Relativistic gravitational potential energy is the result of reformulating Newtonian potential energy using special relativity and the Einstein equivalence principle. The essential difference in the derivation is that a relativistic treatment involves multiplication where quantities would be added classically. The exponential solution amounts to a renormalization of classical potential energy. Mach's principle is incorporated implicitly with the consequence that gravitational potential energy would be limited to the rest energy of a test particle. Space-borne experiments could perform a decisive second-order test using current technology.
shells2010dec29.pdf https://drive.google.com/file/d/1NN69Np36Pbu33-v1sHxptGerknDatWOF/view?usp=sharing
tangent.pdf https://drive.google.com/file/d/1_36I4hzD7KScWCF06nUwwBp92-305Pvr/view?usp=sharing