On their 20th birthday, identical twin
astronauts volunteer for an experiment. Terra will remain on Earth, while Stella
will board a spaceship. Stella’s ship will travel at 86.6% the
speed of light to visit a star that is 10
light-years away, then return to Earth at the same speed. As they prepare to part ways, the twins wonder what will happen
when they’re reunited. Since a light year is exactly the distance
light can travel in a year, Stella’s journey should take 23 years. But from having studied
special relativity, the twins know it’s not that simple. First of all, the faster an object moves
through space, the slower it moves through time
compared to an unmoving observer. This relationship can be quantified with
something called the Lorentz factor, which is defined by this equation. And secondly, the length of a moving
object as measured by an observer at rest will contract by the same factor. At 86.6% of the speed of light
the Lorentz factor is 2, meaning time will pass twice as slowly
aboard the spaceship. Of course, Stella won’t notice
time slowing down. That’s because all time-based processes
in the ship will slow down as well– clocks and electrical devices; Stella’s biological activities including
her rate of aging and her perception of time itself. The only people who could notice time
on the moving spaceship passing slower for Stella would be observers in an inertial,
or non-accelerating, reference frame– like Terra back on Earth. Thus, Terra concludes that when they meet
back on Earth, she’ll be older than Stella. But that’s just one way of
looking at things. Because all movement is relative, Stella argues it would be just as valid to
say her spaceship will stand still while the rest of the universe,
including Terra, moves around her. And in that case, time will pass twice as
slowly for Terra, making Stella the older twin in the end. They can’t each be older than the other,
so which one of them is right? This apparent contradiction is known as
the “Twin Paradox.” But it’s not really a paradox– just an example of how special relativity
can be easily misunderstood. To test their theories in real-time, each of the twins agrees to send
a burst of light to the other every time a year has passed for them. Unlike other objects, the speed of light
is always constant regardless of an observer’s
reference frame. A light burst sent from Earth will be
measured at the same speed as a light burst sent from the spaceship, regardless of whether it’s on its
outbound or return trip. So when one twin observes
a burst of light, they’re measuring how long it took the
other twin to experience a year passing, plus how long it took for light
to travel between them. We can track what’s happening on a graph. The X axis marks distance from Earth,
and the Y axis tracks the passage of time. From Terra’s perspective, her path will
simply be a vertical line, with distance equal to zero and each tick on the line equivalent
to a year as she perceives it. Stella’s path will stretch from the same
origin to a point 11.5 years in time and 10 light-years in distance from Terra… before converging again at zero
distance and 23 years’ time. At her first one-year mark, Terra will send a pulse of light from
Earth towards Stella’s spaceship. Since light takes a year to travel
one light-year, its path will be a 45-degree
diagonal line. And because Stella is
traveling away from it, by the time the light catches up to her, over 7 total years will have passed for
Terra, and over 4 for Stella. By the time Stella observes
Terra’s second burst, she will already be on her return journey. But now, since she’s moving towards the
source of the light, it will take less time to reach her, and she’ll observe the bursts
more frequently. This means that Stella observes Terra
aging slowly for the first half of her journey, but aging rapidly during the return half. Meanwhile for Stella, it seems as though
Terra, the destination star, and the whole universe are
moving around her. And because of length contraction, Stella observes the distance between
them shrinking by a factor of 2. This means each leg of the trip will only
take about six years from Stella’s perspective. When she sends the first signal to Earth,
two years will have passed for Terra. Stella will send four more light bursts
during her outbound journey, each one from farther away. By the time Terra observes the first pulse
from Stella's inbound journey, over 21 years will have passed for her. For the rest of Stella's return home, Terra receives multiple light
bursts each year. Thus, Terra observes Stella aging slowly
for about 90% of their 23 years apart, and aging rapidly during the last 10%. This asymmetry accounts for why the
paradox isn’t really a paradox. Although each twin witnesses time both speeding up and slowing
down for the other, Stella sees an even split, while Terra sees Stella aging slowly for
most of the time they’re apart. This is consistent with each twin’s
measurement of the space voyage, which takes 23 Earth years, but only
11.5 as experienced aboard the ship. When the twins are reunited, Terra will be
43 years old, while Stella will be 31. Where Stella went wrong was her assumption that she and Terra had
equal claim to being inertial observers. To be an inertial observer, one has to
maintain a constant speed and direction relative to the rest of the universe. Terra was at rest the entire time,
so her velocity was a constant zero. But when Stella changed her direction
for the return journey, she entered a different reference frame
from the one she’d started in. Terra and Stella now both have a better
understanding of how spacetime works. And as twins who are eleven
years apart in age, they’re a perfect example
of special relativity.
Explained, yes. Understood by me? No.
Awarded1