非共形映射：分类与奇异动力学
Disformal Maps: Classification and Singular Dynamics
中文总结 AI 辅助
该研究推导了逆非共形度规的显式公式，将非共形变换按霍金斯-埃利斯分类为四类并关联其凯莱-哈密顿次数与塞格雷子类，得到奇异变换对应的类幻影能量-动量张量，可用于研究不同非共形变换的运动学与动力学性质。
中文摘要 AI 辅助
在不明确引力系统场内容的前提下，我们考虑四维洛伦兹流形上度规的一般非共形变换 $g_{\mu\nu}=Ch_{\mu\nu}+Dt_{\mu\nu}$。利用凯莱-哈密顿定理，我们推导了逆非共形度规的显式公式。通过霍金斯-埃利斯（Hawking-Ellis）分类，我们根据可能的若尔当块结构将非共形变换分为四类：I型、II型、III型和IV型。通过研究本征值，我们进一步将每种类型划分为对应的塞格雷（Segre）子类。我们发现非共形张量 $t_{\mu\nu}$ 的凯莱-哈密顿次数、其霍金斯-埃利斯类型与塞格雷子类之间存在明确关联，一旦仅知道凯莱-哈密顿次数，这种关联可限制可能的霍金斯-埃利斯类型；在某些情况下，甚至无需进行完整的若尔当分解即可确定类型。对于奇异变换，当出现新的动力学自由度时，我们得到了其对应的类幻影（mimetic）能量-动量张量 $T^\star_{\mu\nu}$ 的一般形式。我们证明，对于I型，$t_{\mu\nu}$ 和 $T^\star_{\mu\nu}$ 的霍金斯-埃利斯类型始终一致；对于II型和III型，二者可能不同；而IV型通常保持不变，但当复对映射为重复实本征值时可退化为I型。这使得可以直接从 $t_{\mu\nu}$ 的霍金斯-埃利斯类型推断 $T^\star_{\mu\nu}$ 的物理性质。我们将该框架应用于两个具体案例：$t_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi$（其中 $\phi$ 为标量场）和 $t_{\mu\nu}=F^\alpha{}_\mu F_{\alpha\nu}$（其中 $F_{\mu\nu}$ 为规范场的场强张量）。该通用框架可用于系统研究具有不同场内容的各种可逆与不可逆非共形变换的运动学和动力学性质。
英文摘要
Being agnostic about the field content of a gravitational system, we consider a general disformal transformation of the metric, $g_{μν}=Ch_{μν}+Dt_{μν}$, on a four-dimensional Lorentzian manifold. Using the Cayley-Hamilton theorem, we derive an explicit formula for the inverse disformed metric. Implementing the Hawking-Ellis classification, we categorize disformal transformations into four types: Type I, II, III, and IV, based on possible Jordan block structures. By examining the eigenvalues, we further classify each type into its corresponding Segre subclasses. We find explicit links between the Cayley-Hamilton degree of the disformal tensor $t_{μν}$, its Hawking-Ellis type, and its Segre subclass, which can restrict the possible Hawking-Ellis types once only the Cayley-Hamilton degree is known. In some cases, the type can be determined without even performing a full Jordan decomposition. For singular transformations, when new dynamical degrees of freedom emerge, we obtain the general form of their corresponding mimetic energy-momentum tensor $T^\star_{μν}$. We show that the Hawking-Ellis types of $t_{μν}$ and $T^\star_{μν}$ always coincide for Type I. For Types II and III it can differ, while Type IV is preserved generically but can reduce to Type I when the complex pair is mapped to a repeated real eigenvalue. This makes it possible to infer physical properties of $T^\star_{μν}$ directly from the Hawking-Ellis type of $t_{μν}$. We apply our setup to two specific cases: $t_{μν}=\partial_μϕ\partial_νϕ$ and $t_{μν}=F^α{}_μF_{αν}$, where $ϕ$ is a scalar field and $F_{μν}$ is the field-strength tensor of a gauge field. This general framework can be used to systematically study the kinematical and dynamical properties of various invertible and non-invertible disformal transformations with different field content.
