Black-Hole Branches in Massive and Bigravity

Massive and bimetric gravity admit several inequivalent black-hole branches even under spherical symmetry. An algebraic branch factor separates simultaneously diagonal metrics from nonbidiagonal metrics, but solving that factor is only the first consistency condition. A physically admissible solution must also possess a real square-root branch, a regular common exterior and horizon structure, the intended asymptotics, and acceptable perturbations within the effective theory’s regime.

Required background. Variational and Symplectic Geometry supplies boundary-value and perturbative constraint language, and Weak-Field Gravity supplies massive-mode asymptotics and stability diagnostics.

Helpful background. Einstein Equations supplies the GR black-hole branches; Gravitational Actions and Boundary Terms supplies variational boundary conditions; Formulations of General Relativity supplies horizon-penetrating formulations; and Gravity Beyond General Relativity supplies the theory-selection context.

The spherical branch factor

Retain the convention

$$S=\sqrt{g^{-1}f},\qquad \alpha=\frac{m_f}{m_g},$$

and the interaction $-2m^4\sqrt{-g}\sum_{n=0}^4\beta_ne_n(S)$. Choose coordinates in which $g$ is diagonal and write schematically

$$ds_g^2=-U^2dt^2+V^2dr^2+r^2d\Omega_2^2,$$ $$ds_f^2=-A^2dt^2+B^2dr^2+2C\,dt\,dr+D^2d\Omega_2^2.$$

The off-diagonal $g$ equation contains the factor

$$C\left[\beta_1+2\beta_2\frac{D}{r} +\beta_3\left(\frac{D}{r}\right)^2\right]=0.$$

This follows directly from the square-root interaction; see Schmidt-May and von Strauss 2016, § 6.2, eqs. (6.6)–(6.9). It produces two broad possibilities:

• Bidiagonal: $C=0$, so both metrics can be diagonal in the same chart. Proportional metrics $f=c^2g$ are a special subset, not the whole branch.
• Nonbidiagonal: $C\ne0$ and $D/r=u$ is a constant root of $\beta_1+2\beta_2u+\beta_3u^2=0$. The remaining equations can reduce to two Einstein systems with separately induced cosmological terms, but only after further parameter and Bianchi conditions are imposed.

Numerically constructed hairy black holes are generally bidiagonal but nonproportional: their massive spin-2 profile supplies additional radial structure. Thus “bidiagonal,” “proportional,” and “hairy” are not three mutually exclusive labels.

Regularity and boundary data

The algebraic factor does not select a black hole. The following checks do.

1. Square-root domain. $g^{-1}f$ must admit a real, smooth choice of $S$ throughout the exterior and across the chosen horizon chart. Eigenvalue collisions or a branch switch can invalidate the metric potential even when both metric component arrays look finite.
2. Horizon regularity. Test both metrics and $S$ in ingoing Eddington–Finkelstein or another horizon-penetrating chart. For a regular static bidiagonal solution, the two metrics must share the relevant horizon; otherwise a coordinate pole for one metric becomes a physical singularity in the other sector. Regular equilibrium also constrains the two surface gravities. These points are discussed in Babichev and Brito 2015, § 3.1.1, pp. 9–11.
3. Asymptotics. Specify asymptotically flat, de Sitter, or anti-de Sitter behavior for both metrics, eliminate exponentially growing massive modes, and define charges using the metric to which matter couples. “Schwarzschild-like” near the horizon does not fix the asymptotic branch.
4. Perturbation boundary conditions. A mode test needs regular ingoing data at the future horizon and outgoing or normalizable data at the outer boundary. A formal growing solution that violates either condition is not an instability of the boundary-value problem.
5. EFT regime. Curvatures, massive-mode gradients, and frequencies must remain below the scales at which omitted operators compete. Horizon smoothness does not establish EFT control.

For bidiagonal hairy solutions, a useful shooting datum is $u_h=D(r_h)/r_h$. Horizon series leave a restricted set of free coefficients, and asymptotic conditions usually select isolated values rather than a continuum. The construction and its numerical sensitivity are documented in Babichev and Brito 2015, § 3.2, eqs. (46)–(56).

Stability is branch-specific

On a proportional Einstein black hole, away from massless or partially massless degeneracies, the perturbations split into a massless GR sector and a massive field satisfying a Lichnerowicz-type equation. Every derivative and curvature tensor below belongs to the background matter metric $\bar g$, and $\nabla^2=\bar g^{\alpha\beta}\nabla_\alpha\nabla_\beta$ acts on both tensor indices:

$$\left(\nabla^2-m_{\mathrm{FP}}^2\right)\delta M_{\mu\nu} +2R_{\mu\rho\nu\sigma}\delta M^{\rho\sigma}=0, \quad \nabla^\mu\delta M_{\mu\nu}=0, \quad \delta M^\mu{}_{\mu}=0.$$

For Schwarzschild, this is the same radial problem as a Fourier mode of a higher-dimensional black string with wave number $k=m_{\mathrm{FP}}$. It therefore admits a Gregory–Laflamme-type long-wavelength instability in a branch- and convention-dependent low-$m_{\mathrm{FP}}r_h$ regime. The equation-level mapping is given in Schmidt-May and von Strauss 2016, § 6.2.1, eqs. (6.10)–(6.18), and its extension to a multimetric mass spectrum in Wood, Saffin, and Avgoustidis 2024, § V.

