Einstein was not happy with some of the predictions of his theory, notably, as we will see in the next chapter, that the universe is expanding. He therefore looked for a way to modify his theory to permit a static universe.

There is in fact just one way to do so; the only other divergence-free geometric object (of the correct size, that is, with the correct number of components) is the line element itself! Using the fact that \begin{equation} {*}\sigma^i = \pm \sigma^j\wedge\sigma^k\wedge\sigma^\ell \end{equation} for an appropriate choice of $j$, $k$, $\ell$, direct computation shows that \begin{equation} d{*}d\rr = \zero \end{equation} Thus, a generalization of Einstein's equation is \begin{equation} \GG + \Lambda\,d\rr = 8\pi\Tvec \end{equation} where $\Lambda$ is the cosmological constant. In more traditional tensor language, Einstein's equation with cosmological constant is written as \begin{equation} G_{ij} + \Lambda\,g_{ij} = \frac{8\pi G}{c^2} T_{ij} \end{equation}

As we will discuss in the next chapter, suitable values of $\Lambda$ do indeed permit a static universe. After astronomers later observed that the universe is in fact expanding, Einstein referred to the cosmological constant as the biggest mistake of his career. However, recent observations suggest that the cosmological constant is small but nonzero; perhaps Einstein deserves the last laugh after all.