centimeter, and then down one centimeter. Now, if you do that on a flat sheet, the pin is just back where it starts, the vector points in the same direction, and kind of nothing has happened. And that's kind of the definition of flat, actually, in this context. It's while the Riemannian and curvature tensor is zero. It turns out, though, that on a curved sheet, and this might be a bit difficult to actually test on a curved sheet and do it properly,

but if you do this on a curved sheet where you transport or sort of slide the vector across and then up and then back and across and then back down, it actually will end up pointing in a different direction than it did initially. The reason is simply because of the curvature that it's of the space that it's been moved across. So that's the same phenomenon as we saw when we remember we had that guy with the spear who walked from the North Pole down to the equator, or you have your finger on the top of your globe and then you sort of slide it along downwards so it's pointing downwards now. That's the same phenomenon there. A vector will actually change the direction it's pointing in if you move it around in a small

loop on a curved surface. It's a bit hard to visualize, but that is what will happen. That's how curvature works. And so the way the remun curvature tensor measures curvature is basically how much does this vector change in direction as I move it around a very small loop. If it doesn't change at all, then there's no curvature. If it changes a lot, then there's a lot of curvature. And actually, it doesn't just tell you the amount.

It actually tells you the direction that the curvature points in as well. So the Riemann curvature tensor is quite a complex object. In four-dimensional space time, it actually has, it's a 4x4 by 4x4-10. So you remember, the metric tensor is a 4x4-by-4 matrix. So that has 16 elements. But the Riemann curvature tensor is a 4 by 4 by 4. It has 256 components. But thanks to symmetry and various other properties

that we didn't get into here, some of them relating to the fact that it's a tensor, so it must be invariant to certain types of coordinate transformations. Thanks to various mathematical tricks and symmetries, only 20 of these 256 components are independent of each other and non-zero. So theoretically, this giant monster has 256 components, but only 20 of them are actually different from each other. That's still a lot, but at least it's a bit more manageable. Now, the Ream and Curvature tensor describes theoretically kind of all aspects of

curvature that we're interested in. But it turns out that it's a bit more than is necessary. for constructing Einstein's field equations. So we're actually going to simplify the Riemann-Korverger tensor a little bit so that we don't have to worry about this big 4 by 4 by 4-4, which is a bit much. But before we get to that, you might be wondering, okay, so I've introduced this Riemann-Korvich-Tenzo,

but how do you calculate it? How do you know what the numbers are, which will be important later when we get to Einstein's field equations? Well, the answer is, do you remember Christophel symbols? Christophel symbols are these numbers, which describe how a vector changes as, it moves in curved space. We need to use them to fix our derivatives to turn ordinary derivatives into covariant derivatives. And doing that allows us to do calculus properly in curved space time.

It turns out these Christophel symbols are also useful for defining curvature, because the Reamon Curvature tensor is defined in terms of these Christophel symbols and some of their derivatives. I won't read it out fully, but the Rehman Curvature tensor is defined in terms of two partial derivatives of the Christophel symbols and then two product terms of Christophel symbols. And remember each of those Christophel symbols, in turn, is defined as in terms of partial derivatives of the metric tensor. Now, that's really great because that means that the remun curvature tensor, which describes the curvature of a spacetime manifold, can be defined entirely in terms of the metric tensor and some of its derivatives and product terms. The equation is quite long and complicated, but the bits to it are sort of known. It's just the metric tensor, we know that, right?

And so we just have to do a bunch of like multiplying and taking derivatives and stuff and subtracting. In that sense, it's not complicated, right? The actual operations aren't too difficult. It's just there's a lot of terms to it. So we have to compact it a bit. But that's a good result. So we can describe the curvature of a spacetime manifold solely in terms of some relatively simple operations on the metric tensor. As long as they obey, you know, the rules of the remun curveature tensor and we put them together in the right way.

Okay, so that's great. But the remand curvature tensor isn't the correct object to use. use in Einstein's field equations. It's too complicated. It's got too many components to it. It's a 4x4 by 4 by 4 by 4. In a moment we're going to see that the matter and energy content of the universe is described itself in a 4x4 matrix. And remember, we want our two sides of the equation to be equal to each other. So we want the curvature bit to be equal to the mass energy bit and the mass energy bits are 4 by 4 by 4 matrix. So the curvature bit's going to have to be a 4 by 4 as well. And the remun curvature tensor is the wrong, it's too big, it's 4 by 4 by 4.

