Effective geometrostatics of spherical stars beyond general relativity

Abstract

We provide a set of general tools to study the problem of stellar equilibrium in any gravitational theory in which spherically symmetric spacetimes satisfy master field equations taking the form of an equality between an identically conserved tensor, with derivatives of up to second order in the metric, and an identically conserved matter tensor. We derive the most general expression for the Tolman--Oppenheimer--Volkoff equation of stellar equilibrium that is compatible with these minimal requirements. A general discussion of the conditions that guarantee geodesic completeness at the center of symmetry is also presented. The equations of stellar equilibrium are integrated in a subset of the space of allowed deformations of general relativity proposed by Ziprick and Kunstatter, allowing us to illustrate universal aspects associated with the weakening of the strength of gravity, such as the mitigation of the Buchdahl limit obtained in general relativity or the existence of static solutions describing regular black holes with perfect fluid cores.

Figures (6)

• Figure 1: Plot of the $\bm{\beta}_{\text{odd}}$ function with $n=1$ in terms of $r$. This function is constructed to always have a global maximum for $\ell>0$. As $\ell\to0$, this function approaches its general relativity behavior from below. As $n$ is increased, the $r\to0$ divergence becomes stronger, but the $r/\ell\gg1$ behaviour is unaffected.
• Figure 2: Left panel: Pressure profile of constant-density stars with $M=1$ and $R=3$. As $\ell$ is increased, the central pressure decreases. All solutions lie below the general relativity one. Right panel: Pressure profiles of constant-density stars with varying compactness and $M=1$. The dashed lines correspond to the general relativity solutions, and the continuous ones to $\ell=1/5$. The general relativity Buchdahl limit corresponds to $2M/R=8/9$. At the Buchdahl compactness, the general relativity pressure diverges at $r=0$, while $\ell=1/5$ solution is regular.
• Figure 3: Redshift functions of families of constant-density stars of varying compactness. The dark gray curve is the redshift function from the vacuum solution with $\bm{\beta}=\bm{\beta}_{\text{odd}}$ with $n=1$ and $\ell=1$, and the region in between horizons shaded in light gray. Constant-density stars can be placed outside the outer horizon (in blue), leading to an inwards-decreasing redshift that vanishes at $r=0$ (darkest blue curve) for $2M/R\approx0.956$ (its surface is indicated by the dashed, vertical line), or inside the inner horizon (in red), leading to an inwards-increasing redshift that is always finite for any compactness between $0$ and $1$.
• Figure 4: Left panel: Pressure profiles for stars placed inside the inner horizon of the vacuum solution shown in Fig. \ref{['Fig:SmallStars']}. Stars whose surface lies closer to the inner horizon have pressures that decrease faster towards the interior, approaching $p=-\rho_{0}$. Right panel: Pressure profiles for stars place outside the outer horizon. These stars display a maximum compactness bound that depends on the value of $\ell$. The darkest blue curve saturates this bound.
• Figure 5: Left panel: Redshift functions of stars placed outside (in blue) and inside (in red) the global minimum of the redshift. When the vacuum solution (dark gray) does not exhibit horizons, regular fluid spheres can be placed anywhere in the spacetime. Their maximum compactness will be given by $1-e^{2\nu(r_{\text{min}})}$, and can tend to $1$ in the limit where $r_{\text{min}}\to0$. Right panel: Pressure profiles of these stars. As stars are compressed, their internal pressures eventually reach a maximum value.