Universality of Free Fall: Newton vs Einstein

Hawking in Free Fall

Newton's theory of gravitation is described by

• $\rho =\Delta\phi$  (mass density $\rho$ given by gravitational potential $\phi$),      (1)
• $a =-\nabla\phi$  (Universality of acceleration $a$ in Free Fall = UFF),                     (2)

and is extended to Newton's mechanics by a multiplication of (2) by $\rho$ to get

• $\rho a = f$   (Newton's 2nd Law mass x acceleration = force)                        (3)

with force $f$ defined through gravitational force $-\rho\nabla\phi$. Here gravitational mass from (1)-(2) = inertial mass from (3) also named Equivalence Principle EP. 

We see that in Newton's mechanics, UFF serves as a basic postulate, while EP appears as a consequence and does not need to be added as an independent postulate.   

In Einstein's General Theory of Relativity GR, EP serves as a basic postulate, while free fall universality is the same as "geodesic free fall in curved space-time" as the new feature of GR expressed in Einstein's (field) equations.  

We see that both Newton's mechanics and Einstein's mechanics satisfy UFF and EP and one may ask if that is enough to make them effectively express the same thing? 

Now UFF alone would seem to specify motion under gravitation and so Newton's and Einstein's Universa would move the same way under free fall. 

We can also ask what the difference can be between free fall under gravitational force according to Newton (2) and "geodesic free fall" according to Einstein's GR? 

Newton (2) extends by multiplication by mass density $\rho$ and so only applies to bodies having positive mass, and so Newton does not say that massless light is subject to gravitational free fall. 

 

On the other hand, in Einstein's GR light is subject to a form of "geodesic free fall" even if it is massless. 

We understand that Einstein differs from Newton by including light without mass to be subjected to gravitational force acting on bodies with mass. Is this a fair description?