So we need to kind of cut that down. And that's where the Ritchie tensor comes in. The Rishi tensor, that's spelled R-I-Double-C-I, by the way. If you're Googling this, it's a bit counterintuitive, perhaps. The Rishi tensor is found by contracting two of the indices of the Riemann curvature tensor. Basically, this means you sum over a bunch of components of the Riemann curvature tensor, and you reduce it down. You reduce it down from four to only two indices.

So now it goes from a 4 by 4 by 4, that's the Riemann curvature tensor. We reduce it down to just a 4 by 4. So just a 4x4 matrix. That's the Rishi tensor. That's much more manageable. The Rishi tensor describes the curvature of spacetime. You might say, well, that's what the Riemann curvature tensor did. Yes, but it's less information. So we've lost some information, I mean, necessarily, because we've gone down from 256 to only 16 components. But the advantage is that it's in a more manageable form, and it's more relevant as well, and this is sort of critical. The Rishi tensor specifically, what it does is it tells us how the volume of a shape changes

as it moves along a geodesic, like basically a shortest distance, in curved space time. So it tells us how different dimensions change in respect to the other dimension as they move. So that's why it's a four by four. You've got to think about the interactions with each of the four space-time dimensions with each of the four other ones. And overall, it tells us how the volume of a space-time region changes. Importantly, if the Rishi tensor is zero, that doesn't mean that the space is not curved. It doesn't mean that it's flat. it just means that the volume of space-high regions doesn't change as you move across the geodesic

in that region. That's very important. We'll come back to that concept. So we've kind of lost some detail from the Riemann curvature tensor. Riemann tells us sort of everything about curvature. That's not quite true because there are other ways of describing curvature as well. But for our purposes, it sort of tells us quen to quote everything about curvature. The Rishi-Tensor doesn't tell us everything. It actually loses some parts, but it tells us the relevant parts, the crucial parts, and it's sort of more manageable. And it turns out to be the really crucial. one in order to fit into Einstein's equations. Now you may be wondering, well, I just told you that the Riemann curvature tensor is defined in terms of the Christophel symbols and some derivatives.

But what about the Rishi tensor? How do I get that? Well, thankfully, it's kind of more of the same. It's also defined in terms of derivatives of Christophel symbols and some interaction terms like squares of the Christophil symbols. So the equation is slightly different because we've summed over some things from the Riemann tensor, but fundamentally it's defined in terms of the same object, the metric tensor, and so it's all good. We don't have to introduce anything new to get that. Now, before we finish up on the curvature part, I'm just going to introduce one more, one more curvature tensor, and this is called the Ritchie Scalar. Confusingly enough, it's also a tensile, but I won't explain that, but it's usually called the Rishi scalar because it's a single number. It's not a four by four by four, it's not even a four by four. It's just a one by one. It's just the Rishi Scalar. It's a single number. It's denoted as an R. By the way, I should have mentioned that the Rhy tensor is denoted as an R, but with two subscripts. So like little letters to the

bottom, often it's alpha and beta. Those subscripts indicate that it's a 4x4 matrix, but the Rishi scalar is just one number, it's just R, and it's sort of simple. And it describes the total change in the space-time volume of an object, but without giving any information about the direction in which that change occurs or how it occurs. So again, the Rishi Scalar kind of says it the same thing as the Rishi-Tensel, but it gives you less information. It tells you how much an area changes when you move from being in curved to flat space. It might seem a bit redundant to introduce the Rishi-Tensor and the scalar, but you'll see in a moment why we need to do it. that. All right, let's take stock of where we are. So I said that in order to describe the

curvature of space time, we need to introduce a whole bunch of mathematical machinery. We had to introduce this notion of a manifold, which describes curve space time. We need to be able to do calculus on that curve manifold, so we needed to make it a differentiable manifold, and we needed to ensure that the way we calculate velocities and accelerations and so forth is adjusted for the curvature of that manifold. So we introduced covariant derivatives and these special Christophel symbols, which correct for the curvature of space so that we can do calculus and describe trajectories and so forth. We then needed to a way of describing distances in curved spacetime.

