An Analysis of the Resolution of the Michelson-Morley Experiment

From: Tom Roberts (tjroberts_at_lucent.com)
Date: 01/23/05 

Date: Sun, 23 Jan 2005 19:37:01 GMT

Title: An Analysis of the Resolution of the Michelson-Morley
         Experiment
Author: Tom Roberts, tjroberts@lucent.com
Date: January 23, 2005

Introduction
------------

There have been several recent attempts to re-analyze the original
Michelson Morley experiment [1][2]. Cahill[3] has interpreted these
re-analyses as supporting his theory.

Unfortunately, none of these authors understand error analysis, and thus
do not know how silly their analyses actually are. Their basic problem
is that they, like the original authors, attempt to interpret this
experiment as "measuring the velocity of the earth relative to the
lumeniferous ether". While that was a reasonable approach in 1887, today
it is completely ludicrous -- not because of the mention of "ether", but
because today we use experiments like this to _test_theories_, not to
try to make "measurements" on concepts contained in some particular theory.

In this case, this change in outlook of the scientific method is clearly
required because of a simple observation:
   In a hypothetical world in which:
        a) a perfect MMX experiment would yield a truly null result
   and
        b) real measurements are subject to measurement errors
   it is statistically highly unlikely that a real MMX measurement will
   yield a null result. In such a hypothetical world, of course, the
   non-null result is induced purely by the measurement errors. But
   with an error analysis of the measurement, it can be determined
   whether or not the measurement is consistent with a theory that
   predicts a null result.

So when Munera[1] repeatedly proclaims "this is a non-null result" for
various experiments, he is repeating a fundamental error -- sure the
measurements can be interpreted as a non-null result, but the important
question is: are they _consistent_ with the predictions of a given
theory? As we will see below, the actual MMX data are consistent with
the predictions of SR, and with a wide range of theories in which the
earth moves relative to the ether.

Michelson and Morley's data are given, in a reduced form, in their 1887
paper[4]. The above attempts at analysis are based on the data in the
table on page 340 of [4]. Unfortunately these data are not the original
readings, but each row is an average over 6 turns of the interferometer
made over approximately 36 minutes. Note I am discussing only the six
rows for their six runs, not any of the rows containing means.

In performing an analysis on an experiment performed long ago, with only
limited access to the data and no access to the apparatus, we are
limited in our ability to determine the experiment's actual resolution.
I have identified three approaches:
  1. Look into a modern Michelson interferometer and estimate the
     measurement resolution.
  2. Use the original authors' statements to infer their resolution.
  3. Use the original authors' data in a statistical analysis of the
     resolution displayed by the actual data.
There are in increasing order of confidence and accuracy.

Note that it is important to refer to the actual measurements, and not
to averages. Unfortunately, the available data are averages over 6 turns
of the interferometer, not the original readings. So I will assume that
the errors in the individual measurements are uncorrelated, and normally
distributed. While such an assumption is undesirable, the available data
essentially force it -- a competent modern repetition of this experiment
would take pains to accurately measure the actual resolutions.
Fortunately, the presence of a rather large systematic error in the data
implies that this statistical independence is reasonably likely[#]. In
keeping with the assumption of normal errors and with modern practice,
when I discuss "resolution", I mean the sigma of the associated normal
distribution for the original measurement (in this experiment the
location of a fringe).

        [#] During each rotation the reading changed by 15-30
        divisions. This forces the observer to reposition the
        micrometer for each reading. While statistical independence
        is not assured, it is clearly more likely for a system in
        which the micrometer is repositioned for each reading than
        for a system without the systematic error where the readings
        vary by so little that it would be easy for the observer to
        simply leave the micrometer untouched (thus inducing an
        enormous correlation among readings).

When you plot the data given in the table of [4] for each day, it is
quite apparent that there is a large systematic error that dominates the
measurements -- the measurements at mark 16 before and after the turn
are not equal. In fact, for each of the six runs the difference in the
two marker-16 values is larger than the variations among the other
readings. The authors [4][1][2] all subtract off an assumed linear
dependence of this systematic error, and the original authors [4]
mention a "temperature effect". Given the limited availability of
original data, this is the best one can do, and I will do likewise.

Note, however, that this analysis technique _forces_ the data to be
cyclical. That is, the above subtraction ensures that at the beginning
and end of each turn the value will be exactly zero; any non-zero
measurement in between will naturally appear to be "cyclical". Given
non-zero resolution and independent measurements, there will be non-zero
measurements in between. So claims that somehow the "cyclical nature" of
the results implies or supports the "motion of the earth relative to the
ether" are bogus -- _any_ such data will be "cyclical".

Lets' look at the above three estimates of Michelson and Morley's actual
measurement resolution:

1. Look into a modern Michelson interferometer
----------------------------------------------
I believe that anyone who has ever done so will agree that
  a) it is fairly easy to note the location of a fringe to within
     about 1/5 of a fringe width
  b) it is unlikely to be able to locate fringes to better than
     1/10 of a fringe width
Basically the fringes do not have sharp edges, and one must inherently
guess where the center of a fringe is.

So this approach yields an estimate of resolution between 0.1 and 0.2
fringe widths.

2. Use the original author's statements to infer their resolution
-----------------------------------------------------------------
Michelson and Morley[4] state "The width of the fringes varied between
40 to 60 divisions, the mean value being near 50[...]". In keeping with
the assumption that the measurement errors are normally distributed,
I'll assume that this means that 95% of measurements of fringe widths
were contained in the interval from 40 to 60 divisions of their
micrometer. That means their resolution for measuring fringe width is 5
divisions, or 0.1 fringe. As the measurement of a fringe width requires
two measurements of the location of a fringe, their base resolution is
sqrt(2) time this.

