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Description of
the Apparatus and the Operation.
Orig: May 04, 2003 Rev:
Aug 15, 2004
Edward M
Billinghurst
5132 Stone Canyon Ave, Yorba Linda, CA 92886 714 970 7727 <[email protected]>
The purpose of the paper is to discuss the property of congruency of light waves, and then to discuss the Michelson-Morley experimental results on the basis of congruency instead of beam travel time. Michelson’s Interferometer was designed to compare the instantaneous, simultaneous, values of two monochromatic, continuous light beams for equality of relative phase. The experiment is properly analyzed from the aspect of the phase relationship between the experimental beams, that is, in the frequency domain; not from the beam travel time aspect or relationship; the time domain. Congruency is remotely connected to travel time of the beams only in the sense that both beams must have reached the Interferometer for comparison. Travel time has no effect on the beam relative phase relationship.
A monochromatic, continuous, light field consists of successive
congruent sinusoidal cycles of
2 π radians. The instantaneous
vector force amplitude at every instant of time, in each cycle for any φx for which
0 < = φx <
2 π , is the same for every cycle, independent of
time or space location, the speed of light, or any other parameter. Period!
The term φx is the fractional
phase (fraction of one cycle) of a field.
N x 2π + φx is the total phase of the field
with respect to a given length of non-coordinate space.
Phase is in radians, a
dimensionless quantity. It does not
have a ‘time’ dimension.
All the cycles of light fields of the same generated
frequency and intensity (power) are congruent within each field and from
field-to-field. The ‘time’ separation between cycles is set
by the emission process and is invariant after emission. The ‘spatial’ wavelength for all emitted
cycles is dependent on the velocity of the emitter relative to the propagation
velocity of the radiation. The spatial
wavelength, after emission, is invariant in free space, that is, in a
vacuum. The distinction between time
separation and spatial separation is of crucial importance.
The emitted frequency of radiation
(E/h) is termed its ‘Planckian’ frequency in this paper. ‘Doppler’ frequency is
the frequency detected by an object
which moves with some velocity
relative to the radiation propagation velocity. Doppler frequency implies a space-time
coordinate system. If the relative velocity is zero, then the Planckian and
Doppler frequencies are identical. The
Interferometer responds to instantaneous phase only; frequency is not a
consideration, so Doppler frequency shift is not involved in the phase
detector.
Congruency means
that the wavelength and time/wavelength (cycle) are constant, everywhere the
light field exists.
Providing
that the Proposition is actually True.
Continuously
emitted light fields form a pattern in non-coordinate space. The field propagates in space volume. That is, the volume of space filled with a
light beam is continually expanding.
The pattern of the light field, where it exists in the volume, is that
of force at every point which varies in amplitude sinusoidally. This results in a wave-like cyclical cosine
function disturbance at each space-time point.
The disturbances at different points have different fractional phase
amplitudes, unless the different points are separated by exactly an integer
quantity of cycles. Michelson adjusted
to make the latter condition true at the Interferometer.
A fixed length rod, positioned
totally inside an isotropic light volume, sees a constant quantity of cycles
from end to end, for any orientation
in the volume. An observer on the rod does not have any
sense of ‘propagation’ direction, he merely sees a constant quantity of cycles,
with the amplitude (phase) at any point
varying from observation to observation; in space-time. The observer cannot logically infer a speed
of light propagation, since the cyclic variations in amplitude depend solely on
the fact that the emitted Planckian
light cycles in a cosine pattern, not on the speed of propagation.
The speed of propagation of light
is independent of the speed of the emitting source, and is also independent of
frequency. The observer has no way of
knowing where he is in the light field volume. If the position of the rod at a
particular instant is noted and marked, and at a later instant the rod is no
longer at the marked location, then the rod is considered to have moved from
its immediately prior position, but the number of cycles spanned has not
changed. In the light field, the
rod location is not referenced to any absolute or relative position other than
relative to its immediately prior position.
