General Relativity I: The Twin Paradox Part 2

Hello, and welcome back to MPC! Last week, we began discussing the strangeness of the twin paradox. Today, we will continue this discussion and hopefully come up with a logical explanation for the paradox.

As a quick refresher from last week, we have 20-year-old twin brothers, Albert and Richard. Albert goes on a rocket ship, traveling at 0.87c, from Earth to Mars and back.


Figure 1: Our scenario from lat week (from Richard’s perspective)

We calculated the effects of time dilation from Richard’s perspective using our special relativity equations. We found that, when Albert returns, Richard believes that Albert is 21 years old and he (Richard) is 22 years old.

There did not seem to be anything wrong with this. This is exactly how we analyzed time dilation and special relativity in the past, so it should be correct!

Then, however, we decided to take a new perspective on the situation: Albert’s perspective. According to Albert, he is stationary and everything is moving past him:


Figure 2: Our scenario from lat week (from Albert’s perspective)

As a result of this, Albert thinks that Richard experiences time dilation. This means that Albert thinks Richard is 21 years old when he returns and he (Albert) is 22 years old.

Something feels wrong with this. How could Albert and Richard see themselves as different ages? Imagine that Albert went somewhere much farther away, say so far away that it takes Albert 70 years to travel there and back (according to Richard), at the same speed. Our relativistic factor would still be 2 and this means that Albert’s clock will say that it only took 35 years (from Richard’s perspective):


This means that Richard thinks Albert is only 55 years old while he (Richard) is 90 years old, probably with many gray hairs. However, if we flip around the perspectives, we would find that Albert thinks Richard is only 55 years old while he (Albert) is 90 years old, probably with many gray hairs. In other words, Richard thinks he has gray hairs (when he looks in the mirror) and Albert does not, while Albert thinks he has gray hairs (when he looks in the mirror) and Richard does not. Who is right? Richard, for example, cannot have gray hair when he looks in the mirror but not have gray hair when Albert looks at him!

We have made a mistake in our calculations. In order to see this mistake let us draw out the entirety of Albert’s journey:


Figure 3: Albert’s journey

When we made our calculations, we calculated the time dilation as Albert was approaching Mars and as he was coming back (we did this by doubling the distance between the Earth and Mars). However, we did not even consider this part of the journey:


Figure 4: Albert’s journey with his “turn around” emphasized

Yes, that part of the journey does add extra distance that we did not account for. More importantly, though, it marks the point where Albert transitions from going towards Mars to moving away from Mars. Who cares about that? Albert has to turn around at some point, so why does it matter? We have to think, though, what happens when Albert turns around? His velocity changes!

This change in velocity may not be obvious at first — Albert is still traveling at the same speed the entire trip (0.87c). We need to realize, though, that the direction of Albert’s velocity is changing.


Figure 5: Comparison of Albert’s initial velocity and final velocity

In order for Albert to return to Earth, his velocity must change (it may be helpful, in our specific case, to imagine the speed as changing from 0.87c to -0.87c). There is no way around it! Remember from a previous post, if something’s velocity is changing (something does not have constant velocity), it is accelerating. This means that Albert is accelerating.

Still, who cares? Just a few weeks ago we discussed why general relativity is important: it can handle acceleration. Recall that special relativity only works with constant velocities. When we made our calculations for Albert’s journey, we used the time dilation formula — a formula we derived using the principles of special relativity. The time dilation formula assumes a constant velocity on the entire journey, but this is clearly not the case here. We cannot use our simple special relativity formulas. This means that all of our calculations from last week are invalid. If we want to analyze this scenario, we have to use general relativity!

When we use general relativity, we find that Richard and Albert both agree that Richard is older and Albert is younger when Albert returns. In the next few posts, we will try to understand why this is the case. Next week, as a starting point, we will be looking at one of the major concepts of general relativity: the equivalence principle. See you then!

For more information, be sure to check out this resource:

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