Scalarized and Vectorized Compact Objects

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Scalarization and vectorization turn the general-relativistic solution with a vanishing extra field into one branch of a larger boundary-value problem. The decisive phenomenon is a bifurcation: a normal mode of the GR branch reaches zero frequency, a nonlinear field profile grows, and a new family may become energetically or dynamically preferred. This page separates that local onset test from the harder claims that a complete relativistic star or black hole exists, is stable, and lies inside the theory’s effective domain.

Helpful background. Einstein Equations supplies the GR branch to be continued; Gravitational Actions and Boundary Terms fixes the variational problem; Formulations of General Relativity supplies constraint and gauge language; Black-Hole Geometry and Mechanics fixes horizon regularity; Black-Hole Perturbation Theory supplies the zero-mode and stability tests; and Relativistic Matter and Compact Objects supplies stellar structure, equations of state, and conserved baryon number.

A tachyonic zero mode starts the branch

In the Einstein-frame scalar–tensor model, take $\varphi$ dimensionless, with gravitational Lagrangian $[R-2(\nabla\varphi)^2]/(16\pi G)$ and no scalar potential. Choose

A(\varphi)=\exp\!\left(\frac{1}{2}\beta\varphi^2\right), \qquad \widetilde g_{\mu\nu}=A^2(\varphi)g_{\mu\nu}.

Matter follows the physical metric $\widetilde g_{\mu\nu}$ . With $T$ the Einstein-frame matter trace, linearization about the GR branch $\varphi=0$ gives

\square\,\delta\varphi=-4\pi G\,\beta T\,\delta\varphi .

For ordinary stellar matter $T\simeq-\rho$ ; sufficiently negative $\beta$ therefore makes the effective squared mass negative in part of the star. Long wavelengths grow until nonlinearities and backreaction saturate them. Damour and Esposito-Farèse define the asymptotic scalar charge through the sensitivity of the gravitational mass at fixed baryon mass and show how the bifurcation depends on the equation of state and central density Damour–Esposito-Farèse 1996, § II, Eqs. (2.2)–(2.4), pp. 1477–1479. Their plotted convention uses $-\alpha_A\equiv\omega_A/m_A$ ; signs must be translated before comparing charges.

The threshold is a global, EOS-dependent eigenvalue, not the local sign of $T$ alone. For the particular quadratic coupling and polytropic EOS used in that work, the zero-mode onset is not reached before the GR maximum mass when $\beta$ is above $\beta_{\mathrm{cr}}\simeq-4.34$ , while $\beta\leq-5$ puts the critical baryon mass below about $1.5M_\odot$ Damour–Esposito-Farèse 1996, § II, p. 6 (PDF). These numbers characterize that stellar model and normalization; they are neither universal scalarization thresholds nor present-day empirical bounds.

The onset calculation is linear but global. Regularity at the center or future horizon and decay at infinity select discrete parameter values. A negative local effective mass is only a warning; it is not by itself an eigenvalue, a nonlinear solution, or a stability proof.

A controlled spherical branch calculation

The following verification model isolates the mathematics of onset, continuation, and charge extraction. Set the stellar radius to $R=1$ and use a constant-trace interior. Writing $u=r\varphi$ , solve

u''+k^2u-\lambda\frac{u^3}{r^2}=0, \qquad u(0)=0, \qquad u'(0)=A, \qquad u'(R)=0.

The last condition matches continuously to the vacuum exterior $\varphi=Q/r$ , for which $u=Q$ is constant. At $\lambda=0$ , $u\propto\sin kr$ and the first zero mode is exactly $k_cR=\pi/2$ . For $\lambda\gt0$ , fixing the center slope $A$ and solving for $k$ parameterizes a nonlinear branch; its scalar charge is $Q=u(R)$ .

This toy model is not a neutron-star prediction. It is valuable because every numerical output has an analytic or integral cross-check, while retaining the same regular-center, exterior-matching, eigenvalue, and continuation structure as the relativistic problem.

