Equivalence principle and geometry

[checksix]

TL;DR Summary

why does geometric treatment of gravitation require equivalence of inertial and gravitational mass?

In the first sentence of Chapter 2 in Ben Crowell's "General Relativity" he states:

"The geometrical treatment of space, time, and gravity only requires as its basis the equivalence of inertial and gravitational mass".

This is stated as if it's an obvious fact, but I don't understand why. Why does a geometric treatment of gravitation require the equivalence principle? I must be missing something obvious. What am I missing here?

 

[Ibix]

Why does a geometric treatment of gravitation require the equivalence principle? I must be missing something obvious. What am I missing here?

In Newtonian physics $F=m_ia$, where $m_i$ is the inertial mass, and $F=GMm_g/r^2$, where $m_g$ is the gravitational mass. Hence $a\propto(m_g/m_i)$. One could imagine two materials that have different ratios of inertial and gravitational masses and hence that they would follow different paths even if launched from the same place at the same velocity.

But a geometric theory requires that two objects launched from the same place at the same velocity follow the same path. If they don't then it isn't just geometry that matters - what the objects are made of matters also. Hence we require that for a geometric theory $m_g/m_i$ is equal for all objects and we are free to pick units so that it is one.

 

[checksix]

Got it. Thanks!