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

Thought experiments

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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.

Thought experiments

Einstein explained his theory with hypothetical (rather fanciful) high speed experiments, similar to the one we present next. In his thought experiments (or gedanken experiments), the constancy of the speed of light was central. He used light pulses to measure both space and time. His aim was to show the oddities that go along with his assumption and to show that these were strangely consistent with each other.

Even though Einstein knew that traveling at the velocities required by these experiments was completely impossible in his day (as they remain in our day), he did understand relative motion and could project what a constant speed of light would imply. He had his thought experiments involve people in order to make the experiments seem more real to the average person.

If you have trouble accepting Einstein's initial assumption of a constant speed of light relative to the observer no matter how fast the observer is going, you might find it easier to accept Lorentz's approach which is computationally equivalent to Einstein's approach. Lorentz assumes that the phenomena of length contraction and time retardation make the moving party's rulers shorten and their clocks run slowly. Thus because their instruments are warped, the moving party only thinks that their measured speed of light is the standard value whereas in "reality" if they corrected for their instruments' warping, they would calculate the standard Galilean relative speed for the velocity of light. In Lorentz's way of thinking, the speed of light does vary in various moving reference frames (as does the speed of sound and most other waves), but this variation is masked by the warping of human senses and instruments in these moving frames.

If you are using Lorentz's approach, you can substitute the term "locally perceived distance" (LP distance) and "locally perceived time" (LP time) in place of Einstein's distance and time aboard the moving reference frame to refer to the distances and time periods which the moving party perceives by their warped instruments and warped human senses.


Experiment 1

Fig. 10. Animation of our first high speed experiment. Click on the experiment to start it (or restart it from the beginning). Mouse off it to suspend the experiment and back on it to restart it. The buttons at the bottom right hand corner can be used to select whether the experiment is to be shown from the referees' (i.e. ground) point of view or from the space ship's point of view. Everything is tremendously slowed down so that the experiment is observable.

The experiment uses two pulsed lasers, with clocks to record the time it takes for the pulses to travel to a mirror and back. The laser, clock, mirror, and path length of the experiment on the space ship are identical to those on the ground. The paths that the laser pulses travel along are highlighted to help the reader follow the experiment.

From the referee's point of view, the space ship is moving at v = 0.78c  (γ = 1.6)  and is shortened along with everything in it to 63% the original length. The laser pulse of the referees' experiment finishes first and the referee's clock records a time of 20 nanoseconds (the clock dials goes from 0 to 30 nanoseconds). The space ship's pulse has a greater distance to go (along the diagonal) and consequently finishes second. The clock aboard the space ship records a time of 20 nanoseconds in spite of this pulse finishing after the referees's pulse, because the space ship's clock is slow.

From the space ship's point of view, the referees and their equipment are moving in the opposite direction past them and now it is the referees (and their stuff) that are shortened. From the space ship's point of view, their laser pulse finishes first and their clock records 20 nanoseconds. The referees' pulse has further to travel and thus finishes second, but because their clock is slower, the referees' clock indicates the same 20 nanoseconds.

How can both parties think the other's clock runs slower and the other person is shortened? You would think that if one ran slower, the other must run faster. Einstein didn't explain this paradox, but just accepted it as a fact of high speed motion. Lorentz would say that it is an illusion due to the space ship's shortening and slowing of it's clock and other items on board. Both ways of viewing this phenomenon are consistent with the math of special relativity.



Experiment 2

Fig. 11. Animation of our second high speed experiment. Click on the experiment to start it (or restart it from the beginning). Mouse off it to suspend the experiment and back on it to restart it. The buttons at the bottom right hand corner can be used to select whether the experiment is to be shown from the referees' (i.e. ground) point of view or from the space ship's point of view. Everything is tremendously slowed down so that the experiment is observable.

This is similar to the previous experiment above in that it involves timing a laser pulse. It differs in that the laser pulse is traveling in the same direction as the space ship and it does not make a round trip, i.e. it travels in one direction only.

We might be tempted to use two synchronized clocks on board the space ship, one at the source of the light pulse and one at the detector, to measure the time at the beginning and at the end of the light pulse's trip. Einstein pointed out, however, that in these types of experiments, two clocks at different x-positions that seem to be synchronized in one reference frame will not be synchronized in another reference frame. To be clear, we need to use just one clock. Therefore we have include in our animation wires and electrical pulses bringing the starting time and finishing time to a single clock. This same arrangement is duplicated in the referee's experiment.

