From Osher Doctorow
When one treis to frame a Riemannian 4-dimensional manifold in a
Kahlerian structure, in spacetime only BR fulfills the requirement
where BR is a certain solution of the Einstein-Maxwell equations
correspond to a uniform electromagnetic field. The BR solution
involves combining 2 out of 3 fundamental quadrics. The quadrics come
from Beltrami's 1868 paper on surfaces with constant negative Gaussian
curvature realized in a disk on the plane in the more recent indefinite
metrics which have optimal properties. The fundamental quadric is
introduced in a 3-dimensional flat space R^3 with the same signature as
the R^3 metric, yielding surface types with constant Gaussian
curvature: sphere, hyperboloid of 1 sheet, hyperboloid of 2 sheets (the
latter is isomorphic to Beltrami's disk in which latter non-Euclidean
geometry is realized in the plane). The hyperboloid is extended to a
4-dimensional metric with the same signature as spacetime and the
2-sheet case then describes relativistic kinematics of free particles,
while the 1-sheet case is de Sitter cosmology model.
The BR metric is key to hyperbolic geometry in SR, De Sitter's
universe, the physical interpretation of the topological product of 2
fundamental quadrics in Maxwell-Einstein theory that leads to the
Bertotti-Robinson solution and the so-called Already Unified Theory,
the near-horizon limit of the Reissner-Nordstrom black hole, dilatonic
models in which electromagnetism and the dilaton are coupled with
gravity through a 4-dimensional action with both the EM field Fuv and
the scalar field phi acting as sources for gravity, etc.
Osher Doctorow
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