Terra will remain on Earth, while Stella will board a spaceship.

泰拉要待在地球上， 史黛拉則要登上一台太空船。

Stella's ship will travel at 86.6% the speed of light

史黛拉的太空船將會以 86.6% 光速的速度行進，

to visit a star that is 10 light-years away,

目的地是十光年外的一個星球，

then return to Earth at the same speed.

接著，再用同樣的速度返回地球。

As they prepare to part ways,

準備分離時，這對雙胞胎 很納悶在重聚時會發生什麼事。

the twins wonder what will happen when they're reunited.

因為，光年就是 光行進一年的距離。

Since a light year is exactly the distance light can travel in a year,

史黛拉的旅程應該會花二十三年。

Stella's journey should take 23 years.

但，這對雙胞胎學過狹義相對論，

But from having studied special relativity,

她們知道沒有事情那麼簡單。

the twins know it's not that simple.

首先，物體在太空中 移動的速度越快，

First of all, the faster an object moves through space,

相對於不動的觀察者， 它經過時間的速度就越慢。

the slower it moves through time compared to an unmoving observer.

可以用「勞倫斯因數」 將這個關係給量化，

This relationship can be quantified with something called the Lorentz factor,

這是勞倫斯因數的方程式。

which is defined by this equation.

第二，當不動的觀察者 要測量移動物體的長度時，

And secondly, the length of a moving object as measured by an observer at rest

也要考量這個因數的對比。

will contract by the same factor.

在 86.6% 光速時的 勞倫斯因數是 2，

At 86.6% of the speed of light the Lorentz factor is 2,

意即，在太空船上， 時間會慢兩倍。

meaning time will pass twice as slowly aboard the spaceship.

當然，史黛拉不會 注意到時間減慢了。

Of course, Stella won't notice time slowing down.

那是因為，太空船上所有以時間 為基礎的流程也都會減慢——

That's because all time-based processes in the ship will slow down as well–

比如時鐘、電子裝置；

clocks and electrical devices;

史黛拉的生物活動， 包括她的老化速度，

Stella's biological activities including her rate of aging

以及她對時間本身的感知。

and her perception of time itself.

唯一會注意到史黛拉在太空船上

The only people who could notice time on the moving spaceship

時間過得比較慢的人，

passing slower for Stella

就是身處在慣性／非加速 參考座標系的觀察者——

would be observers in an inertial, or non-accelerating, reference frame–

就像在地球上的泰拉。

like Terra back on Earth.

因此，泰拉認為， 當她們於地球再次相見時，

Thus, Terra concludes that when they meet back on Earth,

她會比史黛拉還老。

she'll be older than Stella.

但那只是其中一種觀點。

But that's just one way of looking at things.

因為所有的運動都是相對的，

Because all movement is relative,

史黛拉主張另一個有根據的解釋， 就像是她的太空船原地不動，

Stella argues it would be just as valid to say her spaceship will stand still

而宇宙的其他部分， 包括泰拉，都繞著她運動。

while the rest of the universe, including Terra, moves around her.

這種情況下，對泰拉來說， 時間的速度會是兩倍慢，

And in that case, time will pass twice as slowly for Terra,

最後史黛拉會是雙胞胎中 比較老的那一個。

making Stella the older twin in the end.

她們不可能都比彼此年長， 所以哪種說法才是對的？

They can't each be older than the other, so which one of them is right?

這個明顯的矛盾就是 所謂的「雙胞胎悖論」。

This apparent contradiction is known as the “Twin Paradox.”

但它其實不是個悖論—— 只是一個例子，

But it's not really a paradox–

說明狹義相對論很容易被誤解。

just an example of how special relativity can be easily misunderstood.

為了在真實時間中 測試她們的理論，

To test their theories in real-time,

兩位雙胞胎都同意，她們每過一年

each of the twins agrees to send a burst of light to the other

就要發送一道光給彼此。

every time a year has passed for them.

和其他物體不同， 光的速度永遠是常數，

Unlike other objects, the speed of light is always constant

不論觀察者的參考座標系是什麼。

regardless of an observer's reference frame.

從地球發送的光

A light burst sent from Earth will be measured at the same speed

和從太空船發送的光 會被用相同的速度測量，

as a light burst sent from the spaceship,

不論是去程或是回程。

regardless of whether it's on its outbound or return trip.

所以，當雙胞胎之一觀察到光時，

So when one twin observes a burst of light,

就要測量出另一人 經歷過的一年是多久，

they're measuring how long it took the other twin to experience a year passing,

再加上光在兩人之間 行進要花的時間。

plus how long it took for light to travel between them.

我們可以用一張圖 來追蹤發生的狀況。

We can track what's happening on a graph.

X 軸表示離地球的距離， Y 軸則是經過了多少時間。

The X axis marks distance from Earth, and the Y axis tracks the passage of time.

從泰拉的角度，她的路徑 就只是一條垂直線，

From Terra's perspective, her path will simply be a vertical line,

距離都是零，

with distance equal to zero

線上的每一個標記 就是她所感知到的一年。

and each tick on the line equivalent to a year as she perceives it.

