The time dilation experienced by an object in motion depends on its relative velocity compared to an observer in a different frame of reference. To calculate the specific velocity required for an object in space to experience the same time dilation as being stationary on the surface of the Earth, we need to consider the effects of both relative motion and gravitational time dilation.
Let's assume we have an object in space far away from any significant gravitational fields. In this scenario, the main factor contributing to time dilation would be the relative motion between the object and an observer on the Earth's surface.
According to the theory of relativity, time dilation due to relative motion is described by the equation:
Δt' = Δt / √(1 - (v^2 / c^2))
Where: Δt' is the time experienced by the moving object. Δt is the time experienced by the stationary observer. v is the relative velocity between the object and the observer. c is the speed of light in a vacuum.
To calculate the specific velocity required, we can set Δt' equal to Δt, assuming we want the object to experience the same time dilation as the observer on the Earth's surface:
Δt = Δt / √(1 - (v^2 / c^2))
Simplifying the equation:
1 = 1 / √(1 - (v^2 / c^2))
Squaring both sides:
1 - (v^2 / c^2) = 1
v^2 / c^2 = 0
v = 0
From the equation, we see that the relative velocity (v) required for an object in space to experience the same time dilation as being stationary on the surface of the Earth is zero. This means that, in terms of relative velocity alone, the object would need to be at rest.
However, it's important to note that if we consider the effects of gravitational time dilation, being stationary on the surface of the Earth would result in additional time dilation due to the Earth's gravitational field. To experience the same combined time dilation (from both relative motion and gravitational effects), an object in space would need to account for the gravitational time dilation as well.
In summary, if we only consider relative motion, an object would need to be at rest (relative velocity of zero) to have the same time dilation as being stationary on the Earth's surface. But when factoring in gravitational time dilation, additional considerations would be necessary.