# 230 – The Michelson-Morley experiment

From Tesla, Maxwell, and many others you all have learned that light moves through a medium called aether. A wave that requires a dense medium to propagate is called “elastic” or “mechanic”, because it moves through an elastic or mechanic medium. Michelson and Morley made an experiment to check if the aether is a real entity. It is a light interference experiment. Interference happens when two waves sum up, forming a resultant wave that can have a greater, lower or the same amplitude.

A wave that moves along the x axis is described by an expression that satisfies
the wave equation (D’Alembert equation):

where f is the wave function, v is the speed of the wave. The solution of the equation is the harmonic wave described by the following:
f=Acosα(x,t)=Acos(kx — ωt+φ0)
where A is the amplitude of the wave, k is the wave factor, ω is the pulsation and φ0 is the initial phase. Let’s consider 2 waves and sum them up (interference): f=f1+f2.

The interference is called constructive when 𝛼2 − 𝛼1 = 2𝑛𝜋
In this case the amplitude is:

The interference is called destructive when α2 − α1=π+2nπ
In this case the amplitude is:

With specific instruments (for example Fresnel mirrors) it is possible to visualize interference between two coherent waves that manifest with fringes that are illuminated zones alternated with obscure zones.

A ray of light coming out from the source S is partly reflected in the movable mirror M1 and partly transmitted to the fixed mirror M2. The return light rays from M1 and M2 hit first against the beam splitter and then are cast against the detector that is the focus of the splitter lens. The detector receives two coherent rays of light that are conveyed from the same source. “Coherent” means that these rays have the same phase. These rays, one from M1 and another from M2, interfere or superpose reinforcing or weakening each other, depending on the
optical path that comes from the AM1 and AM2 distances.

By suitably changing the distance AM1, it is possible to produce in O (the detector) interference fringes with a maximum or minimum of intensity. By varying the distance AM1 of λ/4 (being λ the wave length of the casted beam of light) you can pass from a minimum to a maximum. A compensating lens is used to produce exactly the same optical path in the two rays.

In 1881 Michelson and Morley made an experiment to examine if, in the same way sound requires an elastic medium (such as air or water or a solid medium) to propagate, similarly light, to spread out, would need a mechanical medium, called aether.

Aether should be present all over in the intermediate space to allow light to reach Earth from the stars. This implies that space is not empty: vacuum is only a vacuum of air but not an absolute vacuum. Call c the speed of light in the aether. When you move toward the light ray, inside the fixed aether, with a speed v, you shall measure a total speed of light c+v. On the other hand, you will measure c–v when you move in the same verse of the light ray. This expression has much to do with the Galilean relativity.

Michelson and Morley thought that this principle could be used to check if the aether does exist. They thought that an interferometer could be used to evaluate the variation of the interference fringe, due to the speed of the Earth. Their idea was the following: when you put one branch of the interferometer in the direction of the speed of the Earth v and the other branch perpendicular to the first, you will obtain a well precise drawing of interference
fringes.

Then, by rotating the interferometer of 90 degrees, you can invert the two interferometer branches. Since the optical path changes, also the fringes should change. Let’s consider the calculation. The two branches of the interferometer, AM1 and AM2, have the same length. The AM2 branch is rotated in the direction of the motion of the laboratory and relatively to the cosmic aether. When we consider the aether as motionless, fixed to the stars, the direction and the entity of the Earth speed v should depend on the hour of the day and on the day of the year.

According to the law 𝑆 = 𝑣 ∙ 𝑡 (where S stays for space, v stays for speed, t stays for time) of the rectilinear uniform motion, the ray of light going from A to M2 takes a time t=l/(c–v). To return from M2 to A, it takes a time t=l/(c+v). The total time for the branch AM2 is:

l is the length of the segment run by light.
Time t1 of the other branch (AM1) has a different value. For this case you have to remember that during the time t1 the Earth keeps moving. Thus the total trajectory of the ray is triangular. While the ray of light moves from A to M1, the mirror A moves in the direction of the speed of the Earth. This distance AA’ can be calculated taking into account the speed v and the time t1 necessary for the light to reach M1 and to return to A’.

So you have AA’=vt1. The ray of light has thus to travel the distance AM1A’=2AM1 with a speed c. The needed time will be:

The result will be:

These two coherent rays superpose in the O point in a way that depends on t1 and t2. Then, when you rotate the interferometer in order to range the branch AM1 in the direction of the speed of the laboratory and in respect of the aether, t1 and t2 change. So, there should be a difference of phase in the two rays in O with a consequent change of the interference
fringes.

Every time this experiment has been repeated, at different hours of the day and on different days of the year, it has always given the same result: no change in the fringes.
Obviously, when the physicians tried to explain this result, no one supposed the Earth to be motionless. So, Einstein solved the problem according to his famous statement, on the basis of which he later on based his theory of relativity. He postulated light moves with equal
speed c in all directions and in all different reference systems. Moreover, according to Einstein, this would be the maximum reachable speed:

an unbeatable limit that can’t be surpassed. As a consequence scientists stated that the aether, intended as the mechanical mean in which the light moves, can’t exist.
Since, however, there are matter particles definitely able to travel faster than light, the only possible explanation for the Michelson Morley experiment is that the Earth doesn’t move. In this case v=0 and you will notice that t1 and t2 become equal:
t1 = t2 = l/c
that means no change in the interference fringes). This is the main idea: formulas behind this experiment become incredibly simple if we consider the Earth immovable.