__APPENDIX A__

**Relativity
and the Michelson-Morley Experiments**

**Compressible Photon Flow and the Results of
Michelson-Morley Type Experiments**

Bernard A. Power^{1}

*Dorval**, **Quebec**, **Canada** *

August 2002

The interference fringe shifts that are
always observed with experiments of the Michelson-Morley type have never been
completely explained. Special relativity
and isothermal compressible flow theory both predict that no fringe shift
should occur at all. However, with nine
degrees of energy partition, compressible theory applied to photon flows
predicts the fringe shifts that are observed and which appear to reflect the
earth’s basic motion in space.

PACS number(s): 03.30.+p., 01.55.+b., 47.40.Dc.

The
failure of a long series of classical experimental tests to detect any motion
between the earth moving in its solar orbit and some sort of theoretical
substratum or ether, using the Michelson interferometer to detect any
anisotropy in the speed of light in space, eventually led to the abandonment of
ether models and the universal adoption of the special theory of relativity.

With
encouragement from Lorentz and Einstein among others, however, a continuing
test of the Michelson-Morley [1] experiment by D. C. Miller [2] at Cleveland
and at Mt. Wilson was carried on until
the mid -nineteen twenties. All of these
observations refuted the ether drift hypothesis. But, equally, all of them yielded a definite
small fringe shift upon rotation of the interferometer. The tests of D.C. Miller were the most
extensive, spanning several years of time and covering most of the epochs or
seasons of the earth’s orbital year.
Instead of the full expected shift of = 1.12 fringe, he found a smaller average shift (= 0.122) which corresponded to a computed ether drift
velocity of about 10 km/s rather than the expected earth’s orbital speed of 30
km/s. The observations did show a very
small seasonal and azimuth oscillation which, however, could never be put into
any convincing relationship to physical effects.

Later
versions of this type of interferometer experiment [3-7] generally reported
considerably smaller average shifts, and, as these gradually came to be viewed
as being simply statistical or environmental effects, inquiry was eventually
dropped [10].

For
the classical Michelson-Morley experiments, the expected fringe shift,
calculated on the basis of Galilean addition of orbital velocity to the speed
of light, is given by

= 2 *l* (V_{o} /c_{o})^{2}
/
(1)

Here
*l* is the optical path length (32
meters in the Miller apparatus), c_{o} is the speed of light in space,
V_{o } is the mean orbital speed of the earth taken
as 30 km/s, and the wavelength of
light used by Michelson-Morley and by Miller is 5.7 x 10^{-7} m. The expected fringe shift from Eq.1 was 1.12. A zero fringe shift was to be expected from
special relativity, and eventually the smaller fringe shifts (0.122 average)
which were always observed were also taken as being effectively zero. However,
there is a third possibility, which is that of compressibility in the energy
flow. Lorentz [3] speculated that
compressibility might somehow be a factor in reconciling Miller’s results with
the apparently necessary properties of the hypothetical ether then under
discussion: He said* …in the case of Planck’s modification of
Stokes theory. A further possibility
would be a compressible ether. This
would remove even the necessity of having an irrotational ether*.

If,
however, we apply compressibility, not to an ether, but instead to an energetic field - for example a compressible fluid flow, a
compressible gravitational field, a compressible ‘vacuum’ field, or a
compressible photon flow - we have the basic steady state energy equation [11-12]^{}

c^{2}
= c_{o}^{2} - V^{2} /n (2)

Here the mass m is taken as being
unity, as is usual in hydrodynamics; c_{o} for a photon flow is the
static compression wave speed equal to 3 x 10^{8} m/s (when V the
uniform relative motion is zero); c is the reduced wave speed under any uniform
relative flow V; n is the number of degrees of freedom of the field, that is
the number of ways the energy of the system is equally divided. For one-dimensional motions (n = 1) the right
hand side of Eq. 2 becomes (c_{0} + V)( c_{0} – V) which is
also the going-and-coming addition of velocities used in deriving the classical
fringe shift Eq. (1). Since these
predicted shifts are nine or ten times too large, attention historically then
turned to finding a theory that predicts
zero fringe shifts instead. In
the present theory we can rearrange Eq.
(2) with n = 1 to get

c/c_{0} = [1 -
(V/c_{o})^{2}]^{1/2
}(3)

and
the right hand side now becomes, when inverted, identical to the
Lorentz/Fitzgerald contraction factor, which, if it is arbitrarily applied to
shorten the length of the interferometer
arm in the direction of V, will then exactly
cancel the reduction in the speed of light caused by V and give a zero
fringe shift. A second way to employ
compressibility to get a zero shift
would be to take n = ∞ so that in Eq. (2) V^{2}/n becomes zero, and c is always
equal to c_{0}; thermodynamically this would require isothermal flow.