史黛拉的路徑會從同樣的原點 出發，延伸到時間的 11.5 年，

Stella's path will stretch from the same origin to a point 11.5 years in time

距離則是離泰拉 10 光年……

and 10 light-years in distance from Terra…

最後則會在 23 年、 零距離的那個點重聚。

before converging again at zero distance and 23 years' time.

在泰拉的第一年底，

At her first one-year mark,

她會從地球向史黛拉的 太空船送出一道光。

Terra will send a pulse of light from Earth towards Stella's spaceship.

因為一光年的距離，光要走一年，

Since light takes a year to travel one light-year,

它的路徑會是 45 度的對角線。

its path will be a 45-degree diagonal line.

因為當光追上史黛拉時，

And because Stella is traveling away from it,

她是正在遠離光源的過程中，

by the time the light catches up to her,

對泰拉來說，是七年過去了， 對史黛拉則是四年。

over 7 total years will have passed for Terra, and over 4 for Stella.

當史黛拉觀察到 泰拉的第二道光時，

By the time Stella observes Terra's second burst,

她已經是在回程路上。

she will already be on her return journey.

但，這次，因為她是 朝向光源的方向行進，

But now, since she's moving towards the source of the light,

光抵達她那裡要花的時間比較短， 她觀察到光的頻率變高。

it will take less time to reach her,

也就是說，在旅程的前半段， 史黛拉的觀察是泰拉老化得較慢，

and she'll observe the bursts more frequently.

但在後半回程時， 泰拉老化得較快。

This means that Stella observes Terra aging slowly

對史黛拉而言， 似乎泰拉、目的地星球，

for the first half of her journey,

以及整個宇宙都繞著她在運動。

but aging rapidly during the return half.

因為長度縮短，

Meanwhile for Stella, it seems as though Terra, the destination star,

史黛拉觀察到兩地 之間的距離會縮減 2 倍。

and the whole universe are moving around her.

意思就是，從史黛拉的觀點來看，

And because of length contraction,

每段旅程只會花大約六年。

Stella observes the distance between them shrinking by a factor of 2.

當她第一次送出光時， 泰拉已經度過了兩年。

This means each leg of the trip will only take about six years

史黛拉在去程時會再發送四次光，

from Stella's perspective.

每一次都從更遠的地方發送。

When she sends the first signal to Earth, two years will have passed for Terra.

當泰拉觀察到史黛拉 回程的第一道光時，

Stella will send four more light bursts during her outbound journey,

她已經度過了 21 年。

each one from farther away.

在史黛拉返家的剩下過程中，

By the time Terra observes the first pulse from Stella's inbound journey,

泰拉每年會收到好幾次光。

over 21 years will have passed for her.

因此，泰拉觀察到的是，

For the rest of Stella's return home,

在分離的 23 年中， 前 90% 的時間史黛拉老化較慢，

Terra receives multiple light bursts each year.

最後 10% 的時間，會快速老化。

Thus, Terra observes Stella aging slowly for about 90% of their 23 years apart,

這個不對稱的描述，說明了 為什麼這個悖論並不是真的悖論。

and aging rapidly during the last 10%.

雖然兩位雙胞胎都目擊 對方的時間加速和減速，

This asymmetry accounts for why the paradox isn't really a paradox.

史黛拉看到的是一半一半，

Although each twin witnesses time

泰拉看到的是分離的大部分時間 史黛拉都老化得比較慢。

both speeding up and slowing down for the other,

這和兩位雙胞胎對於 太空旅行的測量是一致的，

Stella sees an even split,

該旅行花了 23 年地球年，

while Terra sees Stella aging slowly for most of the time they're apart.

但在太空船上只會經歷 11.5 年。

This is consistent with each twin's measurement of the space voyage,

當雙胞胎團聚時， 泰拉會是 43 歲，

which takes 23 Earth years, but only 11.5 as experienced aboard the ship.

史黛拉則是 31 歲。

When the twins are reunited, Terra will be 43 years old, while Stella will be 31.

史黛拉弄錯的是她假設

Where Stella went wrong

她和泰拉同樣都是 慣性座標系的觀察者。

was her assumption that she and Terra had equal claim to being inertial observers.

慣性座標系的觀察者 相對於宇宙的速度和方向

To be an inertial observer, one has to maintain a constant speed and direction

必須要是常數。

relative to the rest of the universe.

從頭到尾泰拉都沒有動， 所以她的速度是常數，零。

Terra was at rest the entire time, so her velocity was a constant zero.

但史黛拉在回程時改變了方向，

But when Stella changed her direction for the return journey,

此時她便進入了和一開始 不同的參考座標系。

she entered a different reference frame from the one she'd started in.

泰拉和史黛拉現在都更了解 空間時間怎麼運作了。

Terra and Stella now both have a better understanding of how spacetime works.

現在這對雙胞胎年齡相差十一歲，

And as twins who are eleven years apart in age,

她們是證明狹義 相對論的完美例子。

they're a perfect example of special relativity.