Junction conditions and sharp gradients in generalized coupling theories

In this article, we develop the formalism for singular hypersurfaces and junction conditions in generalized coupling theories using a variational approach. We then employ this formalism to examine the behavior of sharp matter density gradients in generalized coupling theories. We find that such gradients do not necessarily lead to the pathologies present in other theories of gravity with auxiliary fields. A detailed example, based on a simple instance of a generalized coupling theory called the MEMe model, is also provided. In the static case, we show that sharp boundaries do not generate singularities in the dynamical frame despite the presence of an auxiliary field. Instead, in the case of a collapsing spherical density distribution with a general profile an additional force compresses over-densities and expands underdensities. These results can also be used to deduce additional constraints on the parameter of this model.

Figure 1

Density profile given by Eq. (100) for the parameters $R_0=10$, $\sigma =1/10$, and $m_0=1$, with ${\rho }_0=4\pi m_0/R_0^3$. The region near $R_0=10$ is plotted here, since away from $R_0$, $\rho$ is approximately constant.

Figure 2

Mass function for the parameters $R_0=10$, $\sigma =1/10$, and $m_0=1$. (a) is the profile in the domain [0, 20] and (b) is the profile in the domain [9.5, 10.5].

Figure 3

Pressure for the parameters $R_0=10$, $\sigma =1/10$, and $m_0=1$. (a) is the profile in the domain [0, 20] and (b) is the profile in the domain [9.5, 10.5].

Figure 4

Differences between Jordan and Einstein frame density profiles for the parameters $R_0=10$, $\sigma =1/10$, $m_0=1$, and $q=-2\pi /3$, plotted with respect to the Jordan frame areal radius $\mathfrak{r}$. (a) is the profile in the domain [0, 20] and (b) is the profile in the domain [9.5, 10.5].

Figure 5

Differences between Jordan and Einstein frame pressure profiles for the parameters $R_0=10$, $\sigma =1/10$, $m_0=1$, and $q=-2\pi /3$, plotted with respect to the Jordan frame areal radius $\mathfrak{r}$. (a) is the profile in the domain [0, 20] and (b) is the profile in the domain [9.5, 10.5].

Figure 6

Differences between Jordan and Einstein frame profiles for the quantity $Y$ in terms of the Einstein frame areal radius $r$. (a) is the profile in the domain [0, 20] and (b) is the profile in the domain [9.5, 10.5].

Figure 7

In (a), the effective potential $V(r)$ (107) for geodesics is displayed, with the parameter choices $R_0=10$, $\sigma =1/10$, $m_0=1$, and $q=-2\pi /3$, with $e=1$ and $l=0$ for the respective energy and angular momentum parameters. In (b), the effective force per unit mass $F$ is displayed. These plots are displayed in terms of the Einstein frame areal radius $r$, since it differs mildly from the proper radius $\mathbf{r}$ for our parameter choices (one can verify that $Y\sqrt{h(r)}\approx 1$), so these plots still capture the qualitative behavior of the effective potential and force.