Abstract
An integral kernel representation for the commutative star -product on curved classical spacetime is introduced. Its convergence conditions and relationship to a Drin’feld differential twist are established. A star -Einstein field equation can be obtained, provided the matter-based twist’s vector generators are fixed to self-consistent values during the variation in order to maintain star -associativity. Variations not of this type are non-viable as classical field theories. star -Gauge theory on such a spacetime can be developed using star -Ehresmann connections. While the theory preserves Lorentz invariance and background independence, the standard ADM 3+1 decomposition of 4-diffs in general relativity breaks down, leading to different star -constraints. No photon or graviton ghosts are found on star -Minkowski spacetime. star -Friedmann equations are derived and solved for star -FLRW cosmologies. Big Bang Nucleosynthesis restricts expressions for the twist generators. Allowed generators can be constructed which account for dark matter as arising from a twist producing non-standard model matter field. The theory also provides a robust qualitative explanation for the matter-antimatter asymmetry of the observable Universe. Particle exchange quantum statistics encounters thresholded modifications due to violations of the cluster decomposition principle on the nonlocality length scale sim 10^{3-5} ,L_P. Precision Hughes–Drever measurements of spacetime anisotropy appear as the most promising experimental route to test deformed general relativity.
Highlights
Λ is a dimensionful real constant, θ AB is a real symmetric numerical matrix, and the X A are a set of smooth vector fields called twist generators labelled by A
While X is constructed from GM matter, it plays a nonstandard role in determining geometry, aside from GM matter fields entering the stress-energy tensor in the usual way
Where are we? What has been learned about deformed general relativity (DGR), what remains obscure? DGR has been outfitted with a curved spacetime compatible integral kernel, whose convergence and equivalence to an extended Drin’feld differential twist (DDT) have been clearly delineated
Summary
Λ is a dimensionful real constant, θ AB is a real symmetric numerical matrix, and the X A are a set of smooth vector fields called twist generators labelled by A. These expressions for the twist generator themselves contain the -product and not the standard one because the DDT (1) will insert arbitrary powers of X into the action, and for the action to be -diff invariant means a composite object like X must incorporate. Section 4: Simple dynamics and the -Einstein equation Section 5: -gauge theory: -fiber bundles, -Bianchi identity, -holonomies Section 6: Photon and graviton -ghosts banished from Minkowski spacetime Section 7: Canonical structure of DGR: No ADM (3+1) decomposition of -4-diffs, classical non-viability of δ X = 0 variations, the self-consistency process (deformation flow) Section 8: -FLRW cosmologies, Big Bang nucleosynthesis constraints on X and new generators that satisfy them, Page 3 of 26 786. Section 9: Experimental tests of DGR: violations of the cluster decomposition principle, spin-statistics and CPT theorems, and Pauli exclusion principle; Michelson–Morley and Hughes–Drever measurements
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