General gravitational properties of neutron stars: curvature invariants, binding energy, and trace anomaly

TL;DR

This paper systematically investigates curvature invariants inside neutron stars by constructing a large, theory- and observation-constrained EOS ensemble and solving the TOV equations for static NSs. It reveals that negative Ricci curvature in the interior is common for stiff, highly compact stars, while the Kretschmann scalar remains a more faithful curvature indicator; it also tightens quasi-universal relations between gravitational and baryonic masses and derives a direct link between the Ricci scalar and trace anomaly, establishing bounds on the central curvature and anomaly. The results yield practical tools, such as improved MB–M and binding-energy–compactness relations, enabling EOS-consistent inferences from observed NS masses and providing insights into gravity under extreme conditions. Together, these findings enhance our understanding of NS interior geometry and its connection to fundamental physics and gravity.

Abstract

We investigate the behavior of curvature invariants for a large ensemble of neutron stars built with equations of state (EOSs) that satisfy constraints from nuclear theory and perturbative QCD, as well as measurements of neutron-star masses, radii, and gravitational waves from binary neutron-star mergers. Surprisingly, our analysis reveals that stars with negative Ricci scalar $\mathcal{R}$ are rather common and about $\sim 50\%$ of our EOSs produce one or more stars with Ricci curvature that is negative somewhere inside the star. The negative curvature is found mostly but not exclusively at the highest densities and pressures, and predominantly for stiff EOSs and for the most compact and most massive stars. Furthermore, we improve the quasi-universal relation between the stellar gravitational mass $M$ and the baryonic mass $M_\mathrm{b}$, which allows us to express analytically one in terms of the other with a maximum variance of only $\sim 3\%$. Finally, using the relation between the Ricci scalar and the trace anomaly $Δ$, we determine the conditions under which $Δ$ vanishes or becomes negative in neutron stars.

Figures (8)

• Figure 1: Normalized PDFs of the radial profiles of the normalized Ricci scalar $\mathcal{R} \, M^2$ for fixed NS masses. The red lines mark the median values of the distribution, while black lines correspond to $1$-$\sigma$ confidence limits. From left to right: the profiles refer to NSs with masses $1.1 \, M_\odot$, $1.4 \, M_\odot$, $2.1 \, M_\odot$ and $M_{_{\mathrm{TOV}}}$, respectively.
• Figure 2: Left panel: Normalised PDFs of the $M$-$R$ relations for the entire EOS ensemble. The green/red colormap represents the distribution of NSs with non-negative/negative Ricci scalar, everywhere/somewhere inside the star. $1$-$\sigma$, $2$-$\sigma$ and $100\%$-confidence intervals are shown from darker to lighter colours. The grey solid line contours the $100\%$-confidence interval for every stable NS. Shown with a black solid is the "zero-curvature" compactness relation $\mathscr{C}_0$, while black dashed and dotted lines mark two reference EOSs (${\rm EOS}_{1,2})$. The top and right parts of the panel report the $100\%$-confidence intervals of both distributions in the relevant direction. Right panel: the same as in the left but for the $p$-$e$ relations. Marked with black filled circles are the largest energy densities for stable NSs with ${\rm EOS}_{1,2}$.
• Figure 3: Left panel: distribution of the baryonic mass $M_\mathrm{b}$ as a function of gravitational mass $M$. Shown respectively with red, black and green solid lines are the fits from our work, from Timmes1996, and from Gao2020. Right panel: distribution of the binding energy $E_{\rm bin}/M$ as a function of the stellar compactness $\mathscr{C}$. Also in this case, the red line shows our fit to the data, while the black and green lines reports the fit by Lattimer01 and Breu2016.
• Figure 4: Distribution of the compactness as a function of the gravitational mass. Shown with a red solid line is the functional behaviour of Eq. \ref{['eq:comp_vs_M']}, which provides a first approximation to the compactness, with an uncertainty of $15$-$20\%$ for low-mass NSs.
• Figure 5: Different estimates of the baryonic mass for pulsar B in the binary system J0737-3039. Shown with different shadings are the estimates by Podsiadlowski2005 (red), i.e., $M_{\rm b} = 1.366 - 1.375\, M_\odot$, by Kitaura06 (yellow), i.e., $M_{\rm b} = 1.360 \pm 0.002\, M_\odot$ and in this work as taken from the 100%-confidence interval of the distribution in the left panel of Fig. \ref{['fig:fig3']} (blue). The predictions by Whittenbury2014 are shown as the green stripe.