Black holes and neutron stars in massive Hellings-Nordtvedt theory

This paper demonstrates that while the massive Hellings-Nordtvedt theory's asymptotic vacuum structure restricts viable solutions to specific single-coupling sectors, the $A^2{\cal R}$ sector uniquely supports asymptotically flat Schwarzschild metrics with nontrivial vector fields, offering a viable framework for studying compact objects that satisfy weak-field constraints while exhibiting significant strong-field deviations from general relativity.

The Gist

Imagine the universe as a giant, stretchy trampoline. In our standard understanding of gravity (Einstein's General Relativity), this trampoline is smooth and follows strict rules. But what if there were invisible "wind" blowing across the trampoline, or if the fabric itself had a hidden tension that changed how it reacted to heavy weights? This is the world of vector-tensor gravity, a family of theories that adds extra "wind" (a vector field) to the fabric of space.

This paper investigates a specific version of this theory called Massive Hellings-Nordtvedt theory. The researchers wanted to solve a mystery: When this theory predicts strange, "monopole-like" shapes for space around black holes and neutron stars, is that shape caused by the "wind" itself, or is it caused by a specific way the wind interacts with the trampoline's curves?

Here is the breakdown of their findings using simple analogies:

1. The Two Ways the Wind Can Push

The theory has two main ways the "wind" (the vector field) can interact with the curvature of space:

• Interaction A ($A^2R$): The wind pushes based on the total "amount" of wind squared.
• Interaction B ($A^\mu A^\nu R_{\mu\nu}$): The wind pushes based on how it aligns with specific directions of the curve.

Previous studies looked at a restricted version where only Interaction B existed. They found that space around black holes and neutron stars in this version looked like a sphere with a tiny slice missing (like a beach ball with a wedge cut out). This is called a "monopole-like" structure.

2. The Big Discovery: You Can't Have Both

The authors asked: "What happens if we allow both interactions to exist at the same time?"

They did the math and found a surprising rule: Nature doesn't allow both interactions to be active simultaneously if the "wind" has a non-zero value in empty space (a "vacuum").

• It's like trying to drive a car with two different steering wheels that fight each other; the car simply won't move in a stable way.
• The equations force the theory to split into two separate, allowed "lanes":
  • Lane 1: Only Interaction A ($A^2R$) is active.
  • Lane 2: Only Interaction B ($A^\mu A^\nu R_{\mu\nu}$) is active.

The Conclusion on Shape: The "wedge-cut-out" (monopole) shape is only found in Lane 2. In Lane 1, space remains perfectly smooth and flat (like a standard beach ball), even though the invisible wind is still blowing. This proves that the strange shape isn't just caused by the wind existing; it's caused specifically by how the wind pushes on the curves in Lane 2.

3. The "Stealth" Black Hole (Lane 1)

In Lane 1 (the $A^2R$ sector), the black hole looks exactly like the ones in Einstein's General Relativity. If you just looked at the shape of the space, you couldn't tell the difference. The authors call this a "stealth" solution.

However, the paper reveals a hidden trick. While the shape looks the same, the weight (mass) of the black hole is different.

• Analogy: Imagine two identical-looking suitcases. One is empty, and one is full of lead. They look the same, but if you try to lift them, the heavy one feels different.
• The researchers calculated the "Noether mass" (a precise way to measure the weight of the system). They found that the invisible wind adds a tiny bit of "extra weight" to the black hole.
• Because of this, the theory isn't truly "hidden." By measuring the mass of objects in our solar system (like Mercury's orbit or how light bends around the Sun), scientists can set limits on how strong this invisible wind can be. They found the wind must be very weak (a tiny fraction of a percent) to fit our current observations.

4. Neutron Stars: The Heavyweights

The most exciting part of the paper is what happens with neutron stars (ultra-dense stars that are the size of a city but weigh more than the Sun).

Even though the "wind" in Lane 1 is so weak that it barely affects the solar system (the "lightweight" tests), it has a huge effect on the heavyweights.

• The Analogy: Think of a spring. If you push it gently (solar system), it barely bends. But if you sit on it (neutron star), it compresses significantly.
• The researchers built models of neutron stars in this theory. They found that even with the tiny, allowed amount of "wind," the stars behave differently than Einstein predicted:
  • Low-density stars: They become slightly smaller and lighter than expected.
  • High-density stars: They become slightly larger and heavier.
  • Spinning: The way these stars spin (their moment of inertia) also changes noticeably.

Summary

The paper concludes that:

1. The "Monopole" shape is specific: It only happens with one specific type of interaction, not just because the invisible wind exists.
2. Two separate worlds: The theory splits into two distinct versions, and they behave very differently.
3. Stealth is broken: Even if a black hole looks like a normal Einstein black hole, its weight tells a different story, allowing us to test the theory.
4. Neutron stars are sensitive probes: Even if the theory passes all the easy tests in our solar system, it leaves a big fingerprint on the most extreme objects in the universe. Neutron stars are the perfect place to look for these hidden forces.

The authors suggest that future studies should check if these strange neutron stars are stable and look at other properties, like how they ripple when they collide, to see if this theory holds up.

Technical Summary

Technical Summary: Black holes and neutron stars in massive Hellings-Nordtvedt theory

Problem Statement
The Hellings-Nordtvedt theory is a vector-tensor gravity model where a vector field $A_\mu$ couples nonminimally to spacetime curvature via two independent interactions: $A^2 R$ and $A_\mu A_\nu R^{\mu\nu}$. Recent studies in a restricted sector of this theory (retaining only the $A_\mu A_\nu R^{\mu\nu}$ coupling) supplemented with a "bumblebee-type" potential (where the potential minimum occurs at a nonzero vector invariant $A^2 = b^2$) have revealed black hole and neutron star solutions with a monopole-like asymptotic structure (a solid-angle deficit).

