Fluid interpretation, Hawking--Ellis classification, and energy conditions of the proper kinetic gravity braiding stress tensor

Abstract

We investigate the fluid interpretation, Hawking--Ellis classification, and energy conditions of the proper kinetic gravity braiding contribution to the stress tensor of minimally coupled scalar fields for timelike, spacelike, and open-region null scalar gradients. Using a $2+1+1$ decomposition adapted to the gradient, we derive the associated effective fluid variables. Unlike k-essence, braiding generates heat fluxes and pressure anisotropies: radial and tangential heat fluxes for timelike gradients, and a radial heat flux plus mixed radial--tangential anisotropies for spacelike gradients. These quantities depend only on the normal fundamental scalars and the two-dimensional accelerations, despite the larger set of variables appearing at intermediate stages. For open-region null gradients, the proper-braiding stress tensor has null-dust form. We obtain the complete Hawking--Ellis classification: the timelike sector is Type I for positive discriminant, Type II on the nontrivial discriminant hypersurface, and Type IV for negative discriminant; the spacelike sector has the same generic branches and additional Type II and Type III degeneracies when the squared radial heat flux equals the squared mixed anisotropy. The open-region null sector is Type II for nonzero null-dust density and vanishes in the zero-density case. Finally, the standard energy conditions further restrict the Type I and generic Type II sectors. The null energy condition excludes Types III and IV and, for spacelike gradients, every nonzero radial heat flux or mixed anisotropy. Thus admissible spacelike proper braiding is diagonal, whereas in the timelike sector the energy conditions bound the total heat flux.