So we introduced the metric tensor, describes the overall shape, and therefore distances between different parts of space time, in a way that's invariant of the coordinate system that's used. We then introduced a way to describe the curvature of space time. And so we talked about the remand curvature tensor, the Rishi tensor, and the Rishi Scalar. And it's the Rishi Tensor and Scalar are the two important ones. Now there's only one ingredient left, And that lies now on the other side of the equal signs.

Remember, there's the left-hand side of Einstein's field equations, which is the geometry part, the curvature of space-time part. We've finished with that now. We've described everything that we need for that. We're just moving to the right-hand side and talking about the other part, which is the matter and energy part of Einstein's field equation. We need a way of describing the matter and energy content of the whole universe.

Now, you might be wondering, okay, how on earth can you do that? How can you describe the matter and energy content of the whole universe in like one mathematical object? In fact, you may recall that I said a bit earlier that, way that this is described as in a 4x4 matrix. How can you describe the whole matter and energy content of the universe in a 4x4 matrix? Well, it turns out that you can, but it's sort of the same issue as we raised when we talked about the metric tensor. In practice, we make a lot of approximations and assumptions that sort of smooth over all of the specific lumpiness of this proton is here and this

star is there and whatever. You know, we approximate that away and just talk about the very general overall distribution of matter and energy in the universe. And I'll talk about that in more detail in a future episode where we go through how this is how this is our actually applied in cosmology. But here we're more talking about the mathematics. And so you can imagine if you had the like the true description of all of, like truly all of the mass and energy in the universe, then again, it could be a very complicated function of of the four coordinates in space time, but which says, you know, this blip here and this blip here and this blip here and this smudge here and so forth. So in theory you could do it. It's just it would be very, very complicated.

In practice, of course, we don't work with that. We work with approximations that generally get the job done. But anyway, the object that describes the matter and energy content of space time is called the stress energy tensor. There's this word tensor again. So this again means that the energy and matter content of the universe shouldn't be dependent. It shouldn't change if I change my coordinate system, if I change my way of describing it. This object is a four by four matrix, or can be written as a four by four matrix, which describes different aspects of the energy and matter in the universe. Now you might be wondering, well, why is it four-dimensional though? Like surely wouldn't it just be like one number which changes depending on where you are in space

and time? Well, no, because in order to describe the entirety of the momentum of a single object, you need something called a four momentum, which is a four vector. It has four components. This is a concept from special relativity and it just says that suppose I want to describe the energy content of a single particle that's moving. I need to know its energy, and then I also need to know the three momentum components. So momentum in classical mechanics is just mass times velocity. It's more complicated in relativistic mechanics, but I won't get into that here. But the point is that in order to fully describe the energy, if you like, of a single particle, you need four numbers, energy and then three momentum components. That's for a single particle. In order to describe

the whole energy momentum of the entire universe, we need more than just four numbers. We need 16 numbers, four by four, because we need to describe how each component of a four momentum vector changes with each of the four space-time coordinates. So how does energy change with the four space-time coordinates? How does the first component of momentum change with the space-time coordinates and so forth? So that's why it's a four-by-four. Now, we can actually apply a more specific interpretation to the different components of the energy momentum tensor. The top left element is the energy density of the universe. The elements below that

on the leftmost column are the momentum density numbers. The numbers along the top row are the energy flux numbers. The diagonal components apart from the very top one are the pressure components and the other ones are momentum flux components. So those things might not necessarily mean very much, but all I'm emphasizing here is that different numbers in this 4x4 matrix mean different things. They correspond to different physical measurements. that you can make obviously at different points in the universe. Remember each of these numbers is actually going to be a function of the coordinate system. So it's not going to be the same throughout the whole universe,

it's going to vary depending on where you are in the universe. But there will be some value at each point in the universe for these different components of the energy momentum tensor. And that's how we describe the mass energy content of the whole universe. It's using this energy momentum 4x4 tensor. And now we come to the culmination of our discussion, which is how do we put these ingredients together to form or to explain Einstein's feel Einstein's field equations. Remember that Einstein's field equations describe the relationship

