论文标题
在弱能量条件下，球形对称的对称反面示例和正质量定理
Spherically symmetric counter examples to the Penrose inequality and the positive mass theorem under the assumption of the weak energy condition
论文作者
论文摘要
在研究数学一般相对论时可以假设的各种能量条件中，最简单的是弱的能量条件$μ\ geq 0 $，这简单地表明观察到的质量能量密度必须是非负的。这种能量条件的关注不如所谓的主要能量条件。当出现弱能量条件下的penrose不平等问题的自然问题时，我们找不到文献中的任何结果，即使在球形对称性中，不平等现象也无法立即清楚。这导致我们构建了一个球形对称的渐近初始数据集，该数据满足了弱能量条件，违反了Penrose猜想的“通常”配方。构造本身是相当基本的。但是，反例的后果非常有趣。 Penrose的不平等是通过某些启发式论证在\ cite {penrose}中猜想的，在我们讨论的情况下，在弱能量条件下继续保持这种论点。但是，存在一个反示例。此外，这些方法自然而然地导致了假设能量条件弱的正质量定理的构造示例。可以将计数器示例构造为差异为$ \ mathbb {r}^3 $，并且不包含最小的表面，也不包含未来的明显范围。
Of the various energy conditions which can be assumed when studying mathematical general relativity, intuitively the simplest is the weak energy condition $μ\geq 0$ which simply states that the observed mass-energy density must be non-negative. This energy condition has not received as much attention as the so-called dominant energy condition. When the natural question of the Penrose inequality in the context of the weak energy condition arose, we could not find any results in the literature, and it was not immediately clear whether the inequality would hold, even in spherical symmetry. This led us to constructing a spherically symmetric asymptotically flat initial data set satisfying the weak energy condition which violates the "usual" formulations of the Penrose conjecture. The construction itself is quite elementary. However, the consequences of the counterexample are quite interesting. The Penrose inequality was conjectured by Penrose in \cite{Penrose} using certain heuristic arguments which, as we discuss, continue to hold in the case of the weak energy condition. Yet, a counter example exists. Moreover, the methods developed naturally led to the construction of a counter example to the positive mass theorem assuming the weak energy condition. The counter example can be constructed to be diffeomorphic to $\mathbb{R}^3$ and to contain no minimal surfaces and no future apparent horizons.
