Conceptualizing Christoffel Symbols in Curvilinear Coordinates

[00:00:00]If you've traveled on an airplane before  you might have noticed that the shortest route to your destination doesn't take  you along a straight line on the map but rather follows a curved path. This is  an example of what's called a geodesic, a route representing the straightest path on  a surface, and to calculate it you need the aid of certain mathematical tools known  as the Christoffel Symbols. Likewise in general relativity these same tools allow you to  calculate the inertial routes of objects through space and time. But where do such tools  come from and how can we understand them intuitively? This is Dialect with  Conceptualizing the Christoffel Symbols. This is Cartesian bear, he lives in Cartesian land, where lines  are all straights and intersect at right angles this is polar bear he lives in  polar land where lines are all straight and intersect at right angles. Okay aren't these  two lands exactly the same thing? Well sure, but not quite. You see Cartesian  Bear lives in the real world, where his grid distances and coordinate angles  match up to actual distances and actual angles, while Polar Bear is living in a matrix,  a false projective world like the map on your airplane screen. What this means is that  when Polar Bear moves in a straight line along his Theta coordinate direction in polar land,  he actually moves along a circle in the real world. And when Polar Bear moves in a straight  line in the r coordinate direction in polar land he actually moves radially outwards in  the real world, as if along the spokes of a wheel. Polar Bear is unaware of all this  because he assumed his coordinates to represent independent dimensions and so to  him Polar Land is perfectly Cartesian. But one day Cartesian Bear decides to Red Pill poor  Polar Bear. "Your idea of reality is actually a map," he tells Polar Bear, a projection  due to the false way you constructed your coordinates. Now Polar Bear descends into an existential crisis. How can  he know what's real and what's not? Before in order to measure distances and  angles in Polar Land he had always relied on his basis vectors ER and E Theta. Across Polar  Land these basis vectors always had unit length one and everywhere were oriented at right angles  to one another. But since realizing he lives in The Matrix, Polar Bear now knows that the lengths  of these basis vectors don't represent true real world lengths, nor does the right angle between  them necessarily represent a real right angle.

[00:03:20]Moreover, the amounts of distance and the angle  these basis vectors do represent could actually change from point to point across Polar Land.  Fortunately for Polar Bear we will provide him with a useful mathematical tool known as the  Metric Tensor. If Polar Land is a map of sorts then this tool is like a bar scale for his  basis vectors. It tells him at every point across Polar Land how much real world distance  his basis vectors represent and what the real world angle between them is. In this case the  first component of the metric tensor is one, so this tells polar bear that everywhere  in polar land his ER basis vector correctly represents one unit of real world length. Thus  if polar bear moves one unit of R coordinate distance in Polar Land, he knows he'll also  have moved one unit of distance in the real world. The second component r-squared tells  Polar Bear that his Theta basis vector represents as many units of real world distance as is  equal to whatever R coordinate he is located at. So if Polar Bear is located at R coordinate 2  and moves one unit of theta coordinate distance, he will move two units of distance in the real  world. And if he's located at R coordinate 4 and moves one unit of theta coordinate distance,  he will move four units of distance in the real world. This should make intuitive  sense for polar coordinates because any change in angle will sweep out a  greater distance farther out from the origin. Finally the last two components of  the metric tensor are both zero, which tell polar bear that the 90° angle between his basis  vectors represents 90 degrees in the real world as well. So it turns out that Polar Bear's theta  ruler essentially grows in length as he moves radially outwards in the real world. This explains  in part why his coordinate construction of Polar Land was false to begin with; however there is  more to the story, for as Cartesian bear can plainly see, when Polar bear moves along a circle  the direction in which he lays out his rulers also steadily rotates and changes. Polar Bear is  still unaware of this however, because although the metric tensor tells him about the real world  lengths of his basis vectors and the relative angle between them it doesn't explicitly specify  anything about the real world orientation of those vectors. But is it still possible for Polar Bear  to deduce such information from the metric tensor alone? Let's have Polar Bear consider just four  infinitesimal pieces of Polar Land at a certain R distance out from the coordinate origin. Each  piece has length DR and height D-Theta. We'll help him out by drawing the ER and E-Theta basis  vectors on each piece. Now keep in mind that the basis vectors have a unit length of one,  even though we are drawing them as being on par with the infinitesimal lengths of our  pieces. Now the r coordinates of the pieces in the first column we'll label as R and the r  coordinates of the pieces in the second column as r + DR. Similarly the theta coordinates of  the rows we'll label as theta and theta plus d- theta. Now let's resize these pieces to match  their real world areas using the metric tensor.

