Vacuum solutions of Einstein equations that depend on one coordinate

In the famous textbook written by Landau and Lifshitz all the vacuum metrics of the general theory of relativity are derived, which depend on one coordinate in the absence of a cosmological constant. Unfortunately, when considering these solutions the authors missed some of the possible solutions discussed in this article. An exact solution is demonstrated, which is absent in the book by Landau and Lifshitz. It describes space-time with a gravitational wave of zero frequency. It is shown that there are no other solutions of this type than listed above and Minkowski’s metrics. The list of vacuum metrics that depend on one coordinate is not complete without solution provided in this paper.

ON TWO-DIMENSIONAL GRAVITY

The field equations that result from the use of R2–the square of the Ricci scalar – in the gravitational action are studied in two dimensions. Some general features of the model are noted; in particular, the existence of a solution R=Λ. The form of the static vacuum metrics is obtained. The problem of cosmology is discussed.