( c² - v² ).For a back-ground explanation to the following, please see my first page on Special Relativity, Part One: The Michelson-Morley Experiment and the Lorentz transformations.
The Michelson-Morley experiment measured the speed of light on the assumption that the universe had an absolute motion, which light could go with or against, like a boat down or up stream. Alternatively, the light could go back and forth across the universal "stream" called the "ether". The experiment split a light beam into two journeys. These were reflected with mirrors, so that one journey was supposed to go up and down stream, the other journey back and forth across the stream.
As explained on the refered page, the distance traveled, one way, is d;
the light speed is c, and the ether velocity is v, which is assumed to be
relative to the earth's velocity, rather as someone in a boat sees the bank (
of Earth ) passing by, tho it is the ( Ether ) stream that is moving. The
combined time, t', for light to travel up and down an ether 'stream' would be
d/( c+ v ) plus d/( c - v ). This works out as 2dc/( c - v )( c + v ), or,
2dc/( c² - v² ).
Unlike the up and down stream journeys, the back and forth journeys take the same time. ( Please see refered page for the working. ) The total distance traveled by a crossing light-speed 'boat' is:
2dc/( c² - v² ).
Dividing this distance, both ways across, by the light-beam's cross-stream
velocity, c, gives its total crossing time, t, as 2d/( c² - v² ).
Thus, Michelson and Morley calculated the two times, t' and t, as a predicted out-come of maximal and minimal ether drag, respectively. But the fame of their experiment rested on the fact that it showed the two times to be equal. The two times could only be made equal, in theory as well as practise, by introducing a factor, F ( the so-called Fitzgerald-Lorentz contraction factor ) so that the shorter time, t, equals the longer time, t', multiplied by the contraction factor.
The contraction factor, F, equals t/t' = { 2d/( c² - v² ) } × { ( c² - v² )/2dc }
= {( c² - v² )}/c = ( 1 - v²/c² ).
However, statistical averaging of the split light beam's two journeys predicts that the two beams will take the same time on their journeys, just as the Michelson-Morley experiment found. There is no need for the ad hoc explanation of a contraction factor to match the Michelson-Morley calculation to the experimental finding.
An average is the measure of the most typical or representative item in a range of values. The experiment's two journeys are both ranges. The times calculated for light to travel down stream and to travel up stream are, respectively the quickest and slowest times, d/( c+ v ) and d/( c - v ). These quickest and slowest times form the limits of a range of times for the down and up stream journey.
A range of values may form an arithmetic series, in which case its average is measured by the arithmetic mean, the most familiar kind of average. But a range that is in a geometric series would be averaged by the geometric mean. My previous pages have discussed the role of the geometric mean in special relativity.
To obtain the geometric mean time for the down and up journey, multiply the quickest and slowest times and take the square root of the product:
{d/( c+ v ) x d/( c - v )} =
d/ ( c² - v² ).
Now for the geometric mean time for the back and forth light-beam journey. Well, the time is the same for the back crossing as the forth crossing. That is:
d/( c² - v² ). That is the
quickest and the slowest times, the upper and lower ends of the range, are
the same time. If you take their geometric mean, by multiplying them together
and taking this product's square root, you get the same time again. And it so
happens that this geometric mean time for the back and forth journey is the
same as the geometric mean time for the up and down stream journey.
That implies that taking the geometric mean times for the two light beam journeys correctly predicts that light will take the same time on both journeys, in the Michelson-Morley experiment. The suitable statistical averaging, of the split light beam's two journeys' ranges of travel times, does away with the need for a supposed contraction factor.
If the geometric mean time is multiplied by the speed of light, c, then a distance value is obtained, which is the familiar contraction factor, if the (one way) travel distance d is set at unity:
c/( c² - v² ) = 1/(1 - v²/c² ).
Apparently, the statistical meaning of the contraction factor is that it is the geometric mean distance. It somewhat shortens, here, distance, d, the standard ( unitary ) travel distance. Indeed, it is equal to the one-way distance traveled, either back or forth, by the light beam supposedly crossing an ether stream.
The Michelson-Morley estimation for a longest delay time taken, by the up and down stream beam, after halving the time for the two journeys, is:
t' = d/c(1 - v²/c²).
The estimation for the time, with the least delay back and forth across stream, after halving the two times, which are the same, is:
t = d/c{(1 - v²/c² }.
One of the two Lorentz transformations for these two times, in terms of each other, is:
t = t'(1 - u'v/c²){(1 -
v²/c²)}.
Therefore, we can equate the Michelson-Morley calculation to the Lorentz transformation as follows:
d/c = t'(1 - u'v/c²). This equation means that the Michelson-Morley case is a restricted version of the Lorentz case, which we lift by no longer assuming that u', in the Lorentz transformation, equals v, as it, in effect, does in the M-M calculation. The logic of this is that velocity u, which was zero, in the M-M case when u' = v, makes its appearance in the Lorentz transformations.
