Why is Inertia not included as one of the fundamental interactions, yet gravity is included as one of the four known interactions, and there is an equivalence established between gravity and inertia.
Inertia and gravity are same at fundamental level and therefore, if gravity is a fundamental interaction, then so is inertia even though we may not recognize it that way.
Inertia - A body due to its mass (energy), creates a dip of space around it. That dip makes a force be required to make a change in state of the body. Hence causes inertia.
Gravity - Same dip (curve) due to mass (energy) of the body manifests as gravity for other bodies.
So, their origin is same and that is curving of space.
Inertia is nothing but gravity of the body acting on itself against any change of state. Therefore gravitational and inertial mass are same.
My view is that gravity and inertia are same phenomena. They are two sides of same coin.
Curving of space by mass/energy of a body manifests as gravity for other bodies.
Same curving of space manifests as inertia of the body, when we try to change its state of rest, or uniform motion.
gravity is measure of coupling strength of a mass to a known gravitational field.
Inertia is measure of coupling strength of a mass to its own gravitational field.
I would be happy if someone busts this view conceptually, or mathematically.
One can say that the Higgs field is responsible from the inertia since the mass is the quantity of the inertia and Higgs gives the masses of the particles. So, inertia can be interpreted as an interaction where the Higgs field is involved.
This is indeed consistent since for instant photon is massless, meaning that it has no inertia, and as a matter of fact, photon does not interact with Higgs boson.
Of course Higgs boson, being responsible for inertia, is not a gauge boson, it does not give a phase difference to the particles but still it is an interaction and the cause of inertia in the first place.
However, gravity interacts with all kinds of energy even it is not interacting with the Higgs boson, i.e., doesn't have a mass/inertia.
So, inertia and gravity are not equivalent. But of course they are interacting.