The Michelson-Morley Experiment and Special Relativity
How does the Michelson-Morley experiment relate to special relativity?
For SR to explain the MME requires a violation of the second postulate, and to compensate for this violation, the ad hoc hypothesis of length contraction is added in order to salvage the theory. This is why it works: Two wrongs make a right.
Violation of the Second Postulate in the MME
The MME is basically two light clocks aligned perpendicular to each other. One is aligned parallel to the motion of the apparatus, the other is perpendicular, or orthogonal, to its motion. It is in the orthogonal clock where the second postulate is violated, but it was used properly in the parallel arm in the context of SR.
The second postulate requires that light does not acquire the motion of the source, but this has to occur for SR to explain the MME. If we are to rigorously adhere to the second postulate, then two criteria must be met: 1) the light does not acquire the velocity of the source, and 2) light will not acquire the direction of motion of the source.
The reason this ball returns to the cannon, and a ball dropped from a moving ship stays in line with the mast, are the same.
In each case, the ball retains the motion of the source. Without this, the ball in each case would “fall behind” the source because 1) the ball did not acquire the velocity of the source, and 2) the ball did not acquire the direction of motion of the source. For a repeating process, such as in the first gif below, this would mean that a cycle would not be possible, which is demonstrated by the second gif below (“Correct View”).
Careful! The code for the following gif is a little “buggy”, or glitchy, and may slow your device.
The gif above demonstrates how light SHOULD behave by the second postulate, since it does not acquire the motion of the source in terms of velocity or direction. Essentially, a light clock in motion, oriented perpendicular to its motion should not tick based on the second postulate alone.
Some may interject and say: “Light is a wave, and will expand in a circular pattern.” Though this is true, it assumes that light, like waves can not be focused. A laser can be focused to a thin beam, just as a Megaphone can focus sound, to which anyone can attest to if they have stood perpendicular to one, and heard almost nothing. This means, that though the beam may diverge into a cone over larger distances, there will be a relative velocity, below the speed of light, and distance, where the mirror will be outside the area of the cone of light. In other words, the clock would not tick, and switching from a ray approximation for light to a wave model for light does not improve the prospects for the theory.
Besides, if you were to search for an image of a light clock used to explain time dilation, the examples are shown as below:
First note that there is no wave-like description of the light. Just as in the cases with the earlier cannon and ship examples above, it requires that light acquire the velocity of the source, and its direction of travel, since the light remains in-line with the source and mirror at all times. This is clearly a violation of the second postulate based on the criteria listed earlier.
To defend the position that the light does acquire the velocity of the source, which is less obvious than the directional violation, we simply need to break the speed of light into its vector components. The speed of light is constant in a frame of reference by the second postulate, so in this particular case, then, the speed of light, c, is the magnitude of its velocity. For an orthogonal clock, there is an x- and y-component to the velocity. For the clock to operate, it requires that the x-component of the light’s velocity match the source and mirror as shown by the last gif. This leaves less and less of the magnitude of the velocity to be available to the y-component of the velocity. Therefore, a clock slows in the orthogonal direction because the rate at which the light climbs and falls is changed by an amount related to the motion of the source.
The ultimate proof is that if a source reaches the speed of light, time will stop because the x-component of the velocity required to stay in line with the clock leaves no remaining velocity for the y-component. Time both slows, and would stop, because of this violation of the second postulate in these clocks.
Special Relativity
The formula for determining the time dilation for an orthogonally oriented clock is
T = 2L/(c^2 - v^2)^½
In this case, L = “y” , and what is not explicitly stated is that the denominator of this equation is equal to Vy, or the y-component of the light’s velocity. Simply put, T = y/Vy, and that the slowing of clocks is completely dependent on changing only the y - component of the velocity of light while rigidly maintaining the x-component of the velocity, along with the directionality of the source.
Making this change even more obvious is that the denominator, which is equal to Vy, is also equal to c divided by the Lorentz factor, gamma. In a comoving frame, Vy=c, but for the moving frame, Vy’ = c/gamma. Essentially, the Vy’ = Vy/gamma. Stated correctly, the orthogonal clock ticks slower for a moving observer because Vy contracts by the Lorentz factor. This is a direct result of the light acquiring the motion of the source, contracting the y-component of the light’s velocity.
It should be obvious from the explanation above that an orthogonal clock should not tick. The only reason an explanation is possible is by dividing the Vy component of light for an observer at rest to get Vy’. Vy contracts between frames in order to explain the Michelson-Morley. In my opinion, it seems right, by being completely wrong in the context of SR.
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