The principle of equivalence - that, locally, you can't distinguish between a uniform gravitational field and a non-inertial frame accelerating in the sense opposite to the gravitational field - is dependent on the equality of gravitational and inertial mass. Is there any deeper reason for why this equality of "charge corresponding to gravitation" (that is, the gravitational mass) and the inertial mass (that, in Newtonian mechanics, enters the equation $F=ma$) should hold? While it has been observed to be true to a very high precision, is there any theoretical backing or justification for this? You could, for example (I wonder what physics would look like then, though), have the "charge corresponding to electromagnetic theory" equal to the the inertial mass, but that isn't seen to be the case.
4 Answers
This is a tricky question. Fundamentally, this is the motivation of general relativity (and all metric theories of gravity)--if all masses interact with a gravitational field in the same way, then, in a sense, the motion of a particular mass is determined by the local gravitational field, independently of the mass. This then leads you into explaining the gravitational "force" as an emergent property of the local spacetime curvature.
But then, what came first? The explanation of gravity as curvature, or the equivalence of gravitational and inertial mass? In a way, they are just dual pictures of the same thing.
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The fact that the value of inertial mass and gravitational mass of a particle are same is the result of the "equivalence principle" which is an essential principle for any metric theory of gravity like general relativity.
If it were different, then the whole edifice of GR will fall down like a house of cards. Fortunately all experiments up to now have confirmed that they are same.
In other words, the only deeper significance of this fact can be attributed to the fact that gravity is not a "force" like other forces but a consequence of the fact that geometry of the spacetime is not static and unchanging but it is a dynamic arena, responding to the presence of mass/energy.
Here a maybe more illustrative comment:
In general relativity, the free fall trajectories caused by gravity, i.e. the geodesics, come from the metric which is related to the energy–momentum tensor $T$ via the Einstein equations. Then there is a concept of four-force $f$ (and therefore the $m_{inertia}$) used to describe the change of momentum when you model classical point particles. You use it when dealing with electromagnetic forces for example, but in general relativity you certainly don't need it to compute the geodesic. When you take the classical limit, then you lose all the features associated with the stronger versions of the equivalence principle and you suddently have to introduce a new force to explain the effects of gravitation. In this process, where a force $f$ is used to describe gravity, $f$ and its source $T$ are suddently related, artificially if you will. Now in Newton's law of gravitation, the cause of gravity $T$ gets condesed to the the mass $m_G$. Therefore $m_{inertia}$ and this $m_G$ are now related and they are related in such a way that $m_{inertia}$ cancels out because by the equivalence principle it wasn't there to explain free fall to begin with.
Let it put me this way: If an alien species came up with a theory with general relativity-like features from the start, would they ever wonder about a gravitational mass $m_G$? This parameter only emerges in the limit. From what I said above, it's not necessary to take a "two different types of masses" point of view.
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There are no such masses as gravitational and inertial. There is only one physical term, the mass. The mass determines the degree of a local deformation of physical space. but for this we have to understand the structure of space. In quantum mechanics the mass is a classical parameter. In classical mechanics, it is the inertial mass. In gravitational physics, it is involved in gravitational interactions.
