Codazzi tensors and their space-times, and Cotton gravity

Abstract

We study the geometric properties of certain Codazzi tensors for their own sake, and for their appearance in the recent theory of Cotton gravity. We prove that a perfect-fluid tensor is Codazzi if and only if the metric is a generalized Stephani universe. A trace condition restricts it to a warped space-time, as proven by Merton and Derdzinski. We also give necessary and sufficient conditions for a space-time to host a current-flow Codazzi tensor. In particular, we study the static and spherically symmetric cases, which include the Nariai and Bertotti-Robinson metrics. The latter are a special case of Yang Pure space-times, together with spatially flat FRW space-times with constant curvature scalar. We apply these results to the recent Cotton gravity by Harada. The equations have the freedom of choosing a Codazzi tensor, that constrains the space-time where the theory is staged. The tensor, chosen in forms significative for physics, implies the form of the Ricci tensor, and the two specify the energy-momentum tensor, which is the source in Cotton gravity for the chosen metric. For example, the Stephani, Nariai and Bertotti-Robinson space-times solve Cotton gravity with physically sensible energy-momentum tensors. Finally, we discuss Cotton gravity in De Sitter space-times.

19 pages, no figures. We added a discussion of Yang Pure spaces, with references. Eq.45 and the Ricci tensor in the proof of Prop.2.3 (this version) are now correct

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Cited by in corpus (18)

• Gravity at cosmological distances: Explaining the accelerating expansion without dark energy
• A note on Harada's Conformal Killing gravity
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• General spherically symmetric solution of Cotton gravity
• Friedmann equations in the Codazzi parametrization of Cotton and extended theories of gravity and the Dark Sector
• The covariant approach to static spacetimes in Einstein and extended gravity theories
• Wave Metrics in the Cotton and Conformal Killing Gravity Theories
• Cosmological constant as an integration constant
• Vanishing of Conserved Charges in Cotton Gravity
• Constrain the linear scalar perturbation theory of Cotton gravity
• Testing Cotton gravity as dark matter substitute with weak lensing
• Schouten-Codazzi Gravity
• The unification of gravity and the spin-1 field
• Black hole solutions in Cotton gravity coupled to nonlinear electrodynamics
• On Pseudo $B$-symmetric spacetimes and $f(\mathcal{R})$ gravity
• Constraints on the $\gamma$-parameter for the vacuum solution of Cotton gravity with geodesics and shadows
• Thermodynamic formulation of Cotton gravity in the Codazzi parametrization
• Generalized Vaidya Spacetime in Cotton and Conformal Killing Theories