Complexity and structure scalars of Type II matter fields

TL;DR

This work extends Herrera's covariant complexity framework to Type II matter fields in generalized Vaidya spacetimes using a 1+1+2 decomposition. It derives covariant structure scalars $X_T$, $X_{TF}$, $Y_T$, $Y_{TF}$ and the complexity factor $Y_{TF}$, and provides propagation/evolution equations and a Gaussian-curvature wave equation that connect these scalars to the Misner-Sharp mass $ ext{M}$ and heat flux $Q$. A key result is the negative complexity of pure Vaidya spacetimes, contrasted with non-trivial vanishing-complexity solutions characterized by specific mass functions $m(v,r)$; the analysis separates gravitation and matter contributions to complexity and highlights clear differences between Type II and Type I fields. The framework offers a covariant basis to study radiating astrophysical systems and their gravitational entropy implications, while outlining limitations for higher-type matter fields (Type III/IV).

Abstract

A general semi-tetrad covariant approach is adopted to analyse the structure scalars of a Type II fluid in generalized Vaidya spacetime. The relationship between the $1+1+2$ covariant quantities and the structure scalars are obtained. We calculate the complexity factor in terms of the Misner-Sharp mass and the matter variables to obtain a non-trivial class of spacetimes with vanishing complexity. Also the Vaidya spacetime with pure Type II matter field has negative complexity. The differences between the complexity of Type I and Type II matter fields are highlighted. We compute the propagation and evolution equations of the structure scalars, showcasing their interdependency through the kinematical variables. The causal wave equation of the Gaussian curvature of the 2-shell and its dependence on the structure scalars are also studied.