[00:07:28]We start by multiply in R lengths everywhere  by one. Obviously this means the DR lengths of all the pieces as well as the lengths of all  the R bases vectors remain the same. Next, we multiply D-Theta lengths everywhere by their  R coordinate distance out from the origin. This means the D Theta lengths of the pieces in the  first column become r d Theta while the Theta bases vectors are resized to length R. In this  the second column meanwhile the D Theta lengths of the pieces become r + Dr * D-Theta and the  theta basis vector lengths become R plus Dr. Polar Bear now has four properly sized pieces  of Cartesian Land, and so to figure out the real world orientation of his basis vectors, all  he needs to do is fit these pieces together like a puzzle. Indeed, should he find a general method  for fitting pieces together at any arbitrary coordinate, then this would allow him to fit  together every resized piece of Polar Land, so as to reconstruct the whole of Cartesian  Land and be able to determine his geodesics upon it. But how does he go about this using  just the information of the metric tensor alone? Well let's examine each metric component  more deeply to see what it's saying. Now the fact that the first component tells us that our  lengths everywhere are unchanging means there is no one-dimensional curvature in the R direction  in Polar Land. In terms of our resized pieces this means we can thus conjoin our two columns  side by side without bending or rotating them.

[00:09:21]Meanwhile the fact that the second component tells  us that the theta lengths do change across Polar Land means we won't be able to preserve  the straightness of the theta direction, and thus that our rows will have to be bent into  another dimension to be connected. Lastly if we consider the second partial derivative of the  Theta component we see that it's a constant which hints to us that we need only one extra dimension  in order to accommodate this bending or curving, and so won't have to worry about  bringing a third dimension into the picture. Now from visual intuition alone you've  probably already guessed that we need to bend the rows so that they're touching like this, and  if you're familiar with polar coordinates you could reasonably surmise that the angle  through which they should be rotated is D-Theta. But how do we extract this information  from the metric tensor? Well to do that we need to invoke something called the Levi-Civita  connection. For infinitesimals like this, this connection is expressed as a simple  equality stating that the change in our e-theta basis vector when transported along  the radial direction must equal the change of our ER basis vector when transported in the  theta direction. What does this equality mean visually? Well consider these two square  infinitesimal pieces of Polar Land side by side. If we transform the theta  lengths of one of the pieces while keeping the other constant we've now added  extra physical space to our manifold. This amount of new length is tracked by  noting the change in the E-Theta basis vector. Now to keep the manifold continuous  there needs to be a change in the radial basis vector of the next piece atop the  first which will compensate for this new length. Obviously the radial basis vector  just simply growing or shrinking won't compensate for this extra length since  it lies in a wholly different dimension.

[00:11:31]Thus the radial basis vector must be  rotated in order to supply the additional length. Now if the theta growth is small,  then this rotation will likewise be small.

[00:11:50]But if the theta growth is large, then  so too will the required rotation be large. So how quickly theta lengths are growing  along the radial direction will in fact determine how rapidly Polar Bear's basis vectors are  rotating when transported along the theta direction. Indeed it is this Levi-Civita condition  that the growing or shrinking of Polar Land along one dimension be compensated by its curving into  another dimension that ultimately causes the rotation of Polar Bear's coordinates in the real  world. But what is the amount of this rotation precisely? Well returning to our infinitesimal  pieces recall that this lower left piece has a resized theta length R D-Theta while  the lower right piece has a resized Theta length R plus R D-Theta. The difference  in theta lengths between the lower left and lower right pieces is thus R + DR D-Theta  minus R D-Theta which equals DR * d-Theta.

[00:13:02]Now the length produced when a radial segment  R is rotated through a small angle D-Theta is R * D-Theta. Here the length of our  radial segment is DR which means to produce the length DR D-Theta it needs  to be rotated through an angle D-Theta.

[00:13:23]Polar Bear is feeling better about his  reality; the metric tensor gave him the real world lengths of his basis vectors and the  real world value of their adjoining angle, and now the Levi-Civita connection gave him the real  world orientation of these basis vectors. Indeed, armed with such knowledge he now possesses all  the requisite information for how to properly connect his resized infinitesimal pieces of Polar  Land. He simply conjoins the columns in the radial direction side-by-side and then tilts the top row  through an angle D-Theta and voila, he is looking no longer at the false geometry of Polar Land but  rather at the true geometry of Cartesian Land.

[00:14:12]Now it's time for him to formalize this  process by describing precisely how the real world vectors corresponding  to his Polar Land basis vectors will change as they are transported  along different coordinate directions.

[00:14:28]Since he has two basis vectors and there  are two coordinate directions in which these vectors can be transported he will  require four derivative vectors to fully describe such change. Each of these derivative  vectors will in turn have two components, meaning eight total numbers will become involved  and it is these eight numbers which are known as the Christoffel symbols. The Christoffel  symbols are notated with three indices: the upper index indicates which component  of the derivative vector is being referred to, the lower left index indicates which  basis vector is having its derivative taken, and the lower right index indicates in  which coordinate direction we are taking that derivative.

