A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity
Abstract
Buchdahl transformations provide a geometric route from vacuum spacetimes to scalar-field solutions. We formulate this mechanism as a target-covariant solution generator for Einstein gravity coupled to nonlinear sigma models in D>3 dimensions. On any connected region in which a Ricci-flat seed admits a non-null hypersurface-orthogonal cyclic coordinate with fixed-sign nonzero norm, the logarithm of that norm defines a Buchdahl potential Σ. A one-parameter redistribution of the cyclic norm then generates the exact Ricci source RAB=CD(β)∂AΣ∂BΣ, with CD(β)=(D−2)/(D−3)(1−β2), while Σ remains harmonic in the deformed metric. Composing Σ with an affinely parametrized geodesic of a non-degenerate target metric whose squared speed is CD(β) therefore yields an exact Einstein--sigma-model solution. For positive-definite targets, the corresponding pullback condition also implies that the nontrivial scalar map has one-dimensional image; for indefinite targets, the geodesic ansatz remains a sufficient solution-generating sector but need not exhaust all possible realizations of the same rank-one spacetime source. Through the Einstein/Jordan-frame correspondence, the construction gives vacuum tensor--multiscalar solutions as well. We illustrate the unified generator with flat, spherical, hyperbolic and axion--dilaton targets and with Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza--Klein bubble and C-metric seeds. The result separates cleanly the geometric Buchdahl source from its target-space realization and extends this rank-one cyclic sector across static, cosmological, wave and accelerating geometries.
Versions
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BibTeX
@misc{pereira2026buchdahl,
title = {{A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity}},
author = {David S. Pereira and Francisco S. N. Lobo and José Pedro Mimoso},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.20180},
eprint = {2608.20180},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20180},
}
RIS
TY - UNPB
TI - A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity
AU - David S. Pereira
AU - Francisco S. N. Lobo
AU - José Pedro Mimoso
PY - 2026
DA - 2026/08/20
PB - arXiv
DO - 10.48550/arxiv.2608.20180
UR - https://arxiv.org/abs/2608.20180
AB - Buchdahl transformations provide a geometric route from vacuum spacetimes to scalar-field solutions. We formulate this mechanism as a target-covariant solution generator for Einstein gravity coupled to nonlinear sigma models in D>3 dimensions. On any connected region in which a Ricci-flat seed admits a non-null hypersurface-orthogonal cyclic coordinate with fixed-sign nonzero norm, the logarithm of that norm defines a Buchdahl potential Σ. A one-parameter redistribution of the cyclic norm then generates the exact Ricci source R_AB=C_D(β)∂_AΣ∂_BΣ, with C_D(β)=(D−2)/(D−3)(1−β^2), while Σ remains harmonic in the deformed metric. Composing Σ with an affinely parametrized geodesic of a non-degenerate target metric whose squared speed is C_D(β) therefore yields an exact Einstein--sigma-model solution. For positive-definite targets, the corresponding pullback condition also implies that the nontrivial scalar map has one-dimensional image; for indefinite targets, the geodesic ansatz remains a sufficient solution-generating sector but need not exhaust all possible realizations of the same rank-one spacetime source. Through the Einstein/Jordan-frame correspondence, the construction gives vacuum tensor--multiscalar solutions as well. We illustrate the unified generator with flat, spherical, hyperbolic and axion--dilaton targets and with Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza--Klein bubble and C-metric seeds. The result separates cleanly the geometric Buchdahl source from its target-space realization and extends this rank-one cyclic sector across static, cosmological, wave and accelerating geometries.
ER -