between the geometry, the curve geometry of space-time on the left-hand side, and the matter-energy content of the universe on the right-hand side. And it says essentially that they're equal to each other with a constant proportionality in there. Now, first I'm going to talk a bit more about the nature of these equations and what terms are in them and sort of what they mean. And then we're going to talk about kind of where they come from. You can't mathematically derive Einstein's field equations. Well, you sort of can, but you have to make certain assumptions. So what we're going to do is we're going to explain how Einstein came up with them, or some of the things he was thinking about, at least when he came up with them. Because there's different ways to motivate these equations. I'm just going to talk you through one of them, which is its relationship to Poisson's equation. But before we get to that, let's talk about the equations themselves and some of the mathematics behind them. So sometimes people talk about Einstein's equation or Einstein's equation of general relativity. I've generally been talking about them as field equations, plural, because in fact, there is, more than one equation. It's a bit confusing though because they're often written as if they're a single equation, but that's really just a way of summarizing the information in one place.

So one way that they're commonly written out is on the left-hand side a capital G, and then there'll be two subscripts. And remember that just indicates that it's actually a matrix, it's a 4 by 4, and then an equal sign, and then there'll be some numbers. So the version I'm looking at has 8 pi g over c to the power of 4. That's just a proportionality constant. So G there is the gravitational constant, that's Newton's gravitational constant, so we know about that. And C to the power of four is the speed of light to the power of four. That's just basically an adjustment for the fact that the kind of conversion factor, if you like, between space and time to convert between the units is the speed of light. And then the final unit on the right hand side is the energy momentum tensor,

which is usually denoted as a capital T and then two subscripts. So again, those two subscripts indicate that it's a four by four matrix. This form of the equation where it's basically capital G equals some constants, times capital T, that's sort of the simplest way of running out the equation, or equations, because there's actually more than one there. Remember, each of these capital G and capital T is actually a four by four matrix, which has ten independent components. So we can interpret the field equations here as a set of equations, which dictate how the stress momentum tensor determines the curvature of space time. So because there are 10 independent components of the two four by four matrices, there are actually 10 of these equations. So 10,

equations, one for each of the independent components of the matrices. And technically speaking, the Einstein field equations consists of 10 coupled, nonlinear, hyperbolic elliptic partial differential equations. Now, just to indicate sort of the complexity here, some of you may have tried to solve a partial differential equation before. That is, I guess, like first or second year university maths level, but partial differential equations themselves are quite hard to solve. They're equations that involve partial derivatives. Non-linear partial differential equations are extremely hard to solve. Non-linear equations in general are actually quite difficult.

So here we're dealing not just with non-linear partial differential equations, but we've got 10 of them, and they're coupled, which means they interact with each other. We can't just solve them separately. They have to all be sort of solved together as a system. So this is looking like a bit of a mathematical nightmare. 10 coupled non-linear partial differential equations. How are we going to solve these beasts?

I'm not going to give you the answer. we're going to talk about that in the next episode, but I just want you to understand that although it looks simple when you write it out, it's actually very complicated. There's a system of complex equations here. The other thing is that we've used a lot of compact notation in order to write these out. Remember, though, that particularly the left-hand side of this equation, or series of equations, is highly condensed,

because we've written this G here, and I haven't explained what this G is yet. I've said that it's related to the curvature of space time, but what is this G? Well, this G is something called the Einstein tensor. It consists of the Rishi tensor minus the Rishi scalar times the metric tensor. So basically the left-hand side of the system of equations is the Rishi-tensor, which describes the curvature of space-time, and then we've subtracted a bit off. We've sort of adjusted it a little bit. It's minus R times the metric tensor. Sometimes I might just talk about it loosely as if the left-hand side is the Rishy tensor.

because that's just a bit simpler, but that's not technically correct. Whether we're talking about G or we're talking about R, the derishi tensor, both have, both a 4x4 matrices, as is the metric tensor, so they all kind of, they're all the same shape. Now, if you think about, well, what is this Rishi tensor? The Rishi tensor is defined, if you recall, in terms of these Christophel symbols and their derivatives. And Christophel symbols, in turn, are defined in terms of the metric tensor and its derivatives.

