Example of a non-Minkowski empty space solution to the equations of general relativity I read in Brian Greene's book The Fabric of the Cosmos that there are nontrivial (that is, non-Minkowski) empty space solutions to the equations of general relativity. Can someone give me an explicit example of such a solution?
 A: In general relativity, the Einstein equations $G_{\mu\nu}=8\pi GT_{\mu\nu}$ relate the curvature to the matter contents of spacetime, via the stress-energy tensor $T_{\mu\nu}$.  (Following modern conventions, this counts the cosmological constant $\Lambda$ as a form of matter content—the vacuum or "dark" energy.)  So the Einstein tensor $G_{\mu\nu}$ is determined by the configuration of matter in the universe.  However, $G_{\mu\nu}$ does not completely determine the curvature of the spacetime maniform.
The spacetime curvature is described by the four-index Riemann tensor $R_{\mu\nu\rho\sigma}$, which has twenty independent components for a four-dimensional manifold.  The Einstein tensor is related only to a subset of these components,
$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},$$
where the two-index Ricci tensor and the zero-index Ricci scalar are formed by
contracting indices of the underlying Riemann tensor, $R_{\mu\nu}=R^{\alpha}\,_{\mu\alpha\nu}$ and $R=R^{\alpha}\,_{\alpha}$. Since the Riemann tensor satisfies $R_{\mu\nu\rho\sigma}=R_{\rho\sigma\mu\nu}$, the Ricci tensor is symmetric, $R_{\mu\nu}=R_{\nu\mu}$, and a symmetric two-index tensor in four dimension has (like a $4\times4$ matrix) ten independent components. (And obviously, the Ricci scalar then has only a single component.)  The remaining ten degrees of freedom in the Riemann tensor can be collected into something called the Weyl conformal tensor, $C_{\mu\nu\rho\sigma}$.
Because the Einstein equations only invovle the Ricci part of the curvature, there are ten curvature parameters that have nothing to do with the matter content of spacetime.  This means that there are lots of solutions that are locally empty of matter (meaning no Ricci curvature) but still possess important Weyl curvature.  The classical Schwarzschild and Kerr black hole solutions are empty of matter outside the event horizons, $G_{\mu\nu}=0$, but there is still obviously curvature.  The black hole solutions are not vacuum solutions everywhere, of course; they have matter in the black hole interiors.  However, there are also solutions that are completely empty of matter everywhere, yet which still have nonvanishing curvature components.
Let me mention two important types of such nontrivial $T_{\mu\nu}=0$ solutions.  The first is a class of cosmological solutions.  The Milne model describes a homogeneous and isotropic expanding universe with curvature.  As such, it represents a special kind of Friedmann–Lemaître–Robertson–Walker metric (FLRW) solution.  In the FLRW spacetimes, different types of matter content contribute differently to the expansion parameters, and within this framework, the curvature actually appears to behave just as if it were another kind of matter source.  The metric in this spacetime is
$$ds^{2}=dt^{2}-t^{2}\left\{\left[d(\sinh^{-1}r)\right]^{2}+r^{2}\,d\Omega^{2}\right\},$$
using the $(+,-,-,-)$ signature convention.  It is easy to see from the spatial line element $d\Sigma^{2}=t^{2}\left\{\left[d(\sinh^{-1}r)\right]^{2}+r^{2}d\Omega^{2}\right\}$ that the spatial slices with constant $t$ are curved; however, the Einstein tensor still vanishes everywhere in space.
The second important class of nontrivial solutions with no matter content are solutions that involve only gravitational waves. A spacetime that is empty of energy and matter, with $T_{\mu\nu}=0$, may nonetheless have have propagating and interacting gravitational waves present.  Again, it is possible to formulate the theory so that the gravitation radiation looks like a matter source; however, under the strictly defined $T_{\mu\nu}$, the gravitation waves do not contribute to that tensor.  One of the simplest models with graviational radiation in an otherwise empty spacetime is the Ozsváth–Schücking model, which has only a single plane gravitational wave propagating uniformly through space.  The (appropriately scaled) metric is
$$ds^{2}=dt^{2}-4\zeta\,dt\,d\xi-2\,d\xi\,d\eta-2\zeta^{2}\,d\eta^{2}-d\zeta^{2}.$$
Direct calculation of the curvature confirms that this metric satisfies $R_{\mu\nu}=0$
(making in "Ricci flat"), but it is not conformally flat, so $C_{\mu\nu\rho\sigma}\neq0$.
