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.
在二十歲生日時,一對同卵雙胞胎 太空人自願參與一項實驗, 泰拉要待在地球上, 史黛拉則要登上一台太空船。 史黛拉的太空船將會以 86.6% 光速的速度行進, 目的地是十光年外的一個星球, 接著,再用同樣的速度返回地球。
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.
準備分離時,這對雙胞胎 很納悶在重聚時會發生什麼事。 因為,光年就是 光行進一年的距離。 史黛拉的旅程應該會花二十三年。 但,這對雙胞胎學過狹義相對論, 她們知道沒有事情那麼簡單。 首先,物體在太空中 移動的速度越快, 相對於不動的觀察者, 它經過時間的速度就越慢。 可以用「勞倫斯因數」 將這個關係給量化, 這是勞倫斯因數的方程式。 第二,當不動的觀察者 要測量移動物體的長度時, 也要考量這個因數的對比。
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.
在 86.6% 光速時的 勞倫斯因數是 2, 意即,在太空船上, 時間會慢兩倍。 當然,史黛拉不會 注意到時間減慢了。 那是因為,太空船上所有以時間 為基礎的流程也都會減慢—— 比如時鐘、電子裝置; 史黛拉的生物活動, 包括她的老化速度, 以及她對時間本身的感知。 唯一會注意到史黛拉在太空船上 時間過得比較慢的人, 就是身處在慣性/非加速 參考座標系的觀察者—— 就像在地球上的泰拉。 因此,泰拉認為, 當她們於地球再次相見時, 她會比史黛拉還老。
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?
但那只是其中一種觀點。 因為所有的運動都是相對的, 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, 而宇宙的其他部分, 包括泰拉,都繞著她運動。 這種情況下,對泰拉來說, 時間的速度會是兩倍慢, 最後史黛拉會是雙胞胎中 比較老的那一個。 她們不可能都比彼此年長, 所以哪種說法才是對的?
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.
這個明顯的矛盾就是 所謂的「雙胞胎悖論」。 但它其實不是個悖論—— 只是一個例子, 說明狹義相對論很容易被誤解。
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.
為了在真實時間中 測試她們的理論, 兩位雙胞胎都同意,她們每過一年 就要發送一道光給彼此。 和其他物體不同, 光的速度永遠是常數, 不論觀察者的參考座標系是什麼。 從地球發送的光 和從太空船發送的光 會被用相同的速度測量, 不論是去程或是回程。 所以,當雙胞胎之一觀察到光時, 就要測量出另一人 經歷過的一年是多久, 再加上光在兩人之間 行進要花的時間。
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.
我們可以用一張圖 來追蹤發生的狀況。 X 軸表示離地球的距離, Y 軸則是經過了多少時間。 從泰拉的角度,她的路徑 就只是一條垂直線, 距離都是零, 線上的每一個標記 就是她所感知到的一年。 史黛拉的路徑會從同樣的原點 出發,延伸到時間的 11.5 年, 距離則是離泰拉 10 光年…… 最後則會在 23 年、 零距離的那個點重聚。
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.
在泰拉的第一年底, 她會從地球向史黛拉的 太空船送出一道光。 因為一光年的距離,光要走一年, 它的路徑會是 45 度的對角線。 因為當光追上史黛拉時, 她是正在遠離光源的過程中, 對泰拉來說,是七年過去了, 對史黛拉則是四年。 當史黛拉觀察到 泰拉的第二道光時, 她已經是在回程路上。 但,這次,因為她是 朝向光源的方向行進, 光抵達她那裡要花的時間比較短, 她觀察到光的頻率變高。 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, 但在後半回程時, 泰拉老化得較快。
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%.
對史黛拉而言, 似乎泰拉、目的地星球, 以及整個宇宙都繞著她在運動。 因為長度縮短, 史黛拉觀察到兩地 之間的距離會縮減 2 倍。 意思就是,從史黛拉的觀點來看, 每段旅程只會花大約六年。 當她第一次送出光時, 泰拉已經度過了兩年。 史黛拉在去程時會再發送四次光, 每一次都從更遠的地方發送。 當泰拉觀察到史黛拉 回程的第一道光時, 她已經度過了 21 年。 在史黛拉返家的剩下過程中, 泰拉每年會收到好幾次光。 因此,泰拉觀察到的是, 在分離的 23 年中, 前 90% 的時間史黛拉老化較慢,
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.
最後 10% 的時間,會快速老化。 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。 雖然兩位雙胞胎都目擊 對方的時間加速和減速, 史黛拉看到的是一半一半, 泰拉看到的是分離的大部分時間 史黛拉都老化得比較慢。 這和兩位雙胞胎對於 太空旅行的測量是一致的, 該旅行花了 23 年地球年, 但在太空船上只會經歷 11.5 年。 當雙胞胎團聚時, 泰拉會是 43 歲, 史黛拉則是 31 歲。
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.
史黛拉弄錯的是她假設 她和泰拉同樣都是 慣性座標系的觀察者。 慣性座標系的觀察者 相對於宇宙的速度和方向 必須要是常數。 從頭到尾泰拉都沒有動, 所以她的速度是常數,零。 但史黛拉在回程時改變了方向, 此時她便進入了和一開始 不同的參考座標系。
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.
泰拉和史黛拉現在都更了解 空間時間怎麼運作了。 現在這對雙胞胎年齡相差十一歲, 她們是證明狹義 相對論的完美例子。