So what we can do is we can actually write out the Einstein field equations fully in terms of just the metric tensor, which is the fundamental object of interest to us, because it describes the shape of space time. Now, if you do that, if you write out these equations without using R or without even using Christophel symbols, but purely in terms of the metric tensile and its partial derivatives, you end up with an extremely complicated expression, which has 10 different terms in it. And each term here has like a couple of these Gs, so a G is the metric tensor, and then there's some partial derivatives and the partial derivatives can change position depending on what you're

differentiating and, you know, there's some constant terms and fractions and things. You know, I won't try to describe the full equation. But the point is, the way that you often see the Einstein field equations written out is like capital G equals constants times capital T. Or even if you see it written in terms of the Rishi tensor, which is like capital R minus big R times the metric tens. Or even if you see it written out like that, that's still a gross simplification or summary of what's actually going on, which is all of these different partial derivatives

of the metric tensor. The right-hand side, by contrast, is fairly simple. It's just the stress-energy tensor, and you don't have to worry too much about that. It's relatively simple. But the left-hand side, the geometric side, that's very, very complicated. So let's bring this together. What do Einstein's field equations actually do? Like, what are they mathematically? What they are is a series of literally hundreds, if you fully write it out, with all of the components, which I haven't even tried to explain here. If you fully wrote it out, there are literally hundreds and hundreds of different derivatives of the metric tensor and combinations of derivatives added up and subtracted in various complicated ways. Literally hundreds of them. Those hundreds are sort of separated out into 10

interacting but separate equations. So they're coupled equations but they're separate from each other. Ten equations which have hundreds of the of terms of the metric tensor and its derivatives. And those are all equal to different components of the stress energy tensor which describes the matter and energy content of space. So as I sort of mentioned before, actually Solving this, like solving these equations is extremely difficult, and I won't explain how we do that until the next episode, but I just want you to understand what the Einstein's field equation is sort of mathematically and what it kind of means, because it looks sort of simple, but buried under that is actually a lot of complexity. Now, before finishing out, I just want to give you a flavor of how

it is that Einstein came up with this. So you might be wondering, well, I've just been telling you about these field equations, but I haven't really explained how we know that these are true, or like how do we know that these correctly describe the way that universe actually is? Well, one cop-out answer is to say that they've been validated by experiment, and so we just sort of trust the experiments that they've sort of validated the predictions of the equations. That's certainly a big part of it, and we'll talk about experimental evidence in favor of general relativity in a future episode. But there's more to it than that, because there are plausibility arguments that can be given. When Einstein was developing these, he didn't have any experimental evidence for it.

I mean, there were certain observations that he pointed to, but fundamentally his argument was based on his sort of geometric intuition, and arguments by analogy from different areas of physics, and the fact that it kind of made sense of certain phenomena, and it was a generalization of things that were known in Newtonian physics. So let me try to articulate a little bit of that now as to how he came up with this, and why we might even suspect that this equation would be correct? Like, come on, why would you just equate the Rishi tensor with the stress momentum tensor?

Like, what do those things have to do with each other? Why would that have anything to do with the curvature of space time? Well, here's an argument that maybe lends a bit of plausibility to that. It doesn't prove it, but it helps you to understand why, Einstein thought this in the first place. We're going to start with something called Poisson's equation. Poisson's equation is a way of describing, I mean, it can describe different things, but in this context it's Poisson's equation for gravitation.

There's different applications of Poisson's equation, but here what it means is that we have this idea of a gravitational field which describes, in a sense, the potential, you can think of it as the potential energy of any massive object placed in that gravitational field. Poisson's equation says, loosely, it says that the second derivative, So you take the derivative twice. The second derivative of the gravitational field is equal to the density of matter inside that region. This is actually a way of formalizing Newton's theory of gravity. We don't often talk about it by that. Often you talk about it in terms of the force equation. So the gravitational force between two massive objects is equal to the product of their mass is divided

by the square of the distance between them and then there's some proportionality constants. But that's the way it's often taught and the way I discuss it way back in episode one. But another way you can say the same thing is to use Parson's equation. Now, Parson's equation basically says that steeper the slope of the potential energy of the gravitational field, the steeper the slope, the more matter is contained in that region. And that kind of makes sense, right? Because it's just a way of saying that massive objects generate a gravitational field around them, just like charged objects generate a charged field around them.

And the more massive they are, then the steeper the field is. Like the more quickly the field changes. You can imagine like a hill, right? The steeper the hill is, the more. more rapidly your potential energy is changing as you go up the hill because you're not climbing it steeper versus if it's shallower, then your potential energy doesn't change as much. So that's kind of what Parson's equation is telling us here. Again, this is just purely Newtonian, no curvature or anything.

