As discussed in the previous section, geodesics are the solutions to a system of second-order differential equations. These equations are not always easy to solve; solving differential equations is an art form. 1) However, dramatic simplifications occur in the presence of symmetries, as we now show.

A vector $\XX$ satisfying \begin{equation} d\XX \cdot d\rr = 0 \label{Killing} \end{equation} is called a Killing vector. 2) The motivation for this definition becomes apparent if one computes \begin{equation} \frac{d}{d\lambda} (\XX\cdot\vv) = \dot\XX\cdot\dot\rr + \XX\cdot\dot\vv \label{const} \end{equation} The second term in this expression vanishes if $\vv$ corresponds to a geodesic, while the first vanishes if $\XX$ is Killing. Thus, each Killing vector yields a constant of the motion along any geodesic, namely $\XX\cdot\vv$. As we will see, these constants of the motion correspond to conserved physical quantities, such as energy and angular momentum.

In what sense do Killing vectors correspond to symmetries? It turns out 3) that if the vector differential can be written in the form \begin{equation} d\rr = \sum\limits_i h_i\,dx^i\,\ee_i \end{equation} and if all of the coefficients $h_i$ are independent of one of the variables, say $x^j$, that is, if \begin{equation} \Partial{h_i}{x^j} = 0 \qquad\qquad\qquad(\forall i) \end{equation} for some $j$, then $\XX=h_j\,\ee_j$ (no sum on $j$) is a Killing vector. Thus, Killing vectors correspond to directions in which $d\rr$ doesn't change.