Charting Space-Time

Einstein's field equations describe the presence of matter in terms of an energy-momentum-stress tensor (the `source' of the field). In the absence of matter, the form of the equations is simplified by having zero source. One tantalising possibility, related to Hadley's paper, is that there may be solutions to the free-field equations in which a background roughly-flat portion of space-time contains a `tubular' structure - the trajectory of some `particle' - inside which some non-trivial piece of topological structure allows the free-field equation to be satisfied in a way which (as seen by the surrounding flatish space) looks like the tube simply contains a flatish piece of space with some source in it.

Research idea: implement a chart-structured model of a smooth manifold, building up an atlas from some given charts (the boundary conditions) and extrapolation using Einstein's field equation for gravity. Make sure the implementation is happy describing a topologically non-trivial manifold, by suitable stitching-together of chart fragments. Take Kerr's general form of the black hole (with the mass, spin and charge of some familiar particle, like an electron), use what it says far from the hole to create a population of charts for the boundary: interpolate inwards to see what happens within. Explore ways of getting a topologically non-trivial interior which solves Einstein's free-field equations. Tolerate minor deviations from exactly matching Kerr's solution at large radius, but pay attention to them ... alternatively, allow there to be some pulsation going on.

Each chart provides a systematic description of some region of space-time. It also overlaps with other charts: each overlap carries the information the two charts need to transcribe information held by one chart into the other chart's terms. The field equations tell us how to propagate the curvature tensor and metric of space-time: the charts record what we know and provide the computational machinery for performing tensor algebra in the course of extrapolation. If we use some topologically non-trivial structure, we'll be describing it by several patches (since each chart is topologically simple): but if we can invent some more concise description of the non-trivial structure as a single chart, we can use this to provide a simpler description which behaves operationally like a chart (i.e. it knows how to talk to its neighbours about their overlaps). Remember that the field equations are non-linear, so the exact scale may matter: the electron's mass, measured in Planck masses, is of order 10-22, whereas its charge, measured in kindred units, is of order a twelfth (give or take various stray factors of 2 and π which may appear in one's definitions of the units).

Be prepared for the available topological structures to force the `quantisation' of charge, spin and mass: examine solutions with the mass, spin and charge of familiar particles, and see what happens with larger or smaller values - why don't we see other combinations ? Start with electron, proton and neutron: bear in mind the eventual study of nuclei, but focus on the simple cases first !

The natural boundary condition to work with is some large-radius `cylinder' about the trajectory of the `particle', with its outer face's character determined by Kerr and its `end-plate' determined by a cyclic boundary condition (since we don't know what's on it but hope to be able to infer this).

Written by Eddy.
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