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Following the paper “Why the Riemann Curvature Tensor needs twenty independent components” by David Meldgin UC Davis 2011, I understand that with a coordinates transformation, using the first and the second partial derivatives of the coordinates transformation to set the metric Minkowskian and the first partial derivatives of the metric equal to zero, while leaving the second order partial derivatives of the metric unchanged, one may using the 80 third order and the 140 fourth order partial derivatives of the coordinates transformation set the 200 third order partial derivatives of the metric equal to zero.

Therefore, by making an error of the fourth infinitesimal order, the highest derivatives of the metric to appear in the field equations would be the second order partial derivatives of it. The components of the spacetime tensor in the field equations would be functions of the second order partial derivatives of the metric only.

Suppose the exact form of the spacetime tensor in the field tensor equation is not known.

If the components of the stress-energy tensor and boundary conditions are given, it is expected that the components of the metric that will be the solutions derived from the “unknown” field equations have to be just the components of the Minkowski metric. Hence, these equations have unique solutions. This would not be possible if spacetime was represented in the field equations by nonlinear functions of the second order partial derivatives of the metric.

If the above is correct, it is derived that the field equations are linear in the second order partial derivatives of the metric.

Therefore, spacetime is represented in the field equations by a tensor of the form

If the above is correct, I think there would be no need to make reference to the Poisson equation, or to the least action principle to derive the field equations.