What I have
tried to establish in the above discussion is that, when totally inside a light
field, an observer has no knowledge of the volume of the light field, or how
long it has been there, or any legitimate sense of light speed, or any sense of
direction related to the emitter. All
he sees is the number of cycles of light spanned at any instant by his
particular rod. Nothing else. If the
observer knows a priori the particular Planckian frequency involved, and the
experimentally measured speed of light, then he can infer the length of his
rod, in light wavelengths, from the
number of cycles (wavelengths) spanned.
But, he has to know the light speed in advance; he cannot infer it from
his observations inside the light field.
Light speed can be measured by an observer outside of the light field,
as Michelson and others did, from a number of different types of experimental
observations.
I have gone to
this great detail because Michelson and his successors predicted a change in
his display (interferometer) due to the velocity of his apparatus with respect
to the velocity of his light beams.
Michelson calculated that the time-coincidence, simultaneity, of the
light waves depends on the path length involved; the rod length. I am proposing the argument that the beam
travel time is not a factor; only the congruence of the continuous beams, cycle
by cycle, is measured by the interferometer.
This congruence is not affected by the apparatus velocity. An experimental observation.
The following explanation of the results of
the Michelson-Morley experiment is based on the premise of ‘Congruency’ as
related to light field patterns. The
velocity-caused travel time difference
for the two orthogonal fields of light used in the experimental
apparatus described below is relevant only to the extent that it affects simultaneity in space-time, but not congruence. A theory is developed which then agrees with the experimental
results. Any theory contradictory to
experimental observation is considered to be false.
Since the Planckian frequency of light, which relates to cyclic
phase, is not affected by the velocity of light this explanation does not
depend on the independence of the velocity of light from the velocity of the
emitter The explanation does not depend
on Special Relativity, other than the a priori postulate that physical laws are
the same everywhere and everywhen in the universe.
Special Relativity is mainly concerned with the effect of relative velocity between two bodies or
observation points on experimental physical effects. SR is a time-domain theory. In the MM experiment, there is no
relative velocity between the light source emitter and the Interferometer. The
light source is rigidly connected to the interferometer. The experiment
involves the frequency domain, not the time domain.
The velocity of light does not even have
to be constant, and in fact isn’t because of the unequal lengths of refractive
glass in the arms of the Interferometer. The light fields have a difference in
average light speed relative to the Interferometer arms, over the total light
path length for the two fields. That difference affects simultaneity of
the light patterns at the Interferometer, but not the relative phase. A difference in simultaneity causes only an
apparent difference in relative phase
at the space point of the observer. As stated in the Congruency discussion
above, all cycles of the light beams are instantaneously in phase in
non-coordinate space-time, always.
Congruence is used in the geometrical sense. It is a function of shape only. It is not a function of size. Two shapes are congruent if, when one is
overlaid on the other, properly scaled, the two shapes touch at all points.
MICHELSON-MORLEY
EXPERIMENT
II Background and Purpose
The Michelson-Morley experiment was
designed to measure the effect of the Earth’s velocity on the speed of emitted
light. A qualitative prediction of the
effect was made by a theoretical analysis.
If an effect could be observed, then the existence of an ‘ether, or aether’ might be corroborated.
No velocity effect was ever observed, with
the test apparatus employed and for
the particular test conditions. Many tests were made. If there was a velocity effect, it was not
detectable by the resolution of the apparatus.
However, it is shown below that no velocity effect is to be expected to
occur, contrary to initial expectations of Michelson.
The experiment
was very valuable in that it appeared
to refute the ‘ether’ theory which held
that the speed of light depended on the speed of the ether. The ether concept was proposed because of
the belief that light required a mechanical medium to sustain
oscillations. It then followed that the
actual velocity of light would have to be composed with the velocity of the
ether.
In point of
fact, the experiment did not refute the ether theory because that theory is
irrelevant to an explanation of the results.
The results are explained by the concept of congruence of all cycles of
a monochromatic, isotropic continuous light field.
