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Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morely is not shortened, but it is shortened for the co-ordinate system which is at rest relatively to the sun." - page 54 of Albert Einsten's Relativity (chapter 16)
Before we delve into the significance of the infamous late-19th century Michelson-Morley experiment that turned the world of physics on its head, let us for the meantime forget about the speed of light and its esoteric implications. Rather, let us try to make sense of the motions of our everyday world. Once we accomplish that, we can go onward to the Michelson-Morley experiment.
Walking Up & Down the Conveyor Belt
Even the most simple problems can prove to be surprisingly enlightening, so please bear with me, reader, as I submit an elementary yet baffling problem. Let's walk a 10 mile round trip, going 5 miles in one direction and 5 miles back. Given that we walk a brisk 5 miles an hour, that means the 10-mile round trip would take us 2 hours.
Okay, let's up the ante and go on another 10-mile round trip, though this time on a conveyor belt that's moving at one mile per hour. For the first 5 miles, we're walking counter to the conveyor belt's motion. On our return, we're walking with the conveyor belt. It would seem that this 10-mile round trip should once again take 2 hours. It seems as though the one mile per hour loss for the first 3 miles should cancel out the one mile per hour gain on the 3-mile return.
Wrong Answer Corrected!
Alas, this in not the case. You are going 4 miles an hour for the first leg of the trip, taking 75 minutes to walk the first 5 miles. However, you are going 6 miles an hour on the way back, and it takes you 50 minutes on your 5-mile return trip.
That's a total of 125 minutes, or 2 hours and 5 minutes for the 10-mile round trip upon the conveyor belt. That's 5 mintes longer than the round trip on the stationary terrain!
Here is the formula for calculating the time (T) for the 10-mile round trip, with d = distance, V = your velocity and v = velocity of conveyor belt:
T = d/V-v + d/V+v
Given that d = 5 miles, V = 5 miles per hour and v = 1 mile per hour, here is the solution:
1) T = d/V-v + d/V+v 2) T = 5/5-1 + 5/5+1 3) T = 5/4 + 5/6 4) T = 1.25 + 0.83333 5) T = 2.08333 hours for the 10-mile round trip
Derived Equation: T = 1/1-(v/V)2
Dividing 2.08333 hours by 2.0 hours equals 1.04166. The 1.04166 figure tells us that the round trip going up and down the conveyor belt takes 1.04166 times longer than the round trip upon the stationary terrain.
Many texts give an equation that leaves out the distance (d), though it's derived from the T = d/V-v + d/V+v formula listed above. The advantage of the revised equation is that we can figure out the relative increase in time of any round trip, without regard to distance. For instance, when V (your velocity) = 5 miles per hour and v (velocity of the conveyer belt) = 1 mile per hour, you can obtain the relative increase in time (T) for the round trip adventure:
1) T = 1/1-(v/V)2 2) T = 1/1-(1/5)2 3) T = 1/1-(0.2)2 4) T = 1/1-0.04 5) T = 1/0.96 6) T = 1.04166 increase in time relative to round-trip time on stationary terrain
Walking Across the Conveyor Belt
Let's say we're going perpendicularly across a 5-mile wide conveyer belt that is traveling at 1 mile per hour. How long would it take us to walk the 10-mile round trip at the brisk pace of 5 miles per hour? For the answer, take the square root of the above answer (1.04166) to find that the round trip going across the conveyor belt results in a net time increase of 1.02062 over the stationary 2-hour round trip. That's 2.04124 hours (1.02062 x 2 hours) for the 10-mile round trip going across the 5-mile wide conveyor belt.
Notice that the increase in time going across the conveyor belt is less than that of the up-and-down trip (1.02062 verses 1.04166). The going-across time is shorter by the square root of the up-and-down time. The converse is also true: the up-and-down time is longer by the square of the going-across time.
Walking-Across Time Equation: x2 + v2 = V2
If you wish to directly compute the walking-across time, the equation is provided below, with V = your walking speed, v = speed of conveyor belt and x = your speed going perpendicularly to the conveyor belt.
