AI 中文总结
该研究构建了适用于 D>3 维耦合非线性西格玛模型的爱因斯坦引力的统一布赫达尔解生成器，可生成多类时空的精确解，分离了几何源与目标空间实现，扩展了秩一循环扇区的适用范围。
AI 中文摘要
布赫达尔变换提供了从真空时空到标量场解的几何途径。我们将该机制构建为 D>3 维下耦合非线性西格玛模型的爱因斯坦引力的目标协变解生成器。在里奇平坦种子存在非零符号的非类空超曲面正交循环坐标的任意连通区域中，该坐标模的对数定义了布赫达尔势 Σ。对循环模的单参数重分布生成精确里奇源 R_{AB}=C_D(β)∂_AΣ∂_BΣ，其中 C_D(β)=(D-2)/(D-3)(1-β²)，同时 Σ 在变形度规中保持调和。将 Σ 与非退化目标度规的仿射参数化测地线（其平方速度为 C_D(β)）组合，即可得到精确的爱因斯坦-西格玛模型解。对于正定目标，相应的拉回条件还意味着非平凡标量映射具有一维像；对于不定目标，测地线假设仍是充分的解生成扇区，但无需穷尽同秩一时空源的所有可能实现。通过爱因斯坦/Jordan 框架对应，该构造也给出真空张量-多标量解。我们以平坦、球形、双曲和轴子-伸缩子目标，以及 Tangherlini、Weyl、Kasner、Rosen 波、Gowdy、Kaluza-Klein 泡和 C-度规种子示例说明该统一生成器。该结果将几何布赫达尔源与其目标空间实现清晰分离，并将该秩一循环扇区扩展到静态、宇宙学、波和加速几何中。
英文摘要
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 $R_{AB}=C_D(β)\partial_AΣ\partial_BΣ$, with $C_D(β)=\frac{D-2}{D-3}(1-β^2)$, 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.
Comments23 pages