There
is also, however, the possibility of having a compressible photon flow with
multiple degrees of energy partition. When this option is explored there is at
once an improvement in explaining the observed residual fringe shifts. However, there is still no really good fit
obtained until we take n = 9. For steady
flow the fringe shift Eq. (1) becomes

= 2 *l* (V_{o} /c_{o})^{2}
/9 (4)

Relationship
(4) greatly changes the prediction of fringe shifts since these now become only
one ninth of the classical expectations from Eq.(1), while classical ether
drift velocities associated with any observed shift become three times larger.
For example, Miller in his day expected a fringe shift of 1.12 from Eq. (1),
whereas now, with compressible flow and n = 9 assumed, we find that the
expected fringe shift becomes 0.125 which is close to what Miller observed (2), his average
for the four seasons being 0.122. The
flow velocity now becomes three times larger,
so that = 0.122 corresponds
not to Miller’s computed 10 km/s but to √9 x 10 km/s or 30 km/s which is the expected orbital speed.

The
small velocity changes associated with the earth’s cosmic motions are: (a) the change in the earth’s orbital
velocity ΔV_{o} from perihelion on January 3 to aphelion on July
3 of approximately 498 m/s over the half
solar year; (b) changes in the expansion
and contraction velocity V_{r } of the
earth’s orbital radius between
perihelion and aphelion, which amount to about 322 m/s over the 180 days,
and (c) the tangential velocity V_{rev}
associated with the diurnal revolution of the earth on its axis, which is about 385 m/s at 34° N, the latitude
of Pasadena and Mount Wilson. These are not considered here.

In
any assessment of the results of the various interferometer tests, it
should noted that the observed data are
fringe shifts, except where masers were used, in which case the basic
observations are a beat frequency or frequency shift. In the classical
literature the test results are commonly expressed as ether drift velocity
which is just the basic fringe shift observation converted to an equivalent drift
velocity using Eq. (1). Early
experiments concentrated on attempts to detect the orbital speed of the earth
namely 30 km/s. Later, when only about
one third of this velocity was found, the approach changed fundamentally; the
fairly large fringe shifts observed at each rotation through 90° were
discarded, since they were too small to be explained by classical ether drift
theory, and efforts were directed instead at trying to find very small changes
in the fringe shifts at different times of the day so as to sample any effect
of the changing orientation of the instrument in space as the earth turned on
its axis. However, as we shall see, with
compressible flow and n = 9, the large discarded basic fringe shifts observed
at each 90 degree turn substantially
match the orbital speed of 30 km/s that was earlier thought to be
lacking.

Turning
now to other tests than those of Miller, there is first the original
Michelson-Morley test in 1887 at ^{-7 }m. They reported an equivalent
ether drift velocity of not more than 5 km/s to 7.5 km/s in 6 hours of testing
over four days in July. Based on compressibility considerations with n = 9 this
would corresponds to cosmic motions of
about 15 to 22.5 km/s. Miller later recalculated the results using
a harmonic analyzer to retrieve the ether drift periodicity (i.e. the effect
which is periodic in each 180 degree turn of the interferometer) and got 8 to
8.8 km/s, the corresponding compressibility calculation then giving 24 to 26.4 km/s (n = 9).

Morley
& Miller [2] in 1904, also at Cleveland,
used an optical length of 3220
cm, and in October observed a drift of 8.7 km/s (fringe shift 0.12)
corresponding to about 26.1 km/s in compressibility theory with n = 9.

Michelson,
Pease and Pearson [4] used an instrument similar to that of Miller, with an
optical length of 25.9 m, which was installed at

Kennedy
and Thorndike [5] (1929-30) at ^{-7}
m. Their small instrument was completely enclosed in an inert gas and could be
maintained at a near constant temperature over long periods of time. **The instrument was not rotated** but,
instead, observations were taken at a fixed azimuth orientation daily over a
full year, the fringe shifts being photographically recorded. For the case of the diurnal observations
their average fringe shift was 0.000212, which, from Eq. (4) with n = 9, gives
an orbital velocity of 17.1 km/s. They also reported an annual orbital effect
of 10 ± 10 km/s, corresponding to 30 ± 30 km/s with n = 9, a result in which
they did not express much confidence

Illingworth
[6] at

Joos
[7] (1930) at

There
were two tests of space anisotropy and special relativity carried out using
masers shortly after they became available. The basic observations in these
experiments were beat frequency shifts upon rotation of the instrument, this
quantity being proportional either to uV/c_{0}^{2} or (V/c_{0})^{2}. In reassessing these tests with compressible
flow theory and n = 9, these two velocity ratios are to be divided by 9.

