Spacetime encodings. II. Pictures of integrability

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a 2 degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion, and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers’ intuition about how integrability arises, and to summarize some of the known facts about 2 degree of freedom systems. Evidence is given, in the form of an orbit-crossing structure, that geodesics in SAV spacetimes might admit a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter’s fourth constant is quadratic).

Figure 1

Constant $J$ potential surfaces for a geodesic in the Schwarzschild metric with $E=0.95$, $L_z=3$, $\mu =1$. Contour Spacing 0.5. The explicit form of the metric functions can be found in Appendix .

Figure 2

Configuration $(\rho ,z)$ space depiction of a geodesic orbit for the potential $J$ shown in Fig. 1 (Schwarzschild metric with $E=0.95$, $L_z=3$, $\mu =1$).

Figure 3

Three-dimensional energy space depiction of the torus of the orbit displayed in Fig. 2. The orbit is depicted in cyan lines. The surfaces depicted correspond to the surfaces in phase space at which the orbit in configuration space reaches an extremum in curvature.

Figure 4

An orbit in the Zipoy-Voorhees metric with $\delta =2$. Orbital parameters are $E=0.95$, $L=3$, $\mu =1$. The black dots and blue stars indicate extrema in orbital curvature.

Figure 5

Metric components of the Manko-Novikov spacetime. Thick black contours indicate zero values, red contours or regions marked by $>0$ admit positive values, and blue contours or $<0$, negative values.

Figure 6

Two orbits in the Manko-Novikov spacetime. The metric functions for this spacetime are given at the end of Appendix , and the following parameters were used: $\alpha =0.626789$, ${\alpha }_2=11.4708$, and $k=(1-{\alpha }^2)/(1+{\alpha }^2)$. The orbital parameters associated with the orbit in the figure are $E=9.5$, $L=-3$, and $\mu =1$.

Figure 7

Points of extreme orbital curvature in Schwarzschild $E=0.95$, $L=3$, $\mu =1$. Black dots correspond to $n=\pm 1$ surfaces. Blue stars have $n=0$, 2 in Eq. (a3).

Figure 8

The four branches on which solution points with extremal curvature can lie ${\theta }_c$ phase $E=0.95$, $L=3$, $\mu =-1$.

Figure 9

Lines of extreme orbital curvature for several orbits of a given $J$. Schwarzschild $E=0.95$, $L_z=3$, $\mu =1$.

Figure 10

Properties of the Manko-Novikov spacetime.