Twins Paradox

Part of the What Time? series, an exploration in science fiction.

The Twins Paradox is less of a paradox and more of a time puzzle originally stated by Einstein.

Puzzling Twins

Alice and Angela are identical twins born seconds apart on a shiny afternoon. Growing up, they do everything together including dressing alike. Their mother insists they wear different colored bows in their hair, Alice in a red bow and Angela in pink. Teachers and some of their friends depend on the bows for identification, but their closest friends can tell them apart most of the time. Sometimes they like switching bows and pretending to be the other, especially when taking exams. Angela is the whiz at math.

At the age of seventeen years, Angela announces she intends on traveling to nearby Barnard’s Star as part of her astronomy studies. The university has limited room and cannot include another member on the field trip spanning several years. Although the ship can accelerate to near the speed of light, it must spend several years at the constant velocity before decelerating at the destination where the team will spend two years observing. Alice argues that it would tear them apart taking such a long trip. How could they live without each other? Alice tries and tries, but Angela has made up her mind. Alice waves goodbye to her sister and watches the craft depart the space station.

Thirty-nine years later, red bow long lost, Alice takes her two grown children to meet her sister at the space station. Angela steps off the spacecraft wearing the pink bow in her hair. Angela appears younger than Alice’s own children. Angela insists she has only been away for twelve years, not thirty-nine, and she argues with her much older twin.

What happened?

Short Answer

Angela’s trip experiences a time-dilation effect. From my “Quick, Dirty Relativity Review,” we know that time is relative to the observer verified using highly accurate clocks. One consequence is that observers moving at significantly different rates will appear to age differently. Both twins age normally and experience the normal passing of time. Since both twins move at significantly different rates, their frames of time relative to each other differ. Time is relative to the observer.

If motion is relative than why isn’t the time-dilation effect relative?

The Paradox

From Alice’s frame of reference, Angela is moving and her time appears slowed by time dilation. From Angela’s frame of reference, Alice is the one moving. (Recall that a reference frame tells us that science experiments gives the same results in uniform motion as if we were sitting still. This doesn’t apply to accelerating objects.) Why isn’t time-dilation effect relative? The answer is the accelerating part of the trip. Einstein brought up this twins puzzle pointing out it isn’t really a paradox. Acceleration isn’t relative.

Math

Assuming the space craft can accelerate without squishing the passengers to death, let’s try using numbers to see how this works. Angela spent three years at Barnard’s Star, the same in Alice’s reference since Barnard’s Star system and Earth are nearly relative to each other in motion. Travel time for Angela is nine years (four and half each way) while the trip from Alice’s reference is thirty-six years (thirty-nine minus three.) Disregarding time for acceleration, we can use the following formula to find out how fast Angela’s ship travels where td is time dilation and v/c is percentage speed of light:

The time dilation (td) from Angela to Alice is 9 / 36 or 0.25. This gives us a velocity of 0.9825% speed of light. Mighty fast! Getting up to that speed safely would actually take a long time without some kind of anti-squishing technology!