Dynamical systems approach to stellar modelling in $f(G, B)$ gravity

Abstract

The novel proposal to invoke the split of the Ricci scalar into bulk and boundary terms in the gravitational action, opens up a new avenue of investigation into stellar dynamics. The Lagrangian contains functional forms of the bulk while the boundary terms do not contribute to the dynamics. The advantage of the proposition is that the stellar structure equations are up to order two thus the theory is not haunted by ghosts. We obtain explicitly the defining equations for the thermodynamical variables and the geometry for the pure quadratic case since the linear case amounts to general relativity. In trying to establish the vacuum geometry associated with the theory it turns out that two possible metrics emerge through the vanishing of the energy-momentum tensor. Next we analyse the isotropy equation and make the observation that it is autonomous. It is rare that this happens in astrophysical modelling. This behaviour prompted the use of dynamical systems to understand the stability properties of fixed points or fixed manifolds. It was necessary to choose a gauge in order to split the autonomous equation into a system from which we could plot a phase portrait and deduce the stability of solution trajectories. We find that the fixed curves were generally stable with nearby paths approaching the fixed curves.

Figures (1)

• Figure 1: Streamlines of $(\dot U,\dot V)$ overlaid with fixed curves $U=0$ (red), $U+2V=0$ (black), and $Q=0$ (blue).