April 9th, 2021

Derivation of the metric tensor, Christoffel symbols, and covariant derivative from first principles

Originally published at 狗和留美者不得入内. You can comment here or there.

I am going to actually do the “simple calculation” for change of coordinates of Christoffel symbols, aware that simply following texts without actively doing some calculations oneself can be rather self-deceiving. In the process I’ll jot down some notes pertaining to the covariant derivative.

First I shall explain some intuition behind curvilinear coordinates.

Image from the Wikipedia page for "curvilinear coordinates"

The example which L&L Classical Theory of Fields used to illustrate curvilinear coordinates was a rotating coordinate axis, with the [公式] -axis as the axis of rotation. Moving at uniform velocity with respect to the Galiliean spacetime coordinates, after an infinitesimal change in time, the [公式] and [公式] components of the rotating axis will have changed infinitesimally in their directions. From this, it is apparent that parallel transport of a velocity vector on a spacetime manifold must change the value of the velocity vector.

Take any sufficiently differentiable bijective self-map on [公式] , the domain corresponding to spacetime. To visualize this, one can superimpose the blue and red images on the Euclidean metric corresponding black one (of course this would only be on [公式] in a such a way that the origin goes to itself. At any point we can make infinitesimal displacements [公式] and see how much the superimposed coordinates are displaced in each of the two directions, the values of which correspond to the Jacobian of course.

The existence of an invertible Jacobian [公式] of curvilinear coordinates with respect to Galilean spacetime corresponds to an open [公式] of the spacetime manifold [公式] that is diffeomorphic to some subset of [公式] . The metric tensor [公式] defining an inner product on the space of one-forms at point [公式] enables us to define infinitesimal distances on the manifold by assigning a scalar to every pair of elements in the standard basis of [公式] and the angle at the point of intersection of two curves on the manifold is also defined as in the Euclidean case, via [公式] , but with a different inner product.

1. The relationship between the metric tensor and the Christoffel symbols

We can on the tangent space at a given point define orthonormal tangent vectors [公式] , here [公式] indexed, with [公式] being the all ones vector. Associated with that point is of course a chart on the which with distance defining metric tensor is that such that

[公式] which is basically saying that the value of same inner product of two basis vectors (which by linearity induces inner products on entire vector space) does not change upon transport along any direction on the manifold. We let

[公式] which defines the Christoffel symbols. How to interpret this? Well, in curvilinear coordinates, if one moves along solely along one coordinate in a Galilean coordinate frame at the same unit speed, after an infinitesimal displacement, the coordinates of one’s direction in the curvilinear frame can change by some amount along every direction. Since the differential operator is linear, we can assume the change function to be linear over the tangent space, and at a different point on the manifold, the tangent space the dimension of which does not change still has an orthonormal basis. Thus, we can express the rate of change of the basis elements as a linear combination of the basis elements themselves.

Substitution of [公式] into [公式] yields

[公式] Seeing that the contracted out indices can be renamed arbitrarily separately in each of the sums, we rename such that the expression becomes, viewing [公式] as fixed, the application of the sum of three covariant rank 2 tensors to [公式] . Then, [公式] transforms to

[公式] which equates to the following relation between the metric tensor and the Christoffel symbols:

[公式]

2. Metric tensor and Christoffel symbols in the Riemannian manifold defined via polar coordinates

The (bijective and smooth) relationship between rectangular and polar coordinates is

[公式] We define a real, smooth manifold the points of which are [公式] and to satisfy the second countable and Hausdorff condition of a topological manifold, we note that any subspace of [公式] inherit its second countable and Hausdorff properties. As for the atlas, we need only one chart, the definition of which is given by [公式] .

As for the tangent space at [公式] with basis elements [公式] and the associated metric tensor, we first note that we want the metric to satisfy

[公式] gives us [公式] for the metric, wherein the metric varies with [公式] but not with [公式] . Thus, the Christoffel symbols, which are determined by the metric, will also vary in the same way.

Take some differentiable curve [公式] on our manifold passing through a given point [公式] , chart coordinates of which we denote as [公式]. The rate of change of [公式] on our chart

[公式] One who has studied manifolds and actually understood it should be able to easily recall that class of curves passing through the same point [公式] such that the rate of change with respect to the curve’s parameter of the coordinates local at [公式] is equal form a equivalence class well-defined with respect to addition and multiplication by a scalar, which is used to define the velocity of a curve on a manifold at a point.

