It might be better to think of it the other way around. Mass distorts spacetime, causing a sort of drag on the flow of time. An object moving through this region of spacetime will experience a difference in time from one side to its other. Just like driving a car with a bigger wheel on one side, the object must follow a curved path through space. We think of a force causing that deflection as gravity, but it’s actually a straight line in spacetime.
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