The Twin Paradox :)

 Hey y'all, 

I couldn't think of anything else to blog about this week, so I'm gonna share a post I made on the multivariable calculus Google Classroom about the twin paradox. And yes, I realize it's not Friday anymore, but it might be Friday when some of y'all read this, so I'm gonna keep the intro. 

Happy Physics Friday y’all! Today I’m going to *attempt* to explain the real twin paradox, which, as Mr. Bahat pointed out on my post from last week, is neither paradoxical nor the scenario I previously described in which time passes differently for twins living at different altitudes (which is also not paradoxical...There are enough pseudo-paradoxes in theoretical physics to make your head spin haha.) The real twin paradox can be resolved by Einstein's theory of special relativity (1905) and involves a set of twins (people the same age), one of whom travels in a rocket on a space mission while the other stays on Earth. The twin remaining on Earth is older than his space-bound counterpart upon the latter’s return. This pseudo-paradox is based on the idea that the inertial frames of the twins are not the same; the space-bound twin is accelerating at the moment that the spaceship turns back toward the Earth, which necessitates that he/she has a non-inertial frame (“coasting”, or moving at a constant velocity, is required in order to have an inertial frame. Remember that acceleration doesn’t have to be a change in speed; you can just as easily accelerate at a constant speed by changing direction). Since the twins have different inertial frames, this disproves the common misconception that each twin should discover himself to be older than his counterpart (since each twin considers his twin and not himself to be the one in motion). There are some calculations that determine the exact age differential between them which I will try to summarize, although unfortunately I don’t have Greek letters on my keyboard (don’t @ me Hunter). Let’s assume the spaceship trip takes 10 years (that is, the twin on Earth is ten years older when the spaceship returns than when it departed). We are told that the distance the rocket travelled was 4 light years and that it was moving at .8*c where c is the speed of light (it’s a very fast rocket haha). So to find the total time, we use t = d/v, except use 2d = 8 light years because we have to account for the return trip. That’s how we get the time (measured on Earth) to be 10 years. The travelers on the spaceship, however, have to account for time dilation, and they modify their total travel time by multiplying by epsilon = sqrt(1-v^2/c^2), which is the inverse of the Lorentz factor (I’ll have to make another post to elaborate because I’m running out of steam haha). The calculation yields epsilon = 0.6, so the travelling twin is 6 years older upon his return. Another way to calculate the spaceship twin’s age is to multiply epsilon (0.6) by the total distance (8 light years) and divide by the velocity (0.8*c) to get 6 years. (Multiplying by epsilon is necessary to account for length contraction, another aspect of the lovely Lorentz factor, which I will expand upon in another post.) Anyway, I hope y’all enjoyed this post! Hopefully it’s not super confusing. (Also, hopefully I explained it right haha…) I referred to the Wikipedia article on the twin paradox which y’all can find here: https://en.wikipedia.org/wiki/Twin_paradox as well as Einstein’s War: How Relativity Triumphed Over the Vicious Nationalism of World War I by Matthew Stanley. (Great book! I’ve just started reading it, but it’s super interesting.)

I know that was probably super dense, but it's riveting to me. I hope y'all enjoyed! 

Love from 

Clara