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##L^\alpha_\beta## is a function of the Newtonian potential ##\phi##.

My question is: Is this a valid tensor equation?

The transformation is motivated by the idea that when ##\phi## is zero then you already have a locally inertial frame hence ##L^\alpha_\beta## are all zero and ##x^{\alpha '} = x^{\alpha}## which is very understandable. But is the equation ##x^{\alpha '} = (\delta^\alpha_\beta + L^\alpha_\beta)x^\beta## symbolically correct (because the superscripts dont balance out like they normally do in tensor calculus)? But may be if ##L^\alpha_\beta## is not a tensor then it does not have to obey those principles of tensor calculus.