The experimental apparatus used was an
Interferometer, which was invented by Michelson for use in studies of light
(optics). The Interferometer compared
two superimposed, orthogonal, continuous light fields, split from a common
emission source, at a particular instantaneous space-time position. Since the
superimposed beams came from a common source, their individual intensity
amplitudes were equal. The total instantaneous intensity of the superimposed
combination is determined solely by the relative
fractional phase difference between the two fields. When the two fields are in phase, the
intensity is at a maximum, (constructive interference). When the two fields are exactly one half
cycle out of phase, the intensity is at a null, (destructive interference).
Michelson had theorized that the intensity would vary as the apparatus was rotated in azimuth, because of the effect of the velocity of the earth. He based this prediction on the assumption that relative fractional phase difference is affected by travel time. But it is not; the fractional phase of each field is the same because of congruence, as described above.
Emitter velocity, field path length, and the average speed of
light in the path all affect the total
phase and fractional phase of either field at the Interferometer, but have no effect on the difference in
fractional phase between the fields, after initialization of the process. Emitter velocity does not affect propagation
speed but it does affect frequency at any moving target. Phase is a parameter associated with
frequency. The two orthogonal fields
are split from one source, so any velocity effect would be the same for each field. The field path length and the average speed
of light are initialized to a
particular state which does not change with time or velocity.
Michelson and Morley, his colleague,
published two descriptive papers of the experiment in the “American Journal of
Science”. “The Relative Motion of the
Earth”, 1881 by Michelson, and
“On the Relative Motion of the Earth and the Luminiferous Ether”, 1887 by
Michelson and Morley. Michelson also authored a book “Studies in Optics”, which contains much interesting
information associated with light, (optics). Michelson received a Nobel Prize
for his experimental determination of the speed of light, using the
interferometer technique. As a side
note, he was the first American Physicist to receive a Nobel prize.
Planckian
Frequency The frequency of light emitted by Planck’s
relationship:
f
= Energy/h, where ‘h’ is Planck’s constant.
Doppler
Frequency The perceived frequency of a Planckian
oscillator as seen by a detector moving relatively to the Planckian
oscillator. Doppler frequency is the
Planckian frequency modified by a velocity function. Doppler frequency is
relativistic because of the relative velocity between light wave and a
detector. Planckian frequency is not
relativistic. It is invariant.
Physical Laws are the same everywhere and everywhen in space. This is Einstein’s First Postulate for
Special Relativity.
Congruence: Light fields are sequences of identical cycles of 2π radians. The instantaneous, simultaneous, amplitudes at all points of identical phase φx , where 0 <= φx < 2π , in
each of the cycles are equal in sign and magnitude. No matter where each cycle
is in space-time. The term φx is the fractional phase of the field.
The total phase of the field is N
x 2π + φx , but the relative fractional phase between
any two fields is φx1 - φx2 where the subscripts refer to the two
fields.
Congruence involves comparison of cycles,
not comparison of the total fields.
The Planckian frequency and congruency is set at field emission and is not affected by any subsequent action or parameter.
Congruence is not related to spatial
coordinates nor to ‘time’. This, as are
all premises always, is a working premise to be validated as being consistent
with experimental observations. The epistemological basis for this premise
comes from the solution to Maxwell’s equations regarding electromagnetic
radiation.
‘Congruence’ is not synonymous with
‘coincidence’. Coincidence and
simultaneity imply a ‘space time’ relationship. Congruence does not involve either time or spatial coincidence.
The fields can be congruent without being simultaneous in time and position.
Congruence is a purely geometric concept related to ‘shapes’.
‘Congruence’ is selected as a premise
because of its independence of spatial position or time or frequency, or any
other concept. It is a primary concept
- it does not depend on any other premise.<
The Interferometer display responds to the congruence of two waves as
seen at the same space-time position, not on their trajectories or
histories. Speed of light has no
relationship to congruence. Speed of
light, along with path length, does affect simultaneity of time and spatial
position of events. However, path
lengths are not required to be identical.
They can differ by an integer number of cycles, with no effect on the
results.
