The Metric Tensor in 20 Glorious Minutes

hey everyone i am super excited to tell you all about the metric tensor today you've seen the metric before both in lecture and in your readings and in other flipped classroom videos you've seen how to calculate norms of four vectors how to raise and lower indices etc but what i want to tell you about today is what the metric tensor really is and why you should care to be honest though this will necessarily take us into the mathematical realm of differential geometry but i promise the overall conceptual clarity that you'll gain by the end will be well worth it so on that note let's get going i don't want to keep you in the dark so i'm going to give you the punch line right at the outset the metric tensor is the fundamental tool that we use to study curved spaces its basic purpose is to define the concept of distance between points in a curved space and curved spaces are also known as manifolds for our purposes we can just think of manifolds as spaces that are sufficiently smooth meaning they don't have any tears or cuts or sharp edges or punctures two-dimensional examples such as the surface of a sphere or the surface of a hyperboloid are great to keep in mind and most importantly notice that if you zoom in enough a smooth manifold looks flat the surface of the earth for example looks perfectly flat no matter where you are until you get into a shuttle and go into orbit and you see the planet from a larger perspective so that's it that's all a smooth manifold is it's a space that looks flat when you zoom in even though it may be curved when you zoom out and i'm sure at least some of you crave more technical mathematical descriptions of these kinds of things so if you want the more mathy kind of description here you go when we say that a manifold looks flat when you zoom in that really means that at any point a small enough neighborhood of that point can be mapped to a subset of euclidean space in which the map and its inverse are infinitely differentiable meaning they have partial derivatives of all orders so very cool that's pretty cool and if you're into these kinds of very cool awesome mathematical details guess what i've got some awesome references for you at the end of this presentation but for our purposes we're going to go on thinking of manifolds in a visual and intuitive way so the next thing for you to visualize is the set of all vectors tangent to a given point in a manifold in this two-dimensional example each point in this tangent plane can be identified with a vector a position vector reaching from the tangent point p to whatever point is in question so this motivates a generalization to manifolds of any number of dimensions the set of all tangent vectors at a given point is a vector space that we call the tangent space and like any vector space it has a best friend called the dual space or in this context the cotangent space whose elements are therefore called dual vectors and as a side note from special relativity when you have a vector whose components are denoted with an upper index that you have seen called a contravariant vector that is in this context here those are what we would call vectors actual true vectors whereas the other kinds of vectors with lower indices that you have heard called covariant vectors those are actually dual vectors but that aside at this point we've seen that at any point in a manifold there's a tangent space there's a cotangent space so there's two kinds of objects vectors and dual vectors and just like that we're ready to define a third kind of object which is called a tensor and a tensor is just a map that takes a bunch of vectors and dual vectors and maps them all to a single real number a tensor has a type which is an ordered pair of two integers and those integers are just the number of dual vectors and the number of vectors which the tensor acts on so now suppose you have a tensor of type zero two it doesn't take any dual vectors and it takes only two vectors and moreover suppose that it doesn't depend on the order of the vectors in that case this is called a metric tensor and those are the only two requirements for a metric tensor it's a very generic concept and the reason why it becomes important is because a manifold all by itself actually doesn't have a shape it doesn't have a geometry to do geometry you first have to define the concept of distance between points and there is no prescription for doing that in the definition of a differentiable manifold by imposing a metric tensor on a manifold you are actually giving it a geometry so to see how this comes about first we need a bit of jargon we first define the inner product of two vectors as being the number you get when the metric acts on those vectors and then we define the norm of a vector as the inner product of a vector with itself the norm is then what we think of as the length of a vector and with these ideas in mind we can picture this setup you're standing at some point and you've drawn a curve that passes through that point this is a parametrized curve meaning that the coordinates of the points on the curve are given as functions of a continuous real parameter what you're going to do is you're going to calculate the norm of the tangent vector to the curve at that point but then you're going to realize that the norm calculated at each point along the curve is actually a function of the continuous parameter and that function can be integrated along the curve and the result of that integration is a number that we define to be the distance along the curve so you take two points on the curve you