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**Additional info for Analysis and partial differential equations. Dedicated to Mischa Cotlar**

**Example text**

Much more astonishing, one also sees the bottom of the gaseous disk. In fact, the light rays that normally propagate downwards, in a direction opposite to that of the observer, climb back to the top and provide a secondary image, a highly deformed picture of the bottom of the disk. In theory, there is a tertiary image that gives an extremely distorted view of the top after the light rays have completed three half-turns, then an image of order four that gives a view of the bottom that is even more squashed, and so on to infinity.

The distance between two arbitrary points that are sufficiently distant 42 I. The Shape of Space ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ← to escape particular movements of non-cosmological character grows with time. Let us imagine some microbes distributed over the surface of a balloon that is inflating: each microbe might notice that its neighbors move away from it; it would thus have the impression, but only the impression, of being at the center of a universe where all the other galaxies are running away! If these solutions of the relativistic equations naturally take into account the observed expansion of space, it remains to extrapolate the global evolution of the Universe, on the one hand toward the past (every cosmology is thus a cosmogony), and on the other hand toward the future.

Let us return to our elastic fabric. If we dig a sufficiently deep well to simulate a black hole, what will happen at the bottom? Is there a knot that blocks spacetime (what in mathematics is named a singularity, where the curvature becomes infinite) or rather is the material pierced? In the latter case, is the opening a gaping hole, or does it look like a passage to somewhere else? How do these spacetime distortions act on the elastic material? Only the mathematical solutions of general relativity can guide us.