The concept of
‘congruence’ is crucial to this explanation of the Michelson-Morley
experimental results. The Interferometer responds to the
differential phase between two superimposed light fields, so the analysis must
be based on phase, not time. Field
travel time differential is only important as related to simultaneity at the
Interferometer of two cycles being compared, but phase is the governing
quantity of interest. This is a Phase analysis procedure.
Congruence is a necessary condition for the use of the
Interferometer, whereas ‘simultaneity’, or zero differential fractional phase
is desirable but not required. In fact,
the actual differential fractional phase was mechanically adjusted to a
difference of one-half cycle (destructive interference) for measurement
purposes.
Note that no premise as to the independence of the speed of light from
emission source velocity is being made.
This explanation of the Michelson experiment does not require that
premise. There is only one emission
source, which is split into two fields.
I do not see a valid logical reason for inferring the independence of
the speed of light from velocity of the source from this experiment since the
Interferometer only compares two fields split from a common source. Any
velocity effect on the common source would be applied equally to the split
fields. The Michelson experiment
neither confirms nor refutes any conjecture as to the speed of light with
regard to the speed of the source.
The
Ratios of the arm length to the average speed of light for each arm are equal.
The actual ratios of arm length to average
speed of light for each of the two fields are affected by the unequal
refractive indexes and arm lengths of the separate paths, as discussed
later. The length of one of the arms is
permanently adjusted so that
simultaneity at the IF display is achieved. The adjustment
sets the required ratio of arm length to average speed of light for that arm to the same value as that
of the other arm. Note that the two
arms do not have to be of equal length, but the ratio of arm length to the
average speed of light must be the same for each arm. The adjustment assures
simultaneity in space-time at the display for all space and all times, for
every azimuthal orientation of the apparatus.
Once set, this adjustment is not changed by any further operation.
However, the effect of the
adjustment on the Interferometer display would change with azimuthal
orientation, if a concomitant change
in velocity with azimuth unequally affected the speed of light in either of the
fields because of an ether. See
“Adjusting the length of one arm to make the two φFrac terms equal at all times”, below.
The experimental observations are assumed
to be valid.
Experimental observations are always given
priority over any theories or premises.
If there is a conflict between an a
priori theory or premise used, the theory or premise is invalidated
until the conflict is resolved.
In
general, valid experimental
observations can be used to make inferential deductions or theories as to any
parameter effects, but the merits of such deductions are always open to
question and review.
This paper starts from the observed experimental observations,
which did not show a velocity effect, and develops a plausible (to me)
explanation for the observed results.
The explanation is based on analyzing the
relative phase relationship (congruence) between two light fields. The fields are approximately orthogonal, but
orthogonality is not required.
The explanation does not involve light travel-time or Special Relativity theory considerations. (Of course, a singular experiment can always be analyzed on its merits without resorting to a general theory, so recourse to SR may be convenient, but is not required, although it may often be very helpful.)
VI General Comments Regarding the
Experimental Apparatus
A test apparatus, the Interferometer, was constructed to produce
two orthogonal fields of light, split from a common source; propagated toward
and reflected back from mirrors after a suitable travel time, and then
superimposed at a common space-time point. The superimposed fields formed a
display which was viewed through a Telescope.
The effect of the earth’s velocity was
predicted to cause a difference
in round-trip travel times from the source back to the display. Variations in the orientation of the
apparatus with respect to the earth’s velocity vector were expected to cause
intensity changes in the Interferometer display.
.
Light field instantaneous force amplitude
varies as a Cos(φ) function; φ in radians. 0 <= φ < 2π The Cos function comes from the solution of
Maxwell’s differential equations regarding the relationship between Electric
(E) and Magnetic (H) fields. E = A Cos (ωr/(cavg)
- ω t). H = B Cos (ωr/(cavg)
- ω t)
The ωr/(cavg) term represents a fixed phase-offset which depends on the Tr/cavg ratio only. The ωt term
represents the time-varying phase of the cos argument. Since the
ωt term is
in radians, which are not relativistic, the time is not relativistic. It is only locally applicable as a
mathematical artifice to indicate simultaneity
when comparing two cos functions; phase, in radians, is the quantity of
interest in the cos argument.
