Gravitational Mass = Inertial Mass: Einstein or Galileo?

Einstein takes as postulate of his general theory of relativity that

• gravitational mass = inertial mass.  

To Einstein this represented a deep insight into the inner nature of things, which he named the Equivalence Principle. To Galileo the same thing was a most natural consequence of his theoretical insight from experiments of dropping objects from the Tower of Pisa noting that all objects fall in the same way (modulo air resistance) and reflecting over the connection between force and motion.

Let us see if we can understand what to Einstein was beyond comprehension and to Galileo more or less self-evident. Newton's second law states that 

• $m_i\frac{dv_i}{dt}=F_i$, 

where $m_i$ is the inertial mass of a body showing acceleration $\frac{dv_i}{dt}$ with $v_i$ velocity and $t$ time when subject to a force $F_i$. On the other hand, the same body when subject to a gravitational force $F_g$, shows an acceleration $\frac{dv_g}{dt}$ satisfying 

•   $m_g\frac{dv_g}{dt}=F_g$,

where $m_g$ is the gravitational mass. 

To find out if $m_i=m_g$, let us consider the following experiment: Consider two identical bodies,  a body $A$ at rest on a frictionless table and another body $B$ in your hand with the two bodies connected by a weightless string stretched over a frictionless wheel attached at the end of the table, see picture in earlier version of this post. Then remove your hand and observe the action of the two-body-string system.  Observe that $A$ is acted upon by the horisontal string force $F_s$, while $B$ is acted upon by $F_g-F_s$ with $F_g$ the gravitational force acting on $B$. Since A and B have the same acceleration, we have

• $\frac{F_s}{m_i}=\frac{F_g-F_s}{m_g}$.  

If we now observe that 

• $F_s=\frac{F_g}{2}$,    (1)

then we can conclude that $m_g=m_i$ as a simple experimental verification of the Equivalence Principle. We can also argue that (1) must be true according to Leibniz' principle of sufficient reason, since there is no reason that the two-body-string system should not show this form of symmetry. We can also argue that (1) must hold if we re-orient the system to be all horisontal and pull $B$ with a certain force $F$ which must result in a string force $\frac{F}{2}$. 

Summing up, we have given simple evidence that gravitational mass = inertial mass, based on the insight that there is only one type of mass, namely inertial mass as a measure of acceleration vs force. Since gravitation is a force the measure of acceleration vs gravitational force as gravitational mass is necessarily the same as inertial mass. This is captured in the experiment with $B$ subject to (vertical) gravitational force (minus vertical string force) and $A$ to horizontal string force. 

The Equivalence Principle is thus a direct consequence of Newtonian mechanics, and as such a most questionable empty Basic Postulate for general relativity. As usual Einstein managed to create confusion rather than clarification. For more reason to this verdict see earlier posts on the Equivalence Principle.