Volume 107A. number 1 PHYSICS LETTERS 7 January 1985 EXACTPOWERLAWSOLUTIONSOFTHEEINSTEINEQUATIONS* Kjell ROSQUIST institute of Theoretical Physics, University of Stockholm, Vanadisvcigen 9, S-113 46 Stockholm, Sweden Robert T. JANTZEN Istituto di Fisica, University of Rome, 00185 Rome, Italy and Department of Mathematical Sciences, Villanova University, I/il.&ova, PA 19085, USA Received 23 October 1984 An exact power law metric is discussed which arises when one considers the reduced Einstein equations for certain scale invariant variables associated with a spatially homogeneous or spatially self-similar vacuum or nonvacuum space-time. The metric contains a number of new solutions as welJ as many known ones. Recently Rosquist [l] has reformulated the Bogoyavlensky-Novikov qualitative treatment of the Einstein equations for an orthogonal spatially homogeneous perfect fluid [2,3] in a way which ext-nds to the nonortho- gonal (tilted) case while remaining compatible with the symmetry transformations of those equations. Although discussed explicitly for Bianchi type VI, the general spatially homogeneous or spatially self-similar case is easily considered [4] . In this treatment, hamiltonian methods in conjunction with the scale invariance of the Einstein equations are used to obtain a reduced system of equations involving only the geometrical variables. Singular points in the inte- rior of the physical domain of this system of first-order differential equations lead to a class of exact power law solutions of the Einstein equations in the same sense as defined by Wainwright for the orthogonal case [5,6]. The form of the metric for this class of “singular point solutions” is determined completely as a function of the varia- ble values which characterize the singular point. If t is the natural cosmological time for these space-times (proper time along the normal congruence in the spatially homogeneous case and the proper time of the conformally related spatially homogeneous space-time in the spatially self-similar case), then the components of the Riemann, Ricci and Einstein tensors in the natural orthonormal frame (the conformally related spatially homogeneous components in the spatially self-similar case) are polynomials in the constant parameters specifying the metric multiplied by a common factor of tp2. Finding the singular points using the original variables is quite difficult, but one can take the alternative approach of im- posing the algebraic conditions on the metric parameters which are necessary to make the Einstein tensor assume the form of the energy-momentum tensor of a perfect fluid. This was in fact done by Dunn and Tupper [7] for a very special type VI0 orthogonal case (the Taub-like diagonal case, whose solution is the M-asymptote solution of Ellis and MacCallum [8] ); they also considered how the metric parameters could be varied to accommodate an additional electromagnetic source. Similar considerations apply to the present case as well. We consider here the nonsemisimple spatially homogeneous case of Bianchi types I-VII and their spatially self- * This work was supported in part by the Swedish Natural Science Research Council. K.R. was also supported by a 1984 Lennander grant from the University of Uppsaia. 0.375-9601/85/S 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 29