- 1Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, Bonn, Germany
- 2Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing, China
- 3State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing, China
- 4Center for Theoretical Physics, China Institute of Atomic Energy, Beijing, China
Neutron stars are extraordinary astrophysical objects with densities close to and even very far above these in atomic nuclei. Their structure and dynamic observables are governed by the equation of state (EoS). Due to difficulties in both theory and experiments, there exist still big uncertainties on the EoS for neutron stars. From the realistic nucleon–nucleon
1 Introduction
Neutron stars serve as natural laboratories for investigating the properties of matter under extreme densities and strong gravitational fields (Lattimer and Prakash, 2004). Understanding the properties of dense nuclear matter is essential for describing the structure and evolution of neutron stars (Burgio et al., 2021; Sedrakian et al., 2023). At the core of this pursuit lies the nuclear equation of state (EoS), which connects microscopic nuclear interactions to macroscopic observables such as neutron star masses and radii (Lattimer and Prakash, 2000; Lattimer and Prakash, 2007; Oertel et al., 2017; Huth et al., 2022). The EoS essentially encapsulates the relationship between pressure and density in nuclear matter, determining how matter behaves under the extreme conditions found in neutron star interiors. In particular, the recent detection of neutron stars with masses exceeding 2 solar masses and the advent of multi-messenger astronomy have placed stringent constraints on the EoS, highlighting the necessity of developing and refining theoretical models that are not only consistent with laboratory nuclear physics data but also aligned with the latest astrophysical observations (Demorest et al., 2010; Antoniadis et al., 2013; Fonseca et al., 2016; Arzoumanian et al., 2018; Cromartie et al., 2020; Fonseca et al., 2021; Abbott et al., 2017; Abbott et al., 2018; Tong et al., 2020; Han et al., 2023).
Over the years, considerable theoretical efforts have been devoted to determining the EoS of neutron star matter using various nuclear many-body approaches. In general, these approaches can be categorized into two classes: density functional theories (DFTs) employing effective nucleon-nucleon
The RBHF theory provides a self-consistent framework to study the nuclear many-body problem by combining Dirac phenomenology with the in-medium scattering equation. In this approach, the interaction between two nucleons in the nuclear medium is described by the in-medium scattering matrix
In this review, we summarize these recent advances in RBHF theory formulated in the full Dirac space and their implications for the physics of dense matter and neutron stars.
2 Relativistic Brueckner–Hartree–Fock theory and neutron stars
In the RBHF theory, nucleons within the nuclear medium are treated as dressed particles due to their interactions with surrounding nucleons. The single-particle motion of these nucleons is described by the Dirac equation
where
The quantities
the solution of Equation 1 leads to the in-medium positive-energy spinor
where
The Dirac equation can be solved exactly once the single-particle potentials are determined. To this end, three matrix elements of
After obtaining
This approach avoids approximations in the Dirac space with PESs only. The matrix elements
In Equations 7a–7c, the anti-symmetrized
The
where
The inclusion of an infinitesimal
After the solution of
where
It should also be noted that the calculation of the binding energy yields a three dimension integrals over the c. m. momentum
It does not depend on the direction and this value is usually applied in the
One of the motivations for developing a microscopic and fully relativistic theory of dense nuclear matter is its application to neutron star. The neutron star matter here is assumed to be composed of nucleons and leptons (mainly electrons and muons), while neglecting possible phase transitions or the appearance of exotic degrees of freedom at densities above nuclear saturation. The matter is considered to be in beta equilibrium and charge neutrality, leading to the following equilibrium conditions for the chemical potentials of the nucleons and leptons in Equation 12:
where
where
where
For a given density
This yields the EoS of beta equilibrium nuclear matter in the form of
Once the EoS in the form
where
where
with
The differential Equation 20 can be integrated together with the TOV equations with the boundary condition
Here,
The frame-dragging angular velocity
where
It should be noted that, under the slow-rotation approximation, the moment of inertia is independent of the stellar frequency
The quadrupole moment characterizes the degree of rotational deformation of the neutron star away from spherical symmetry (Yagi and Yunes, 2013). It can be computed by numerically solving for the interior and exterior gravitational field of a neutron star in a slow-rotation (Hartle, 1967; Hartle and Thorne, 1968) and a small-tidal-deformation approximation (Hinderer, 2008; Hinderer et al., 2010). To explore the universal dimensionless moment of inertia-tidal deformability-quadrupole moment (
In addition, to describe the rapidly rotating and axisymmetric neutron star configurations in general relativity, the stellar matter is treated as a perfect fluid, characterized by the energy-momentum tensor in Equation 27:
where
where the metric potentials
3 Neutron star mass and radius
In this review, we have focused on recent advances in the study of neutron star properties based on RBHF theory formulated in the full Dirac space.
