The answer lies 12,000 miles over your head

秘密就位於你頭頂上方12,000英里，

in an orbiting satellite that keeps time to the beat of an atomic clock

環繞地球運行的人造衛星上，

powered by quantum mechanics.

搭載著依據量子力學原理設計， 不斷運行的原子鐘。

Phew.

讓我們先喘口氣，聽我娓娓道來。

Let's break that down.

首先，為什麼當我們想要知道位置的時候，

First of all, why is it so important to know what time it is on a satellite

也需要知道衛星上的時間？

when location is what we're concerned about?

第一，智慧型手機得先確定

The first thing your phone needs to determine

它與衛星之間的距離。

is how far it is from a satellite.

每一個衛星持續不斷地發送訊號，

Each satellite constantly broadcasts radio signals

從太空中，以光速傳送到你的手機。

that travel from space to your phone at the speed of light.

而手機會記錄收到訊號的時間，

Your phone records the signal arrival time

藉此計算與衛星間的距離，

and uses it to calculate the distance to the satellite

用這個簡單的公式：距離 ＝ c 乘以 時間。

using the simple formula, distance = c x time,

這裡的 c 是指光速， 時間是指訊號傳遞的時間。

where c is the speed of light and time is how long the signal traveled.

但有個小問題，

But there's a problem.

光速超乎想像的快。

Light is incredibly fast.

如果我們只能以秒為單位來估算時間，

If we were only able to calculate time to the nearest second,

無論地球上任何一個位置，或是太空中其他地方，

every location on Earth, and far beyond,

與衛星之間的距離，計算的結果都一樣遠。

would seem to be the same distance from the satellite.

所以，為了讓測量到的距離， 能精確到幾英尺之內，

So in order to calculate that distance to within a few dozen feet,

我們得用上至今最精準的時鐘。

we need the best clock ever invented.

原子鐘非常地準確，

Enter atomic clocks, some of which are so precise

即使經過了三億年，

that they would not gain or lose a second

他們依然能分秒不差。

even if they ran for the next 300 million years.

原子鐘的運作原理是量子力學，

Atomic clocks work because of quantum physics.

所有的鐘都必須有一致的規律性。

All clocks must have a constant frequency.

也就是說，鐘一定會有重複不斷的運動

In other words, a clock must carry out some repetitive action

用來代表時間的等量累加。

to mark off equivalent increments of time.

就像老爺鐘的鐘擺，

Just as a grandfather clock relies on the constant swinging

會因為重力而來回規律擺盪。

back and forth of a pendulum under gravity,

原子鐘也有類似時鐘滴答聲一樣的規律，

the tick tock of an atomic clock

是根據原子在兩個能階之間的躍遷行為計算而來，

is maintained by the transition between two energy levels of an atom.

這時量子力學便派上用場。

This is where quantum physics comes into play.

根據量子力學，原子帶有能量，

Quantum mechanics says that atoms carry energy,

但原子的能量總額並不是自由變動的，

but they can't take on just any arbitrary amount.

而是受限於固定能階，

Instead, atomic energy is constrained to a precise set of levels.

我們稱之為量子。

We call these quanta.

打個比方，想像你正在高速公路上開車，

As a simple analogy, think about driving a car onto a freeway.

當你加速時，過程中你的車速會持續增加，

As you increase your speed,

例如從時速20英里增加到70英里。

you would normally continuously go from, say, 20 miles/hour up to 70 miles/hour.

現在，想像你有一輛量子車，

Now, if you had a quantum atomic car,

你不會持續地以線性方式加速，

you wouldn't accelerate in a linear fashion.

而是會立即的轉變速度， 從某個速度直接變成另一個速度。

Instead, you would instantaneously jump, or transition, from one speed to the next.

對一個原子來說， 從某一個能階跳到另一個能階，稱為躍遷。

For an atom, when a transition occurs from one energy level to another,

根據量子力學，

quantum mechanics says

能量的差異會等於

that the energy difference is equal to a characteristic frequency,

特性頻率 乘以 常數，

multiplied by a constant,

能量的變化會等於

where the change in energy is equal to a number, called Planck's constant,

某個固定數值(稱為普朗克常數) 再乘以頻率。

times the frequency.

這個特性頻率，就是我們製作原子鐘時所需要的。

That characteristic frequency is what we need to make our clock.

GPS衛星仰賴著"銫"和"銣"原子做為頻率的基準。

GPS satellites rely on cesium and rubidium atoms as frequency standards.

以銫133為例，

In the case of cesium 133,

它的特徵頻率為9,192,631,770 赫茲。

the characteristic clock frequency is 9,192,631,770 Hz.

也就是每秒大約九十億個週期。

That's 9 billion cycles per second.

真是個運作非常快速的時鐘。

That's a really fast clock.

無論再怎麼熟練的鐘錶匠，

No matter how skilled a clockmaker may be,

每個不同的鐘擺、發條和石英晶體，

every pendulum, wind-up mechanism

產生的振動頻率都會有些微差異。

and quartz crystal resonates at a slightly different frequency.

但是無論如何，在宇宙中的 銫-133原子

However, every cesium 133 atom in the universe

皆以完全相同的頻率進行振盪。

oscillates at the same exact frequency.

所以多虧了原子鐘，

So thanks to the atomic clock,

我們可以測量到1億分之一秒的時間，

we get a time reading accurate to within 1 billionth of a second,

以及非常精確地，計算出衛星與物體間的距離。

and a very precise measurement of the distance from that satellite.

現在，我們先忽略你在地球上這件事。

Let's ignore the fact that you're almost definitely on Earth.

我們可以知道你與衛星之間的距離是固定的。

We now know that you're at a fixed distance from the satellite.

換句話說，你是位在某個

In other words, you're somewhere on the surface of a sphere

繞著衛星運行的球體表面。

centered around the satellite.

接著測量你與第二個衛星之間的距離，

Measure your distance from a second satellite

會得到另一個重疊的球體。

and you get another overlapping sphere.

持續做這個動作，

Keep doing that,

只需要經過四次測量，

and with just four measurements,

並利用愛因斯坦的相對論進行修正，

and a little correction using Einstein's theory of relativity,

便可以指出你的位置，就位於空間中的某一點。

you can pinpoint your location to exactly one point in space.

這個過程總共需要：

So that's all it takes:

價值數十億美元的衛星網路、

a multibillion-dollar network of satellites,

振盪的銫原子、

oscillating cesium atoms,

量子力學、

quantum mechanics,

相對論、

relativity,

智慧型手機、

a smartphone,

還有你。

and you.

只要備齊了，一切就都沒問題。

No problem.