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There must be a relation between mass m being affected by gravity and mass m being affected by inertial force.If mass is connected somehow to space-time and this connection enable mass to curve space-time this connection will also enable space-time to affect motion resulting in inertia .
Space-time curvature makes mass to move with acceleration and space-time opposes mass motion with inertia acceleration.
Gravity is a force caused by the gravitational field.Most pushes and pulls- like me pressing the keys on this keyboard- are caused by an electrostatic field.There is no real difference.