Krdytkyu

It is a known fact that Inertial and gravitational mass are the same thing, and therefore are numerically equal. This is not an obvious thing, since there are even experiments trying to find a different between the two kind of masses. What I don't understand is: why is this not obvious? Usually when something that isn't considered obvious seems obvious to me there's something deep that I'm not getting.

Here's my line of thought:

The Inertial mass is defined by

$$
{bf{F}} = m_i {bf{a}} tag{1}
$$

The gravitational mass is derived from the fact that the gravitational force between two objects is proportional to the product of the masses of the objects:
$$
{bf{F_g}} = -G frac{m_{G1} m_{G2}}{|{bf{r}}_{12}|^2} hat{{bf{r}}} tag{2}
$$

Now if the only force acting on the object $1$ is the gravitational force, I can equate equations $(1)$ and $(2)$, and I can always fix the constant $G$ in such a way that the gravitational mass and the Inertial mass are numerically equal.

What's wrong with this line of thought and why the equivalence is not really so obvious?

To make it clear that it is not obvious it is better to stop using the word "mass" in both cases. So it is better to say that it is not obvious that the inertial resistance, meaning the property that scales how different objects accelerate under the same given force, is the same as the "gravitational charge", meaning the property that scales the gravitational field that different objects produce.

Just to make it clear, the problem with the equivalence of masses is not "does $m_i=1$ implie $m_{G}=1$?" in whatever units. The real question is, "does doubling the inertial mass actually double the gravitational mass?". So your procedure of redefinition of G, while keeping it as a constant, is only valid if the ratio $ {m_i}/{m_G}$ is constant, meaning if there is a constant scale factor between the inertial mass and the gravitational mass. By the way, this scale factor would actually never be noticed since from the begginning it would already be accounted for in the constant $G$.

You could do the same thing with electrical forces, using Coulombs law. You can check if electrical charge is the same as mass, since you have:

$${bf{F}} = m_i {bf{a}} tag{1}$$
$${bf{F_g}} = -K frac{k_1 k_2}{|{bf{r}}_{12}|^2} hat{{bf{r}}} tag{2}$$

You can ask, is $k_1$ the same as $m_i$? And it is true that for one specific case you could redefine $K$ such that it would come out that $k_1$ and $m_i$ are the same, but it would not pass the scaling test, since doubling the inertial mass does not double the charge.

Scenario I: I have a white ball and a black ball. In the system of units I've adopted, I discover that:

Gravitational mass of white ball = 2

Inertial mass of white ball = 3

Gravitational mass of black ball = 10

Inertial mass of black ball = 15

But I want each ball's gravitational mass to equal its inertial mass. So I fix the constant $G$, multiplying it by $2/3$ so that all my gravitational masses will be multiplied by $3/2$. Problem solved, just like you said!

Scenario II: I have a white ball and a black ball. In the system of units I've adopted, I discover that:

Gravitational mass of white ball = 2

Inertial mass of white ball = 3

Gravitational mass of black ball = 10

Inertial mass of black ball = 20

But I want each ball's gravitational mass to equal its inertial mass. So I fix the constant $G$, multiplying it by $2/3$ so that the white ball's gravitational mass will be multiplied by $3/2$. Or maybe I should fix the constant $G$, multiplying it by $1/2$ so the black ball's gravitational mass will be multiplied by $2$. Uh oh. Neither solution manages to cover both cases.

The full content of the statement that "gravitational mass equals inertial mass" is that "gravitational mass equals inertial mass provided you choose the right units". This doesn't rule out Situtation 1, but it rules out Situation 2. The fact that Situation 2 can't occur is a substantive statement.

It's kind of difficult since we have so much prior intuition that gravitational and inertial mass are the same thing because we know that something heavy to lift also takes more force to accelerate, but there's not really any inherent reason why this should be the case. I think it helps if you call the quantity that goes into the gravitational force law "gravitational charge", to differentiate it from the one in $F = ma$. The question then is "why is the gravitational charge the quantity that determines how much inertia an object has?" (up to a constant scaling factor that you can, indeed, incorporate into $G$).

You could imagine a universe where the magnitude of the electric charge is what determines an object's inertia, and now Newton's second law becomes $F = Sigma |q_i|a$ for an object made up of a number of charged particles. Now an electron and a proton would accelerate to the same speed if you subjected them to the same potential difference, but an atom of hydrogen (mass ~1 GeV) and an atom of positronium (a bound electron and positron, mass ~1 MeV) would fall at different speeds - the force of gravity would still be the same as it is in our universe for each particle, so it would be larger for the hydrogen, but would no longer be balanced by the larger inertial mass of hydrogen proportionally decreasing its acceleration.

Interestingly, if the physical laws of our universe suddenly switched to this behaviour it actually wouldn't be immediately obvious. Since regular matter is made up of equal numbers of protons and electrons, doubling the amount of stuff in something would double both this "electrical inertia" and the regular inertia due to mass, I think the most obvious effect would be plasmas behaving very strangely. Note that I'm ignoring neutrons in this because I didn't think of them and quark charges until now.

Try it like this; "All objects follow the same trajectory in a gravitational field".

That means that the force of gravity on an object, which alters its momentum, scales exactly with its inertial mass, which resists that alteration of momentum.