In Einstein's theory of general relativity, there is a connection between inertia and time dilation, which arises due to the curvature of spacetime caused by the presence of mass and energy. This connection is described by the principle of equivalence.
The principle of equivalence states that in a small region of spacetime, the effects of gravity are indistinguishable from those of an accelerated frame of reference. In other words, an observer in a gravitational field cannot locally distinguish whether they are experiencing the effects of gravity or are in an accelerated reference frame.
Let's focus on the absence of gravity in a small region of spacetime, where only inertial forces are present (e.g., in free-fall). According to special relativity, time dilation occurs when an object is moving relative to an observer at a different velocity. However, in the context of general relativity, there can be time dilation even in the absence of relative motion.
When an object is at rest in a gravitational field, it experiences gravitational time dilation. This means that time for the object passes slower compared to an observer located at a higher gravitational potential (farther from the source of gravity). This effect is a consequence of the curvature of spacetime caused by the mass of the planet or massive body the object is near.
The curvature of spacetime around a massive object like a planet causes clocks in lower gravitational potential (closer to the planet's center) to tick slower than clocks in higher gravitational potential (farther from the planet's center). This is because spacetime is stretched near massive objects, leading to a slowing down of time.
So, the connection between inertia and time dilation in the absence of gravity can be understood as follows: Inertial motion and gravitational time dilation are intrinsically linked by the curvature of spacetime. When an object is in free-fall, it follows a geodesic, which is the straightest possible path through curved spacetime. As it moves along this geodesic, its path through spacetime is affected by the curvature due to the presence of mass, leading to time dilation. The stronger the gravitational field (higher curvature), the more significant the time dilation.
In summary, in the context of general relativity, inertia and time dilation are connected through the principle of equivalence, which links inertial motion and the effects of gravity. In the absence of gravity, free-falling objects experience time dilation due to the curvature of spacetime caused by the presence of mass and energy, as described by Einstein's theory of general relativity.