Final answer:
To determine the time experienced by an astronaut traveling at 99.999% the speed of light over 100 light years, time dilation effects must be accounted for, resulting in a significantly shorter duration on the astronaut's clock compared to an Earth-based observer.
Step-by-step explanation:
To calculate how long it would take an astronaut going with 99.999% the speed of light to traverse the distance from Galaxy A to Galaxy B according to his clock, we need to consider the effects of time dilation as described by the theory of special relativity. The formula for time dilation is t' = t / √(1 - v²/c²), where t is the time in a stationary frame (such as Earth), t' is the time in the moving frame (the astronaut's clock), v is the velocity of the moving frame, and c is the speed of light. Given that Galaxy A is 100 light years away and the astronaut is traveling at 0.99999c, we first calculate t in the Earth frame, which is t = distance/speed = 100 light years / 0.99999c. Converting light years to the time it would take light to travel (since c is the speed of light, it would take 1 year to travel 1 light year), t is simply 100 years. Now applying the time dilation formula, t' becomes significantly reduced due to the high velocity.
To get the dilated time, t', we plug the values into the time dilation formula: t' = 100 / √(1 - (0.99999c)²/c²), which results in a very small value due to the denominator being very close to zero. This immensely shortens the apparent time experienced by the astronaut compared to the stationary observer on Earth. Performing this calculation will give us the astronaut's time to travel between the galaxies.