The TOV equation appears as the relativistic counterpart of the classical condition for hydrostatic equilibrium. In the present work we aim at showing that a generalised TOV equation also characterises the equilibrium of models endowed with other symmetries besides spherical. We apply the dual null formalism to spacetimes with two dimensional spherical, planar and hyperbolic symmetries with a perfect fluid as the source. We also assume a Killing vector field orthogonal to the surfaces of symmetry, which gives us static solutions, in the timelike Killing field case, and homogeneous dynamical solutions in the case the Killing field is spacelike. In order to treat equally all the aforementioned cases, we discuss the definition of a quasi-local energy for the spacetimes with planar and hyperbolic foliations, since the Hawking-Hayward definition only applies to compact foliations. After this procedure, we are able to translate our geometrical formalism to the fluid dynamics language in a unified way, to find the generalized TOV equation, for the three cases when the solution is static, and to obtain the evolution equation, for the homogeneous spacetime cases. Remarkably, we show that the static solutions which are not spherically symmetric violate the weak energy condition (WEC). We have also shown that the counterpart of the TOV equation for the spatially homogeneous models is just the familiar equation \r{ho} + P = 0, defining a cosmological constant-type behaviour, both in the hyperbolic and spherical cases. This implies a violation of the strong energy condition in both cases, added to the above mentioned violation of the weak energy condition in the hyperbolic case. We illustrate our unified treatment obtaining analogs of Schwarzschild interior solution, for an incompressible fluid $\rho = \rho_0$ constant.

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从双零方法看 TOV 均衡的新视角

TOV 方程表现为流体静力平衡的经典条件的相对论对应物。在目前的工作中，我们的目标是表明广义 TOV 方程还表征了除球面之外还具有其他对称性的模型的平衡。我们将双零形式主义应用于具有二维球面、平面和双曲对称性的时空，并以完美流体为源。我们还假设了一个与对称面正交的 Killing 矢量场，它给出了类时 Killing 场情况下的静态解，以及类时 Killing 场情况下的齐次动力学解。为了平等对待上述所有情况，我们讨论了具有平面和双曲叶理的时空的准局域能量的定义，因为霍金-海沃德的定义只适用于紧凑的叶面。在这个过程之后，我们能够以统一的方式将我们的几何形式主义翻译成流体动力学语言，找到广义 TOV 方程，对于三种情况下的解是静态的，并获得演化方程，对于齐次时空案件。值得注意的是，我们表明非球对称的静态解违反了弱能量条件 (WEC)。我们还表明，空间均匀模型的 TOV 方程的对应物只是熟悉的方程 \r{ho} + P = 0，定义了双曲线和球面情况下的宇宙常数类型行为。这意味着在两种情况下都违反了强能量条件，添加到上述双曲线情况下对弱能量条件的违反。我们说明了我们的统一处理获得 Schwarzschild 内部解的类似物，对于不可压缩流体 $\rho = \rho_0$ 常数。