For the Lorentz case, the geometric mean time, with respect to the
averaged ( that is halved ) journey t', now becomes: d/( c² - u'v ).
The geometric mean time, for the journey, one way, t, is:
d/( c² - v² ).
If u' = v, the Lorentz case reduces to the Michelson-Morley case, where the geometric mean times are equal.
E = mc² is the most famous equation in science, derived by Henri Poincaré, and independently by Albert Einstein.
The subsequent century has been called the nuclear age for the Pandora type story of the release of nuclear energy on the world, whether in the form of nuclear arsenals' over-kill potentials, or for "peacable" purposes of permanently polluting the planet with radioactive waste products from fission energy. Commercial interests keep pressing their poisonous fission plants on the world before they become obsolete and their investments worthless, with the coming of the first fusion energy plant.
E = mc² applies to an increase in mass with motion. The mass, m, in the equation can be regarded as a difference in motion of mass M' from mass, M. The energy, E relates to the energy of motion or kinetic energy. In classical physics, this is recognised as: mv²/2.
The divisor, of two, means that one of the velocity multiples is an average of a velocity starting from zero to a final velocity. The other velocity multiple is just the final velocity. Such calculations apply when the kinetic energy is equated to a classical quantity, called the Work, meaning the work done to set a body in motion.
The multiple of the mass, m, by c² signifies the enormous energy, E, in any given mass.
The kinematic and dynamic forms of the Lorentz transformations, respectively for time and mass, are similar. So the Lorentz transformation for mass can be simplified as shown for the time transformation, in the previous section, reduced to its Michelson-Morley special case.
t' = t/{(1 - v²/c² }.
Substituting M' for t' and M for t:
M' = M/{(1 - v²/c² } = M(1 -
v²/c²)^-1/2.
The far right side of the equation merely restates the square root denominator as the power ( denoted by a circumflex ) of minus one-half. This may be expanded or put in a series of terms, by the binomial theorem. Thus:
(1 - v²/c²)^-1/2 = 1 + v²/2c² + 3v^4/8c^4 + 5v^6/16c^6 +....
The series has an infinite number of terms but only the first two terms count for much, because of the one-sided weight of the two terms in the factor to be expanded. If both terms were unity, instead of unity and a fraction, then the series would form a balanced distribution, gradually increasing and symetricly decreasing in importance of terms. Instead, the one-sided expansion results in a steep falling off in importance of terms from the start, or a skewed distribution like exponential decay.
Consequently, the energy equation is derived as an approximation that only uses the first two terms of the series. Hence:
M' = M( 1 + v²/2c²).
At this point Energy is equated to Mv²/2. And m is equated to the difference between mass M, and mass M'. Hence:
m = M' - M = E/c². That is: E = mc².
It wouldnt trouble Einstein that the equation is an approximation, because its actual value can be calculated to any degree of accuracy. The equation isnt an approximation for want of knowing a better value. So, this was no concession of the determinist philosophy of classical physics to statistical approximation.
Nevertheless, this equation boils down to an approximate expression of the contraction factor expanded to its first two terms, or (an approximate) tautology.
One can show this with the equivalent equation for times, to the energy equation for masses. Thus:
t' - t = tv²/2c².
But from the previous section, t' = d/c(1 - v²/c²). And t = d/c{(1 - v²/c² }.
Therefore:
d/c(1 - v²/c²) - d/c{(1 - v²/c² }
= (d/c{(1 - v²/c² })v²/2c².
Dividing thru by the value for t:
{(1 - v²/c² } - 1 = v²/2c².
This is back to the approximate expansion of the contraction factor. It wouldnt matter whether we did this reduction for times or for masses. The only difference is that masses use a different constant to the times' constant of d/c. Whichever constant is used, it cancels out, to reduce to the above equation.
A Michelson-Morley type experiment should be possible not just for the speed of light but for the mass of light. Classicly, light was variously calculated to be slowed down or sped up. On aggregate it was supposed more or less slowed down by the two journeys. Analgously, light of more or less shorter wavelengths produces more or less greater wave frequencies. Greater frequency means more energy. Altho light has no rest mass, it always has energy in motion, which must have a mass equal to the energy divided by the square of the speed of light. The more energetic photon or light particle associated with a gamma ray has about the mass of an electron.
A member of the Special Relativity e-mail group ( signed Gary ) suggested
a testable consequence of the Michelson-Morley experiment as light beam
journeys taking the same geometric mean time.
Possibly there is also a test whether the light beams, differently massed M'
and M, have the same geometric mean mass.
Richard Lung.
8, 10, 23 august 2006.