The experiment starts out with the laser launching a light pulse towards the detector. At the same time it launches a "start timing" signal down the wire to the clock. A short time later, the clock receives the "start" signal and begins timing the experiment. Still later, the light pulse reaches the detector and the detector sends out a "stop timing" signal down a wire towards the clock. Later still, the "stop" signal reaches the clock and the clock stops. A similar set of events occurs in the referees' experiment, albeit at a different rate because of time dilation.

From the referee's point of view, the space ship is moving at v = 0.7c  (γ = 1.4)  and is shortened along with everything in it to 71% the original length. The laser pulse of the referees' experiment finishes first and the referee's clock records a time of 25 nanoseconds (the clock dials go from 0 to 30 nanoseconds). The space ship's pulse has a greater distance to go (the motion of the space ship adds to its distance) and consequently finishes second. The clock aboard the space ship records a time of 25 nanoseconds in spite of this pulse finishing after the referee's pulse, because the space ship's clock is slow.

From the space ship's point of view, the referees and their equipment are moving in the opposite direction and now it is the referees (and their stuff) that is shortened. From the space ship's point of view, the referees' pulse has a shorter distance to go because now the motion of the referees subtracts from the distance travelled by the pulse allowing the referees' laser pulse to finish first. However the referees' stop signal is going the opposite direction and the relative motion of the referees greatly slows down its arrival at the clock. The net effect of all this is that both clocks stop at about the same time. The referees' clock starts first; however, because of time dilation due to its motion, it progresses more slowly making both clocks show the same time (25 nanoseconds) when they stop.

Again we are faced with the same dilemma, how can both parties think the other party's clock is slow, while both end up with the same velocity for the light pulse?

The results

The description of the experiments and the observations made by the two parties are found in the above captions. The point to take away is that there is something strange going on that makes the two parties disagree as to what really happened.

What's happening here?

The first instinct for most scientists (those who have not been trained in relativity) is to want to resolve this illusion. What is really going on that both parties think the other party has shrunk in the direction of relative motion and that the other party is operating at slower than normal speed? Which party's length is really contracted and which party's clock is really going slower. It's just good science to ask this sort of question.

Einstein's view

Everyone agrees that Maxwell's equations "conspire" to hide the truth from us as to which party is really moving and which is really at rest (compared to some background "ether" or fabric of space). Einstein's view was that because of this "conspiracy" or quirk of the equations, we will never know which party is "at rest" and which is "moving" or which is really shortened and which is not. Thus, in his mind, there is no point to asking those questions. These questions have no answer, at least not to us. He felt that this conspiracy is a fundamental law and should actually be the starting point and not something to be explained. He felt that there is therefore only "relative motion" and no "absolute motion".

Einstein felt that this conspiracy was connected to an even more absolute truth, that "physics should be the same in all uniformly moving reference frames". This concept was important to Galileo and was important in the development of his laws of motion. It means, we can toss a ball around inside the cabin of a modern jet liner traveling at 600 miles per hour and the ball behaves the same as it would if we were on the ground. Einstein felt it should be true even for objects traveling at speeds near the speed of light. Thus he felt that the crew on a spaceship should measure the same speed of light as should a person at rest.

If this conspiracy is an absolute law, ether is then nonexistent or at least unknowable. His approach also avoids the question of what is an electric or magnetic field.

Without any way to determine which reference frame was going along with the ether, all reference frames were equal. All is "relative". Einstein believed in the democracy of reference frames. His view is that this argument was like debating which time zone around the Earth should be used as the official world wide time. Scientifically they are all on equal footing.

Einstein also felt that if all physical processes slowed down on a rapidly moving reference frame (rapidly moving compared to someone else), we might as well say that time itself slows down. This is similar to saying time slows down in some areas of the world where life progresses at a slower, more relaxed rate. Einstein also felt that if all objects became shorter on a rapidly moving frame, we might as well say that distance itself shrank, i.e. the metric of space on that reference frame is shorter.

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Posted by P. Ceperley at 10:17 AM