In Newtonian theory, the gravitational potential exerts a force F = dp/dt = m on a particle with mass m and momentum p. Before

In Newtonian theory, the gravitational potential Φ exerts a force F = dp/dt = −m∇Φ on a particle with mass m and momentum p. Before Einstein formulated general relativity, some physicists constructed relativistic theories of gravity in which a Newtonian-like scalar gravitational field Φ exerted a 4-force F(vector) = dp(vector)/dτ on any particle with rest mass m, 4-velocity u(vector), and 4-momentum p(vector) = mu(vector). What must that force law have been for it to 

(i) Obey the Principle of Relativity, 

(ii) Reduce to Newton’s law in the nonrelativistic limit

(iii) Preserve the particle’s rest mass as time passes?