Consider a situation where immediately after birth, twins are separated. One continues to live on earth, while the other is whisked away in a space ship at 90% of the speed of light. After 65 years they are reunited. Will they look the same? Show proof of your answer through mathematics.
To determine whether the twins will look the same after being separated and one traveling at 90% the speed of light for 65 years, we can utilize the theory of special relativity. Let's examine the scenario step by step.
First, we need to calculate the time experienced by each twin during the 65-year period. The twin on Earth will experience the full 65 years, as time passes normally for them. However, for the traveling twin (let's call her Twin A), time will pass at a different rate due to time dilation.
According to the theory of special relativity, the time experienced by Twin A (t’) can be calculated using the time dilation formula:
t’ = t / sqrt(1 - v^2/c^2)
Where:
t’ = time experienced by Twin A
t = time experienced by Twin B (65 years, in this case)
v = velocity of Twin A (90% of the speed of light, so v = 0.9c)
c = speed of light in a vacuum (approximately 3 × 10^8 meters per second)
Plugging in these values, we get:
t’ = 65 / sqrt(1 - (0.9c)^2/c^2)
t’ = 65 / sqrt(1 - 0.81)
t’ = 65 / sqrt(0.19)
t’ = 65 / 0.4359
t’ ≈ 149.14 years (approximately)
So according to the calculations, while 65 years pass for the twin remaining on Earth, approximately 149.14 years pass for the traveling twin. This time dilation effect occurs because as an object approaches the speed of light, time slows down relative to a stationary observer.
Therefore, when the twins are reunited, the traveling twin (Twin A) will have aged more than the twin on Earth (Twin B). Twin A will look older than Twin B, based on the difference in the experienced time during their separation.
So, they will not look the same. Twin A will appear older than Twin B due to time dilation caused by the high speed of the space ship.