arXiv 1908.087782019-08hep-thgr-qc

四维标度不变引力的临界引力

Critical gravity from four dimensional scale invariant gravity

Ariel Edery, Yu Nakayama

备注：: 14 pages; v2: updated references
DOI：: 10.1007/JHEP11(2019)169

摘要：

证明了由纯二次作用量$β\,C_{μνσρ} C^{μνσρ} + α\,R^2$给出的四维标度不变引力存在一个临界条件，其中$C^μ_{\,\,νσρ}$是Weyl张量，$R$是Ricci标量，$β$和$α$是无量纲参数. dS或AdS背景中的临界条件是$β=6 α$。这导致了临界引力，在那里，大质量的自旋2物理幽灵变成了无质量的自旋2引力子。与最初关于临界引力的工作相反，没有将具有宇宙学常数的爱因斯坦引力明确地加入到高阶导数作用量中。临界条件是在两个独立的方式获得。在第一种情况下，我们证明了初始作用量与包含爱因斯坦引力、宇宙学常数、无质量标量场和外尔平方引力的作用量之间的等价性。标度不变性被自发地破坏。关于dS或AdS背景的线性化Einstein-Weyl方程产生临界条件$β=6α$。在第二种情况下，我们直接使用原始的二次动作。经过适当的领域重新定义，其中的度量扰动是无痕和横向的，我们得到线性方程的dS或AdS的背景，产生的临界条件$β= 6α$。和第一种情况一样，我们也得到一个传播的无质量标量场。将$β=6α$代入史瓦西和克尔AdS或dS黑洞在高导数引力下的能量和熵公式，得到的值为零，与在临界引力的原始工作中得到的值相同。讨论了边界条件在放宽$β=6α$条件中的作用。

英文摘要：

We show that a critical condition exists in four dimensional scale invariant gravity given by the pure quadratic action $β\,C_{μνσρ} C^{μνσρ} + α\,R^2$ where $C^μ_{\,\,νσρ}$ is the Weyl tensor, $R$ is the Ricci scalar and $β$ and $α$ are dimensionless parameters. The critical condition in a dS or AdS background is $β=6 α$. This leads to critical gravity where the massive spin two physical ghost becomes a massless spin two graviton. In contrast to the original work on critical gravity, no Einstein gravity with a cosmological constant is added explicitly to the higher-derivative action. The critical condition is obtained in two independent ways. In the first case, we show the equivalence between the initial action and an action containing Einstein gravity, a cosmological constant, a massless scalar field plus Weyl squared gravity. The scale invariance is spontaneously broken. The linearized Einstein-Weyl equations about a dS or AdS background yield the critical condition $β=6α$. In the second case, we work directly with the original quadratic action. After a suitable field redefinition, where the metric perturbation is traceless and transverse, we obtain linearized equations about a dS or AdS background that yield the critical condition $β= 6α$. As in the first case, we also obtain a propagating massless scalar field. Substituting $β=6α$ into the energy and entropy formula for the Schwarzschild and Kerr AdS or dS black hole in higher-derivative gravity yields zero, the same value obtained in the original work on critical gravity. We discuss the role of boundary conditions in relaxing the $β=6α$ condition.

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