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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. The direction this derivative vector is pointing
meanwhile is in the negative radial direction. 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. 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. 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. 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. 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. 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. 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. Indeed from this knowledge he can begin
constructing something which will prove extremely important to him -- his
geodesic paths across Polar Land. 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.
