BGV定理与零收敛条件
The BGV Theorem and the Null Convergence Condition
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中文总结 AI 辅助
该研究探讨NCC与BGV定理的关系，证明特定条件下NCC和非正空间曲率可导出BGV定理，一般情况下二者不足以保证BGV定理成立，并提出适用于过去渐近de Sitter永恒暴胀的更通用BGV条件。
中文摘要 AI 辅助
我们研究零收敛条件（Null Convergence Condition, NCC）与Borde-Guth-Vilenkin（BGV）定理之间的关系。首先证明，对于正交于具有零剪切和涡度的类时测地线同余的膨胀时空，当NCC成立且空间曲率非正（³ℛ ≤ 0）时，可导出BGV定理。当存在剪切或时空的非测地线穿入时，情况更复杂：此时BGV构造涉及的局域膨胀不仅依赖于局域标量膨胀，还包含剪切与加速度和局域定义的空间单位矢量收缩的项。一般情况下，零收敛与非正曲率不再足以保证BGV定理成立。我们在共动曲率扰动存在的永恒暴胀背景下讨论该结果，并给出与过去渐近de Sitter永恒暴胀相关的更通用BGV条件版本。
英文摘要
We examine the relationship between the Null Convergence Condition (NCC) and the Borde-Guth-Vilenkin (BGV) Theorem. We first show that, for an expanding spacetime foliated orthogonally to a timelike geodesic congruence with vanishing shear and vorticity, the BGV Theorem follows when the NCC holds and the spatial curvature is non-positive, ${}^3\mathcal{R} \leq 0$. The situation becomes more complex in the presence of shear, or non-geodesic threading of the spacetime. In these cases, the local expansion that enters the BGV construction depends not only on the local scalar expansion, but acquires terms given by the contraction of the shear and acceleration with locally defined spatial unit vectors. In the general case, null convergence and non-positive curvature are no longer sufficient to guarantee that the BGV Theorem holds. We discuss the result in the context of eternal inflation in the presence of comoving curvature perturbations and state a more general version of the BGV condition relevant to asymptotically past-de Sitter eternal inflation.
