從狹義相對論到廣義相對論：另一種愛因斯坦方程式於真空中的解與史瓦西解的解釋及其應用於重力場如何影響光的傳遞

關於光如何在具重力場處（例如太陽、木星或者其他行星等）傳遞已經困惑人們數百年之久，牛頓基於其重力理論，計算了光通過太陽的偏折角度大約是 0.85 角秒。然愛因斯坦基於其重力理論（或稱廣義相對論），算得兩倍的結果，也就是 1.7 角秒。目前 1.7 角秒這個理論值，已被諸多實驗所驗證，也因此被科學界所接受，但事實上愛因斯坦並非一開始就得到這個正確的理論值，而且甚至也僅基於狹義相對論發表過與牛頓相同的偏折值 0.85 角秒（1911, On the Influence of Gravitation on the Propagation of Light, Einstein），五年後，愛因斯坦導出了場方程式（現通稱愛因斯坦場方程式），這被視為廣義相對論的先河，並且在那之後，史瓦西發表了關於愛因斯坦的場方程式的解（現通稱史瓦西解），於是愛因斯坦引用了史瓦西解，計算得 1.7 角秒這個正確的理論值（1916, The Foundation of the Generalized Theory of Relativity, Einstein）。在這之中旭夫發現了狹義相對論到廣發相對論之中，仍有可橋接之處，於是表了一篇如何僅用狹義相對論即得 1.7 角秒的方法（1959, On Experimental Tests of the General Theory of Relativity, Schiff）。本文將首先對愛因斯坦在狹義相對論的工作重新以現代的符號與現代的理解重述之（第二、第三章），接著，我們修正了一些旭夫僅用狹義相對論計算偏折角度的論文（第三、第四章），主要有以下幾個部分：一、對一些細節處與估計的部分，尤其是被旭夫所跳過之處，做更仔細的解釋。二、我們選用了一些更清晰的座標系、符號、推導與假設，使讀者不致在繁瑣的計算中迷失，尤其在計算光通過太陽的偏折角度時，由地球觀測與由太陽附近觀測的兩座標系，在旭夫的文章中有幾處是混用的，我們做了相當程度的調整。三、我們檢討愛因斯坦僅用狹義相對論計算偏折角度的錯誤，以及審視旭夫的計算，以比較出何處是愛因斯坦所思慮不周的。最後，我們探討史瓦西解所得重力場附近的光速與僅由狹義相對論所推得重力場附近的光速有何關係，並試圖了解愛因斯坦有沒有可能僅由狹義相對論即推導出愛因斯坦場方程式的解，而非如同史瓦西的作法：在給定諸多假設後，經由複雜的計算才導出史瓦西解。最後，本文的附錄將附上馬克士威方程組在相對性原理之下的張量形式，雖然這並非本文重點亦非原創，但這卻是愛因斯坦思考相對性原理的起點，這對相對論的發展確實至關重要。

關鍵字

相對論；狹義相對論；廣義相對論；重力理論；勞倫茲變換；愛因斯坦場方程式；史瓦西解；相對性原理；等效原理；光的偏折；馬克士威方程組

並列摘要

How light propagates especially through a gravitational field such as sun, Jupiter or other planets has been questioned for many centuries. In Newton's theory of gravitation, it can be calculated that the angle of bending of light passing through sun is about 0.85 arc seconds. In Einstein's theory of gravitation, also called theory of relativity, shows twice quantities of previous: 1.7 arc seconds. This value had been proved many times by observational experiment. But Einstein did not calculate the right value at beginning, actually he also published the value 0.85 arc second (1911, On the Influence of Gravitation on the Propagation of Light, Einstein) by using merely special relativity (SR). Five years after, Einstein derived the Einstein's field equation which is regarded as beginning of general relativity(GR). After, he used solution to this equation now called Schwarzschild solution to modify the value to 1.7 arc seconds (1916, The Foundation of the Generalized Theory of Relativity, Einstein). From above, Schiff published how to derive the value 1.7 arc seconds by only using SR (1959, On Experimental Tests of the General Theory of Relativity, ‎Schiff). In this essay, we will first revise some Einstein's work on SR and re-explain in modern notations and expressions (Chapter 2,3). Next, we modify the Schiff's paper about many portions (Chapter 4,5): First, we give some explanations on Schiff's work more detailed and estimate some formulations clearly which is skipped in Schiff's arguments. Second, we choose some appropriate frames, notations and assumptions to avoid misleading from complicated derivations. Especially for the speed of light observed from different coordinate systems, this is not only a way to understand how light propagates in the gravitational field but a key to calculate the right value of bending of light. Third, we revise both Einstein's wrong derivations and Schiff's derivation to see what is missed. Finally, we also discuss the relation between the Schwarzschild's solution and speed of light derived by SR. In the end, we question whether Schwarzschild's solution can be derived for given SR and Field equation or not. Finally, we attach the appendix of Maxwell's equation under the principle of relativity in tensor form. Although this is neither the point of this essay nor original concept, this do important for constructing the theory of relativity. Since it is the beginning of thinking what is real physics by Einstein.

並列關鍵字

theory of relativity ； special relativity ； general relativity ； gravitational theory ； Lorentz’s transformation ； Einstein’s field equation ； Schwarzschild’s solution ； principle of relativity ； equivalence principle ； bending of light ； Maxwell’s equations