Maxwell’s original differential eqns do
not directly include radiation-emitter velocity. They are based on the differential equations of the E and H
fields as they are thought to exist, without regard to how they were generated.
The equations were later modified by Einstein to include Special
Relativity velocity
information. However, the application
of Special Relativity is not required to explain the experimental results,
since any velocity effect would be the same for each field, and
therefore would not affect the
differential phase at the Interferometer.
If a velocity effect did occur, it would not affect congruency, only
simultaneity. The mechanical adjustment
of the arm length eliminates any velocity effect.
.
Field intensity is linearly proportional to the square of the
vector force amplitude. Amplitude is
related to the fractional phase of the light field, not its travel time. Of
course, phase and hence amplitude itself, as seen by an observer, may vary in
coordinate space-time with time or other parameters, but it is only the actual
instantaneous value of the sum of the
squares of the amplitudes that determines the Interferometer display. In the experimental apparatus, the congruent
(coordinate independent) wave cycles are mechanically adjusted to be coincident
from field to field. This is done by
adjusting the length of the light path of one field so that its Tr/cavg ratio is the same as that of the other
field. This ratio does not depend on
apparatus velocity. It only depends on
the fixed mechanical construction of the apparatus. It does not require that the two arm lengths be identical, only
the ratio above.
The instantaneous amplitudes do not give
any information as to prior history or trajectory or travel time or effects of
path conditions or velocity of the apparatus or any other parameters. The
Interferometer display represents the ‘here’ and ‘now’ status only.
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Description of
the Apparatus and the Operation.
Refer to Figure 1 above, the mechanical
schematic of the apparatus.
A monochromatic, continuous light source
is split into two fields directed towards the mirrors M1 and M2 and reflected
back and superimposed at the Telescope,
T, which then serves the function of an Interferometer display. No assumption is made as to the dependence
of the speeds of the light fields on the speed of the source, since that is
what this experiment is trying to determine.
The speed of light is however
dependent on the refractive index of
the medium at any space-point occupied by the light. The speed is assumed to be different in free space from what it
is in the glass field splitter and in the compensating glass which is inserted
to equalize the refractive index effect in the two paths. The light speed on either side of a
refracting medium is that of free space. (I don’t know why, it just is). It
changes in the medium, but immediately assumes the speed corresponding to free
space when out of any refractive medium.
For a light path containing portions at different refractive indexes, an
average speed, (cavg)
for the total path can be calculated.
The average speed is a function of the total path and the portions of
the total path occupied by free space and
refractive material. Due to inequality in the total lengths of
refractive material, even with the compensating glass, (cavg) is not the same in the two paths. (cavg) is slightly slower than c in space, but it is constant in each path
throughout the experiment.
The argument of the Cos function - E = ACos (ωr/(cavg)
- ωt) represents the total value of the radians associated with the light path. The display is only concerned with the relative phase between the
two light fields, not the total number of cycles. The relative phase is based on the difference in fractional phase of each field.
φFrac = the fractional remainder of the ((ωr/(cavg)
- ωt)/2 π)) argument.
The phase term of one of the fields is:
(ωr/(cavg) - ωt)/ 2 π) = M +( φFrac)1, where M is an integer. The phase term for the other field is N + (φFrac)2.
Relative or differential phase between the two fields is the
difference between the two φFrac terms. Since M and N are
integer numbers of cycles, their actual values are not important. That is, the total arm lengths do not have
to be the same.
Adjusting the
length of one arm to make the two φFrac terms equal at
all times.
The position of Mirror M1 on its arm is adjustable by means of a
lead screw. This sets the length of the
associated arm to make the relative Fractional
phase shift difference between the two fields equal to zero, (actually to pi
radians, for destructive interference). After the initial adjustment is made,
the mirror position remains fixed for the remainder of the experiment.
