Second-order hyperbolic Fuchsian systems: Asymptotic behavior of geodesics in Gowdy spacetimes

Recent work by the authors led to the development of a mathematical theory dealing with “second-order hyperbolic Fuchsian systems,” as we call them. In the present paper, we adopt a physical standpoint and discuss the implications of this theory which provides one with a new tool to tackle the Einstein equations of general relativity (under certain symmetry assumptions). Specifically, we formulate the “Fuchsian singular initial value problem” and apply our general analysis to the broad class of vacuum Gowdy spacetimes with spatial toroidal topology. Our main focus is on providing a detailed description of the asymptotic geometry near the initial singularity of these inhomogeneous cosmological spacetimes and, especially, analyzing the asymptotic behavior of timelike geodesics—which represent the trajectories of freely falling observers—and null geodesics. In particular, we numerically construct Gowdy spacetimes which contain a black hole–like region together with a flat Minkowski-like region. By using the Fuchsian technique, we investigate the effect of the gravitational interaction between these two regions and we study the unexpected behavior of geodesic trajectories within the intermediate part of the spacetime limited by these two regions.

Figure 1

Illustration of the approximation scheme.

Figure 2

Orthogonal geodesics in the homogeneous polarized case for $k=0.5$, $k=1.8$.

Figure 3

Schematic diagram for the solution considered here.

Figure 4

Contour plot of the absolute value of the Kretschmann scalar.

Figure 5

Causal geodesics from the intermediate point.

Figure 6

Contour plot of $a^x$ as defined in the text.