By 苏剑林 | October 15, 2016
Geodesics
Riemannian metrics should not be difficult to understand. In differential geometry textbooks, we have already studied the "First Fundamental Form" of surfaces. In fact, the two are the same thing; the only difference is the perspective. Differential geometry views a surface as a two-dimensional subset of three-dimensional space, while Riemannian geometry studies geometric problems intrinsically from the perspective of the two-dimensional surface itself.
What does geometry care about? In fact, geometry is concerned with "objective entities" that are independent of transformations (or things that remain invariant under transformations); this is the definition of geometry. According to the "Erlangen Program" proposed by Klein, geometry is the study of properties that remain invariant under a certain group of transformations. If the transformations are limited to rigid transformations (translation, rotation, reflection), it is Euclidean geometry; if the transformations are general linear transformations, it is affine geometry. Riemannian geometry, on the other hand, is concerned with objective entities that are independent of all coordinate systems. For example, if I have a vector with a fixed direction and magnitude, it is $(1, 1)$ in a Cartesian coordinate system and $(\sqrt{2}, \pi/4)$ in a polar coordinate system. Although the components differ in the two coordinate systems, they both refer to the same vector. That is to say, the vector itself is an objectively existing entity, regardless of the coordinates used. From an algebraic perspective, as long as objects can be transformed into one another through a coordinate transformation, we consider them to be the same thing.
Therefore, when learning Riemannian geometry, it is always beneficial to think in the direction of "objective entities."
Geodesics on a plane
With a metric, one can naturally introduce the entity of a "geodesic." In a narrow sense, it is the shortest line between two points—an extension of the concept of a straight-line segment in flat space (actual geodesics are not necessarily the shortest globally, but let's not worry about details for now, as this doesn't hinder our understanding; a geodesic is at least locally shortest). It is not hard to imagine that once two points are fixed, the shortest line between them is determined regardless of the coordinates used. Thus, this is clearly an objective entity. There is a simple analogy: no matter how you transform coordinates, the local extrema of a function $f(x)$'s graph are always fixed—whether you change them or not, they are there, unbiased.
Geodesics on a sphere
Geodesics on an isothermal surface
From a mathematical point of view, the distance between two points $\boldsymbol{x}^1$ and $\boldsymbol{x}^2$ is naturally:
\[s = \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} ds = \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \sqrt{g_{\mu\nu} dx^{\mu} dx^{\nu}} \tag{16} \]
Therefore, a geodesic is found by identifying the function among all functions passing through $\boldsymbol{x}^1$ and $\boldsymbol{x}^2$ that minimizes the above integral. This is a problem of the calculus of variations. Regrettably, many mathematics students have not studied the calculus of variations, but I will still use this approach because it is a very natural line of thought. As we will see later, it also provides a simplified method for calculating connections.
The core idea of variation is quite simple, much like taking a derivative, with the added step of integration by parts. When we find the extremum of a function, we take its derivative and set it to zero; for the extremum of a functional, we take its variation and set the variation to zero. (Refer to the "Natural Extrema" series on this blog).
\[\begin{aligned}
&\delta s\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \delta\sqrt{g_{\mu\nu} dx^{\mu} dx^{\nu}}\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \frac{\delta(g_{\mu\nu} dx^{\mu} dx^{\nu})}{2ds}\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \left(\frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} \delta x^{\alpha} + g_{\mu\nu}\frac{dx^{\mu} }{ds} d \delta x^{\nu}\right)\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} \delta x^{\alpha} + \left.g_{\mu\nu}\frac{dx^{\mu} }{ds} \delta x^{\nu}\right|_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} - \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} d\left(g_{\mu\nu}\frac{dx^{\mu} }{ds}\right) \delta x^{\nu}\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} \delta x^{\alpha} - \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} d\left(g_{\mu\nu}\frac{dx^{\mu} }{ds}\right) \delta x^{\nu}\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \left[\frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} \delta x^{\alpha} - \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\alpha}\delta x^{\nu} - g_{\mu\nu}d\left(\frac{d x^{\mu} }{ds}\right)\delta x^{\nu}\right]\\
