Gödel and Rotating Homogeneous Spacetimes

Gödel spacetime is an exact homogeneous universe filled with rigidly rotating dust and a negative cosmological constant. Its curvature and matter variables are constant, yet sufficiently large azimuthal circles become timelike. Homogeneity then carries a closed timelike curve through every event. The example shows that regular local invariants and exact matter support do not guarantee causal well-behavior.

Required background. Einstein’s Equation supplies the dust and cosmological terms; Continuum Kinematics and Balance Laws supplies vorticity, expansion, and shear.

Helpful background. The Riemann Curvature Tensor supplies invariant checks; Exact Solutions of Einstein’s Equations supplies the homogeneous-family context; Killing Fields and Isometry Groups supplies the transitive isometry action.

Metric and matter support

One standard coordinate form is

$$ds^2=a^2\left[ -(dt+e^xdy)^2+dx^2+\frac12e^{2x}dy^2+dz^2 \right],$$

with $a>0$. The dust four-velocity is

$$u^a=a^{-1}(\partial_t)^a,$$

which is unit timelike despite the off-diagonal $dt\,dy$ term. Direct substitution into

$$G_{ab}+\Lambda g_{ab}=8\pi G\rho\,u_au_b$$

gives

$$8\pi G\rho=\frac1{a^2}, \qquad \Lambda=-\frac1{2a^2}.$$

These relations fix the normalization: rescaling coordinates or writing the metric with a rotation parameter changes the displayed coefficients. Gödel’s original solution and its physical interpretation appear in Gödel 1949, pp. 447–450.

Tracing the field equation independently gives

$$R=4\Lambda+8\pi G\rho=-\frac1{a^2}.$$

Indeed the Ricci tensor in this convention is $R_{ab}=a^{-2}u_au_b$, so $R_{ab}R^{ab}=a^{-4}$. Both scalars are constant. Their signs and powers provide a quick translation check against sources using the opposite Riemann convention or a different overall metric scale.

The dust congruence is geodesic, expansion-free, and shear-free, but has nonzero constant vorticity. A convenient rotation parameter $\Omega$ satisfies $a^2=1/(2\Omega^2)$ in a common convention. Rotation is therefore not coordinate dragging added to an FLRW metric; it is an invariant part of the fluid kinematics and Weyl curvature.

Circular causal test

Introduce cylindrical coordinates adapted to the homogeneous rotation so that

$$ds^2=-\big[dt+H(r)d\phi\big]^2 +D^2(r)d\phi^2+dr^2+dz^2,$$

where

$$H(r)=\frac{4\Omega}{m^2} \sinh^2\!\left(\frac{mr}{2}\right), \qquad D(r)=\frac1m\sinh(mr), \qquad m^2=2\Omega^2.$$

For a circle at fixed $t,r,z$, the squared line element is

$$ds^2=g_{\phi\phi}d\phi^2, \qquad g_{\phi\phi}=D^2-H^2.$$

Writing $s=\sinh(mr/2)$ gives

$$g_{\phi\phi}=\frac{4s^2}{m^2}(1-s^2).$$

Thus the azimuthal circle is null at

$$\sinh\!\left(\frac{mr_c}{2}\right)=1$$

and timelike for $r>r_c$. Because $\phi$ is periodic, those timelike circles are closed. They are not created by periodically identifying the time coordinate.

Why every event lies on a closed timelike curve

The displayed circles surround the coordinate origin, so they do not visibly pass through an arbitrary inner event. Gödel spacetime is homogeneous: its isometry group acts transitively. Choose one closed timelike curve and one event on it; an isometry can map that event to any prescribed event while preserving timelikeness and closedness. The image curve is therefore a closed timelike curve through the target event.

The threshold $r_c$ is not a curvature singularity or material boundary. Scalar polynomial curvature invariants are constant, as expected in a homogeneous geometry. The sign change of the norm of the periodic axial generator is instead a causal diagnostic. General causal hierarchy and chronology protection belong to global-structure analysis; this page establishes the exact family-specific curve.

Coordinate forms can obscure the result. The correct check is to identify a genuinely periodic orbit of a Killing generator and compute its invariant norm. A negative $g_{\phi\phi}$ matters only because the corresponding $\phi$ orbits close; an off-diagonal time term alone proves nothing.

The dust density is positive, but the spacetime is not globally hyperbolic and admits no global time function. Energy conditions and causal hierarchy are logically distinct: satisfying a standard local matter inequality cannot rule out global closed timelike curves.

Exercise

Derive the critical radius from the functions $H$ and $D$ above.

Solution

Use $\sinh(mr)=2sc$ with $c^2=1+s^2$. Then $D^2=4s^2c^2/m^2$ and, since $m^2=2\Omega^2$, $H^2=8s^4/m^2$. Their difference is $4s^2(1-s^2)/m^2$. It vanishes at $s=0$ on the axis and again at $s=1$; beyond the second zero it is negative, making the closed azimuthal orbit timelike.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models.” In Theoretical and Observational Cosmology, edited by Marc Lachièze-Rey, 1–116. Dordrecht: Springer, 1999. DOI; Open PDF.
• Gödel, Kurt. “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation.” Reviews of Modern Physics 21 (1949): 447–450. DOI.