The sort of steepness of your field, the rate at which your gravitational field changes, is directly proportional to the mass density in that area. Now, what's interesting about that is that if we take this equation and compare it to the Einstein field equations that we've been working with, turns out that Poisson's equation for gravity is actually one of these ten equations that we've been looking at. So in other words, this mass density component is actually the top left most component of the Rishi tensor under certain additional assumptions. This is part of something called new turn carton theory, which we're not going to get into. It involves some extra

simplifications which treats time as separately from space. I won't try to explain that here. All I'm trying to get across is that with certain extra assumptions and approximations, you can actually show that the Poissonzance's Gravitation equation from pure Newtonian mechanics is kind of like one bit of the overall 4x4-by-4 matrix, one like component of the 4x4 by 4 matrix of Einstein's field equations. So Einstein was sort of looking at this and thinking, huh, that's interesting. What if instead of saying that it's just these one component of the Rishi-Tensor, which equals one component of the stress momentum tensor, what if I say the whole tensor is equal to, so the whole Rishi tensor is equal to the stress momentum tensor. So instead of saying one bit of the

tensor is equal to another bit of the tensor, say that the whole tensor is equal to the whole other tensor. There's numerous advantages of this, apart from sort of generalizing it, describe the interaction over all of space time. It also allows it to be invariant because a single component of a tensor is not invariant. You can think of time dilation and length contraction as an example of this. Space and time individually can change depending on your reference frame. But the whole tensor itself, the whole Rishi tensor or stress momentum tensor, those tensors don't change. They're invariant. And so if you equate the whole tensor to the other whole tensor, then that actually can describe something that will be invariant across reference frames and across coordinate systems as well.

And that's important in describing something that's going to be true, you know, for all reference frames and for all coordinate systems. So this was Einstein's key insight, or one of his key insight, is that instead of basically looking at one component with one component, which is what Poisson's equation of gravitation was saying, He said, no, let's take all of them, the whole tensor, the whole 4x4, and equated to the whole 4 by 4, the Rishi tensor, and equated to the whole 4 by 4 of the energy momentum tensor. Now, that's not quite right. The Rishi tensor is not proportional to the energy momentum tensor. You need to make this adjustment, and this is the adjustment I mentioned before. You have to add in the Rishi scalar times the metric tensor.

That's the extra bit that I mentioned. Remember I said it's the Rishi tensor minus Rishi scalor, which is R, times the metric tensor. So you have to add this extra little bit in. But when you do that, then you get Einstein's field equations. So to sort of summarize that, what Einstein sort of did is he looked at Poisson's equation for gravity under Newton, and he imagined generalizing that to using objects which could be relevant to describing curvature and also would be invariant across different reference frames and commodity systems.

So he sort of was thinking about curvature and thinking about geometry and thinking about the need invariance as well across like reference frames and he combined those ideas together and imagine hmm what if instead of saying just this little bit of the richy tensor is equal to this bit of the engine momentum tensor what if i say the whole tensor is equal to the whole other tensor and that's a big leap like there's no guarantee that that's true it's just that was his idea based on his sort of intuition and arguments as we talked about before about the equivalence of inertial and gravitational mass and so forth he was thinking about maybe this whole idea of gravity is a force is actually wrong. Maybe what looks to us as a gravitational force is actually just an object

moving along a geodesic, moving along a path in curved space. And that explains why the inertial and gravitational masses are the same, because there's no actual force that's exerted here. It's sort of all just down to the geometry of the situation. And that led him to then extrapolate from this Poisson's equation to then the more general Einstein's field equations. And that was his postulate. He postulated these field equations and they've subsequently been tested experimentally. I should say that there are other ways to derive Einstein's field equations, which make different assumptions, but I won't go through these here because they're even more complicated than what I've already described. One final thing you might be wondering is, well, you know, how do we know that these are the right equations?

Even if Einstein made an informed guess, and even if we've, you know, validated experimentally, maybe there are other equations that give us similar predictions in some cases, but like different in others. Maybe it's incomplete and things like that. And that's a very valid thing to worry, to wonder about because there have been many, many alternative proposals to replace or augment Einstein's field equations. And some of these have been tested over the years, and some of them haven't been tested, like some of them can't be tested yet. But so far, every modification of Einstein's field equations that has been tested has been proven false. And Einstein's theory of general relativity has stood the test of time. It's been over 100 years now.

