The Universality of vacuum Einstein equations with cosmological constant
The Universality of vacuum Einstein equations with cosmological constant
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DOI：
10.1088/0264-9381/11/6/015
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发表时间：
1994-06
影响因子：
3.5
通讯作者：
M. Ferraris;M. Francaviglia;I. Volovich
中科院分区：
文献类型：
--
作者：
M. Ferraris;M. Francaviglia;I. Volovich
它表明，对于广泛的一类解析拉格朗日，这只取决于标量曲率的度量和连接，所谓的“Palatini形式主义”的应用，即治疗的度量和连接作为独立变量，导致“通用”方程。如果时空的维数n大于2，这些普遍方程是真空爱因斯坦方程与宇宙学常数为一个通用的拉格朗日方程，并适当地取代其他普遍方程在退化点。我们表明，退化发生在特别是共形不变的拉格朗日，我们证明了他们的解决方案是共形等价的爱因斯坦方程的解决方案。相反，对于二维时空，我们发现普适方程总是常数量曲率方程;在这种情况下，联络是外尔联络，包含度规的列维-奇维塔联络和由共形不变性产生的附加向量场。作为一个例子，我们详细研究了一些多项式拉格朗日函数，并讨论了它们的退化点。
It is shown that for a wide class of analytic Lagrangians, which depend only on the scalar curvature of a metric and a connection, the application of the so called `Palatini formalism', i.e. treating the metric and the connection as independent variables, leads to `universal' equations. If the dimension n of spacetime is greater than two these universal equations are vacuum Einstein equations with cosmological constant for a generic Lagrangian and are suitably replaced by other universal equations at degenerate points. We show that degeneracy takes place in particular for conformally invariant Lagrangians and we prove that their solutions are conformally equivalent to solutions of Einstein's equations. For two-dimensional spacetimes we find instead that the universal equation is always the equation of constant scalar curvature; in this case the connection is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vector field ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their degenerate points.