arXiv 1602.043972016-02gr-qchep-thmath-phmath.DGmath.MP

$f(R)$引力理论的宇宙学奇点定理

Cosmological singularity theorems for $f(R)$ gravity theories

Ivo Alani, Osvaldo Santillan

备注：: An improved version is published in JCAP 05 (2016) 023
DOI：: 10.1088/1475-7516/2016/05/023

摘要：

在目前的工作中，霍金奇异性定理的范围内的$f(R)$理论的一些推广。这些推广定理的假设是：对于任何一般的单位时间类场，物质场满足条件$\bigg(T_{ij}-\frac{g_{ij}}{2} T\bigg)k^i k^j\geq 0$，标量子在其演化过程中取有界正值，并且由此产生的时空是全局双曲的。然后，如果存在一个柯西超曲面$Σ$，且与$Σ$正交的测地同余的展开参数$θ$满足某些特定条件，则可以证明所得到的时空是测地不完备的.参考文献\cite{fewster}的一些数学结果对于证明这一点是非常重要的。本文给出的推广定理直接应用于某些特定的模型，如Hu-Sawicki或Starobinsky的\cite{especif3}，\cite{capoziello4} .然而，对于其他场景，应该实施一些额外的假设，以使测地线不完整性发生。然而，否定这些结果的假设并不一定意味着不存在奇点，而是应该考虑其他数学结果来证明这一点。

英文摘要：

In the present work some generalizations of the Hawking singularity theorems in the context of $f(R)$ theories are presented. The assumptions are of these generalized theorems is that the matter fields satisfy the conditions $\bigg(T_{ij}-\frac{g_{ij}}{2} T\bigg)k^i k^j\geq 0$ for any generic unit time like field, that the scalaron takes bounded positive values during its evolution, and that the resulting space time is globally hyperbolic. Then, if there exist a Cauchy hyper surface $Σ$ for which the expansion parameter $θ$ of the geodesic congruence emanating orthogonally from $Σ$ satisfies some specific conditions, it may be shown that the resulting space time is geodesically incomplete. Some mathematical results of reference \cite{fewster} are very important for proving this. The generalized theorems presented here apply directly some specific models such as the Hu-Sawicki or Starobinsky ones \cite{especif3}, \cite{capoziello4}. However, for other scenarios, some extra assumptions should be implemented for the geodesic incompleteness to take place. However, the negation of the hypothesis of these results does not necessarily imply that a singularity is absent, but that other mathematical results should be considered to prove that.

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