calculate the integral of this function between those two points and the number you get by doing that is what you define to be the distance along that curve between those two points also this expression is conventionally written in this form down here because the parameterization of a curve is arbitrary you are free to choose whatever parameterization you like and so we write the expression for the what you can think of as the infinitesimal distance between points in this form which is called a line element and when you've written the distance in this form this is something that you can integrate to get distances along curves now i realize that at this point everything i've said is very abstract in general so you're probably wanting an example so let's take the simplest example the euclidean metric how do you calculate the distance between two points in euclidean space you use the pythagorean theorem and then you make a small change of notation replace x with x1 y with x2 etc dx1 squared is dx1 times dx1 etc and then you recognize that the coefficient of each of these terms is just a one but then you also realize that you can also write cross terms whose coefficients necessarily have to be zeros and then you take this entire expression and you compact it down into a double sum and then you invoke the summation convention to write it in this final nicer looking form so we find that the pythagorean theorem is essentially contained in this three by three matrix which for euclidean space is the three by three identity matrix back in the 19th century there was this awesome mathematician named bernard riemann who realized that this way of writing the pythagorean theorem was actually the key to studying curved spaces the pythagorean theorem follows from the axioms of euclid and that relationship is an if and only if relationship but that means you could start with the pythagorean theorem as your axiom and derive all of euclidean geometry from there so now imagine that in this formula instead of using the three by three identity matrix suppose you use a different matrix restricted only in that it has to be a symmetric matrix then you would have a different type of pythagorean theorem it would no longer be valid for calculating distances in euclidean space but it would be valid for describing distances in some other space with some other geometry so bernard riemann actually generalized the pythagorean theorem and in doing so open the door to studying curved spaces of any number of dimensions and all you have to do is specify a matrix bearing in mind of course that this matrix is really just a representation of the metric tensor in a given coordinate system but as long as you remember that you shouldn't get into too much trouble if you just think of the metric as a matrix so now that we know what the metric is we can have a little fun finding out what the metric can do for us and what it can teach us first note that there are two types of metrics metrics whose eigenvalues are all positive such as the euclidean metric those are called romanian metrics and the main consequence is that distances are guaranteed to always be positive and that's great as we learned in class because then you can have nice things like the koshi schwarz inequality and the triangle inequality however there's also a different type of metric where some eigenvalues can be positive and some can be negative and the main consequence of that is that distances can be positive negative or even zero best example of a lorentzian metric is of course the minkowski metric which describes the geometry of space time according to the special theory of relativity as a side note the placement of the minus signs is a convention this particular convention is used in particle physics and in field theory and the other convention is used in general relativity and studying black holes and curved space time and gravitational waves in cosmology and in this other convention we put the minus sign in the top left corner but regardless of the sign convention the fact the minkowski metric is lorenzion is the is the single reason why special relativity is so cool why we have time dilation and length contraction and relativity of simultaneity and light cones and e equals mc squared but since this is a video on the metric we're not going to focus on all these cool predictions of special relativity for now we're just going to focus on the fact that there are three types of distances and consequently there's also three types of vectors depending on whether the norm of a vector is positive negative or zero and also you can have what are called time-like trajectories or space-like trajectories or null trajectories these are paths in space time in which the tangent vector is always a time-like vector or space-like vector or a null vector you may recall that in special relativity there's a law of physics that you have a particle that has mass and if the particle is a free particle meaning that it moves under its own inertia with no forces acting on it then its path is a time-like path which maximizes the proper time between any two points in space-time the path of longest proper time is called a geodesic now in a space with a romanian metric a geodesic would actually be a path of shortest distance like a straight line in euclidean space but in special relativity with this lorentzian metric we have to speak instead of longest proper time so the question is how do we figure out which paths are geodesics well you guessed it the metric comes to the rescue by taking a bunch of derivatives of the metric plugging them into this formula