To make ( φFrac )1 equal to ( n Frac) 2 , it is only necessary to make a
slight adjustment in r1. The wavelength
of the sodium light field used is about 0.6 microns. The length of each the two arms, from Interferometer to
reflecting mirror is approximately 1 meter, or 1,700,000 wavelengths of the Na
light source. The arm length accuracy
requirement is 6:10,000,000, a precision of great difficulty to obtain without
some sort of Vernier adjustment. But a lead-screw adjustment range of 2 microns or less is enough to get the
equality desired. This adjustment is
cyclic, so continued movement of the adjusting screw will repeatedly bring
about the desired equality in the fractional phase values.
Michelson took great care to shield the arms with boxes to reduce any
temperature effect on arm (brass) length due to air convection, and maintained
the experimental room at as close to constant temperature as he could.
The value of (cavg)1 depends on the value of r1, but its dependency is much less than
that of the length increment, so the desired criterion for zero phase
differential can be achieved at some setting of the lead screw.
The arm length is actually adjusted to
make the fractional phases of the two fields equal in amplitude but opposite in
sign. This makes the fields one half
cycle out of phase. This sets up
‘destructive’ interference between the two fields, so that the total intensity
becomes zero at all times and for all azimuth positions of the apparatus. This is referred to as a ‘null’ condition.
The superimposed reflected fields are
observed through the Telescope. The
observed effect is due to the intensity (instantaneous square of the
vector sum of the force amplitudes) of the two light fields. The fields are superimposed in time, at the
same space point, that of the position of the Interferometer display. The display is dimensionally a ‘point’
(positional) quantity. The display
itself does not give any velocity information.
(According to Heisenberg, one
cannot measure position and velocity in one experiment.)
The apparatus was rotated in the
horizontal plane by eight steps (first apparatus) or 16 steps (second
apparatus). This effectively rotated the fields around the resultant of the earth’s rotational and orbital vectors. This process
was performed many times and the
resulting field intensity values were recorded and statistically
evaluated. The mean and standard
deviations of the readings were a very small fraction, and were well within
expected experimental error of the
theoretically predicted display changes made according to preliminary
calculations based on field travel time difference. The statistical changes that were seen were assumed to be due to
random experimental effects, since no correlation was found with the earth’s
velocity.
No change in Interferometer display was
observed during this process, indicating that orientation with respect to the
earth’s velocity vectors has no effect within the resolution of the equipment.
Several analyses and hypotheses were
proposed to account for the ‘null’ results, but none of them that I have
reviewed in the many books or publications or as results of Web searches, have explained the experimental
results on the basis of the differential relative Fractional phase between the
two fields. Of course, the fact that I
have not found such an analysis does not mean that it has not been done before.
But, I did look for it.
The ‘null’ result of the Michelson-Morley
experiment has been explained by reference to the congruency of light
waves. The congruency concept is based
on Planckian frequency as set at emission, and Maxwell’s solution to
electromagnetic radiation theory.
I argued that the two light fields, having
been split from the same emission source, were congruent. This further implies that the frequencies of
the split fields were identical, and were unchanged during the experiment.
If the frequency of one field had changed
by a minute amount, then the Interferometer display would have detected
this. The Interferometer display is
capable of detecting a difference in frequency of less than one part in a
million, at least. It may actually be
much better, by using more accurate detection equipment than was available to
Michelson at the time.
Planck’s theory says that the frequency of
emitted light is given by f = E/h, (frequency, energy, and Planck’s
constant. According to my limited understanding
of quantum theory, the energy of a particular radiator is determined by the
allowable energy states of emitters in an atom, and these allowable states are
presumed to be independent of the chemical compound where the particular
emitting atom is contained. Michelson
used Sodium light in the experiment. He
also did experiments using light from other elements, such as Mercury.
