扩展引力理论中的Komar超势

中文摘要 AI 辅助

本文研究了拉格朗日量对黎曼张量具有任意依赖且包含任意阶导数的情况。诺特第二定理分别应用于${\cal L}_{G}$和${\cal L}_{M}$，其中物质拉格朗日量依赖于一个标量场。对于${\cal L}_{G}$，微分同胚不变性要求广义爱因斯坦张量满足$\nabla_{\alpha}E^{\alpha\beta}=0$，并且存在一个流满足$0=\partial_{\alpha}(\sqrt{g}\\, J^{\alpha}_{G2})$。主要结果是给出了超势$\Phi^{[\alpha\mu]}$的显式公式，该超势满足$\partial_{\mu}\Phi^{[\alpha\mu]}=\sqrt{g}\\,J^{\alpha}_{G2}$。文中讨论了在$f(R)$引力和各种二次引力理论中的一些应用。当物质场与度规的耦合包含度规导数时，物质部分也有类似的结果，包括超势。

英文摘要

This paper investigates Lagrangians with arbitrary dependence on the Riemann tensor and any number of derivatives. Noether's second theorem is applied separately to ${\cal L}_{G}$ and ${\cal L}_{M}$, where the matter Lagrangian depends on a scalar field. For ${\cal L}_{G}$ diffeomorphism invariance requires that the generalized Einstein tensor satisfy $\nabla_αE^{αβ}=0$ and that there is a current satisfying $0=\partial_α(\sqrt{g}\, J^α_{G2})$. The main result is an explicit formula for the superpotential $Φ^{[αμ]}$ that solves $\partial_μΦ^{[αμ]}=\sqrt{g}\,J^α_{G2}$. Some applications to $f(R)$ gravity and various quadratic gravity theories are discussed. There are analogous results, including superpotentials, for the matter sector whenever the couplings to the matter fields contain derivatives of the metric.

发表机构

• West Virginia University（西弗吉尼亚大学）

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