The Gödel universe: Exact geometrical optics and analytical investigations on motion

In this work we derive the analytical solution of the geodesic equations of Gödel’s universe for both particles and light in a special set of coordinates, which reveals the physical properties of this spacetime in a very transparent way. We also recapitulate the equations of isometric transport for points and derive the solution for Gödel’s universe. The equations of isometric transport for vectors are introduced and solved. We utilize these results to transform different classes of curves along Killing vector fields. In particular, we generate nontrivial closed timelike curves from circular closed timelike curves. The results can serve as a starting point for egocentric visualizations in the Gödel universe.

Figure 1

Lightlike geodesics in the ($xy$) subspace depicted in pseudo-Cartesian coordinates (cf. Fig. 2 in 2). Photons emitted at the origin propagate counterclockwise into the future, reach a maximum radial distance of $r_G$, and then return to the origin. It can be shown that there exists no causality violation for arbitrary lightlike or timelike geodesics starting at the origin.

Figure 2

Chronological structure of Gödel’s universe. In this $xyt$ diagram, a possible timelike worldline is depicted. A traveler $\mathbf{T}$ could move on this curve, propagating in his own local future at any given point. Beyond the horizon (gray cylinder), he travels into the past of an observer located at the origin. The worldline itself is a CTC, because the traveler departs from and returns to the origin at the same coordinate time $t$. For an observer at the origin, coordinate time and proper time coincide. The figure illustrates Gödel’s original idea to prove that there exist CTCs through every point in spacetime [1].

Figure 3

Lightlike geodesics starting at the origin. For better orientation, the Gödel horizon and three planar geodesics (compare Fig. 1) are provided in Fig. 3a. The nonplanar geodesics are solid, dashed, or densely dotted curves. The angle between neighboring geodesics is 10°. Figure 3b: Radial coordinate as a function of coordinate time. Nonplanar geodesics do not reach the Gödel radius. Timelike geodesics are of similar shape but reach smaller maximal radial distances from the origin.

Figure 4

Planar lightlike geodesics resulting from the general analytical solution to the geodesic equations. In Fig. 4a, nine geodesics at different initial positions are shown. Position $\mathbf{A}$ is at $r_0=r_G/2$, position $\mathbf{B}$ shows geodesics starting on the horizon $r_G$, and in $\mathbf{C}$ we have $r_0=3r_G/2$. At each starting point, geodesics propagate in the $+{\mathbf{e}}_{(1)}$ or $\pm {\mathbf{e}}_{(2)}$ direction, respectively. Nonplanar geodesics show a behavior similar to the special solution, Fig. 3, i.e. are smaller in radial extent. Figures 4b, 4c, 4d show the radial coordinate of the geodesics as a function of coordinate time.

Figure 5

Time travel on geodesics. Figure 5a shows how far a traveler $\mathbf{T}$ (photon or massive particle) can travel into the past for a given initial radial coordinate $r_0$ (maximum time travel $\Delta t_{\min }$ for fixed $r_0$). Figure 5b explains under which direction the traveling particle has to start, where ${\xi }_0$ denotes the angle to the local ${\mathbf{e}}_{(1)}$ axis in the $\{{\mathbf{e}}_{(1)},{\mathbf{e}}_{(2)}\}$ subspace. Starting in another direction can also result in time travel, but results in $\Delta t>\Delta T_{\min }$. Note that the time axis in Fig. 5a is magnified by a factor of 20 for $\Delta t<0$.

Figure 6

Testing if causality can be violated on geodesics. A traveler $\mathbf{T}$ moves partially beyond the horizon of an observer $\mathbf{O}$ [Fig. 6a]. He leaves the horizon passing observer $\mathbf{A}$ and reenters it passing $\mathbf{B}$. The observer $\mathbf{B}$ sends a light pulse to $\mathbf{O}$, informing him on the arrival of $\mathbf{T}$. Then, $\mathbf{O}$ signals $\mathbf{A}$ this information. If this information arrived before $\mathbf{T}$ passes the horizon, then $\mathbf{B}$ could stop the traveler, resulting in a paradox. Figure 6b reveals that no causality violation arises, because the information arrives in the future light cone of the event “$\mathbf{T}$ passes $\mathbf{A}$”. Note that we use a lightlike geodesic for the path of $\mathbf{T}$ as the limiting case $v\to c$.

Figure 7

Isometrically transporting $\mathbf{A}$ to the spatial origin yields the situation from the point of view of observer $\mathbf{A}$ [Fig. 7a]. Both observers $\mathbf{O}$ and $\mathbf{B}$ are located on the horizon of $\mathbf{A}$. The traveler’s movement is restricted to the interior of this horizon, and no time travel arises. From this perspective, the signal from $\mathbf{B}$ to $\mathbf{O}$ travels back in time with respect to $\mathbf{A}$ [Fig. 7b].

Figure 8

Projection of Killing vector field ${\xi }_1^{\mu }$ onto the $xy$ plane ($t=\mathrm{const}$ and $z=\mathrm{const}$ in Cartesian coordinates). The left half plane shows the vector field itself, and the right half plane depicts the integral curves of finite isometric displacements along ${\xi }_1^{\mu }$. The Gödel horizon appears as the dotted circle.

Figure 9

Isometrically transporting points on the $x$ axis along the Killing vector field ${\xi }_1^{\mu }$, showing the lines of finite isometric transport. Solid lines indicate transport to positive $t$ values, and dotted lines illustrate negative time values. The Gödel radius appears as partially dashed circle.

Figure 10

Three horizons of three different observers $\mathbf{O}$, $\mathbf{A}$, and $\mathbf{B}$. Crosshatched regions mark common causality regions. Note that $\mathbf{O}$ and $\mathbf{B}$ do not share a common region.

Figure 11

Isometrically transporting a circular CTC of radius $R=3r_G/2$ (densely dotted circle) along the Killing vector field ${\xi }_4^{\mu }$ in positive $x$ direction. In Fig. 11a, sparsely dotted curves represent the isometric transport, and each of these dotted curves is a CTC itself. Arrowheads indicate the traveler’s flight direction. The resulting CTC (solid curve) represents a traveler starting at the origin, moving beyond the Gödel horizon—indicated as dashed circle in Fig. 11a or dashed line in Fig. 11b—and returning to the origin at the coordinate time of departure. Figure 11b: Observer time as a function of the radial coordinate of the CTC. Note that coordinate time coincides with the proper time of a resting observer $\mathbf{O}$. Therefore, the traveler $\mathbf{T}$ is moving back in time from the resting observer’s perspective. Also note that $dt/d\lambda >0$ as long as $r<r_G$.

Figure 12

Three-dimensional presentation of Fig. 11a. Isometrically transporting a circular CTC does not only result in a different spatial appearance but affects the time coordinate as well. Dashed or sparsely dotted curve segments are below the $xy$ plane, and solid or densely dotted curve parts have $t={\tau }_{\mathrm{obs}}\ge 0$. The Gödel horizon is shown as solid circle.