If it is correct that the frequency of
light emitted from a particular atom, in any compound containing the atom, is
identical under any and all conditions of, in particular temperature and
velocity and gravitational or other forces, then I would suggest that
Michelson’s Interferometer technique with different light sources containing
sodium or mercury or whatever could be used to test the theory or the range of
accuracy of the frequency from different sources, but the same frequency
emitter.
It might also be used to compare light
from a terrestrial source with same-frequency light from a stellar source using
the Interferometer to determine if the frequencies from the two sources are
identical, and if not, what the possible range of variation is.
I would also note that, given the ability
of the Interferometer to detect a difference in frequency to better than one
part per million, further tests to detect a velocity effect with the technology
available today would be very interesting.
A difference in frequency between the two fields would result in ‘beat’ frequencies equal to the sum and the
difference between the two field frequencies.
A tiny difference frequency of as little as one Hz should be readily
detectable by eye, or at least by precision technology available today.
Comparing two frequencies of the same
Planckian value from two different sources could be done with one emitter
stationary, and the other mounted on a spinning disk to achieve a velocity
effect. If the spinning emitter
frequency is affected by its velocity, then the ‘beat’ frequency phenomenon
mentioned above would occur, and should be detectable. If no effect is observed, then it would
refute any premise that the Planckian frequency is affected by the velocity of
the Planckian emission source.
It is worth emphasizing that Michelson’s
Interferometer design represents a significant contribution to Physics
investigations for other useful purposes, even though it did not actually
refute the idea of an ‘ether’. It did
not test that existence or non-existence.
But the Interferometer design is applicable to other useful
purposes. For instance, the IF
technique can be used to measure the length of the ‘standard meter’ in terms of
the number of wavelengths of a particular light source. That is, the IF can be used to measure
spatial wavelength of a moving radiation source. That wavelength is a function of the relative velocity of the
source with respect to the propagation velocity (invariant) of the radiation.
See Michelson’s “Studies in Optics” for
further interesting remarks relating to the Interferometer technique, and other
related experiments that Michelson performed, such as measuring the speed of
light for which he received a Nobel
Prize.
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Figure 2 is deliberately drawn to show a
difference in the leading edges of the two reflected fields. The Interferometer is located at a position
where the two fields both exist. Note
that the two congruent, coincident-in-space-time cycles were not emitted at the
same time. But they are still congruent. If the arms were exactly equal in length, and if the average speeds of light were
exactly equal, then the two
leading edges would still be incoherent, due to the difference in path lengths
for the two fields. Light path length
is not equal to the arm length, because of the arm velocity. The times
at which the leading edges of each field are reflected back to the common
space-time point of the Interferometer are not the same. This time difference was predicted,
incorrectly, by Michelson to cause a difference in fractional phase between the
two fields at the common point of the Interferometer.
The fields schematically shown in Fig 2
can be interpreted to illustrate the earlier statement that the number of
cycles occurring in a rod of a given length, or the number of cycles spanned by
a rod of a given length, is always the same, if you mentally visualize arms
between the mirrors and the common Interferometer location. Because the light source is continuous, the
field patterns exist as a succession of congruent cycles in space from the emitter to the leading
edge, with amplitude variations of a
Cos function. The leading edges of the fields propagate in space, without any
reference to a coordinate system. A
rod or arm moving in space moves
through the continuous field pattern, or the field pattern moves along the
rod. The motion between rod and light
field pattern is relative; either or both can be considered as moving, but it
makes no difference as to which is moving because the motion is relative.
(Excuse the use of the two terms, ‘rod’ and ‘arm’. The term ‘arm’ relates to the Interferometer as described by
Michelson. The term ‘rod’ was generally
used by Einstein in his derivation of Special Relativity. So, in this presentation, ‘arm’ and ‘rod’
are used as synonyms, indiscriminately.)
The two arms, joined together at one end
as they are in the Interferometer, and having a mirror at the other end of each
arm, see continuous, congruent
patterns. The total number of cycles, not an integer number, along each rod
remains the same at all times, but
the fractional phase of the cos wave at any particular point on a rod is
variable. The number of cycles
may be different for the two arms, depending on arm length.