1) Cedarholm and Townes [8] (1959) used
an ammonia maser to look for an ether drift effect in the Doppler shift. They compared the frequencies of two such
masers with their opposing beams run in parallel, and when the apparatus was
rotated through 180°. Their tests were run at intervals over a year. The expected beat shift f, based on an ether
drift effect, was f = [4uV/c_{o}^{2}]ν where V = 30 km/s, u the speed of the ammonia
molecules is 600 m/s, and ν the
frequency of the excited molecules is 2.387 x 10^{10} /s. The expected
beat shift was 19.8 cps, so that the actual expected frequency shift was about
10cps. The observed shift was only 1.08 ± 0.02 cps. Their conclusion was, since
the observed shift of 1.08 was far less than the predicted 10 cps, that the
speed of light was thereby shown to be unaffected by the orbital motion of the
earth in space of 30 km/s.

However, if compressible flow with n = 9
is included, then the predicted
frequency shift becomes instead f
= 10/9 = 1.11 cps , which agrees well with the observed value of 1.08.

(2) Jaseja, Javan, Murray and Townes [9]
(1964), using two rotating masers positioned at right angles to one another,
tested for the ether drift second order effect in (V/c)^{2}. They reported on a short six hour test in
January. Upon rotation through 90° they
got a basic, repeating frequency shift of 275 kc/s plus a variation of not more
than 3 kc/s. They discarded the observed repeatable frequency shift on rotation
of 275 kc/s as presumably due to magnetostriction, and concluded that there was
no evidence of any frequency shift effect arising from the from the earth’s
orbital motion of 30 km/s

Their expected frequency shift equation
was 2Δν ≈ (V/c_{o})^{2} , which for V = 30 km/s
gives 3000 kcs expected frequency shift on rotation. Only about 275kcs shift
was observed and was discarded. However,
when the compressibility parameter n is included, we get 2Δν ≈
(V/c_{o})^{2} /9_{ }which gives 3000/9 = 333 kcs
predicted versus 275 observed or only about 20%
high. Expressed in relative motion terms this corresponds to about 27.2
km/sec.

Compressible
photon flow theory is self-consistent and moreover it matches the experimental
data, thereby fulfilling the two essential requirements for viability. We can then, from this new standpoint,
evaluate classical Galilean addition of
velocities, the Lorentz/Fitzgerald contraction factor and the special
relativity theory of Lorentz
transformations.

Classical
or Galilean addition of velocities uncritically extended the addition of mass
particle velocities to the case of addition of particle velocity to the wave
speed of light c in the formulation of the classical fringe shift equation for
the Michelson-Morley experiment. Fortuitously, the two-way going-and-coming
design of the experiment involves forming the quantity ( c^{2} - V^{2})
= (c + V ) (c- V) which, if c is identified with the static speed wave speed c_{o},_{
}agrees _{ }correctly with
the compressible flow Eq. (2) for n =1. It is important to realize that with n
= 1 all of the velocity adds to or subtracts from the wave speed c, whereas
with n larger than unity only a fraction of V adds to or subtracts from c which gives
(c + V/√n)( c - V/√n) = (c_{o}^{2} - V^{2}/n).

The
classical formulation with n =1 predicted the impossibly large fringe shifts
which posed a major problem to the science of the day. The Lorentz/Fitzgerald
solution was not just to reduce the
fringe shift but to eliminate it entirely. This was accomplished by inserting
into the velocity transformations an ad hoc reduction factor [1- (V/c_{o})^{2}]^{-1/2}
which simply cancelled the large shift completely.

The
special theory of relativity succeeded
in putting the ad hoc formulation on a theoretical basis by assuming, first,
the classical Galilean addition of particle and wave velocities as before, and
second by postulating the constancy of the speed of light in all relatively
moving inertial coordinate systems. When this is done the Lorentz/Fitzgerald
contraction factor emerges automatically. The price of an apparent
inconsistency was eventually felt to be
justified since the unacceptably large fringe shift predictions were now
eliminated. On compressible flow theory, such an addition of the entire relative
velocity V to the wave speed c would be
interpreted as being an adiabatic photon flow with n = 1, while the postulated
constancy of c = c_{0} would necessarily be seen as an isothermal flow
(n = ∞); the two assumptions taken together now become physically
inconsistent.

