Polar modulation

Summary

This figure depicts, in the Born chart, the desynchronization of clocks carried by observers riding on a rigidly rotating disk. The z coordinate is inessential and has been suppressed in the figure.

Consider integral curves of the third spacelike Langevin frame vector

${\overrightarrow{p}}_3=\sqrt{1-{\omega }^2{r}^2}\frac{1}{r}{\partial }_{\varphi }+\frac{\omega r}{\sqrt{1-{\omega }^2{r}^2}}{\partial }_t$

which pass through $\varphi =0,t=0$. As the figure shows, we can use these to form a spatial hyperslice. But two things go wrong.

First, the spatial hyperslice we obtain only contains the second spacelike Langevin frame vector

${\overrightarrow{p}}_2={\partial }_r$

along one radius. By the Frobenius integrability theorem, it is in fact impossible to find a spatial hyperslice to which the unit vector fields ${\overrightarrow{p}}_2,{\overrightarrow{p}}_3$ are tangent. The Frobenius integrability theorem can be viewed in terms of the failure of our two vector fields to form a two-dimensional Lie algebra of vector fields, so this is Lie theoretic obstruction to defining spatial hyperslices for the Langevin observers.

Second, as the figure shows, our spatial hyperslice is not only non-orthogonal to most of our Langevin observers, it also leads to a discontinuity (the coral colored vertical "jump"). If we tried to define a slicing of Minkowski spacetime into a family of spatial hyperslices isometric to this one, we would be forced to allow multiple valued time. This obstruction is global and arises from our inability to synchronize the clocks of Langevin observers riding even a ring, much less a disk (which we can think of as formed from concentric rings).

This figure was created by User:Hillman using Maple to export a jpg image and eog to convert this to a png image.

Licensing

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