General Relativity I: The Twin Paradox Part 1

Hello, and welcome back to MPC! Last week, we spoke about why special relativity is so “special.” We also briefly discussed general relativity. Today, we will talk about a famous scenario that illustrates why general relativity is so important.

Let’s imagine a simple scenario. Say we have twin brothers, Albert and Richard:


Figure 1: Richard and Albert, twin brothers

Being that these two brothers are twins, they are, of course, the same age (we’ll say that they are 20 years old). Let’s also say that Albert is an astronaut and he is going to be heading off on a round trip to Mars and back. Albert takes off on his rocket ship, traveling at 0.87c, while Richard remains on Earth:


Figure 2: Albert in his rocket ship, heading off to Mars

From what we learned about special relativity, we know that something interesting will happen: with motion at such a high speeds (0.87c), time dilation is going to come into effect (significantly). Who cares? Well, this means that, when Albert returns, him and Richard are going to be different ages! So what? They are twins! Twins with different ages!

Alright, maybe that does not sound too interesting for some of you, but it is actually very intriguing. Let’s try doing some calculations. First things first, let’s calculate the relativistic factor for this situation:


Let’s now think about what this means. Recall from our time dilation discussion the following equation:


Also, recall from our proper time discussion that t0 represents the time measured by the person who is stationary to the scenario. In our case, we are discussing Albert speeding away in a rocket ship. This means that the proper time would be measured by Albert (he is stationary with respect to the rocket ship because both he and the rocket ship are traveling at 0.87c). So, t0 represents the amount of time it takes for an event to occur (say, for Albert to travel to Mars and back) as measured by Albert (technically, as measured by Richard looking at Albert’s clock; we will see why this precise definition is important later in the post, but you can ignore it for now), and t1 represents the amount of time it takes for that same event to occur as measured by Richard. Let’s think about how long it will take Albert to travel back and forth according to Richard. To do this, we must know how far away Mars is from Earth. Let’s say that the distance is:


This means that the total distance Albert travels is:


**Remember: Albert is going on a round trip**

Using our definition of velocity, we can calculate how long Richard thinks it takes for Albert to go to Mars and back:


**VERY IMPORTANT NOTE: We have calculated the amount of time it takes for Albert to travel back and forth according to Richard. How do we know that this measurement is for Richard and not for Albert? For our distance, we used twice the distance from Earth to Mars. This is the distance Richard sees Albert traveling. Albert, on the other hand, experiences length contraction. If we wanted to calculate the amount of time it takes for Albert to travel back and forth according to Albert using the velocity equation, we would have to use the length contraction equation to find the appropriate distance. This would not be too hard, but we will instead calculate Albert’s time measurement using the time dilation equation. Both techniques result in the same answer, using the time dilation technique is just simpler in this specific scenario**

That was not too bad. Now we can calculate Albert’s time measurement with Equation 1:


So, according to Richard’s clock (you can think of this as an “internal clock”), Albert’s journey took 2 years, but according to Albert’s clock, his journey only took 1 year. This means that, by the end of Albert’s journey, Richard will be 22 years old and Albert will be 21 years old (time passed more slowly for Albert than Richard). The two twins are no longer the same age. Cool!

But wait, we have a problem. This whole time, we have been making these calculations from Richard’s perspective. What does this mean? We stated that Albert was zooming off to Mars at 0.87c, but this statement is only true according to Richard. According to Albert, he (Albert) is sitting in a rocket ship and Richard, Earth, and Mars are zooming past him at 0.87c:


Figure 3: Our scenario from Albert’s perspective

That’s true, but does it really matter? The calculations will turn out the same from this new perspective, right? Not exactly. In our original calculations, Albert was moving so time dilation occurred to him (i.e. his clock was slower, making the trip for him seem shorter and making him end up younger than Richard). In our new scenario, though, Richard is moving so time dilation occurs to him. This means that Richard’s clock runs more slowly, which means that Richard will end up younger than Albert. From this perspective, Albert is 22 years old and Richard is 21 years old!

Something does not seem right. We are discussing one scenario from two different perspectives, but we are getting wildly different results. When Albert returns, one of them has to be older. They can’t both be older! Who is right and why? Is Albert younger or is Richard younger?

There is an answer to this “paradox,” officially called the twin paradox, but we will save this answer for next week. For now, try reviewing what we did and see if you can find any mistakes. Next week, we will go over what we did wrong and how we can correct it. See you then.

For more information (and a headstart on next week’s post), be sure to check out these resources:

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