And one interesting addendum here, there's something called Lovelock's theorem, which was proved many decades after Einstein, so he didn't know about this. But it's now known that Einstein's field equations are the only possible equations which describe gravity in four dimensions of space and time that use just the first and second derivatives of the metric and that ensure that local energy conservation is maintained. So there's a couple of criteria there that I mentioned. One is that it's four dimensions. Another is that we only use the first and second derivatives of the metric tensor. And the final one is that local energy conservation is assured. If you make those stipulations, which are all, I mean, obviously we want local energy conservation. That's sort of something with

think is physically important. We think there are four dimensions in space time, so we'll go with that. As for using only the first and second derivatives, well, apart from being simpler, there's sort of reasons why we might expect it to be the first two derivatives, because that's kind of common in physics elsewhere. We very often see first and second derivatives. Seeing third derivatives or more is actually quite rare. Now, that's a heuristic argument. I'm sure other people might be able to say a bit more why we'd expect it to be those first two derivatives. But the point is, if you make these stipulations, it turns out that Einstein's field equations are the only possible equations you can have. They're the most general and only type of equation.

Now, that doesn't mean that they're true. It just means that if you make this sort of starting point of assumptions about the type of things that we're going to stipulate, then you're going to end up with Einstein's field equations. And so that's sort of strong evidence that the field equations are kind of correct, at least insofar as we don't then start talking about things like a quantum theory of gravity. And for those who maybe know, Einstein's field equations cannot be reconciled currently with quantum mechanics. We need a new theory for that. So that's a whole other kettle of worms.

But in terms of being the right equations just for gravity itself, there seems good reason to think that Einstein's field equations are the right once. But of course, we never know that for sure because we can only test things up to a certain amount of precision. So continual experiment and theorizing will go on to see if we can continue to validate it or maybe refute Einstein's equations. That will conclude us for this episode. It's very dense. And so I hope that even if you didn't get every detail, you sort of got the overall picture. we talked about how we want to explain force of gravity in terms of the geometry of curved space time and objects kind of moving on a geodesic, on a sort of a straight line bit in a curved space, so to speak.

And we then introduced all of the mathematical objects necessary to do that. So we talked about manifolds. We talked about differentiable manifolds and covariant derivatives. We talked about Christophel symbols and how they allow us to fix differentiation so that it sort of works in a curved space. We introduced the metric tensile, which allows us to describe distances. and also the overall shape in our curved space. We then talked about how to define curvature in our curved space time. So we talked about the remun curvature tensor, the Rishi tensor,

the Rishi Taser, and the Rishi Scalar. We talked about how to describe the distribution of matter and energy in our four-dimensional space-time using the energy momentum tensor, which is also called the Stress Energy Tensor. And then we just explained how these elements are put together to form Einstein's field equations with the Rishi Tensor minus half, the Rishi Scalar times the metric tensor. being the geometric left-hand side of the equations,

and then the right-hand side being some constants times the energy momentum tensile, which describes the mass and energy content of the universe. And that's Einstein's field equations. We talked a bit about what that means and how complicated they are, and a little bit about sort of how Einstein came up with this and why you might expect these to be correct equations. In the next episode, we will talk about how we actually solve these equations

to actually get some solutions out and to actually have a description of specific phenomena in space and time. We'll also talk about the experimental evidence for the correctness of Einstein's theory of relativity and some of the predictions that it's made. We'll also talk about black holes at some point because that's a very important phenomena which can only be described using general relativity. So stick around for that. Hopefully that will be coming out soon. I don't know if it'll be the next episode or maybe a couple afterwards. We'll see how we go. Thanks very much for listening. I hope you enjoyed this episode. If you did consider leaving a positive review on iTunes or Spotify or the podcast aggregator that you prefer.

If you'd like to support the show financially, you can consider making a contribution either by becoming a Patreon supporter, or you can make a one-off donation via PayPal. The links to those are in the show notes, so click through to that if you're interested. You can also send me an email. My address is Fods12 at gmail.com. That's F-O-D-S-1-2 at gmail.com. For free to send any questions, suggestions, or other feedback. And once again, thanks very much for listening. I'll talk to you next time.