In this post, I am returning to the subject of relative motion, that I first described in posts 16.4 and 16.12. But we shall see that this description is complicated by what I wrote in post 26.6.

According to post 26.6, the speed, c, of an electromagnetic wave (for example, light), in a vacuum, is given by

where ε₀ is the permittivity of free space and μ₀ is the permeability of free space. Since we consider ε₀ and μ₀ to be fundamental constants, we would then expect c to be a fundamental constant. Let’s think about this in a bit more detail. The constants ε₀ and μ₀ enable us to calculate forces arising from electromagnetic phenomena. These forces determine the state of motion of an object. According to the Galilean principle of relativity, the state of motion of an object (for example the time period of a pendulum) is the same in all inertial frames of reference. Since I first described this principle, in post 16.4, I have covered a lot of mathematics which help us to understand it more clearly – see appendix 1.

But, in post 16.4, we saw that measurement of a speed depends on the state of motion of the observer. So, according to this idea, c cannot be a constant. We appear to have a problem.

In 1905 Einstein proposed that c was indeed a constant in all inertial frames of reference. We could rephrase the Galilean principle of relativity to state that the laws of mechanics (that is the laws governing the motion of objects) are the same in all inertial frames of reference. If we extend this idea to include electromagnetism we get that the laws of physics are the same in all inertial frames of reference. This is one way of stating Einstein’s special theory of relativity although it is more often explained by the equivalent statement that the speed of light is independent of the motion of an observer.

Einstein’s proposal could explain some experimental results obtained in 1887 by the American physicists Albert A Michelson (1852-1931) and Edward W Morley (1838-1923). They measured the difference of the speed of light parallel and perpendicular to the surface of the earth. But they couldn’t detect any difference. This was surprising at the time. The reason was that, as the earth rotated, the observers moved in the same direction as the parallel beam but almost perpendicular to the perpendicular beam. So they would have expected a difference between the two speeds. But no difference would be expected if c were a constant and, so, the same in all inertial frames.

If c is a constant it has implications for our understanding of the concepts of length and time that I first described in post 16.12. I hope to discuss these implications in my next post.

Related posts

26.6 Maxwell’s equations and electromagnetic waves
16.12 Measuring movement
16.9 Does the sun move around the earth?
16.4 Movement

Appendix

Let’s think about two inertial frames of reference, Φ and Φ’ that moves with a velocity V with respect to Φ. We’ll use the direction of V to define an x-axis.

Now consider an object in Φ’ moving in the direction of the x-axis with a velocity v’. Its velocity in Φ will be

Its acceleration in Φ’ is given by differentiating v’ with respect to time, t, so that

and in Φ by

The final step arises because V is, by definition, a constant for an inertial frame of reference.

The result is that a and a’ are identical. This means that the motion of the object changes in exactly the same way in the two inertial frames.

If we consider force acting on the object to be its mass multiplied by its acceleration, we consider the force acting on the object to be the same in the two inertial frames since we have no reason to suppose that the mass is not the same in both.