We generalize Birkhoff’s Theorem in the following fashion. We find necessary and sufficient conditions for any spherically symmetric space-time to be static in terms of the eigenvalues of the stress-energy tensor. In particular, we generalize the Tolman–Oppenheimer–Volkoff equation and prove that Birkhoff’s theorem holds under the weaker hypothesis of no pressure (with respect to an appropriate frame.) We provide equations that show how the coefficients of the metric relate to the eigenvalues of the stress-energy tensor. These involve integrals that are simple functions of those eigenvalues. We also determine among all static spherically symmetric space–times those that are asymptotically flat. A few examples are presented taking advantage of the results. The calculations are done by viewing the space–times as warped products and the computations are done using Cartan’s moving frames approach.

我们以以下方式推广伯克霍夫定理。我们发现，就应力-能量张量的特征值而言，任何球对称时空都是静态的必要和充分条件。特别是，我们推广了Tolman–Oppenheimer–Volkoff方程，并证明Birkhoff定理在无压力的较弱假设（相对于适当的框架）下成立。应力能量张量。这些涉及积分，这些积分是这些特征值的简单函数。我们还确定了所有静态球对称空间-渐近平坦的时间。列举了一些利用这些结果的例子。