We develop a generic spacetime model in general relativity which can be used to build any gravitational model within general relativity. The generic model uses two types of assumptions: (a) geometric assumptions in addition to the inherent geometric identities of the Riemannian geometry of spacetime and (b) assumptions defining a class of observers by means of their four-velocity \(u^{a}\) which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the \(1+3\) decomposition of tensor fields in spacetime. The \(1+3\) decomposition results in two major results. The \(1+3\) decomposition of \(u_{a;b}\) defines the kinematic variables of the model (expansion, rotation, shear and four-acceleration) and defines the kinematics of the gravitational model. The \(1+3\) decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and defines the dynamics of the model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the four-velocity of the observers \(u_{a}\) there exists a second universal vector field \(n_{a}\) in spacetime so that one has a so-called double congruence \((u_{a},n_{a})\) which can be used to define the \(1+1+2\) decomposition of tensor fields. The \(1+1+2\) decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables. After presenting and discussing the results in their full generality we show how they are applied in practice by considering in a step by step approach the case of a string fluid in Bianchi I spacetime for the comoving observers.

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广义相对论的通用模型

我们在广义相对论中开发了一个通用的时空模型，该模型可用于在广义相对论内建立任何引力模型。通用模型使用两种类型的假设：（a）除了时空黎曼几何的固有几何恒等式之外的几何假设，以及（b）通过四速度\（u ^ {a } \），它是一个单位时间形式的矢量场。几何假设通常涉及对称假设（所谓的归类）。后者介绍了时空中张量场的\（1 + 3 \）分解。的\（1 + 3 \）分解导致两个主要结果。\（u_ {a; b} \）的\（1 + 3 \）分解定义模型的运动学变量（扩展，旋转，剪切和四加速度），并定义重力模型的运动学。代表所有引力物质的能量动量张量的\（1 + 3 \）分解引入了模型的动态变量（能量密度，各向同性压力，动量传递或热通量矢量和各向异性压力的无痕张量），由定义的观察者测量并定义模型的动力学。模型假定的对称性对模型的运动学和动力学变量都具有约束作用。作为通用模型的第二个进一步发展，我们假设除了观察者\（u_ {a} \）的四速度外在时空中存在第二个通用矢量场\（n_ {a} \），因此其中一个具有所谓的双重同余\（（u_ {a}，n_ {a}）\），可用于定义\张量场的（1 + 1 + 2 \）分解。的\（1 + 1 + 2 \）分解导致有关这两个字段构建双一致性和涉及多个物理变量的更精细动力学扩展运动学。在对结果进行了全面介绍和讨论之后，我们将通过逐步研究比安奇一世时空中串流流体的情况，来向共同观察者展示如何将其实际应用。