There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Saturday, May 28, 2011

The Michelson-Morley experiment

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history of relativity
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This posting includes flash animations showing the physics discussed. Most computers have a flash player already installed, but if yours does not, download the free Adobe flash player here.

The Michelson-Morley experiment

In the 1880's Albert Michelson and Edward Morley set out to measure the apparent difference in speed of light traveling at different angles with respect to the motion of the Earth through the space, i.e. through the ether. Their assumption was that because light traveled so rapidly, it was hard to detect changes in apparent speed because of an observer's motion. But using the extremely accurate Michelson interferometer they expected to be able to detect the differences in the speed of light due to variations in the velocity of the Earth relative to the ether (see the animation in Fig. 3 of the previous posting). After all, the Earth spins on its axis and orbits the sun and we would expect to experience differences in the ether velocity compared with that of the Earth at different times of the day and at different seasons of the year. Amazingly, the result of their experiment showed that the apparent speed of light was not influenced by the direction that the light traveled relative to the direction of motion of the Earth around the sun. This result was very confusing.

image of Albert Michelson
illustration of motion of the Earth through the ether illustration of Michelson interferometer
Fig. 5a. Albert Michelson (1852-1931) was a Polish-American experimental physicist who is best known as the inventor of the Michelson interferometer and for experiments measuring the relative velocity of light at various orientations.

Fig. 5b. Because the Earth orbits the sun and spins on its axis, and the entire solar system moves in orbit around our galaxy's center, we would expect that we, living on the surface of this moving Earth, are moving with respect to the fabric of space, i.e. with respect to the ether. The speed of our motion through space is estimated to be roughly 250 kilometers per second or 150 miles per second, mostly due to our motion around the galaxy's center.

Furthermore, we would expect this speed would vary on a daily basis, to be maximum when the motion due to our spinning on axis adds to our orbital velocity and to the solar system's velocity, and minimum when it subtracts. We would also expect yearly variation on top of this since our orbital velocity would add to our solar system's velocity at times during the year and subtract at other times.

Fig. 5c. A Michelson interferometer splits a light beam into two beams, has the two beams travel different paths, and then recombines them. The diagonal mirror labeled "c" is only partially silvered to reflect half the light beam and transmit half. The relative phase (alternate reference) of the two beams upon recombination greatly changes the observed light intensity at the detector.

Not shown are components of the beams that are directed back at the lamp. When the two beams destructively interfere at the detector, there is constructive interference for the beams directed back at the lamp. Also, we have shown the light beams reflected by the mirrors "a" and "b" to be slightly angled for clarity. In fact, they are usually directed exactly back on top of the oncoming beams.

Because the wavelength of light is so small (a few hundred nanometers) a tiny change in length of either path (greater than about 100 nanometers or 4 millionths of a inch) will be clearly detectable. Minute changes in the speed of light in one path relative to the other will also be detectable.

Today this and similar interferometers are often combined with a laser to very precisely measure lengths and very small distance changes. Measuring minute length changes in structures before mechanical failure and tiny oscillations associated with vibration analysis are typical modern applications.

There were numerous theories in the nineteenth century that attempted to explain this null result, such as ether dragging, i.e. that the Earth dragged the Ether inside itself; however, attempts to find evidence of this around rotating massive astronomical bodies failed.

A constant velocity of light

Both Maxwell's equations and the Michelson-Morley experiment suggested that the speed of light is constant, independent of the velocity of the reference frame in which the measurement is performed. Normally a moving object's velocity is dependent on the reference from which you measure it (as we discussed above and illustrated in Fig. 3). Thus an automobile may be traveling at 60 miles per hour relative to the ground, but relative to another automobile moving at 50 miles per hour the first automobile is moving only at 10 miles per hour. In fact if the second automobile speeds up and matches the speed to the first one, the relative speed of the two cars is zero.

