At the top of the Wikipedia page on the Equivalence Principle is this quote attributed to Einstein:
A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:
(Inertial mass) (Acceleration) = (Intensity of the gravitational field) (Gravitational mass)
It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.
I read this statement to imply that two statements are equivalent:
- Acceleration due to gravity is independent of the mass of the body
- Inertial mass and gravitational mass are numerically equal.
I also logically understand "equivalent" to mean, "if and only if."
I understand one direction of this argument. If gravitational mass is numerically equal to inertial mass, then acceleration due to gravity will be the same for all bodies (because is the same for all things on Earth). But I can't understand how the independence of the mass of the body and acceleration due to gravity implies that inertial mass and gravitational mass are the numerically equal. For instance, that would be true if inertial mass were exactly twice gravitational mass--we could just choose a different value for G.
Am I misreading Einstein's statement? Does it not mean what I think it means?