Black holes with regular scalar hair in Brans-Dicke gravity via the Herglotz variational principle

Technical Summary: Black Holes with Regular Scalar Hair in Brans–Dicke Gravity via the Herglotz Variational Principle

Problem Statement
Classical no-hair theorems in General Relativity (GR) assert that stationary black holes are uniquely characterized by mass, charge, and angular momentum. While scalar-tensor theories, such as Brans–Dicke (BD) theory, theoretically allow for "scalar hair," standard formulations of BD theory generally forbid nontrivial scalar hair for black holes, except for the Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) solution. However, the BBMB solution suffers from a scalar field divergence at the event horizon, rendering its physical interpretation problematic. Furthermore, recent extensions to evade no-hair theorems in more complex theories (e.g., Horndeski or Einstein-scalar-Gauss-Bonnet gravity) often introduce numerous additional degrees of freedom (DOF) or arbitrary coupling functions, which can diminish the predictive power of the theory. The central problem addressed is whether the simplest non-minimally coupled scalar-tensor theory (Brans–Dicke) can support regular, nontrivial scalar hair without introducing excessive DOF or singularities, by reformulating the theory within the Herglotz variational principle (HVP).

Methodology
The author reformulates Brans–Dicke theory within the Herglotz variational principle, a framework historically developed for classical dissipative systems. Unlike the standard action principle, the HVP defines the action $S$ via a differential equation where the Lagrangian density $L$ acts as a source for the divergence of an action-density vector field $s_\mu$:
$$\nabla_\mu s_\mu = L(g_{\alpha\beta}, \partial_\mu g_{\alpha\beta}, \phi, \partial_\mu \phi, s_\mu, x_\mu)$$
A key feature of this formulation is the introduction of an arbitrary closed one-form, the Herglotz vector $\lambda_\mu$, which couples to the action density. The Lagrangian density is constructed as:
$$L = \phi R - \frac{\omega_0}{\phi}(\nabla_\mu \phi \nabla^\mu \phi) - V(\phi) + \lambda_\mu s^\mu$$
The study focuses on the simplest case where the Brans–Dicke parameter $\omega_0 = 0$ and the scalar potential vanishes ($V(\phi) = 0$). The field equations are derived by varying the action with respect to the metric and the scalar field. Crucially, unlike in standard scalar-tensor theories where the metric equations imply the scalar equation, the dissipative nature of the HVP means these equations are independent; consistency must be verified explicitly.

To find solutions, the author imposes a static, spherically symmetric ansatz for the metric and assumes the Herglotz vector has only a radial component $\lambda_\mu = (0, \eta(r), 0, 0)$. The analysis proceeds in two stages:

1. Weak-field limit: The equations are linearized to determine the behavior of $\eta(r)$ required to reproduce the Newtonian potential.
2. Exact solution: The full non-linear field equations are solved for the metric functions and the scalar profile $\phi(r)$.

Key Contributions and Results

• Derivation of Field Equations: The paper derives the modified Einstein and scalar field equations for Brans–Dicke theory under the HVP. The resulting equations include a "Herglotz tensor" $H_{\mu\nu}$ and a scalar equation of motion that depends on the trace of this tensor.
• Stealth Schwarzschild Solution: For the case $\omega_0 = 0$ and $V(\phi)=0$, the author obtains an exact solution where the metric is exactly the Schwarzschild metric (a "stealth" solution), while the scalar field $\phi(r)$ is nontrivial and strictly positive everywhere.
  $$ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \left(1 - \frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2$$
  $$\phi(r) = \phi_0 \exp\left( \int \frac{dr}{r} \frac{r-2M}{r} \eta(r) \right)$$
  Unlike the BBMB solution, this metric is not modified by the scalar field, yet the scalar field itself is non-constant.
• Regularity via Herglotz Vector Choice: The scalar profile depends on the arbitrary function $\eta(r)$. By analyzing the weak-field limit, a specific form for $\eta(r)$ is motivated:
  $$\eta(r) = \eta_0 \frac{(r-2M)^k}{r^{k+2}}$$
  where $\eta_0$ is a constant and $k$ is a positive parameter.
  • For integer $k > 1$, the scalar field $\phi(r)$ is regular ($C^\infty$) at the horizon ($r=2M$).
  • For non-integer $k$, the field remains regular but its differentiability is limited ($C^{p-1}$ where $p$ is the smallest integer greater than $k$).
    This choice allows the evasion of standard no-hair theorems while maintaining a regular scalar field at the horizon, a feat not achieved in standard BD formulations.
• Stability Analysis: A linear perturbation analysis of the scalar field around the background solution reveals an effective potential $V_{eff}$. The dissipative term inherent to the HVP introduces a first-order derivative in the perturbation equation. The paper identifies a condition for potential tachyonic instabilities based on the integral of the effective potential, suggesting that stability depends on the specific choice of parameters (mass $M$ and constants $\eta_0, k$).

Significance and Claims
The paper claims that the Herglotz variational principle provides a novel mechanism to evade standard no-hair theorems in scalar-tensor theories. By introducing a non-dynamical degree of freedom (the Herglotz vector) through a dissipative framework, the theory supports regular, nontrivial scalar hair on a Schwarzschild background without requiring the complex arbitrary functions found in extended Horndeski theories or the singularities of the BBMB solution.

The author posits that this framework offers a natural setting for modeling the irreversible coupling between classical gravitational fields and quantum matter. The work suggests that HVP-type gravity theories could be a promising direction for further investigation, potentially extending to Einstein-Gauss-Bonnet theories without arbitrary coupling functions. The paper concludes modestly, noting that a full stability analysis requires metric perturbations and that the physical relevance of rotating (Kerr) solutions and charged (Reissner-Nordström) extensions remains to be explored.