Figure 1 from Ref. Qu et al. (2025) illustrates the gravitational mass of both static and rotating neutron stars as a function of their equatorial radius. The left panel presents results for fixed spin ratios,
Figure 1. The gravitational mass
Specifically, the radii of a canonical neutron star with mass
4 Universal relations
In the multimessenger era, the tidal deformability
Beyond tidal deformability, universal relations among neutron star observables offer an additional, largely EoS-independent avenue for cross-checking theoretical models against observations. Figure 2 examines the EoSs derived from RBHF theory in the full Dirac space, with the projection method, and the momentum-independence approximation with Bonn potentials–in light of the universal
where the fitting coefficients are summarized in Table 4 of Wang et al. (2022c). These coefficients closely agree with those obtained in Ref. Yagi and Yunes (2017), based on a broad ensemble of EoSs. The bottom panels of Figure 2 present the absolute fractional deviations between the data and the fit, which remain below 1% over the entire mass range examined. The universal relation between
Figure 2. (Top panel) The universal
5 Summary and perspectives
We have reviewed recent developments in RBHF theory within the full Dirac space, with particular emphasis on their implications for the properties of dense nuclear matter and neutron stars. This relativistic ab initio calculations enhance the internal consistency of relativistic many-body calculations and represent a significant advancement in the microscopic description of dense matter under extreme conditions. Further progress in the RBHF theory is anticipated through the inclusion of higher-order many-body correlations, in particular by extending beyond the two-hole-line expansion currently employed in standard RBHF theory. The incorporation of three-hole-line contributions and other higher-order terms is essential for achieving a more complete and quantitatively accurate description of in-medium nuclear interactions at supranuclear densities. In parallel, while a leading order and next-to-leading order covariant chiral nuclear forces have recently been applied within RBHF calculations under the momentum-independence approximation (Zou et al., 2024; Zou et al., 2025b; Zou et al., 2025a; Zheng et al., 2025; Shen et al., 2025), a natural next step is to implement the high-fidelity chiral nuclear forces (Ren et al., 2018; Lu et al., 2022) in the full Dirac space. Such an extension would enable a more consistent and comprehensive treatment of relativistic effects, thereby improving the predictive power of relativistic ab initio calculations for the EoS and neutron star properties.
Author contributions
HT: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review and editing. SW: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review and editing. JM: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing.
Funding
The author(s) declare that financial support was received for the research and/or publication of this article. HT acknowledge funding by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (AdG EXOTIC, grant agreement No. 101018170), and by the MKW NRW under the funding code NW21-024-A. SW is supported in part by the National Natural Science Foundation of China (NSFC) under Grants No. 12205030. JM is supported in part by the National Natural Science Foundation of China under Grants No. 12435006, and the National Key Laboratory of Neutron Science and Technology NST202401016, and by the High performance Computing Platform of Peking University.
Acknowledgments
The authors would like to thank Xiaoying Qu for reading of the manuscript.
Conflict of interest
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The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Keywords: relativistic ab initio calculations, full Dirac space, nuclear matter, equation of state, neutron star
Citation: Tong H, Wang S and Meng J (2025) Relativistic ab initio calculations for static and rotating neutron stars. Front. Astron. Space Sci. 12:1666331. doi: 10.3389/fspas.2025.1666331
Received: 15 July 2025; Accepted: 06 October 2025;
Published: 21 October 2025.
Edited by:
Armen Sedrakian, University of Wrocław, PolandReviewed by:
Tsuyoshi Miyatsu, Soongsil University, Republic of KoreaCopyright © 2025 Tong, Wang and Meng. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jie Meng, bWVuZ2pAcGt1LmVkdS5jbg==