We study the existence of gradient conformal Killing vectors (CKVs) in the class of locally rotationally symmetric (LRS) spacetimes which generalizes spherically symmetric spacetimes, and investigate some implications for the evolutionary character of marginally outer trapped surfaces. We first study existence of gradient CKVs via the obtention of a relationship between the Ricci curvature and the gradient of the divergence of the CKV. This provides an alternative set of equations, for which the integrability condition is obtained, to analyze the existence of gradient CKVs. A uniqueness result is obtained in the case of perfect fluids, where it is demonstrated that the Robertson–Walker solution is the unique perfect fluid solution with a nonvanishing pressure, admitting a timelike gradient CKV. The constant mean curvature condition for LRS spacetimes is also obtained, characterized by three distinct conditions which are specified by a set of three scalars. Linear combinations of these scalars, whose vanishing define the constant mean curvature condition, turn out to be related to the evolutions of null expansions of 2-spheres along their null normal directions. As such, some implications for the existence of black holes and the character of the associated horizons are obtained. It is further shown that dynamical black holes of increasing area, with a non-vanishing heat flux across the horizon, will be in equilibrium, with respect to the frame of the conformal observers.

中文翻译：

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LRS 时空中的梯度共形平稳性和 CMC 条件


我们研究了局部旋转对称（LRS）时空类中梯度共形杀伤向量（CKV）的存在，该类概括了球对称时空，并研究了边缘外俘获表面演化特征的一些含义。我们首先通过获得Ricci曲率和CKV散度梯度之间的关系来研究梯度CKV的存在性。这提供了一组替代方程，获得了可积条件，以分析梯度 CKV 的存在性。在完美流体的情况下获得了唯一性结果，证明了 Robertson-Walker 解是唯一具有非零压力的完美流体解，允许类时梯度 CKV。还获得了 LRS 时空的恒定平均曲率条件，其特征在于由一组三个标量指定的三个不同条件。这些标量的线性组合（其消失定义了恒定平均曲率条件）与 2-球体沿其零法线方向的零膨胀的演化有关。因此，获得了黑洞存在和相关视界特征的一些含义。进一步表明，相对于共形观察者的框架，面积不断增加的动态黑洞，在地平线上具有不消失的热通量，将处于平衡状态。