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Comment on “Emergence of the Cotton tensor for describing gravity”

Recently, it has been claimed by J. Harada [Phys. Rev. D 103, L121502 (2021) that the Cotton tensor can describe the effects of gravity beyond general relativity. In this short comment, we show that the Cotton tensor is already present in general relativity. Moreover, we show that Harada's proposal is equivalent to general relativity concerning the equation of motion both for the sources and for the trace-free part of the curvature. In essence, what remains completely free in Harada's theory are the field equations relating the part of the curvature which is locally determined by the energy-momentum distribution.