So this approach yields an estimate of resolution of 0.14 fringe widths.

3. A statistical analysis of the resolution displayed in the data
-----------------------------------------------------------------
The key to doing this is to find instances in the data where they
measured the same value multiple times; then a histogram of the multiple
measurements will give a distribution of the errors, and the resolution
can be obtained from the distribution.

In an idealized Michelson interferometer, the interfering light rays
travel both directions along each path, so there is exact 180 degree
symmetry. In the actual apparatus, the ray paths are indeed
out-and-back, so this symmetry should apply to the measurements. The
original authors applied this symmetry in their analysis. Here we will
use it to estimate their resolution.

        [In fact for perpendicular arms there is an additional
         90-degree symmetry, unexploited by all authors including
         me.]

The idea is to first subtract the linear systematic from each of the six
rows of data, thus forcing the two measurements at mark 16 to be equal
for each run. Then histogram the eight differences for measurements 180
degrees apart, for all six rows, and determine the resolution of the
measurement from the histogram. This was done in an Excel spread***,
but it is not feasible to display the details in this ASCII medium. The
histogram does not look very Gaussian, but is rather flat between -5 and
+7 divisions. The likely source of this non-Gaussian behavior is the
systematic error that was _assumed_ to be linear, and nonlinearities due
to either non-uniform behavior or non-uniform spacing of the
measurements could cause this. The sigma of the histogram is 3.0
divisions, corresponding to 0.060 fringe widths; as each point is an
average of 6 turns, the resolution of the original measurements is
sqrt(6) times this value.

So this approach yields an estimate of resolution of 0.15 fringe widths.

Discussion
----------
None of the above estimates are particularly compelling, mainly because
the histogram of method 3 is not really Gaussian; this does not destroy
that approach, but makes it less compelling that it would be with
Gausssian errors. But their agreement indicates they are not crazy.
Certainly there is no support for any resolution estimate much lower
than 0.14-0.15 fringe width.

To compare to the original authors' data table, each row is an average
of 6 turns, and so should have a resolution of 0.14/sqrt(6) = 0.057
fringe width. After subtracting the assumed-linear systematic from the 6
rows, the largest deviation from zero is 0.132 fringe widths, or 2.3
sigma; of the 96 data points, only 1 point exceeds 2 sigma, and 11
exceed 1 sigma. Clearly the readings are not Gaussian distributed. But
equally clearly they are consistent with a null result, and provide only
equivocal support for the notion that there is a non-null result.

Interestingly, when one histograms the data with the assumed-linear
systematic subtracted, the deviations from zero are roughly Gaussian
distributed with a mean of -0.01 fringe and a sigma for individual
measurements of 0.1 fringe. While this is _not_ an error plot, when
compared to the above resolution estimates it solidly demonstrates that
the measurements are consistent with the hypothesis of a truly null result.

When Consoli and Costanzo [2] display a graph of the July 9 PM data,
they drew error bars approximately 0.005 fringe -- more than a factor of
ten too small. They give no indication whatsoever how they arrived at
this value; certainly the original authors gave no error bars. The above
estimate of 0.057 fringe is larger than their entire plot, and indicates
their fit is meaningless. Their fit has 10 parameters for 16 data
points, so it is not surprising that they can draw a line through most
of the points, even with tiny error bars. They do not mention any
chi-squared tests for goodness of fit, and without that and realistic
error bars their estimates on the errors in their parameters are
completely bogus. It is clear that with the above error estimate a
zero-parameter flat line fits the data as well as their 10-parameter
Fourier decomposition.

Munera[1] correctly points out that for a velocity relative to the ether
the MMX only displays the projection of the velocity vector onto the
plane of the interferometer, and this implies that it is unlikely that
such a signal will be a pure cosine. He goes on to claim that even the
intra-session average of 6 turns is invalid as during 36 minutes there
is a change in this projection. While true, that is not important,
as his values show it changes by at most ten percent -- this is wildly
exceeded by the resolution of the measurement.

Cahill[3] has interpreted this as a positive observation of motion
relative to his ether, with a value consistent with the CMBR dipole=0
frame. As mentioned above, he is performing an invalid comparison, and
is basically imposing his hopes and dreams onto the data. A proper
analysis would take his formulas with an unknown speed and direction of
motion relative to the ether, and _predict_ the results of the
measurement. Presumably this could then determine the speed and
direction of that motion. Had he done so, it is clear that with the
above resolution estimate his formula would fit the data for any speed
between zero and several thousand km/s and any direction whatsoever.

Conclusion
----------
The recent attempts to "re-analyze" the Michelson Morley
experiment[1][2] are woefully incomplete, and do not include an accurate
consideration of the experiment's actual resolution. If considered as a
measurement of the motion of the earth relative to some ether, the value
depends upon the details of the theory used to model such motion. For
the ether theory used by the original authors, an upper limit of 5 km/s
is appropriate, but might be reduced by a careful modern analysis. For
Cahill's theory an upper limit of several thousand km/s is appropriate.

In any case, the experiment is indeed solidly consistent with the
prediction of SR -- a null result.

[1] H.Munera, APEIRON _5_ (1998), p37.
[2] Consoli and Costanzo, http://arxiv.org/abs/astro-ph/0311576
[3] Cahill, http://arxiv.org/abs/physics/0501051
     Cahill and Kitto, http://arxiv.org/abs/physics/0205070
[4] Michelson and Morley, Am. J. Sci., _XXXIV_ (1887), p333.
     http://www.aip.org/history/gap/PDF/michelson.pdf