And if you use 'light clocks' you will see that time dilations and Lorentz contractions aren't solely connected to inertia, or gravity. It's a result of the fact that 'c' is a constant invariant factor, belonging to all 'inertial frames of reference', aka uniform motions. It's always 'c', doesn't matter how fast your uniform speed (geodesic/velocity) is relative some other reference frame. So describing it as a proportionality to gravity solely becomes a tough proposition.

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What I personally think is that there must be a connection to 'c' for gravity too though. And 'c' is a description of displacements over time, but, as locally defined. If you want to be strict we define a proper mass from uniform motion too btw. Accelerations plays havoc with your scale, as you weight yourself, so we need to include proper mass there too. And then we have transformations and 'energy' which are the ones making it make sense, although not locally measurable, except as gravity/inertia under accelerations.

So not only 'c', but 'local'. The last, to me, means that we need to wonder what 'motion' is, and how we define a clock, and so also a 'distance'. All of those are local definitions made from a uniform motion.

And then there is relativistic mass, which is the sort of mass you can't measure locally in a uniform motion, or acceleration, but becomes very tangible in any collision, expressed in kinetic energy. And that is a added problem as it neither is measurable from the vacuum, nor locally in your uniformly moving, or accelerating, rocket. What you can measure locally is a blue/redshift, and a gravity/inertia under a acceleration but it's not proportional to your relativistic mass, as you easily can see thinking of accelerating at one uniform gravity, considering the kinetic energy expressed in a collision, at different times. And if in a uniform motion this too will be absent, locally.

Einstein gave us one description, QM gives us one more, with Higgs defining inertia under accelerations. None of them seems wrong, but none of them seems to tell the whole story either. Not to forget, we have classical physics too, as Newtons and Maxwell's that works really nice under normal circumstances. Einstein said himself that it was Maxwell that gave him the idea of 'c', if I remember right.