In the scenario you described, where Person A and Person B are both moving at 90% of the speed of light in opposite directions around an event horizon, time dilation would manifest between them according to special relativity. Let's analyze the situation from the perspective of each observer:
From Person A's perspective: Person A, who is moving around the event horizon, would observe time dilation between themselves and Person B. This is because when two observers are in relative motion, each observer sees the other's clock as running slower. As Person A moves at a high velocity relative to Person B, Person A would perceive time passing more slowly for Person B.
From Person B's perspective: Person B, who is also moving around the event horizon, would observe time dilation between themselves and Person A. Since Person A and Person B are moving at the same speed in opposite directions, they are in a symmetrical configuration. According to the principle of special relativity, each observer would perceive the other's clock as running slower. Thus, Person B would perceive time passing more slowly for Person A.
It is important to note that the actual time dilation factor between Person A and Person B would depend on the specific velocities involved. The time dilation formula in special relativity, known as the time dilation factor γ, can be calculated as γ = 1 / sqrt(1 - v^2/c^2), where v is the relative velocity between the observers and c is the speed of light. By plugging in the appropriate values, you can determine the exact time dilation factor between Person A and Person B in this scenario.
In summary, both Person A and Person B would observe time dilation between each other due to their relative velocities. Each observer would perceive the other's clock as running slower, in accordance with the principles of special relativity.