[00:15:31]All right, now let's figure out what  these symbols are by calculating each derivative vector. We'll start with the easy one: we'll take the R-basis vector and transport it  a coordinate distance DR in the R direction in Polar Land. Here there is no change in this  basis vector in the real world whatsoever, so its derivative vector is zero, meaning our  first two Christoffel symbols are likewise both zero. Next let's take the theta basis vector and  transport it along a coordinate distance D-Theta in the theta direction in Polar Land. When we do  this, the real world Cartesian vector is rotated counterclockwise through an angle D-Theta. The  change in this vector equals the length produced by this rotation and since the real-world vector  had a length R the length produced by its rotation should equal R D-Theta. Now in this infinitesimal  picture it's easy to see that this rotation occurs relative to the prior piece, but let's take  a moment to look at the continuous picture in Cartesian Land. Here we have the transformed  Theta Vector scaled to its proper length, and as we transport it through a small angle D-Theta  the vector rotates accordingly. But what do we measure its rotation relative to? Well in this  case, we'll transport a second vector alongside it which maintains a constant orientation in the  direction in which the theta vector was initially pointing before it was transported. Then we  can see that the change in the theta vector is a change relative to this second vector.  This is essentially the same thing we did in the infinitesimal case but in the continuous  case it's referred to as parallel transport.

[00:17:32]Returning to our calculations, Polar  Bear's theta basis vector thus changes by an amount R D-Theta in the real world per  coordinate distance D-Theta in Polar Land, so the relevant derivative here is R  D-Theta / D-Theta which equals simply R.

[00:17:52]The direction this derivative vector is pointing  meanwhile is in the negative radial direction.

[00:18:00]Now at different points across Cartesian  Land such a vector will look like this; while at different points across  Polar Land it will look like this. If we were to express this vector  in its Cartesian basis we would get the components minus r cosine of theta in the  X hat direction and minus r sine of theta in the Y hat direction. In Polar Land  however we see that the components of this vector are minus r in the r hat  direction and zero in the theta hat direction. Our next two Christoffel  components are therefore minus r and zero.

[00:18:49]Next, let's take the radial basis vector  and transport it a coordinate distance D-Theta in the theta direction in  Polar Land. In Cartesian Land this vector is again subjected to a rotation  D-Theta and since its length is one, this rotation produces a length of simply D-Theta.  D-Theta / D-Theta is 1 so the derivative vector everywhere in Cartesian Land has length one  and points in the Theta coordinate direction.

[00:19:21]However these vectors do not have length  one like this everywhere in Polar Land, because if Polar Bear is expressing  them in terms of his basis vectors, we must remember that his theta ruler  length is growing across Cartesian Land.

[00:19:39]At a certain R distance out in polar  land Polar Bear's theta basis vector represents r Cartesian basis vectors, so if  the derivative vector has the length of one Cartesian basis vector it must be shrunk  in Polar Land to a distance of 1 / r. The components of the derivative vector are  therefore zero in the r-hat direction and 1 divided by r in the theta-hat direction.  These are our next two Christoffel symbols.

[00:20:17]Last but not least, let's take our theta basis  vector and transport it a length DR in the radial direction in Polar Land. In Cartesian Land the  vector changes by length DR so its derivative is a vector with length DR / DR or 1 which points in  the theta direction. Again normalizing this theta distance in Polar Land we see that our final  two Christoffel components are again 0 and 1 / R. Now of course we could have short-cutted  this process by remembering the Levi-Civita connection requires that that the derivative of  the theta basis vector when transported in the r direction equal the derivative of the radial basis  vector when transported in the theta direction.

[00:21:14]With all eight Christoffel symbols  calculated Polar Bear now has a much stronger grasp on reality. These symbols  will tell him precisely the rate at which his real-world basis vectors are changing  with respect to changes in his coordinate system. So for instance if he traverses an  infinitesimal length D-Theta in Polar Land he knows his real world theta basis  vector will shift to the left due to picking up a radial component equal to the  Christoffel component there time D-Theta.

[00:21:49]If he wants to counter this shift  and continue traveling as straight as possible in the real world he will  need to veer rightwards in Polar Land by an amount which will compensate for  this component, in this case R D-Theta.

[00:22:06]Indeed from this knowledge he can begin  constructing something which will prove extremely important to him -- his  geodesic paths across Polar Land.

[00:22:21]Yes, it's a happy ending for Polar Bear who has  finally made his way to the real world. But for us, we have only begun to scratch the surface of  the Christoffel symbols. In ensuing installments we will need to make the relationship between  basis vectors, the Christoffel symbols and the metric tensor explicitly precise; we  will need to begin training for the index olympics; we will need to move  from flat surfaces to curved ones; and lastly we will need to understand how to swap physical manifolds for spacetime manifolds.  But to really truly understand relativity, there is only one revelation you need to make. You  are Polar Bear -- you are the one living in The Matrix. This has been Dialect, thanks for watching.