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Projective General Relativity

Michael Connolly; University of Iowa

Projective Special Relativity studies conic sections of regular quadrics in flat spacetime tangent spaces which are adapted to accommodate points at infinity. Projective General Relativity permits a dynamical determination of a fundamental quadric constrained by a warped “picture plane” varying in the projective parameter. This setting is nicely described by the nonlinear realization of the projective linear group with center restored. The nonlinear Cartan connection of the associated homogeneous space contains all the information necessary to describe the spacetime geometry: A rotational spacetime connection, a solder form, a central dilation potential, and a symmetric bilinear. The symmetric bilinear is identified with the Diffeomorphism field of Thomas-Whitehead (TW) theory and is used to construct general quadrics. The development of a curve in spacetime, lifted to the curved projective spaces yields the general projective geodesic equation. This system is shown to contain a Riccati equation in the warped picture plane with general quadric as potential. Setting the central connection to pure gauge reduces the set of equations to TW theory. The focus of this presentation is the visual representation of these ideas.