淺問相對論．．．．

其實坊間有很多所謂科普書或網上很多資料對「相對論」有錯誤的理解.它們有時嘗試用大眾容易理解的角度(即「牛頓力學」)去解釋「相對論」的種種推論.這種解釋只會令人似明非明,掌握錯誤的慨念.用「牛頓力學」去解釋「相對論」是從根本錯起.但是若我們先放下「牛頓力學」,由「相對論」的基本公設開始,一切便變得容易理解.

「相對論」分狹義(special relativity)及廣義(general relativity).「狹義相對論」建立在如下的兩個基本公設上
1.狹義相對性原理（狹義協變性原理）：一切的慣性參考系都是平權的，即物理規律的形式在任何的慣性參考系中是相同的。
2.光速不變原理：真空中的光速在任何參考系下是恆定不變的.

為甚麼會有這兩條原理呢?
1.狹義相對性原理是很基礎的原理,這條原理代表了宇宙沒有一個絶對的慣性參考系.
2.光速不變原理. 甚麼?光速會不變嗎?

根據「牛頓力學」,一件物件的速度在以不同速度移動的觀察者來說是不同的.例如
我騎著單車以x km/hr前進,在我的前方有一件跑車以y km/hr 行駛,那麼對我來說跑車是以y-x km/hr 的速度遠離我.那怕跑車行得多快,它相對我及地面的速度是必定不同.

但是在19世紀,所有對光速進行的實驗都發現無論怎樣量度,光速都是一樣,科學家開始用不同的方法解釋,但都沒有一個滿意的答案.在當時根本沒有人想像到把「牛頓力學」慨念推翻,以光速不變作基本理論,去發展一套全新的理論.但是愛因斯坦卻有天才的智慧,勇敢踏出這一步.

既然光速不變,那麼我們必要放棄某些「牛頓力學」的慨念.很明顯,在以上的例子中,跑車遠離我的速度 v 不再是 y-x km/hr . 因為若果 y = c, (c即是光速) 及 x=10 的話, v = c -10,違背了光速在任何觀察者都是不變的假設.

那麼有甚麼需要改?速度是衍生出來的物理量,它的定義是物體在每單位時間裏面所移動的距離.所以在這新的理論,我們應該繼續用這個定義.所以,若果我們要光速在任何觀察者都是不變,我們必要修訂不同觀察者量度每一事物的時間及長度的慨念.

Time dilation

假設一架火車裏面有一組儀器,他可以垂直向上發射一團光束到火車天花板的一面鏡子,然後再反射回儀器那裏.鏡子距離儀器有 s 米. 設光的速度為 c1. 那麼對火車裏面的觀察者來說,
光從儀器發出再反射回來的時間為 t = 2s / c (式1)
而根據狹義相對性原理,無論火車是否正在移動,火車裏面的觀察者都會量度出 t = 2s/c .

現在火車正在以速度v行駛,一位在地面的觀察者會看見那光束所有的路線長了,因為光束除了垂直移動外,它還跟隨火車向前移動.
設在地面的觀察者所量出的光束來回的時間為 t1.

光束向前移動的距離 = v t1

光束總行走距離 = 2 [( (v t1/2)^2 + (s)^2) ^ (1/2) ] (畢氏定理)
= ((v t1)^2 + 4 s^2) ^ (1/2)

光的速度c1 = ((v t1)^2 + 4 s^2) ^ (1/2) / t1 (式2)

直到這裏,以上的計算是適用於「相對論」及「牛頓力學」.「牛頓力學」會說 t = t1, 所以光的速度會不同, c1 > c.
「相對論」則說光速在任何觀察者都是不變, c = c1, 所以 t 不等於 t1 !

從 (式1) s = c t / 2, 代入 (式2), 得出
t1 = [ 1 – (v /c)^2 ] ^(-1/2) t
就是著名的time dilation轉換公式.


其實以上的例子不可說是一個正式的推算,因為它假設了 s 在兩個不同的觀察者是一樣的.但是這個假說在「相對論」來說是成立的.以上的例子目的是說明「相對論」與「牛頓力學」的主要分別.希望帶給你對「相對論」多一些了解.

今次寫到這裏,沒有時間再討論 length contraction. 若你有甚麼回應,歡迎你提出,再作討論.

(資料來源: 以上全部自己寫 (除了「狹義相對論」的兩個基本公設上外 http://zh.wikipedia.org/w/index.php?title=%E7%9B%B8%E5%B0%8D%E8%AB%96&variant=zh-tw)

2007-03-17 16:54:20 補充：
另一位回答者 matt_aca
把 GR and SR 調轉咗

First a background of relativity before answering your specific question:

BACKGROUND OF RELATIVITY
Relativity is a study of light and about its effect on big heavenly things, such as stars, planet, but not everyday life objects like the movements of bullets, trains or a moving car. General relativity is advocated before Special relativity and GR skip gravity in consideration the velocity of light; while SR considers also the impact of gravity of light.

By why study light? Because light is only thing that is unchanged in different observation frames/environments. Observation frames are the situations you are in for example in the moving car, standing still in street, or travelling a light speed spacecraft. Relativity predicts that if a flying space ship is about the speed of light, the size of the flying object will appear to be shorter than actual size to an observer standing on the ground. Under the general relativity, when objects are flying around the speed of light, time and distance are no longer constant! While it also states that speed of light is still a CONSTANT under any observation frame, whether you are standing on earth, or travelling around the speed of light in a spacecraft. Because time not a constant, you have the situation that if one of the Sola twin brothers is travelling on Earth, while the one brother travels in a light-speed space-ship for a number of years, when they again meet each other, the one travelling in light speed will be much younger than the one on Earth.

The basic law for general relativity is that the speed of light is a CONSTANT, no matter what frame of reference you are in (while time and space can be variables, not constant.)

The GR ignores gravity, e.g. the big gravity of the sun of the path of the light. The newer version of relativity, the special relativity (SR), considers also the impact of gravity of big heavenly objects, such as the sun, on the path of the light. The SR predicts that the gravity of large objects, such as the sun, will cause the time and space around the sun to distort... e.g. a iron ball standing in your cotton pillow, this is how the gravity of the sun or other big heavenly objects cause to the space and time to distort. This distortion of time and space around the big object consequently cause any light rays travelling around the sun to BENT (i.e. not travelling in straight line.) This SR theory was proved to be correct by accident during a total of the sun, when a star normally should be hidden under the sun suddenly become visible during a sun eclipse: The sun curved the light rays from the star, and made itself visible to observers on Earth, even though the star is actually directly hidden behind the star.

ANSWER TO YOUR QUESTION
You now know that relativity predicts that light travels at a constant speed, that is,
speed of light =distance travelled/ time elapsed, or c = d / t in all observation frames, this means the left hand of the light equation has to be a constant (this is the backbone of the general relativity theory, if you believe in relativity);
so the distance and time can be variables, but not necessarily constants, in order to accommodate the theory that c=constant.

Remember now that space and time are no longer constants in relativity. For an ordinary object such as a spacecraft travelling at a speed of v, the equation is:
speed of spacecraft, s = d / t If you deliberately put time t=constant to consider the relationship of s and d, you will see speed s is directly proportional to distance. Hence, when you travel in your spacecraft in very high speed s close to light, the distance will be prolonged (when time is a constant).

You can use the same technique to assume distance d is a constant, to establish that s is inversely proportional t, so when you travel at a speed s close to light, the time seems to slow down.