If partial derivatives of chart coordinates with respect to local coordinates on the manifolds are prescribed at every point, which is done implicitly by assigning a metric tensor to every point, then the partial derivative operators of first order ( [公式] in our concrete example), which form a basis of covariant vectors, also span the partial derivative operators of second order via the Christoffel symbols. Being a two dimensional vectors, each of

[公式] is spanned by

[公式] Here we note that the basis vectors have to non-zero determinant everywhere in order to linearly independent everywhere, and moreover, that the symmetry of partial derivatives suggests symmetry in the Christoffel symbols. As for explicit computation of the Christoffel symbols for our polar coordinate manifold, it is very straightforward and will be left to the reader.

3. Transformation of Christoffel symbols under change of coordinates

In section 1, we used [公式] to represent basis elements of the tangent space, except these are contravariant vectors when basis elements of the tangent space as partial derivative operators are supposed to be covariant. I personally did so, then being influenced by [公式] for the distance induced by infinitesimal displacements along each of the coordinate axes. If our basis here spanning the space of infinitesimal displacement is [公式] , then the above [公式] is simply the inner product of the all ones vectors with itself. As for this inner product, it is uniquely determined by [公式] or more like [公式] inner products due to symmetry, which are pairs of values of projections of the all one vectors onto some basis element.

The reader might feel of course as did myself that using [公式] in that case to represent the all ones contravariant vector and operating along that was somewhat a bastardization of notation. From now on, I will define the covariant basis [公式] and a velocity field along a Riemannian manifold as represented by [公式] . At every point on the manifold, there exists by definition a local chart that is smooth. We are interested in the rate of change of the velocity field at a given point along each of the local chart directions, which we shall denote via [公式] . Per the metric tensor for the given chart at the given point, we can find Christoffel symbols compatible with [公式] (with a tweak that alters a contravariant component to a covariant one) at that point, satisfying

[公式] This is the covariant derivative with respect to an arbitrary basis tangent vector as applied to any arbitrary basis element of [公式] , which lies in the dual space of the tangent space that is denoted as [公式] . The magitude of the projection of tangent vector [公式] onto the [公式] th basis element is given by the scalar [公式] , from which is it easy to see that the elements of [公式] are linear maps from tangent vectors to scalars.

From the partial derivative in [公式] , which exhibits linearity, it is easy to see that the covariant derivative along a specified direction must also be linear. Thus if we assume that the tangent vector [公式] does not vary when transported along [公式] , the formal statement of which is [公式] , then

[公式] In the general case, this becomes

[公式] Now, we proceed to derive the transformation of the Christoffel symbols under a change of coordinates wherein [公式] . We will in the transformed frame use [公式] in place of [公式] and a bar above the tensor. [公式] in the transformed frame is then written as

[公式]

This is a covariant vector, as is the value of [公式] , and covariant vectors should transform as follows:

[公式] Thus we can establish the relation between the values of [公式] and [公式] .

[公式]

Taking out the same partial derivative operator on both sides gives followed by some evaluations on the left hand side gives[公式] In the product of four partial derivatives on the left hand side wherein includes the product of the Jacobian with its inverse, we have

[公式]

Thus, we have

[公式]

which then equates to

[公式] the formula for transformation of the Christoffel symbols under change of coordinates. One notices that because of the second term with the second derivative, which violates the law of transformation of tensors, the Christoffel symbols do not constitute a tensor.

Rearranging [公式] , one obtains

[公式] which is the same as the formula given in L&L Classical Theory of Fields ([1]). Sometime to note here is that [公式], which involve two different coordinates bases certainly does not transform as a tensor. Therefore, the “argument” that if one rearranges [公式] to express [公式] in terms of [公式] , one would get a negative sign for the second order term that makes the transformation formula ill-defined is clearly invalid.

When the coordinate change is linear or the partial derivatives of second order associated with it are all zero, [公式] does transform like a tensor. [公式] , which cancels out the second term visibly transforms like a tensor as well, and this is called the “curvature tensor” of space.

By the equivalence principle, there must be at the given point a “galilean” coordinate system at each in point in which the Christoffel symbols and thus also [公式] are zero. Since [公式] transforms as a tensor, it must be zero in any coordinate system if it is in some specific coordinate system. This shows that the Christoffel symbols are symmetric with respect to the lower indices. As for this final paragraph, it relates the physics or more specifically the general relativity to the math. I will, hopefully, write in more detail about the physics of this once I gain a deeper understanding of it. In any case, I believe that the mathematics and differential geometry behind the covariant derivative I have explained quite thoroughly and intuitively in this very article.

References