Gödel metric

Example

Use units where the gravitational constant and the speed of light equal $1$. Fix $a > 0$. On $\R^4$ with coordinates $(t, x, y, z)$, the Gödel metric is $$g = a^2\left(-\left(dt + e^x dz\right)^2 + dx^2 + dy^2 + \frac{1}{2} e^{2x} dz^2\right).$$ Check of the signature: the $(t, z)$ block of $g$ has determinant $-\frac12 a^4 e^{2x} < 0$, so it has one negative and one positive direction. With $dx^2$ and $dy^2$ positive, $g$ has signature $(-, +, +, +)$.

Let $u = a\,(dt + e^x dz)$, a unit timelike covector. A computation of the Ricci curvature gives $$R_{ab} = \frac{1}{a^2} u_a u_b, \qquad R = -\frac{1}{a^2}.$$ Hence $G_{ab} + \Lambda g_{ab} = 8\pi\rho\, u_a u_b$ with $$\Lambda = -\frac{1}{2a^2}, \qquad 8\pi\rho = \frac{1}{a^2}.$$ So $g$ solves the Einstein field equations with pressureless dust of constant density $\rho$, moving along $u$, and a negative cosmological constant.

The metric is homogeneous: the isometry group acts transitively on spacetime. The dust rotates: the vorticity of $u$ is nonzero. Through every point there pass closed timelike curves.