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

Marek Wazny

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arXiv · 2606.30091

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

Marek Wazny

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Abstract

Brans-Dicke theory is reformulated within the Herglotz variational principle (HVP), and an exact black hole solution with scalar hair is obtained for $\omega_{0}=0$ and vanishing potential $V(\phi)=0$. The scalar profile is strictly positive and the resulting stealth Schwarzschild solution arises without fixing the otherwise arbitrary Herglotz function $\eta(r)$. Motivated by the weak-field limit, the explicit choice $\eta(r)=\eta_{0}(r-2M)^k/r^{k+2}$, with $\eta_0$ a constant of dimension length, produces a scalar field configuration remaining regular at the black hole horizon. Consequently, the HVP provides a new mechanism for evading standard no-hair theorems in scalar-tensor theories.

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Marek Wazny. 2026-06-29. Black holes with regular scalar hair in Brans-Dicke gravity via the Herglotz variational principle. https://arxiv.org/abs/2606.30091

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