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Jordan-frame reconstruction of Einstein-Rosen-type geometries in generalized Brans-Dicke theory

László Árpád Gergely ^1,2,*, Bence Racskó ¹, and Kata Karácsonyi ^1,2

¹Department of Theoretical Physics, University of Szeged, Tisza Lajos krt. 84–86, H-6720 Szeged, Hungary
²Department of Theoretical Physics, HUN-REN Wigner Research Centre for Physics, Konkoly-Thege Miklós út 29–33, H-1121 Budapest, Hungary

^*Contact author: gergely.laszlo@wigner.hu

Phys. Rev. D 114, 104023 – Published 6 November, 2026

DOI: https://doi.org/10.1103/v1f8-cwtp

Abstract

We investigate vacuum generalized Brans-Dicke theory with scalar-dependent coupling $ω (ϕ)$ in an Einstein-Rosen-type metric. Let $W$ be the area density of the two-dimensional Killing orbits, $ϕ$ the Brans-Dicke scalar, and $Ω = W ϕ$ their composite area density. On regions where the gradient of $Ω$ is non-null, we impose $W = W (ϕ)$ and local invertibility of $Ω (ϕ)$ . The scalar equation admits the exact first integral $(3 + 2 ω) [(Ω / ϕ) (d ϕ / d Ω)]^{2} = κ$ , where $κ$ is a signed integration constant, giving direct and inverse reconstruction of the coupling. For $ϕ = b (Ω / Ω_{*})^{a} e^{q Ω}$ one obtains $3 + 2 ω = κ / (a + q Ω)^{2}$ . The choices $Ω = ρ$ and $Ω = t$ define the spacelike- and timelike-gradient branches, each yielding a closed exact solution family. Their mode functions obey the cylindrical Einstein-Rosen and Euler-Poisson-Darboux equations, respectively. In the Einstein frame the normalized scalar is $σ = σ_{0} + \sqrt{| κ |} \ln (Ω / Ω_{0})$ , while $q$ cancels from the metric. In the spacelike branch the scalar contributes to the radial C-energy profile separately from the tensor flux, and the reconstructed smooth-mode class has no curvature-regular axis. In the static subclass, $q = 0$ reproduces the circular-speed law of a singular isothermal dark-matter halo, while $q < 0$ gives a local correspondence with the rising inner NFW rotation curve. The homogeneous timelike subclass is locally Bianchi I, reduces in the Einstein frame to an exact scalar-Kasner geometry, and has a Jordan-frame effective Planck mass that evolves monotonically with Jordan-frame comoving proper time on each connected time interval on which the reconstructed scalar profile is locally invertible.

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Jordan-frame reconstruction of Einstein-Rosen-type geometries in generalized Brans-Dicke theory

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