In this scenario, we have two objects moving at speeds close to the speed of light (0.99c) in the same direction, with one object being inside the other. Let's consider the time dilation experienced by the smaller object from the perspective of an observer inside the larger object and an observer outside.
From the perspective of an observer inside the larger object: When the smaller object is inside the larger object, both objects are moving at a high velocity relative to the observer. According to the principles of special relativity, time dilation occurs when objects move at high speeds relative to each other.
As the smaller object moves at 0.99c relative to the observer inside the larger object, the observer would perceive time dilation for the smaller object. The time experienced by the smaller object would appear to be dilated or slowed down compared to the time experienced by the observer inside the larger object.
From the perspective of an observer outside both objects: An observer outside both objects would also perceive time dilation for the smaller object due to its high velocity. According to special relativity, time dilation occurs for objects moving at high speeds relative to an observer.
Since the smaller object is moving at 0.99c relative to the observer outside, the observer would see time dilation for the smaller object. The time experienced by the smaller object would appear to be dilated or slowed down compared to the time experienced by the observer outside both objects.
It's important to note that the exact amount of time dilation experienced by the smaller object would depend on the relative velocities and the Lorentz factor, which takes into account the speeds of the objects involved. The formula for time dilation is given by:
Δt' = Δt / γ
where Δt' is the time experienced by the moving object, Δt is the time measured by a stationary observer, and γ (gamma) is the Lorentz factor, given by:
γ = 1 / √(1 - (v^2 / c^2))
In this case, both the observer inside the larger object and the observer outside would perceive time dilation for the smaller object, but the exact amount of time dilation would require specific velocity values to be calculated accurately.