A critical ambiguity remains: Is this monopole-like asymptotic structure a generic consequence of the nonzero vector vacuum (spontaneous Lorentz symmetry breaking) itself, or is it an artifact specific to the $A_\mu A_\nu R^{\mu\nu}$ coupling? Resolving this is essential for understanding the physical roles of different curvature-vector couplings and for determining the viability of the full theory in strong-field regimes.

Methodology
The authors analyze the full massive Hellings-Nordtvedt theory, including both nonminimal couplings ($\gamma_1 A^2 R$ and $\gamma_2 A_\mu A_\nu R^{\mu\nu}$) and a potential $V(X)$ with a zero-energy minimum at $X=b^2$.

1. Asymptotic Analysis: The authors examine the field equations near spatial infinity for static, spherically symmetric configurations. They expand the metric functions and the vector field profile in powers of $1/r$ and impose the asymptotic vacuum condition $X \to b^2$.
2. Black Hole Solutions: They investigate whether the asymptotic consistency allows for generic nonzero values of both coupling constants ($\gamma_1, \gamma_2$). They identify two distinct single-coupling sectors that satisfy the asymptotic conditions.
3. Mass Calculation (Noether Charge): For the $A^2 R$ sector (Case I), where the metric appears identical to the Schwarzschild solution in terms of integration constants, the authors employ the Wald covariant phase-space formalism to compute the Noether mass. This distinguishes the physical mass from the geometric integration constant.
4. Solar-System Constraints: Using the derived physical mass, the authors re-evaluate weak-field constraints (Mercury's perihelion advance, light deflection, and Shapiro time delay) to bound the Lorentz-violating parameter $\ell_1 = \gamma_1 b^2$.
5. Neutron Star Modeling: The authors construct slowly rotating neutron star solutions in the $A^2 R$ sector using the SLy equation of state. They solve the coupled system of nonlinear ordinary differential equations (Tolman-Oppenheimer-Volkoff equations extended to include the vector field) and the first-order rotational perturbation equation. They compare the resulting mass-radius and moment of inertia relations against General Relativity (GR) and the previously studied $A_\mu A_\nu R^{\mu\nu}$ sector.

Key Contributions and Results

• Decoupling of Asymptotic Structure from Potential: The analysis demonstrates that the monopole-like asymptotic structure is not a generic consequence of the nonzero vector vacuum. Instead, the field equations, combined with the asymptotic vacuum condition, strictly forbid generic nonzero values for both couplings simultaneously. The theory separates into two allowed single-coupling sectors:

  • Case II ($A_\mu A_\nu R^{\mu\nu}$ only): Reproduces the known monopole-like asymptotic structure (solid-angle deficit).
  • Case I ($A^2 R$ only): Admits asymptotically flat vacuum solutions. The metric is exactly Schwarzschild when expressed in terms of the integration constant, but the vector field remains nontrivial (a "stealth" solution).

• Noether Mass and Weak-Field Constraints: In the $A^2 R$ sector, the physical mass $M_1$ differs from the geometric integration constant $m/2$ by a factor dependent on the coupling: $M_1 = \frac{1}{2}(1+\ell_1)m$. This breaks the apparent degeneracy with GR. Consequently, the Lorentz-violating parameter $\ell_1$ enters the weak-field observables. The authors derive constraints on $\ell_1$ using Solar-System tests, finding the range $-10^{-5} \lesssim \ell_1 \lesssim 10^{-6}$. This bound is several orders of magnitude weaker than the bound on $\ell_2$ in the monopole sector, primarily because the $A^2 R$ sector is asymptotically flat.

• Strong-Field Deviations in Neutron Stars: Despite the stringent weak-field bounds requiring $\ell_1 \approx 10^{-6}$, the authors find that neutron stars in the $A^2 R$ sector exhibit appreciable deviations from GR in the strong-field regime.

  • Mass and Radius: Relative to GR, the mass and radius are reduced at low central densities but enhanced at high central densities.
  • Moment of Inertia: The moment of inertia follows a similar trend (reduced for low-mass stars, increased for high-mass stars).
  • Comparison with Ricci-Tensor Sector: While the qualitative trends in the $A^2 R$ sector mirror those in the $A_\mu A_\nu R^{\mu\nu}$ sector, quantitative differences become significant at high densities and high masses. The $A^2 R$ coupling leads to a faster growth of stellar mass with central density, resulting in maximum-mass configurations at lower central densities compared to the Ricci-tensor sector.

Significance
The paper establishes that the specific form of the curvature-vector coupling is essential in determining the asymptotic geometry of compact objects in massive Hellings-Nordtvedt theory; the nonzero vector vacuum alone does not dictate a monopole-like structure.

Crucially, the work identifies the $A^2 R$ sector as a viable framework for studying strong-field compact objects with a nonzero vector vacuum. Even though this sector is compatible with weak-field Solar-System tests (due to its asymptotic flatness and the specific mass correction), it allows for significant deviations in neutron star structure (mass, radius, and moment of inertia). This reinforces the role of neutron stars as sensitive probes of strong-field gravity, capable of distinguishing between different nonminimal coupling mechanisms that might otherwise appear indistinguishable in weak-field or black-hole (stealth) limits. The authors note that while the solutions are compatible with current observational mass and radius constraints, the stability of these solutions remains an open issue for future investigation.