Interpretation of the Weyl tensor

According to folklore in general relativity, the Weyl tensor can be decomposed into parts corresponding to Newton-like, incoming and outgoing wavelike field components. It is shown here that this one-to-one correspondence does not hold for space-time geometries with cylindrical isometries. This is done by investigating some well-known exact solutions of Einstein’s field equations with whole-cylindrical symmetry, for which the physical interpretation is very clear, but for which the standard Weyl interpretation would give contradictory results. For planar or spherical geometries, however, the standard interpretation works for both static and dynamical space-times. It is argued that one reason for the failure in the cylindrical case is that for waves spreading in two spatial dimensions there is no local criterion to distinguish incoming and outgoing waves already at the linear level. It turns out that Thorne’s local energy notion, subject to certain qualifications, provides an efficient diagnostic tool to extract the proper physical interpretation of the space-time geometry in the case of cylindrical configurations.

Figure 1

Plots of the metric function $\alpha$ and all the independent components of the Weyl tensor for the outgoing Einstein-Rosen wave solution (57). The solid black lines correspond to $t=0$, whereas the dotted and dashed lines are evaluated at times $t=1/\omega$ and $t=2/\omega$, respectively. Evidently, all functions behave like waves traveling outwards. All the terms have different falloff behavior with $r$, ${\Omega }^{(\mathrm{out})}$ being the only component with the same behavior $\propto 1/\sqrt{r}$ as a linear cylindrical wave, but they are all nonvanishing.

Figure 2

Plots of the nonvanishing components of the C-energy flux vector for the outgoing Einstein-Rosen wave solution (57). The solid black lines correspond to $t=0$, whereas the dotted and dashed lines are evaluated at times $t=1/(2\omega )$ and $t=1/\omega$, respectively. The gray lines are the corresponding time-averaged quantities, which (asymptotically) fall off like $1/r$.