Given that V = 5 miles per hour and v = 1 miles per hour, we solve for x:
1) x2 + v2 = V2 2) x2 + 12 = 52 3) x2 + 1 = 25 4) x2 = 24 5) x = 4.8989795 miles per hour at a right angle to conveyor belt
Take 10 miles, then divide by the above figure to find the round trip takes 2.0412415 hours (or 1.0206207 times longer than the stationary round trip).
Enter The Michelson-Morley Experiement
Of course, Professors Michelson and Morley knew all about this and more when they performed their mind-blowing experiment in 1887. Professor Michelson used a set of mirrors to split a beam of light into two, then send the perpendicular beams of light on equally-distant round trips. One light beam traveled against and then with the direction of the Earth's orbital motion (analogous to up-and-down trip on conveyor belt), while the other light beam traveled crosswise to the Earth's motion (analogous to round trip across conveyor belt).
The Michelson Interferometer would determine the difference in the return time of the two light beams. Graphic courtesy of Wikipedia Commons.
Like the walker going crosswise to the conveyor belt, it was presumed that the light beam traveling crosswise to the Earth's motion would win the race over the light beam that travels counter and with the Earth's motion. Much to Michelson's surprise and no doubt chagrin, the round trips for the two beams of light took the same amount of time. For a video demonstration of the Michelson-Morley experiment, click here.
Wordsworth: Which having been must always be
Careful repetitions of the experiment only verified the original result. What were the implications of perhaps the most mindboggling experiment in the history of physics? Is the Earth actually at rest, as was claimed by the religious zealots who persecuted poor Galileo? Or perhaps the ancient Greek philosopher Parmenides was right after all, in asserting that motion and change are not real but only an illustion.
Answer: Space Contracts!
No less difficult than accepting the tenets of Parmenides' philosophy, modern-day physicists tell us that the results of the Michelson-Morley experiment are due to the fact that space itself contracts just enough to cancel out the expected increase of travel time. In other words, the distance between the mirrors contracted in the right amount to guarantee that the race between the competing light beams would always end up in a tie. (How's that for leveling the playing field?) This presumed shrinking of space is known as the Lorentz-Fitzgerald contraction hypothesis.
Earth's orbital velocity is about 1/10,000th that of light. But it wouldn't really matter how fast or slowly the Earth was moving through space. The two beams of light would always return to the finish line at the same time.
Imaginary Scenario: Earth Orbiting at Half the Speed of Light
Let's say, for instance, that the Earth's orbital velocity equals 1/2 the speed of light. If I understand things correctly, Michelson and Morley would have expected the round trip for the beam of light bouncing back and forth along Earth's orbital path to take 1.1547 times as long as the beam of light bouncing crosswise to the Earth's orbital direction. However, that's not how it works. Apparently, both beams of light would return simultaneously.
The beam of light riding the "headwind" and then the "tailwind" would not take the expected 1.1547 times longer than the beam of light riding back and forth through the "crossbreeze". Given the Earth's orbital motion at half the speed of light, the length of the round-trip journey against and with the Earth's motion would shrink to 86.6% of that traveled by the crosswise beam of light.
Therefore, the beam of light that's suppose to place second in the round-trip race would take only 86.6% of the predicted 1.1547 increase of time. And wouldn't you know it: 0.8660254 x 1.1547005 = 1. Length contraction saves the day for the Michelson-Morley experiment!
Fitzgerald-Lorentz Contraction Equation
The formula for the Fitzgerald-Lorentz contraction: the square root of 1 - v2/c2, where v = speed of body and c = speed of light. So if v is traveling at 90% the speed of light, the contraction of the body in direction of motion would be 43.59%.
1) length contraction = square root of 1 - 0.92/12 2) length contraction = square root of 1 - 0.81/1 3) length contraction = square root of 1 - 0.81 4) length contraction = square root of 0.19 5) length contraction = 0.4358899 or 43.59%
Theory of Relativity to the Rescue
According to Einstein's Theory of Relativity, the velocity of light remains constant, no matter the frame of reference of any observer. I've heard reports that Albert Einstein came up with his Theory of Relativity before knowing of the Michelson-Morley experiment. Whether the story is true or not, the results of the Michelson-Morley experiment are exactly what the Theory of Relativity would have expected.
copyright 2010 by Bruce McClure
March 2010 Feature * May 2010 Feature