To determine whether an object will still be inside a takeoff point's cosmological event horizon when a traveler arrives, you need to consider the concept of the cosmological event horizon and the effects of cosmic expansion. The equation that describes this situation is the cosmological event horizon formula:
d = c * t
where: d is the proper distance of the receding object from the takeoff point, c is the speed of light (299,792,458 meters per second), t is the time it takes for the traveler to reach the object.
However, this equation alone does not provide the complete picture, as it does not account for the expansion of the universe. To incorporate the effects of cosmic expansion, you need to consider the scale factor of the universe, denoted by "a."
The equation can be modified as follows:
d = c * (t - t0) / a(t)
where: t0 is the time at which the light was emitted from the object, a(t) is the scale factor of the universe at the time the light was emitted.
In this modified equation, we subtract t0 to account for the time it takes for the light to reach the traveler, and we divide by a(t) to consider the expansion of the universe.
If the proper distance, d, is smaller than the cosmological event horizon at the traveler's time of arrival, then the object will still be inside the event horizon. Conversely, if d is greater than the event horizon, the object will be outside the event horizon.
It's important to note that calculating precise distances and time intervals in cosmological contexts can be complex due to various factors, such as the geometry of the universe, the presence of dark energy, and the specific cosmological model being considered.