The
development of compressible theory in the last half-century permits the
alternative solution outlined here, which not only explains the partial success
of previous attempts, but which, instead of incorrectly eliminating the fringe
shift entirely, reduces it to the experimentally observed values by postulating a relative
photon flow with n = 9.

The
transformations between relatively moving inertial coordinate systems in the
new theory are made by replacing the static speed of light c_{o} with
the reduced local speed of light c required from Eq. (2), by computing c = c_{o
}[1- 1/9(V/c_{o})^{2}]^{1/2} = c_{o} . Alternatively, if desired or more familiar,
the compressibility could be ignored, with the wave speed arbitrarily taken
at the static speed of light c_{o} ( 3 x 10^{8} m/s) in all coordinate systems as before, but then
the factor γ of special relativity
theory ( = [1- (V/c_{o})^{2}]^{-1/2})
must be replaced by the factor where 1/ = c/c_{o} = [1-
(1/9)(V/c_{o})^{2}]^{1/2} in the transformation
equations. If this computation method is chosen, the space and time coordinates
and the various physical quantities must then be altered to: = l'/l_{o} =
dt_{o}/dt' = m'/m_{o,}
etc.

In
the case of a material gas, the energy partition is among random kinetic motion
of molecules in three space dimensions plus perhaps rotation and vibration, and
these random motions give rise to pressure and compressibility. For photons, however, there are no obvious
equivalents to thermodynamic pressure, random kinetic motion, and so on, and
therefore the precise physical nature of a photon flow that could give rise to
the observed agreement with compressible flow theory remains to be established.
However, it does appear possible that three space dimensions, each with
associated states, plus vibration states
might provide the nine-dimensional energy partition required to explain
the results summarized in Table 1.

**Table 1.
Observed interference fringe shifts and corresponding computed flow velocities.**

---------------------------------------------------------------------------------------------------------

Observer Observed “Ether drift” __Compressible flow (n = 9__)

fringe shift
velocity V ^{a}
Orbital

() (km/s) velocity V ^{b}

(km/s)

---------------------------------------------------------------------------------------------------------

**Michelson-Morley** 0.01 5
- 7.5 15 - 22.5

(1887)
^{}

**Morley & Miller** 0.09 8.7 26.1

(1905)

**Miller** 0.122 (avg) 10.05 30.1 ^{}

(1925-6)

**Michelson et al.** 0.01 (max) ≤6
≤18

(1929)

**Kennedy-Thorndike** 0.000212 (daily) 17.0 ^{c}

(1929-30)
(annual) 10 ± 10 30 ± 30^{}

**Illingworth** 0.0094
(rotational) 32.6 ^{c}

(1927)

**Joos** 0.083
(est.) (rotational) 29.56 ^{c}

(1930)

---------------------------------------------------------------------------------------------------------

**Cedarholm & Townes** Maser 1.08
± 0.02 cps (30 km/s ± 556 m/s)

(1959)

**Jaseja et
al.** Maser 275 ± 3 kc/s 27.2 km/s (29

(1964)

^{a} Computed
from Eq. 1; ^{b} V = (‘ether
drift’)x√9; V computed
from Eq. 4;

**References**

[1] A.A.
Michelson and E.W. Morley, Am. J. Sci*.***
34**, 333 (1887).

[2] D.C.
Miller, Rev. Modern Physics, **5**, 203 (1933).

[3] H.A.
Lorentz, Astrophys. J*.* **68**,
395 (1928).

[4] K.K
Illingworth, Phys. Rev*.***
30**, 692 (1927).

[5] A.A.
Michelson, F.G. Pease, and F.
Pearson, Nature, **123**, 85 (1929).

[6] G.
Joos, Ann. d. Physik,** 7**, 385 (1930).

[7] R.J. Kennedy and
E.M Thorndike, Phys. Rev. **42**, 400 (1932).

[8] J. P.
Cedarholm and C.H. Townes, Nature, **184**,
1350-51 (1959).

[9] T.S.
Jaseja, Javan, J. Murray, and C.H. Townes*, *Phys.
Rev*.*** 133**, A1221 (1964).

[10] R.S. Shankland, *et al.* Rev. Modern Physics*.* **27**,
167-178 (1955).

[11] A.H. Shapiro,
*The Dynamics and Thermodynamics of
Compressible Fluid Flow. *(John Wiley & Sons, New York, 1954).

[12] R. Courant and K.O. Friedrichs, *Supersonic
Flow and Shock Waves. *(Interscience, New York, 1948).

** Note:
**For a review
of latest laser/maser/oscillator tests of Lorentz invariance see Main Page

**Copyright © 2004
Bernard A. Power, Consulting Meteorologist (ret.)**