In this scenario, we can analyze the time dilation effect predicted by the theory of special relativity. Time dilation refers to the difference in the passage of time between two observers moving relative to each other at significant speeds.
Let's assume that person 1, who is stationary in space, is our reference frame. Person 2 is moving at 99% of the speed of light in a doughnut-shaped spaceship, which itself is moving at 99% of the speed of light relative to person 1.
According to special relativity, the time dilation factor (γ) can be calculated using the formula:
γ = 1 / √(1 - (v^2/c^2))
Where: γ is the time dilation factor. v is the relative velocity between the observers (person 2) and the reference frame (person 1). c is the speed of light in a vacuum.
Let's calculate the time dilation factor for person 2 in this scenario:
v = 0.99c (99% of the speed of light) c = 299,792,458 m/s (speed of light in a vacuum)
γ = 1 / √(1 - (0.99^2))
γ ≈ 7.088
This means that from the perspective of person 1, person 2's clock appears to be running slower by a factor of approximately 7.088. In other words, person 1 will perceive that less time has passed for person 2 compared to the time that has passed for themselves.
Conversely, from the perspective of person 2, who is in the moving reference frame, person 1's clock will also appear to be running slower by the same factor. This is a consequence of the relativity of simultaneity and the symmetry of time dilation.
It's important to note that time dilation effects become more pronounced as velocities approach the speed of light. In this scenario, with person 2 moving at 99% of the speed of light, the time dilation factor is significant.