Inertia: the hidden field

I was travelling in a bus on the streets of São Paulo, a living laboratory of inertial forces, and I was wondering about the weirdest of all Newton’s laws: the first.
Imagine the following situation: a closed room with a subject inside (let us call him the Schrödinger’s cat for simplicity’s sake). Without his knowledge this room is in fact a vehicle, which is perfectly insulated from vibration and all forms of sound. The room starts moving and begins a curve.
The cat, inside the room, will sense mysterious force field acting over his body.
The strange nature of inertia arises, in my opinion, that this “field” has no source. Gravity is generated by mass, the Coulomb field is generated by charge, even the nuclear strong qnd weak forces have their own souces. Inertia doesn’t. However, this becomes ever weirder.
Back in college I wrote my first paper (for the Physics Lab., equivalent to Experimental Physics 101) on the subject of the identity between inertial and gravitational mass.
Inertial mass is the ratio berween a force acting over a body and the acceleration it produces, which can be precisely determined in an elastic colision experiment. Gravitational mass is the ratio between gravitational force and gravity acceleration, which is related to the spacetime curvature, which can be precisely determined in a free fall experiments. There is no sensate reason to assume that both things are the same.
I remember when I wrote that paper (30 years ago) someone (I believe it was Einstein, but I am not sure) tried to justify this equality using Mach’s principle, which, as far as I could understand, meant that inertia is created by the gravitational attraction of all mass im the Universe (the reference frame of the fixed stars).The problem is, after relativity we know that the fixed stars are not even there where we see them. I see another problem with this idea: there is no time delay between the change in movement and the onset of inercial forces. If inertia originates from the interaction between the body and faraway masses, shouldn’t this imteraction obey the restriction of the light speed?
I know I’m writing about quite specialized things I didn’t study, and worse, writing with the memory of a 1st year physics bachelor student of 30 years ago, but anyway I believe there is something very fundamental about Nature hidden in these weird properties of inertia.

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