Black holes and neutron stars in massive Hellings-Nordtvedt theory

Abstract: Hellings-Nordtvedt theory is a vector-tensor theory in which a vector field $A_μ$ is nonminimally coupled to curvature through two independent interactions $A^2{\cal R}$ and $A^μA^ν{\cal R}{μν}$. When supplemented by a potential whose zero-energy minimum occurs at nonzero $A^2$, the restricted $A^μA^ν{\cal R}{μν}$ sector is known to admit black-hole and neutron-star solutions with a monopole-like asymptotic vacuum structure. We examine whether this structure is a generic consequence of the nonzero vector vacuum or instead relies on the special Ricci-tensor coupling. By analyzing the field equations near spatial infinity, we show that the asymptotic vacuum condition is incompatible with generic nonzero values of both couplings and instead selects two allowed single-coupling sectors. The $A^μA^ν{\cal R}_{μν}$ sector reproduces the known monopole-like asymptotics, whereas the $A^2{\cal R}$ sector admits an asymptotically flat Schwarzschild metric with a nontrivial radial vector field. We further compute the Noether mass in the $A^2{\cal R}$ sector, derive the corresponding Solar-System constraints, and construct neutron-star configurations. Although the weak-field deviation is constrained to be small, neutron stars can still show appreciable departures from both general relativity and the Ricci-tensor-coupling sector in their masses, radii, and moments of inertia. Our results identify that the $A^2{\cal R}$ sector of massive Hellings-Nordtvedt theory as a viable and useful framework for studying strong-field compact objects with a nonzero vector vacuum while remaining compatible with weak-field tests.

• The paper demonstrates that nonminimal vector-tensor couplings lead to two distinct sectors, with one yielding standard Schwarzschild metrics and the other a monopole-like asymptotic structure.
• It applies Noether charge methods to reveal that Lorentz-violating parameters modify physical observables, with solar-system tests constraining these deviations relative to GR.
• Numerical analysis of neutron stars shows significant strong-field effects on mass, radius, and moment of inertia, highlighting potential observable departures from general relativity.

Black Holes and Neutron Stars in Massive Hellings-Nordtvedt Theory

Theoretical Framework and Motivation

The massive Hellings-Nordtvedt (HN) theory extends standard vector-tensor gravity by introducing a vector field $A_\mu$ nonminimally coupled to curvature through two independent terms: $A^2 {\cal R}$ and $A^\mu A^\nu {\cal R}_{\mu\nu}$. When augmented by a potential driving the vector field to a nonzero vacuum, this structure aligns with spontaneous Lorentz symmetry breaking and shares conceptual features with bumblebee gravity models. Previous investigations within the restricted $A^\mu A^\nu {\cal R}_{\mu\nu}$ sector revealed black holes and neutron stars exhibiting monopole-like asymptotic geometry with a solid-angle deficit. The present work interrogates whether this monopole-like structure is inherent to the nonzero vector vacuum or contingent on the precise form of nonminimal coupling.

Asymptotic Structure and Sector Separation

A detailed examination of the field equations and asymptotic vacuum conditions reveals that generic nonzero values of both couplings are incompatible with the required vector vacuum at spatial infinity. The theory bifurcates into two allowed single-coupling sectors:

1. $A^2 {\cal R}$ sector ($\gamma_2 = 0$): Admits asymptotically flat Schwarzschild geometry, with the vector field nontrivial but metric remaining standard.
2. $A^\mu A^\nu {\cal R}_{\mu\nu}$ sector ($\gamma_1 = 0$): Reproduces the previously established monopole-like asymptotics, characterized by a Lorentz-violating parameter and a solid-angle deficit.

Thus, spontaneous Lorentz breaking via the vector vacuum does not generically induce monopole-like asymptotics; it is a feature exclusive to the Ricci-tensor coupling.

Figure 1: Numerical solutions of $(f, \phi^{-2}, \rho, \lambda', w/\Omega)$ for neutron stars with SLy EOS in the $A^2 {\cal R}$ sector; strong-field deviations occur with weak-field-compliant $A^2 {\cal R}$0.

