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.

Same as your last question really

check:
https://en.wikibooks.org/wiki/Special_Relativity/Simultaneity,_time_dilation_and_length_contraction#The_twin_paradox
The "twin paradox" :)

To determine whether the twins will look the same after being separated for 65 years, we can apply the concepts of time dilation and length contraction from special relativity. Let's analyze the scenario step by step:

1. Define the properties of the situation:
- Twin A stays on Earth.
- Twin B travels away in a spaceship at 90% of the speed of light (0.9c).
- The separation lasts for 65 years as measured by Twin A.

2. Find the time elapsed for Twin B:
According to time dilation, the time experienced by a moving observer (Twin B) is dilated or stretched relative to the time experienced by a stationary observer (Twin A).
The time dilation formula is given by: Δt' = Δt / √(1 - (v^2 / c^2))
Here, Δt' represents the time experienced by Twin B, Δt is the time measured by Twin A (65 years), v is the velocity of Twin B (0.9c), and c is the speed of light (3x10^8 m/s).

Substituting the values into the equation:
Δt' = 65 / √(1 - (0.9^2))
Δt' = 65 / √(1 - 0.81)
Δt' = 65 / √(0.19)
Δt' = 65 / 0.435
Δt' ≈ 149.43 years

Therefore, Twin B experiences approximately 149.43 years while traveling at 90% of the speed of light.

3. Find the distance traveled by Twin B:
The distance traveled by Twin B can be found using the equation: d = v * Δt
Here, d represents the distance traveled, v is the velocity of Twin B (0.9c), and Δt is the time experienced by Twin B (149.43 years).

Substituting the values into the equation:
d = 0.9c * 149.43
d = 0.9 * 3x10^8 * 149.43
d ≈ 40293870000 km

Therefore, Twin B travels approximately 40.3 billion kilometers during the separation.

4. Analyze length contraction:
According to length contraction, an object traveling at high velocities appears shorter relative to a stationary observer.
The formula for length contraction is given by: L' = L * √(1 - (v^2 / c^2))
Here, L' represents the contracted length, L is the proper length (distance on Earth), v is the velocity of Twin B (0.9c), and c is the speed of light (3x10^8 m/s).

Substituting the values into the equation:
L' = 40.3 * 10^9 km * √(1 - 0.9^2)
L' = 40.3 * 10^9 km * √(1 - 0.81)
L' = 40.3 * 10^9 km * √(0.19)
L' ≈ 23.81 billion km

Therefore, the length contracted distance between the twins is approximately 23.81 billion kilometers.

5. Reuniting the twins:
After 65 years, Twin A aged 65 years, while Twin B aged approximately 149.43 years. Both twins will meet on Earth again.

6. Conclusion:
While Twin B will be older than Twin A due to time dilation, they would share the same genetic makeup and physical characteristics. So, in terms of appearance, they would look the same when they are reunited.