$$\require{cancel}$$
1. Example 3 gave the Killing vectors $$\partial_{z}$$ and $$\partial_{\phi}$$ of a cylinder. If we express these instead as two linearly independent Killing vectors that are linear combinations of these two, what is the geometrical interpretation?
1. Show that the metric $$ds^{2} = e^{2gz} dt^{2} - dx^{2} - dy^{2} - dz^{2}$$from section 7.5 has constant values of R = 1/2 and k = 1/4. Note that Maxima’s ctensor package has built-in functions for these; you have to call the lriemann and uriemann before calling them.
2. Similarly, show that the Petrov metric $$ds^{2} = -dr^{2} - e^{-2r} dz^{2} + e^{r} [2 \sin \sqrt{3} r d \phi dt - \cos \sqrt{3} r (d \phi^{2} - dt^{2}]$$has R = 0 and k = 0.
4. Section 7.5 presented the Petrov metric. The purpose of this problem is to verify that the gravitational field it represents does not fall off with distance. For simplicity, let’s restrict our attention to a particle released at an r such that cos $$\sqrt{3}$$r = 1, so that t is the timelike coordinate. Let the particle be released at rest in the sense that initially it has $$\dot{z} = \dot{r} = \dot{\phi} = 0$$, where dots represent differentiation with respect to the particle’s proper time. Show that the magnitude of the proper acceleration is independent of r.