Save the Python fence as scalarization_branch.py. The script must produce the three rows shown below and exit nonzero if its independent onset estimate or nonlinear residual fails.

import sys
import numpy as np
from scipy.integrate import solve_bvp

R, LAMBDA, K_CRIT = 1.0, 1.0, 0.5 * np.pi

def branch_solution(amplitude, n, tol, seed=None):
    r = np.linspace(0.0, R, n)
    if seed is None:
        guess = np.vstack((amplitude*np.sin(K_CRIT*r)/K_CRIT,
                           amplitude*np.cos(K_CRIT*r)))
        p0 = np.array((K_CRIT + 0.02,))
    else:
        guess = seed.sol(r) * (amplitude / seed.y[1, 0])
        p0 = seed.p.copy()

    def equations(x, y, p):
        cubic = np.divide(y[0]**3, x**2, out=np.zeros_like(x), where=x > 0.0)
        return np.vstack((y[1], -p[0]**2*y[0] + LAMBDA*cubic))

    def boundary(ya, yb, p):
        return np.array((ya[0], ya[1] - amplitude, yb[1]))

    sol = solve_bvp(equations, boundary, r, guess, p=p0,
                    tol=tol, max_nodes=16*n)
    if not sol.success:
        raise RuntimeError(sol.message)
    mismatch = np.max(np.abs(boundary(sol.y[:, 0], sol.y[:, -1], sol.p)))
    return sol, mismatch

def variational_onset(n):
    r = np.linspace(0.0, R, n)
    u = np.sin(K_CRIT*r)
    du = np.gradient(u, r, edge_order=2)
    return np.sqrt(np.trapezoid(du**2, r) / np.trapezoid(u**2, r))

def main():
    print("N final_nodes k_branch scalar_charge onset_error "
          "charge_mismatch ode_residual boundary_mismatch")
    previous_error = None
    for n, tol in ((80, 1e-6), (160, 1e-8), (320, 1e-10)):
        seed = None
        for amplitude in (0.05, 0.15, 0.25):
            seed, mismatch = branch_solution(amplitude, n, tol, seed)
        kval, charge = seed.p[0], seed.y[0, -1]
        sample = np.linspace(0.0, R, 8*n + 1)
        u = seed.sol(sample)[0]
        phi = np.divide(u, sample, out=np.full_like(sample, 0.25),
                        where=sample > 0.0)
        charge_int = np.trapezoid(
            sample**2 * (kval**2*phi - LAMBDA*phi**3), sample)
        charge_mismatch = abs(charge_int - charge)
        onset_error = abs(variational_onset(n) - K_CRIT)
        residual = np.max(seed.rms_residuals)
        print(f"{n:d} {seed.x.size:d} {kval:.12f} {charge:.12f} "
              f"{onset_error:.3e} {charge_mismatch:.3e} "
              f"{residual:.3e} {mismatch:.3e}")
        if previous_error is not None and onset_error >= previous_error:
            raise SystemExit("variational onset did not converge")
        previous_error = onset_error
    if (abs(variational_onset(320) - K_CRIT) > 3e-5 or
            charge_mismatch > 1e-7 or residual > 2e-10):
        raise SystemExit("independent onset or residual benchmark failed")

if __name__ == "__main__":
    main()

The reproducible environment is CPython version 3.9.6 with NumPy 2.0.2 and SciPy 1.13.1:

python3 -m pip install numpy==2.0.2 scipy==1.13.1
python3 scalarization_branch.py
python3 scalarization_branch.py > scalarization_branch.txt

The reported calculation is:

grid $N$	final nodes	branch $k$ at $A=0.25$	charge $Q$	onset error	charge mismatch	ODE residual
80	80	1.583596537822	0.159446346260	$9.957\times10^{-5}$	$8.241\times10^{-8}$	$1.648\times10^{-8}$
160	160	1.583596536566	0.159446346298	$2.507\times10^{-5}$	$2.054\times10^{-8}$	$2.025\times10^{-9}$
320	639	1.583596536552	0.159446346300	$6.288\times10^{-6}$	$5.136\times10^{-9}$	$3.139\times10^{-11}$

The onset error decreases by nearly four on each refinement. More importantly, integrating the differential equation gives the independent Gauss-law identity

Q=\int_0^R r^2\!\left(k^2\varphi-\lambda\varphi^3\right)dr,

whose mismatch also converges. The fitted branch value alone would not provide this check.

From scalarization to vectorization

Replacing the scalar by a massive vector is not a change of labels. A matter-dependent effective mass can again destabilize the zero-field branch, but the vector equation also carries a divergence constraint and a longitudinal mode. Ramazanoğlu derives both the vector equation and its constraint, then constructs static neutron-star profiles Ramazanoğlu 2017, § III, Eqs. (3)–(7), pp. 2–3. The scalar script above therefore tests a shared bifurcation mechanism, not the constraint propagation, degree-of-freedom count, or stability of a vector theory. The published static vectorized families were explicitly left in need of time-evolution or perturbative stability tests Ramazanoğlu 2017, § IV, pp. 3–5.

