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 65 years of separation, we need to consider the effects of time dilation caused by one twin traveling at 90% of the speed of light relative to the other twin on Earth.
According to Einstein's theory of special relativity, time dilation occurs when an object moves relative to another at speeds approaching the speed of light. The moving twin, traveling at 90% of the speed of light, experiences time passing more slowly compared to the stationary twin on Earth.
Let's denote:
- v as the velocity of the traveling twin, which is 90% of the speed of light (v = 0.9c, where c is the speed of light)
- tEarth as the time experienced by the twin on Earth, which is 65 years
- tTravel as the time experienced by the traveling twin
The formula for time dilation is given by:
tTravel = tEarth / √(1 - (v^2 / c^2))
Plugging in the values:
tTravel = 65 years / √(1 - (0.9c)^2 / c^2)
tTravel = 65 years / √(1 - 0.81)
tTravel = 65 years / √(0.19)
tTravel = 65 years / 0.43589
tTravel ≈ 149.35 years
According to the time dilation formula, the twin traveling at 90% of the speed of light would experience approximately 149.35 years. This means that when they are reunited after 65 Earth years, the traveling twin would have aged less compared to the twin on Earth.
Therefore, they would not look the same. The traveling twin would appear younger than the twin who stayed on Earth.