That result does not transfer mechanically to nonbidiagonal or hairy branches. Nonbidiagonal perturbation equations need not have Fierz–Pauli form; some modes that are nondynamical at linear order may instead signal a special constraint or strong coupling. Hairy solutions can bifurcate near an instability threshold, suggesting a possible endpoint, but bifurcation is not a nonlinear time evolution and does not prove endpoint selection or mode stability.

Results and open questions

The literature status below is fixed at 2026-08-19. It records theoretical existence and stability evidence, not observational bounds on a graviton mass.

Branch/result	Evidence status at the cutoff	Scope and limitation
Proportional Schwarzschild–(A)dS branch	Exact background family; the massive perturbation has a Gregory–Laflamme-type channel Schmidt-May and von Strauss 2016, § 6.2.1, eqs. (6.10)–(6.18)	Linear result on a real proportional branch; nonlinear endpoint and generic rotating evolution remain unresolved
Nonbidiagonal Einstein branch	Exact analytic backgrounds exist after the polynomial and remaining equations are satisfied Schmidt-May and von Strauss 2016, § 6.2.2, eqs. (6.23)–(6.25); radial boundary conditions can exclude formal growing modes	Does not establish full nonradial, rotating, nonlinear, or EFT stability
Bidiagonal massive-hair branch	Numerical asymptotically AdS and selected asymptotically flat solutions exist in bigravity Babichev and Brito 2015, § 3.2, eqs. (46)–(56)	Existence is parameter-, asymptotic-, and shooting-branch dependent; a complete mode-stability classification is unresolved
General multimetric non-/partially proportional branches	A linear analysis classifies nonrotating four-dimensional sectors Wood, Saffin, and Avgoustidis 2025, § III, and § IV gives coupled ODEs and Yukawa asymptotics for hair	Results depend on the interaction graph and do not automatically establish a global bimetric solution or nonlinear endpoint

The recent multimetric results sharpen rather than close the question. Wood, Saffin, and Avgoustidis 2024, §§ III and V construct the branch classes and analyze proportional stability. Wood, Saffin, and Avgoustidis 2025, §§ III–IV analyze non-/partially proportional linear sectors and give a general hairy-ODE prescription with asymptotic massive modes. In the latter work, the global numerical solution is not completed for every proposed multimetric model, so an asymptotic series must not be reported as an existence theorem.

Two interpretations remain in tension. One regards hairy branches as natural nonlinear endpoints of the proportional instability; the other notes that no general well-posed evolution has yet selected them and that some branches may instead terminate, become singular, or leave the EFT regime. Likewise, absence of an admissible radial growing mode on a nonbidiagonal background is evidence for radial mode stability, not proof of full viability.

Open questions. The conclusions could change if a nonlinear initial-value evolution in ghost-free bimetric or multimetric gravity identifies the endpoint of the proportional instability; if a complete mode analysis establishes or excludes stability of an asymptotically specified hairy branch; if a theorem changes the common-horizon or real-square-root regularity conditions; or if either 2024 primary analysis is corrected, superseded, or withdrawn. A new observational bound alone would not change these existence statements.

Exercises

1. Branch diagnostic. Solve the nonbidiagonal polynomial for general $\beta_3\ne0$. Then take $(\beta_1,\beta_2,\beta_3)=(1,-1,0)$ and classify the candidate with $C\ne0$.

Solution

For $\beta_3\ne0$,

$$u_\pm=\frac{-\beta_2\pm\sqrt{\beta_2^2-\beta_1\beta_3}}{\beta_3}.$$

A real candidate requires $\beta_2^2-\beta_1\beta_3\ge0$; a positive angular scale convention further selects an allowed sign of $u$. In the stated linear case, $1-2u=0$, so $u=1/2$. With $C\ne0$ and $D/r=1/2$, it is a nonbidiagonal candidate. The factor alone does not prove a solution: one must still satisfy the diagonal equations and Bianchi condition, choose a real smooth $S$, and impose horizon and asymptotic regularity.

2. Transfer the instability diagnostic. Explain why a massive perturbation of a proportional Schwarzschild solution can inherit a black-string instability, and why the conclusion does not decide the fate of a hairy branch.

Solution

The transverse-traceless massive equation is identical to the four-dimensional equation obtained by separating a five-dimensional black-string perturbation with $e^{ikz}$, after identifying $k^2=m_{\mathrm{FP}}^2$. Hence any admissible long-wavelength negative mode of that operator transfers to the proportional massive sector. A hairy background changes both the coefficients and generally the mode decomposition, so it is governed by a different spectral operator. Its stability and nonlinear selection require a new boundary-value or evolution calculation.

References

• Babichev, Eugeny, and Richard Brito. “Black Holes in Massive Gravity.” Classical and Quantum Gravity 32 (2015): 154001. doi:10.1088/0264-9381/32/15/154001; Open PDF.
• Schmidt-May, Angnis, and Mikael von Strauss. “Recent Developments in Bimetric Theory.” Journal of Physics A: Mathematical and Theoretical 49 (2016): 183001. doi:10.1088/1751-8113/49/18/183001; Open PDF.
• Wood, Kieran, Paul M. Saffin, and Anastasios Avgoustidis. “Black Holes in Multi-Metric Gravity II: Hairy Solutions and Linear Stability of the Non- and Partially Proportional Branches.” Physical Review D 111 (2025): 024057. doi:10.1103/PhysRevD.111.024057; Open PDF.
• Wood, Kieran, Paul M. Saffin, and Anastasios Avgoustidis. “Black Holes in Multi-Metric Gravity.” Physical Review D 109 (2024): 124006. doi:10.1103/PhysRevD.109.124006; Open PDF.