1. I Ozsváth and E Schücking, An anti-Mach metric, in: Recent developments in general relativity (Pergamon Press, Oxford, 1962), p. 339 [1a] The same is not true for the complete Einstein’s equations which require the speculative dark matter and dark energy in order to explain the cosmological observations. On the one hand, the theory predicts that about 27% of the total content of the Universe is made of non-baryonic dark matter particles, which should certainly be predicted by some extension of the Standard Model of particle physics. However, there is no indication of any new physics beyond the Standard Model which has been verified at the LHC. On the other hand, the dark energy, which according to the theory, constitutes about 68% of the total content of the Universe, poses serious confrontation with particle physics. [1b] Although other field equations, for example Maxwell’s equations, possess source-free solu tions and do not pose any consistency problem, the case of gravitation is different wherein the existence of a source-free solution goes against the requirements of Mach’s principle. [1c] Though the presence of a singularity in a solution can be asserted unanimously by the diver gence of the Kretschmann scalar K, the flatness of a solution cannot be asserted by the vanishing of K. For this we have to depend on the vanishing of the Riemann tensor. We shall come through some curved solutions for which K will be vanishing!
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