| Let's consider the nonvanishing Weyl tensor case. Instead of a dustball or a spherical glob of loose coffee grounds, drop a very loose "coarse dust" of extremely precise but low-mass radar-equipped clocks onto a relatively low-radius large mass, non-rotating (or very slowly rotating), like a small planet. Clocks at precisely the same altitude will report the same time and the same phase, but clocks at different altitudes will report differences (lower = slower); this is a metric effect, and can be represented using nothing but gravitational time dilation. However, the upper clocks have much further to travel in spacetime than the lower clocks (assuming the clocks are collectively moving non-relativistically, it takes many nanoseconds for them to fall a light-nanosecond closer to the planet), the North/South/East/West radar distances between any pair of clocks all decrease but the Up/Down radar distances all increase between any pair of clocks not at the same altitude. This stretch-squash increases with proximity to the massive object the clocks are falling towards. Additionally, in general, free-falling objects have a vanishing Ricci curvature, so the volume of the boundary around the stretch-squashed cloud of clocks remains constant, and this can be confirmed by radar within the cloud as well as by external observers. As we increase the density of the object that the cloud of clocks is falling toward, but keeping the object's mass constant, these altitude-dependent results become more stark. While one might be tempted to think that some sort time distortion alone can account for the radar distance effects, if the clocks are large enough and the focusing of their geodesics strong enough, the clocks will outright collide. How would you explain that as a time distortion, rather than a spatial distortion? You might recognize this as a gentle case of (animated gif) http://en.wikipedia.org/wiki/File:Spaghettification.gif Indeed, in the extreme case, the internal structure of the clocks themselves might not be strong enough under the squash-strain, and they will break before smashing into the surface. Probably anything that's not a black hole that is dense enough to spaghettify an atomic clock made of metal and plastic would also have sufficiently intense magnetic fields to induce comparable distortions on the clocks' molecules, though. Here's the a rough diagram of the squash-strain distortion of a hydrogen atom in a strong magnetic field (left, B=0; right B > 100 000 Tesla) like one would find near a neutron star, so you can consider a very rough gravitation-magnetism analogy between this image and the animation above: https://gravityandlevity.files.wordpress.com/2015/01/distort... This sort of rough analogy has been explored, tightened, and formalized as gravitoelectromagnetism (GEM). GEM is far from exact, but it is useful in studying the kinetics of small things moving near large bodies. |
