The geodesic equation

Guth starts with the simplest case: a two-dimensional space with coordinates $x$ and $y$ and some metric, the kind of space Gauss, Bolyai and Lobachevsky studied. Everything carries over to spacetime unchanged.

A geodesic between two points is a path whose length is stationary: change the path slightly, keeping the endpoints fixed, and the length does not change to first order. In an ordinary space a geodesic is usually the shortest path. In spacetime, where the metric is not positive definite, geodesics can also be maxima or saddle points. The derivation only uses "the first-order change vanishes", so it covers all of these at once.

[Guth suggests that in two-dimensional spaces the stationary paths are always minima. Not quite: on a sphere the long way round a great circle between two points is also stationary, but it is not the shortest path: it is a saddle point.]

Two improvements make the metric easier to handle.

Number the coordinates. Write $x^1, x^2$ instead of $x, y$. The superscripts are labels, not powers — an unfortunate clash, tolerated because general relativity needs both upper and lower indices.

Sum with indices. The metric becomes $\sum_{i=1}^{2}\sum_{j=1}^{2} g_{ij}\,dx^i dx^j$. The $(1,1)$ and $(2,2)$ terms are the old $dx^2$ and $dy^2$ terms; $g_{12}$ and $g_{21}$ are equal and both multiply $dx\,dy$. Then comes Einstein's summation convention: an index that appears twice, once up and once down, is summed automatically, so the sum signs can be dropped.

In four dimensions this one line stands for 16 terms (10 different ones, since $g_{ij} = g_{ji}$).

Describe a path from $A$ to $B$ by two functions $x^i(\lambda)$ of a parameter $\lambda$, which runs from $0$ to $\lambda_f$:

Cut the path into small pieces from $\lambda$ to $\lambda + d\lambda$. Over one piece the coordinates change by $dx^i = (dx^i/d\lambda)\,d\lambda$; putting this into the metric gives

For a space, $ds^2$ is positive, so the square root is the length of the piece. Adding up all pieces:

To test whether a path is a geodesic, compare it with every nearby path between the same endpoints. This is the calculus of variations, which slide 4 dates to 1696. Write a nearby path as

Here $w^i(\lambda)$ is any ordinary function (with values like 2 or 5) and $\alpha$ is a small number that controls how far the new path strays. The conditions on $w^i$ keep the endpoints fixed — otherwise you could always find a shorter path by moving them. The path $x^i(\lambda)$ is a geodesic if

Name the integrand of the varied path's length:

$\alpha$ appears in three places. The metric does not depend on $\alpha$ directly, but it is evaluated at $\tilde x$, which does; by the chain rule

and each $d\tilde x^i/d\lambda$ gives $dw^i/d\lambda$. The derivative of a square root brings $1/(2\sqrt A)$. After differentiating, everything else is set to $\alpha = 0$, so the tildes disappear. The two terms with $dw/d\lambda$ are equal once the summed ("dummy") indices $i$ and $j$ are renamed, using $g_{ij} = g_{ji}$, which gives a factor 2:

The first term is proportional to $w$, the second to its derivative, so it is not yet clear when the sum vanishes for every $w$. Guth calls this the only real difficulty in the derivation; the fix is to integrate the second term by parts, $\int u\,dv = -\int v\,du + [uv]$, with

• $u = \dfrac{1}{\sqrt A}\,g_{ij}\,\dfrac{dx^j}{d\lambda}$ and $dv = \dfrac{dw^i}{d\lambda}\,d\lambda$, so $v = w^i$.

The boundary term $[uv]$ contains $w^i$ at the two endpoints, where it vanishes: no boundary term survives. Now $w$ appears undifferentiated in both terms. Renaming dummy indices so it carries the same index $i$ in each, it can be taken outside:

This must vanish for every $w^i$. One integral can be zero by luck — positive in some places, negative in others. But if the expression in curly brackets were not zero, we could choose $w^i$ equal to it; the integrand would then be a perfect square and the integral could not vanish. So the brackets must vanish identically:

There is one equation for each value of $i$. Both sides depend on the metric and on the path, through $dx/d\lambda$. A continuous path is a geodesic if and only if it satisfies this equation. Lecture 14 simplifies it and tries it on examples.

• A geodesic is a path of stationary length between fixed endpoints: usually a minimum in space, possibly a maximum or saddle in spacetime.
• Index notation: $ds^2 = g_{ij}\,dx^i dx^j$, summed over repeated upper–lower indices (Einstein convention).
• A path $x^i(\lambda)$ has length $S = \int\sqrt{g_{ij}\,\frac{dx^i}{d\lambda}\frac{dx^j}{d\lambda}}\,d\lambda$. Compare it with $x^i + \alpha w^i$, $w^i = 0$ at the ends, and demand $dS/d\alpha = 0$ at $\alpha = 0$ for all $w^i$.
• Integration by parts, with no boundary term, leaves $\int\{\cdots\}\,w^i\,d\lambda = 0$ for all $w^i$, so $\{\cdots\} = 0$: the geodesic equation $\frac{d}{d\lambda}\left[\frac{1}{\sqrt A}g_{ij}\frac{dx^j}{d\lambda}\right] = \frac{1}{2\sqrt A}\,\partial_i g_{jk}\,\frac{dx^j}{d\lambda}\frac{dx^k}{d\lambda}$.