In that case, the two cycles being compared were not emitted at the same time, but
because of congruency, that has no effect on the display. The difference in field travel times, as
explained below and as calculated by Michelson, causes the simultaneous
comparison of two congruent cycles which were emitted at different times.
At all instants of viewing, each arm length measure based on the number of cycles spanned is constant. The time between zero crossings (or two points on any a wave cycle shape of identical amplitude and sign) is set by the constant Planckian wavelength and the speed of light at time of emission. It then becomes a fixed, unvarying absolute standard for time, because all cycles are congruent. For a given Planckian oscillator, the time/wavelength is fixed. This time is independent of any observer’s position or velocity. It is constant. Total time of light propagation along the length of the rod is then calculated from the total number of pairs of zero crossings, that is, the number of cycles, spanned by the arm. Note: the spatial wavelength is a function of emitter velocity, but that fact is not relevant to this experiment. Spatial wavelength and time/cycle are both invariant after emission.
The time per cycle does not vary with any
Doppler effect. The cycle is determined
by the space between zero crossings, not the perceived Doppler frequency. The frequency is only the Planckian
frequency.
For this reason, a light field can be used
as a velocity-independent clock, anywhere and anytime, by counting the number
of cycles. A detector which puts out one ‘tick’ for a
fixed number of cycles will produce ticks at a fixed rate, depending only on
the invariant characteristic of the emitted Planckian frequency. Since motion is relative, we can consider
the Planckian emitter as being a fixed reference point and thus its Planckian
frequency is fixed. Doppler frequency
is irrelevant. Any relative motion is
considered as pertaining to the rods. I
throw this in because Einstein opined that the tick rate of moving clocks
depends on their velocity. Einstein “On
The Electrodynamics of Moving Bodies”, paper in “The Principle of Relativity”, a collection of papers by Dover
Publications.
Michelson, in his book “Studies in Optics”
details his measurement of the ‘standard meter’ using the Interferometer to
actually count the number of cycles spanned by the standard meter; on the order
of 2 million cycles in one meter, to an accuracy of about 1:2,000,000.
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Prediction as
made by Michelson using Travel Time Calculations . See “Six Not So Easy Pieces”, by Richard P. Feynman, Perseus
Books.
Travel time calculations are the same as
those in Michelson’s 1881 paper, although using slightly different terminology.
The round trip travel times are different due
to the fact that one field is propagated parallel to the earth’s velocity
vectors, and the other is perpendicular to the earth’s velocity vectors.
These travel
times are the actual times for the light beams to traverse the light-path
lengths, and return. Light path length
is not equal to rod, or arm, length, because of the velocity of the apparatus
with respect to the velocity of light. But, the number of light
cycles spanned by each arm is independent of apparatus velocity.
Considering Equation 2 above:
t1 +
t2 = (2L/c)/(1 - v2/c2) = T
This is the total round trip time for the
light field to travel from the emitter to the mirror and back to the
Interferometer. This time is related to
the total number of cycles spanned by the total light path. The number of cycles spanned by the arm from
the mirror back to the Interferometer is the ratio of arm length (L) to the
total path length (in cycles) x total path length. N is proportional to (L/TPL)*TPL, where TPL is total path
length.
In other words, the number of cycles spanned
by the arm depends on the arm length only and is independent of the total
travel time.
However, as mentioned earlier, the two
particular cycles superimposed at the Interferometer may not have been emitted simultaneously,
but that is of no consequence because of congruence. Also, it was noted above
that the simultaneity of the two fields is set by the arm length adjustment and
once set it does not change. So field
simultaneity at the Interferometer is preserved with a constant fractional
phase difference of pi radians, as set by the micrometer adjustment.
The error made was in using travel time differences to predict
that the Interferometer display would vary as a function of azimuth rotation
due to the earth’s velocity. Congruence
was not considered, nor was the fact that different travel times only result in two cycles that were emitted at
different times being simultaneously located at the Interferometer.