In contrast, all evidence suggests that a pulse of light will travel at 300,000 kilometers per second, independent of the velocity of the platform from which that speed is measured. Thus, a super rocket that is made to travel at 290,000 kilometers per second in the same direction as a light pulse would measure that light pulse as traveling 300,000 kilometers per second relative to the rocket, and not 10,000 kilometers per second as you might expect! Very, very puzzling!

image of Hendrik Lorentz
contraction of length in direction of motion

no effect on length perpendicular to the direction of motion

no effect on length perpendicular to the direction of motion

dilation of time due to motion

definition of the gamma faction in relativity
Fig. 6a. Hendrik Lorentz (1853-1928) was a Dutch physicist who initially derived the transforms for relativity that were later used by Einstein. He and Pieter Zeeman also discovered and explained the Zeeman effect for which they received the Nobel prize in physics. Fig. 6b. The Lorentz transforms for the three coordinates x, y, z, and time, t. The primed coordinates including time are assumed to be the location of an object in the moving frame (such as on a coordinate system aboard a fast space ship). V is the velocity of that coordinate system which is assumed to be in the x-direction. γ is the contraction factor and is between 1 (slow speeds) and infinity (velocities approaching c, the speed of light).

Lorentz transformations

In 1895 and 1904, Hendrik Lorentz came up with a set of equations indicating how time and space must transform in order that electromagnetic phenomena remain the same in a moving reference frame, similar to physics behaving the same inside a moving airplane (flying at constant speed along a straight path). If the distance and time transformed as per these equations, then the standard Maxwell's equations would be valid for electric and magnetic phenomena in moving reference frames. These transforms insure that the observed speed of light be the same, independent of reference frames from which it was measured. Exactly what "transformed distance" and "transformed time" meant was unclear.

These transforms are shown in Fig. 6b. To repeat: these transforms allowed the electric and magnetic fields to obey Maxwell's equations even in a reference frame that was moving with respect to the ether. They also allowed for slowly moving objects to have the normal relationship with respect to a stationary reference frame. That is to say, for slowly moving reference frames, the transformation approximated the Galilean transformations which governing relative motion between objects as experienced by people in their daily lives.

Length contraction and time dilation

stationary ball moving ball
illustration of time dilation illustration of time dilation
Fig. 7a. A round ball made of electromagnetic material moving at very high speed will appear squished, i.e. shorter, in the direction of motion. The ball on the left as seen when it is stationary, will appear foreshortened when moving extremely rapidly, as seen on the right. Fig. 7b. A very rapidly moving clock made of electromagnetic material will appear to have its ticking slowed down relative to a stationary clock. The clock on the left is as seen when not moving, while that on the right is moving. Shown in orange is the angle swept by the minute clock hand during a time period of 25 minutes as measured by a stationary person in the two cases (not moving and moving).
Fig. 7c. Animation of two balls pulsing electromagnetic balls, one stationary and one moving. They are identical when both are made stationary. The pulsations are controlled by an internal clock (or a internal physical process) in each ball and clocks are set to about 0.6 seconds per pulse. This animation illustrates length contraction and time dilation. Comparing the two balls, we see that the moving ball has been shortened in the direction of motion and the pulsations are slower, from the stationary observer's point of view. The speed of the moving ball is 0.95c for which γ = 3.2 . We have greatly slowed down this velocity to make it visible in the animation.

Mouse over the animation to start it (or click on it if it is unresponsive).

Lorentz and FitzGerald also showed that Maxwell's equations indicated that electric fields would not be symmetrical around very rapidly moving charges and further thinking about Maxwell's equations showed that high speed objects held together by electric and magnetic forces will be shorter in the direction of the motion. It was also shown that physical phenomena occurring inside these moving objects would appear to progress slower when viewed in the stationary frame. This means, for instance, that the ticking of a clock inside a high speed space ship would be slowed down (the speed of the whole space ship would not be affected). These effects of shortening and slowing down are known as length contraction and time dilation. Figures 7a, 7b, and 7c illustrate these effects. Both these effects can be derived from the Lorentz transforms.

