Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid
Spherically symmetric similarity solutions of the Einstein field equations for a perfect fluid
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完美流体爱因斯坦场方程的球对称相似解
DOI：
10.1007/bf01646482
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发表时间：
1971
影响因子：
2.4
通讯作者：
A. Taub
中科院分区：
文献类型：
--
作者：
M. Cahill;A. Taub
研究了球对称时空，其中存在由向量φ μ生成的单参数共形变换群，使得φ μ;v+ φ μ;μ=2gμv.结果表明，这种时空的度规系数本质上取决于单变量z =r/t，其中r是径向坐标，t是时间。爱因斯坦场方程然后减少到常微分方程。这些方程的解类似于经典流体力学理论的相似解。如果场的源是一种理想流体，其比内能仅是温度的函数，则场方程的解通过指定类时超曲面z =常数上的数据唯一地确定，并且是相似解。拟合的相似性解决方案的另一个解决方案的场方程的超曲面=常数所描述的冲击的问题进行处理。本文给出了描述Robertson-Walker时空的一个特殊的相似解，其中w = 3 p。这个解适合于爱因斯坦场方程的一个特殊的静态解，它具有奇异性atr=0。爱因斯坦场方程的解在除atr=0 λ t和激波外的任何地方都是正则的。特殊的Robertson-Walker度规也适合于一类特殊的坍缩尘埃解（也是相似解）。除了atr=t=0和激波外，所得解在任何地方都是正则的.
Spherically symmetric space-times which admit a one parameter group of conformal transformations generated by a vectorξμ such thatξμ;v+ξv;μ=2gμv are studied. It is shown that the metric coefficients of such space-times depend essentially on the single variablez=r/t wherer is a radial coordinate andt is the time. The Einstein field equations then reduce to ordinary differential equations. The solutions of these equations are analogous to the similarity solutions of the classical theory of hydrodynamics. In case the source of the field is a perfect fluid whose specific internal energy is a function of temperature alone, the solution of the field equations is uniquely determined by specifying data on the time-like hypersurfacez=constant and is a similarity solution. The problem of fitting a similarity solution to another solution of the field equations across a shock described by the hypersurfacez=constant is treated. A particular similarity solution for whichw=3p obtains is shown to describe a Robertson-Walker space-time. This solution is fitted to a special static solution of the Einstein field equations which has a singularity atr=0. The resulting solution of the Einstein field equations is shown to be regular everywhere except atr=0≧t and the shock. The special Robertson-Walker metric is also fitted to a particular class of collapsing dust solutions (which are also similarity solutions) across a shock. The resulting solution is regular everywhere except atr=t=0 and on the shock.