=& \int_{\boldsymbol{x}^1}^{\boldsymbol{x}^2} \left[\frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} - \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}\frac{dx^{\mu} }{ds} dx^{\nu} - g_{\mu\alpha}d\left(\frac{d x^{\mu} }{ds}\right)\right]\delta x^{\alpha}
\end{aligned} \tag{17} \]
The term resulting from integration by parts disappears because we specified "from all functions passing through $\boldsymbol{x}^1$ and $\boldsymbol{x}^2$," meaning at the boundaries we have $\delta x^{\nu}(\boldsymbol{x}^1)=\delta x^{\nu}(\boldsymbol{x}^2)=0$. Finally, since $\delta x^{\alpha}$ is arbitrary, for $\delta s=0$ to hold, we must have:
\[\frac{1}{2}\frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}\frac{dx^{\mu} }{ds} dx^{\nu} - \frac{\partial g_{\mu\alpha}}{\partial x^{\nu}}\frac{dx^{\mu} }{ds} dx^{\nu} - g_{\mu\alpha}d\left(\frac{d x^{\mu} }{ds}\right)=0 \tag{18} \]
After minor rearranging, we obtain:
\[\frac{d^2 x^{\mu} }{ds^2}+\Gamma_{\alpha\beta}^{\mu} \frac{d x^{\alpha} }{ds}\frac{d x^{\beta} }{ds}=0 \tag{19} \]
where:
\[\Gamma_{\alpha\beta}^{\mu}=\frac{1}{2}g^{\mu\nu}\left(\frac{\partial g_{\alpha\nu}}{\partial x^{\beta}}+\frac{\partial g_{\nu\beta}}{\partial x^{\alpha}}-\frac{\partial g_{\alpha\beta}}{\partial x^{\nu}}\right) \tag{20} \]
is called the Christoffel symbol (of the second kind), also known as the connection coefficient, a name whose meaning we will soon understand. And $g^{\mu\nu}$ is the inverse of the matrix $g_{\mu\nu}$, i.e.,
\[g^{\mu\alpha}g_{\alpha\nu}=\delta_{\nu}^{\mu} \tag{21} \]
Additionally, the variation of $s$ is equivalent to the variation of the following function $S$ using $s$ as the parameter (refer to the blog post "A Trick in the Calculus of Variations and Its 'Misuse'"):
\[S=\frac{1}{2}\int g_{\mu\nu} \frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}ds \tag{22} \]
Because $S$ has no square root and a simpler form, it can be directly substituted into the Euler-Lagrange equations for calculation, which is sometimes more convenient than directly varying the original $s$.
A Powerful Calculation Tool
Equation $(20)$ already provides the method for calculating the connection coefficients $\Gamma_{\alpha\beta}^{\mu}$. It involves partial derivatives, calculation of inverse matrices, and summation over indices, making it a very complex term. If readers attempt to calculate it, they will feel the pain involved. However, sometimes after very complex calculations, we find that many terms of $\Gamma_{\alpha\beta}^{\mu}$ are zero. That is, the calculation process is complex, but the result is simple. This encourages us to seek a simplified technique.
In fact, we derived the geodesic equation from a variational approach, and this approach itself is a powerful tool for calculating $\Gamma_{\alpha\beta}^{\mu}$, as discussed in Chapter 14, "Calculation of Curvature," of the famous gravity "bible," MTW's Gravitation. (This is provided the result is simple; if the result itself is complex, there is no simplification trick). For example, consider the spherical coordinate case:
\[ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2, \quad x^1 = r, x^2 = \theta, x^3 = \phi \tag{23} \]
This is equivalent to the variation:
\[s = \int \frac{1}{2}\left[\left(\frac{dr}{ds}\right)^2 + r^2 \left(\frac{d\theta}{ds}\right)^2 + r^2 \sin^2\theta \left(\frac{d\phi}{ds}\right)^2\right] ds \tag{24} \]
Using the Euler-Lagrange equations, we can quickly write:
\[\left\{\begin{aligned}&\frac{d^2 r}{ds^2}=r \left(\frac{d\theta}{ds}\right)^2 + r \sin^2\theta \left(\frac{d\phi}{ds}\right)^2\\
&\frac{d}{ds}\left(r^2 \frac{d\theta}{ds}\right)=r^2 \sin\theta \cos\theta \left(\frac{d\phi}{ds}\right)^2\\
&\frac{d}{ds}\left(r^2 \sin^2\theta \frac{d\phi}{ds}\right) = 0
\end{aligned}\right. \tag{25} \]
Rearranging gives:
\[\left\{\begin{aligned}&\frac{d^2 r}{ds^2}=r \left(\frac{d\theta}{ds}\right)^2 + r \sin^2\theta \left(\frac{d\phi}{ds}\right)^2\\
&\frac{d^2\theta}{ds^2}=-\frac{2}{r}\frac{dr}{ds}\frac{d\theta}{ds}+\sin\theta \cos\theta \left(\frac{d\phi}{ds}\right)^2\\
&\frac{d^2\phi}{ds^2} = -\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds}-\frac{2\cos\theta}{\sin\theta}\frac{d\theta}{ds}\frac{d\phi}{ds}
\end{aligned}\right. \tag{26} \]
Comparing with the geodesic equation $(19)$, we obtain:
\[\begin{aligned}&\Gamma_{22}^1 = -r,\quad \Gamma_{33}^1=-r \sin^2\theta\\
&\Gamma_{12}^2=\Gamma_{21}^2=\frac{1}{r},\quad \Gamma_{33}^2=-\sin\theta \cos\theta\\
&\Gamma_{13}^3=\Gamma_{31}^3=\frac{1}{r},\quad \Gamma_{23}^3=\Gamma_{32}^3=\frac{\cos\theta}{\sin\theta}
\end{aligned} \tag{27} \]
All others are zero. As can be seen, if one is proficient in the variational method (which does not require much effort), it helps us quickly identify the connection coefficients without getting bogged down in various index summations.
Of course, in the computer age, few people calculate the various connection coefficients of complex metrics by hand. However, for certain metrics that are not complex, calculating them personally allows for a deeper understanding of them.