The physicists of the late 1800's assumed that all matter was electromagnetic. Thus in their minds, all matter traveling at high speed would be contracted in length and all temporal processes would be slowed down. We now know there are other types of matter besides the electromagnetic variety. Nuclear matter, for example, is responsible for most of the mass of our bodies and objects around us and is governed by a different set of forces than specified by Maxwell's equations. Electromagnetic fields are responsible for most of the interaction forces between the atoms that make up our world. It has turned out we now believe that all matter, including nuclear matter, is in fact subject to the same length contraction and time dilation as is electromagnetic matter.

What is the internal mechanism that causes length contraction and time dilation?

Einstein, who we are getting to next, didn't worry about this question, but Lorentz would probably say something like this if he were asked this question today.

"Because the lower ball in Fig. 7c is moving at such high speed, the electromagnetic interactions between the atoms in this ball are altered. This is similar to the airflow being altered around an airplane traveling near the speed of sound such that the interaction forces cannot keep up with the extreme velocity. The result is a pile up of atoms on the airplane's leading surfaces which is called a shock wave, a phenomenon that does not occur at lower velocities when atoms have time to properly interact and keep themselves spaced the normal distance. In the case of the ball in the animation, because the electromagnetic interactions between parts of the atoms and between whole atoms only travel at the speed of light and the ball is moving almost that fast, these interactions are impaired, resulting in a partial collapse, i.e. shortening, of the structure in the direction of motion. This also makes the structure of the ball less rigid and more dense and sluggish. Thus the moving ball (or anything moving that fast) will be shortened in the direction of motion and its temporal processes slowed down."

Mathematically, Lorentz demonstrated that the electromagnetic interactions would be the same, i.e. obey the same Maxwell's equation for the high speed atoms as they are for the low speed ones, provided two things were true:

  • all distances between atoms and parts of atoms are reduced in the direction of motion as specified by the length contraction formula, and
  • their interactions are slowed down by the amount specified by the time dilation formula.

1921 image of Albert Einstein
Fig. 8. Albert Einstein. His theory of relativity assumed that the velocity of light was constant and from that derived many interesting properties of objects moving at high speeds.

Einstein starts with a constant light velocity

In 1905 in his paper: "On the Electrodynamics of Moving Bodies", Albert Einstein took the matter a step further by proposing a theory that started off assuming the observed speed of light be the same for all reference frames. Based on this one assumption, he went on to derive Lorentz's transforms as well as length contraction and time dilation. In addition, he interpreted these transforms. He said they meant that "time" itself slows down in all reference frames moving with respect to a "stationary" reference frame and also that "space" in the moving frame became shortened when measured by a stationary observer. These ideas and equations started the strangeness of modern physics which was developed in the first part of the twentieth century. We will discuss this more later on.

From this assumption, Einstein went on to derive many other equations having to do with very high speed motion. His most famous equation,  E = mc2  logically explained that equations of energy would be simpler, more logically complete, and consistent with the rest of his theory if the mass of an object, m, was considered to be a form of energy, E, and this equation gives the amount of energy that it equals. Physicists now call this the rest mass energy of an object. The arrival of the atomic bomb and nuclear power has seen the conversion of rest mass energy into the more conventional forms for energy, validating this concept. Studies of nuclear processes and also studies with particle accelerators (which accelerate tiny charged particles to velocities very near the speed of light) have repeatedly verified Einstein's equations.

Fig. 9a. Nuclear power converts a fraction of the mass of its fuel to usable forms of energy, as predicted by Einstein. Fig. 9b. Einstein's theory of relativity has been invaluable in understanding and making technical use of energetic particles traveling at or near the speed of light. X-rays for medical imaging are produced by bombarding energetic electrons into a tungsten target in a vacuum tube. Fig. 9c. Huge particle accelerators push back the frontiers of physics. This is an image of the explosion of particles created by a very high energy collision at CERN laboratory. CERN is a nuclear research laboratory located on the Franco-Swiss border, and is supported by 20 member states in Europe.

All postings by author previous:
history of relativity
contents of relativity
thought experiments