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The "Light" Problem
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6.3 The "Light" Problem
As stated by the Galilean principle of relativity, Newton's laws hold in all inertia frames and these inertia frames can be related by Galilean transformation. However, Physics is not all about mechanics. What about the laws of electromagnetism? As we shall soon see, unlike the laws of mechanics, the laws of electromagnetism change under Galilean transformation and are different between inertia frames.
A fundamental law of electromagnetism, as revealed from Maxwell's equations, is that light (and, more generally, any electromagnetic wave) propagates through vacuum in any direction with speed, c=2.998\times 10^{8}\text{ m/s}.
Suppose a light beam travels at speed u=c in the x direction, as measured in frame S. Now, consider a second frame S' travelling at a constant speed v along the x-axis of S, and imagine the same beam of light travelling in the same direction. Using the classical velocity addition formula Eq. GR (3), the speed of light as measured by S' will be
\begin{align}
u'=c-v
\end{align}in the x direction. Similarly, if the light beam is travelling in the opposite direction, then according to Eq. GR (3), S' will measure the speed of light to be
\begin{align}
u'=c+v
\end{align}Depending on the direction of the light beam, S' will measure a speed of u' that varies anywhere between c-v and c+v. Therefore, while the speed of light is constant in any direction in frame S, it is not so in a different inertia frame S' that moves at constant velocity relative to S. It seems that there is a unique frame in which light will travel at the same speed in all directions. This supposed frame is sometimes called the ether frame.