According to Einstein's theory of general relativity, the connection between inertia and time dilation is established through the concept of gravitational time dilation. In the absence of gravity, the principle of equivalence states that the effects of gravity are indistinguishable from the effects of acceleration. Therefore, when an object is accelerated, it experiences a similar effect on time as it would in a gravitational field.
Inertia, in this context, refers to the resistance of an object to changes in its motion. When an object is accelerated, it experiences a force that opposes this change in motion. Now, according to general relativity, the acceleration of an object causes it to move through spacetime along a curved trajectory. This curvature of spacetime affects the flow of time, leading to time dilation.
In the presence of acceleration or gravity, the rate at which time flows depends on the strength of the gravitational field or the magnitude of the acceleration. More specifically, time runs slower in regions with stronger gravitational fields or higher accelerations. This means that clocks in these regions will tick slower compared to clocks in weaker gravitational fields or regions with lower accelerations.
Therefore, inertia, which resists changes in motion, is connected to time dilation. An object experiencing inertia due to acceleration will have its local time run slower compared to an object in a less accelerated or non-accelerated state. This effect is similar to what happens in gravitational fields, where time dilation occurs due to the curvature of spacetime caused by mass or energy distributions.
It is important to note that the connection between inertia and time dilation is a consequence of the general theory of relativity and is not solely limited to the absence of gravity. The theory describes the relationship between acceleration, gravity, and time dilation in a unified framework.