Transcript Module 2
Module 2 Light and Newtonian Relativity Three Scenarios We said in the last module that the laws of mechanics are invariant in Newtonian relativity. By ‘mechanics’, we mean macroscopic push and pull forces. But what about the laws of electricity and magnetism? Recall that Maxwell’s equations are a nice summary of these laws and that, from Maxwell’s equations, we can derive the wave equation for the electromagnetic wave (light wave) in free space. The wave equation for the electric field component of a linearly polarized wave looks like 2 E ( x, t ) 1 2 E ( x, t ) 0 (2.1) x 2 c2 t 2 for a wave traveling in the x-direction with its electric field oscillating in the y-direction. The speed of the wave c is found to satisfy c 1 / oo where o and o are the permeability and permittivity of free space, respectively. When the values of these free space constants are substituted, we find that c = 3x108 m/s. Now in Newtonian relativity we know that the measured speeds of objects are relative to the motion of the observers. Here we have a speed c popping out of Maxwell’s equations. But the motion of the observer should affect the value of this speed, right? Let’s say that it does. Then perhaps this value of c is measured only by the observer for which Eq. (2.1) is valid. This frame of reference is a special frame in which light travels at c in vacuum. In any other inertial frame, the speed of the light wave is measured to be different than c. Fig. 2-1 shows one such possibility. Module 2 Light and Newtonian Relativity 1 y’ y S S’ v ux= c x O’ O z u’x= c - v z’ x’ Fig. 2-1 Suppose a laser pointer is at rest in frame S and S is the special frame for which Unprime measures the light to travel at c. Then Prime in frame S’, moving as shown at speed v relative to Unprime, will measure the laser light to travel at c-v according to Newtonian relativity. Perhaps this is the way things are. This one special frame for which light is measured to travel at c is called the ether frame. The reason for this is that the prevalent thought at the time of the development of Maxwell’s equations was that the light wave needed some medium to support the oscillations of the electric and magnetic fields, much like a water wave needs water or a sound wave needs air. This medium was called the ether. While similar in function to water or air as a wave medium, the ether is a very different kind of material. It doesn’t have the usual properties of normal matter but that doesn’t mean it could not exist. We see then that the frame S Fig. 2-1 is the ether frame. And we say further that Maxwell’s equations (the laws of electricity and magnetism) as we know them work fine in this ether frame but they change form in any other frame. Thus, perhaps the laws of electricity and magnetism are not invariant in Newtonian relativity. Module 2 Light and Newtonian Relativity 2 What we have just described is one possible scenario for how the laws of electricity and magnetism behave in a proposed relativity theory. But there are two other possible scenarios to consider. Perhaps the laws of electricity and magnetism are invariant under Newtonian relativity but the laws, as written by Maxwell, are incorrect. Or perhaps Maxwell is correct, the laws of mechanics and the laws of electricity and magnetism are invariant in a relativity theory, but that theory is not Newtonian relativity. This last scenario implies that the Galilean transformation equations are incorrect. It also implies that Newton’s laws are most probably incorrect. His laws are invariant in Newtonian relativity and most likely would not be invariant in a different relativity theory. Let’s summarize these three scenarios* in Table 2-1. Table 2-1 Scenario 1 Scenario 2 Scenario 3 Newtonian Newtonian Non-Newtonian Laws of Mechanics Correct? yes yes probably not Laws of Mechanics Invariant? yes yes yes (once corrected) Laws of Elec. & Mag. Correct? yes no yes Laws of Elec. & Mag. Invariant? no yes (once corrected) yes Relativity Theory Comments ether frame Galilean transformation equs. incorrect * I suppose we could perform another permutation and argue that there is a fourth scenario that goes like this. Maxwell’s equations are correct and there is a new relativity theory in which the laws of electricity and magnetism are invariant. Newton is correct and the laws of mechanics are not invariant in the new theory. This has the downside that the laws of mechanics are invariant. This goes against our observations of mechanical interactions. Module 2 Light and Newtonian Relativity 3 The Michelson-Morley Experiment We see now that the famous Michelson-Morley experiment was a method to test the validity of Scenario 1. If the experiment had revealed a fringe shift, then one could have more confidence in stating that the measured speed of light is a relative quantity and that there is an ether frame. The details of the experiment are discussed in the supplemental document. Read this document. You will read not only about the result of the experiment but also about some hypotheses put forth at the time that explained the result and that also were consistent with the idea of an ether frame. These hypotheses, however, have been abandoned by all but a few and the general consensus is that Scenario 1 is incorrect. So what about Scenario 2? Well, the laws of electricity and magnetism seem to work pretty well. Before we roll up our sleeves and try to fix something that may not be broken, let’s look at Scenario 3. Newtonian relativity sure seems to work in our everyday world. And the laws of mechanics seem to be okay, too. If there are flaws in these, they must be “minor” flaws. And who wants to find a new relativity theory? That doesn’t seem trivial. Well, as you probably know, Scenario 3 is the generally accepted winner. The laws of mechanics and Newtonian relativity do work well when the relative speed v is small compared to c. That’s why they seem fine in our everyday, macroscopic world. But they need to be adjusted to a more general form that includes high relative speeds. That’s why they are “incorrect”. The fix to Newtonian relativity, which boils down to generalizing the Galilean transformation equations, is made in the next module. The fix to the laws of mechanics follows in subsequent modules. Module 2 Light and Newtonian Relativity 4