Black Hole Solutions: Noether Mass and Weak-Field Constraints

Black holes in the $A^2 {\cal R}$1 sector possess metrics identical to Schwarzschild when parametrized by the integration constant, but the Noether charge computed via Wald’s formalism yields a physical mass $A^2 {\cal R}$2, explicitly correcting for the Lorentz-violating parameter $A^2 {\cal R}$3. The physical observables are thus sensitive to the coupling strength, breaking apparent metric degeneracy with GR at the level of conserved charges.

Solar-System tests—perihelion advance, light deflection, and Shapiro delay—impose bounds $A^2 {\cal R}$4, several orders of magnitude weaker than the monopole sector ($A^2 {\cal R}$5). Notably, in the $A^2 {\cal R}$6 sector, the Noether mass is essential for extracting observable constraints; the coupling enters weak-field geometry only via the mass redefinition.

Neutron Star Structure and Strong-Field Effects

Construction of slowly rotating neutron stars (SLy EOS) in the $A^2 {\cal R}$7 sector demonstrates that even with $A^2 {\cal R}$8, significant deviations from GR emerge in the strong-field regime. Numerical solutions indicate:

• Mass and Radius: At low central densities, both are reduced relative to GR. At high densities, they increase sharply, with the maximum mass occurring at lower $A^2 {\cal R}$9 compared to GR and the monopole sector.
• Moment of Inertia: Behaves similarly—decreased for low-mass stars and enhanced for high-mass stars; sensitive to the interplay between mass, radius, and matter distribution.

  Figure 2: Mass-radius and mass-central density relations for SLy EOS; $A^\mu A^\nu {\cal R}_{\mu\nu}$0 and $A^\mu A^\nu {\cal R}_{\mu\nu}$1 sectors diverge notably at high densities.

  

  Figure 3: Moment of inertia versus mass for SLy EOS; deviations from GR and between sectors amplify for high-mass stars.

Observational constraints from pulsars and GW170817 are satisfied for representative choices of $A^\mu A^\nu {\cal R}_{\mu\nu}$2 and the potential parameter $A^\mu A^\nu {\cal R}_{\mu\nu}$3; deviations from GR in mass-radius and $A^\mu A^\nu {\cal R}_{\mu\nu}$4-$A^\mu A^\nu {\cal R}_{\mu\nu}$5 relations remain within allowed ranges but are future targets for refinement.

Implications and Prospects

Strong-field deviations are prominent in neutron-star observables, supporting the role of compact stars in probing nonminimal vector-tensor gravity, even when weak-field bounds are satisfied. The qualitative similarity to scalar-tensor theories with nonminimal coupling underscores the utility of the $A^\mu A^\nu {\cal R}_{\mu\nu}$6 sector as a viable framework for exploring spontaneous Lorentz breaking and vector vacua in the context of compact objects.

Future developments should address:

• Stability and well-posedness of solutions in both sectors, as dynamical pathologies have been reported for self-interacting vector fields in related models.
• Extension to additional neutron-star observables (quadrupole moment, tidal Love numbers), enabling tests of universal I-Love-Q relations and improved constraints on coupling strengths and potential forms.
• Embedding within broader theoretical contexts (e.g., Einstein-frame analyses), ensuring clarity in matter sector interactions and observational signatures.

Conclusion

The massive Hellings-Nordtvedt theory, with its sector separation governed by the asymptotic vector vacuum, delineates sharply between monopole-like and flat asymptotics. The $A^\mu A^\nu {\cal R}_{\mu\nu}$7 sector allows phenomenologically rich compact-object solutions, where weak-field constraints still permit substantial strong-field modifications. The formal distinction between metric and physical observables necessitates careful use of Noether charges for astrophysical predictions. Stability analyses and further nonlinear observable studies are crucial for theoretical viability and for leveraging neutron stars as empirical probes of Lorentz-violating modified gravity (2605.14711).