In a flat spacetime, time dilation occurs when there is a relative velocity between two observers. In this scenario, if observer A is orbiting observer B at 99.9999% the speed of light, observer B would perceive time dilation effects when observing observer A.
According to the theory of relativity, as an object moves closer to the speed of light, time appears to slow down for that object from the perspective of a stationary observer. This phenomenon is known as time dilation. In this case, observer B would see observer A's time running slower compared to their own.
From observer B's perspective, it would appear that observer A is experiencing time at a slower rate. Therefore, observer B would perceive that observer A takes a longer time to complete an orbit compared to what observer A would experience from their own reference frame.
However, it's important to note that no object with mass can reach or exceed the speed of light in a vacuum according to our current understanding of physics. As an object with mass accelerates toward the speed of light, its energy requirements would become infinitely large. Therefore, achieving a velocity of 99.9999% the speed of light is currently considered impossible for objects with mass.
In summary, if observer A were somehow able to orbit observer B at 99.9999% the speed of light, observer B would see observer A's orbit taking a longer time to complete due to time dilation effects. However, it's worth emphasizing that reaching such velocities for objects with mass is beyond our current technological capabilities and theoretical limits.