Why vectorization needs a separate stability test

The 2017 existence calculation is not the last stability result for that vector model. Later work finds ghost and gradient problems in the longitudinal sector, including on the constructed vectorized stars: Demirboğa, Coates, and Ramazanoğlu 2022, §§ II.C–III, Eqs. (18)–(31). This excludes interpreting those studied spherical solutions as healthy endpoints of a scalar-like instability. The result applies to the specified matter-coupled vector action and branches, rather than every possible vector–tensor theory.

A simple local calculation explains the danger. Take a standard Proca Lagrangian $-\tfrac14F_{\mu\nu}F^{\mu\nu}-\tfrac12\mu^2 A_\mu A^\mu$ with constant effective $\mu^2$ on Minkowski space. For a spatial Fourier mode $A_i=ik_i\chi$ with $k=\lvert\boldsymbol k\rvert$ , eliminating the nondynamical $A_0$ gives

L_{\rm long} =\frac12\frac{k^2\mu^2}{k^2+\mu^2}\lvert\dot\chi\rvert^2 -\frac12\mu^2k^2\lvert\chi\rvert^2.

For $\mu^2\lt0$ and $k^2\gt\lvert\mu^2\rvert$ , the time-kinetic coefficient is negative: the longitudinal mode is a ghost. A scalar’s negative mass squared need not change its kinetic sign; a vector’s mass also enters the constraint used to eliminate $A_0$ . The constant-coefficient example explains that distinction, while a stellar calculation must use the full position-dependent effective metric and constraint, especially where a coefficient vanishes.

Completing the relativistic branch calculation

A production stellar calculation must solve the modified TOV and extra-field equations together, use a thermodynamically consistent causal equation of state, impose regular central data, locate the pressure-zero surface, match to the correct exterior, and read all charges in a fixed asymptotic frame. Comparisons along a family must hold baryon mass fixed when evaluating sensitivities. Central density, equation of state, node number, asymptotic field value, and coupling parameters are distinct branch labels; projecting them onto a mass–radius curve can hide folds and disconnected families. Damour and Esposito-Farèse give the full interior/exterior matching and charge extraction in Damour–Esposito-Farèse 1996, § III, Eqs. (3.6)–(3.13), pp. 1481–1484.

For black holes, replace center regularity by ingoing or smooth-horizon conditions and verify the physical metric remains regular. In every case, existence and lower energy do not prove dynamical selection. Radial, nonradial, and extra-field modes must be checked, together with hyperbolicity, ghosts, gradients, and the effective-theory cutoff. Failure to find a branch can mean a poor continuation parameter, an unresolved narrow profile, or the absence of a regular solution; these possibilities require different diagnostics.

Exercise: locate the onset and charge

Set $\lambda=0$ in the verification model. Find every value of $kR$ admitting a nonzero solution, and compute $Q$ in terms of the center slope $A$ on the fundamental branch.

Solution

Regularity and $u'(0)=A$ give $u=(A/k)\sin kr$ . Exterior matching requires $u'(R)=A\cos(kR)=0$ , hence

k_nR=\left(n+\frac12\right)\pi, \qquad n=0,1,2,\ldots .

The fundamental mode has $kR=\pi/2$ and no interior node. Since the exterior is $u=Q$ , continuity gives

Q=u(R)=\frac{A}{k}=\frac{2AR}{\pi}.

Substitution into the integral identity gives the same answer. Higher $n$ are separate nodal branches, not refinements of the fundamental family.

References

Damour, Thibault, and Gilles Esposito-Farèse, “Tensor-scalar gravity and binary-pulsar experiments,” Physical Review D 54 (1996) 1474–1491. DOI; Open PDF.
Demirboğa, Ekrem S., Andrew Coates, and Fethi M. Ramazanoğlu (2022), “Instability of Vectorized Stars,” Physical Review D 105, 024057. DOI. Open PDF.
Ramazanoğlu, Fethi M., “Spontaneous Growth of Vector Fields in Gravity,” Physical Review D 96 (2017) 064009. DOI; Open PDF.