we can calculate these functions which are known as christopher symbols which act as coefficients in this so-called geodesic equation which is the differential equation whose solutions are the paths of longest proper time so this is how you calculate the trajectories of free particles but wait there's more the metric also lets you determine whether a space is curved or not it doesn't do so directly but its first and second derivatives come together to define the riemann tensor which is itself what actually defines the intrinsic curvature of a space intrinsic curvature means things like do parallel lines always remain parallel or do they converge or diverge do the interior angles of a triangle always add up to 180 degrees etc so if all of the components of the riemann tensor vanish then the space has no curvature but if they don't all vanish then the space is intrinsically curved so we use the word flat as a synonym for the riemann tensor being equal to zero so euclidean space euclidean space is obviously flat and minkowski's base is flat but now we come to the best part of this presentation where we ask the question even though special relativity tells us that the geometry of space-time is described by the flat minkowski metric we now ask the question is it actually true that space-time is actually flat so we know from experiments that special relativity is true and therefore that the geometry of space-time is described by the minkowski metric but on the other hand when we do experiments that test special relativity we do them in laboratories and the physical size of human laboratories even really big ones like the large hadron collider are tiny compared to the size of the earth or the size of the moon's orbit or the size of the solar system so by doing experiments in laboratories all we're verifying is that space-time is described by the minkowski metric over distance scales that are of the same size as our laboratories but suppose you were to go all the way out to the orbit of the moon and ask is the geometry of space-time over the huge region surrounding earth all the way out to the moon's orbit is spacetime still described by the minkowski metric just like surface of a sphere looks perfectly flat when you zoom in even though it's actually curved when you zoom out as scientists we have an immediate obligation to ask is the geometry of space time and cow skin over all distance scales well to make a long story short we can write down a theory described by this lagrangian in which the minkowski metric is no longer the metric of space time but rather the true metric is something else which could perhaps vary from location to location and whose derivatives would therefore no longer be zero in which case the riemann tensor might no longer be zero so you can write down this lagrangian which later on in the course you'll learn about relativistic field theory and relativistic lagrangians and you'll learn that just like in classical mechanics you take a lagrangian you plug it into the euler lagrange equations and you get a set of equations which in this context are called field equations which are partial differential equations that tell you how the components of the metric tensor depend on the distribution of matter and the distribution of matter is encoded in this matrix called the stress energy tensor so now suppose you want to know the geometry of space time on the scale of for example the solar system the sun is the primary body in the solar system but it is tiny compared to the entire solar system so we can treat it as a point particle so that's our mass distribution a single point particle and we can then solve the field equations to find the metric which according to this particular theory ought to describe space-time in our solar system and that metric has a name it's called a short shield metric and now that we have it we can do things like calculate time like geodesics so suppose the planets are to be regarded as free particles we can then calculate their orbits as time-like geodesics in the short shield space-time and absolutely remarkably the predicted orbits of the planets match the measured orbits to a stunning degree of precision far surpassing the predictions of newtonian gravity and what i really want you to take away from this is that whereas newton had to imagine that the sun is somehow exerting a force on planets that are millions of miles away this relativistic theory we've been describing actually explains the orbits of planets and the orbits of the moons and even the falling of an apple as geodesic motion in a curved space geodesic motion is nothing more than motion under your own inertia with no forces acting on you so just by allowing for the possibility that space-time can be curved that is to say that the metric can deviate from the minkowski metric we get gravity for free and i'm sure you probably remember the name of the awesome dude who discovered this and this is really quite possibly the most mind-blowing and unexpected discovery in all of physics and i never cease to be amazed at the ever growing capacity of the human mind so on that note i would like to thank you all for sticking with me through this award-winning video you now know what the metric tensor is and why the metric tensor matters and i hope you've also enjoyed getting a first taste of differential geometry and curved space time so to end here are some resources for further reading see you later

Channel: Jacob Sprague

Views: 6,062

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Id: L9WR78xvCPY

Length: 19min 43sec (1183 seconds)

Published: Mon Feb 01 2021