Chern-Simons modified gravity and closed timelike curves

We verify the consistency of the Gödel-type solutions within the four-dimensional Chern-Simons modified gravity with the nondynamical Chern-Simons coefficient for different forms of matter including dust, fluid, scalar field, and electromagnetic field and discuss the related causality issues. We show that, unlike general relativity, a vacuum solution is possible in our theory. Another essentially new result of our theory having no analogue in general relativity consists in the existence of the hyperbolic causal solutions for a physically well-motivated matter.

Figure 1

CS solution $\omega (k)$ and $m^2(k)$ for dust with $\rho =1$.

Figure 2

The $w$ and $m^2$, for $e=1$