# Why is the Michelson-Morley experiment considered a null-result?

The Michelson-Morley experiment did not seem to be a null-result. Here is what Michelson and Morey say in their 1887 paper:

the displacement to be expected was 0.4 fringe. The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part.

In the absence of further constraints on the upper limit, how is a 0.01 or 0.02 displacement in the fringes a null-result?

Did the following later experiments constrain Michelson and Morley's upper bound on the displacement of the fringes?

• Sagnac 1913
• Miller 1921-26
• Tomaschek 1924
• Kennedy 1926
• Piccard 1926
• Illingworth 1927
• Zeeman 1927
• Michelson 1926-29
• Joos 1930
• I suppose the key is the sentence from the paper: "It appears, from all that precedes, reasonably certain that if there be any relative motion between the earth and the luminiferous ether, it must be small; quite small enough entirely to refute Fresnel's explanation of aberration.". – Ruslan Aug 21 '17 at 17:44
• When you are seeking a non-zero result not being able to do better that setting an upper limit us exactly what a null result looks like: you can't reliable tell your result from the null hypothesis. – dmckee --- ex-moderator kitten Nov 5 '17 at 20:34

First a general remark: no experiment can achieve an exact null result, as there is always some noise. In case of Michelson and Morley experiment, the arm lengths were not exactly equal, the light rays hit each mirror at different angles, etc. What matters is whether the result is zero within the experimental uncertainty.

Then, Michelson and Morley experiment was designed to test the dragged Ether hypothesis of Fresnel. The result they found did falsify that as @Ruslan pointed out. But since the outcome was zero within experimental limits, it became seen as a null result.

More importantly, the only relevance of Michelson-Morley type of experiment is to test Special Relativity (SR) but then they are not the only type of such experiments. You also have the Kennedy-Thorndyke type, the Ives–Stilwell type, the Mössbauer type, Doppler, etc. Thus one needs a meaningful way to compare them all. Looking at fringe shifts is not general enough: first some experiments don't measure that and then even among all the optical experiments, the fringe shifts depend on the experimental setup and they cannot directly be compared to each other. As a result, physicists have developed so-called test theories, allowing for departure from SR, within which one can model any experiment. The most used one is that of Mansouri and Sexl [1,2,3]. This is conceptually so simple that I will describe it here before I can then go an compare Michelson-Morley and Joos. I haven't studied all your other references.

One assumes that there is an inertial frame $\Sigma$ in which the speed of light is isotropic, and in which clocks are synchronised with Einstein's procedure, the Ether frame, usually chosen to be the "CMBR frame". Let $X$, $Y$, $Z$ and $T$ be the space and time coordinates in $\Sigma$. Then one considers another frame $S$ moving at a speed $v$ along the x-axis, where coordinates are $x$, $y$, $z$ and $t$ (thus $v\approx 300\, \mathrm{km}/\mathrm{s}$). Finally one assumes the following transformations:

\begin{aligned} t &= aT+\epsilon x + \epsilon_2 y + \epsilon_3 z\\ x &= b(X - vT)\\ y &= dY\\ z &= dZ \end{aligned}

where $a$, $b$, $d$, $\epsilon$, $\epsilon_2$, and $\epsilon_3$ are functions of the speed $v$. Those last 3 functions define the synchronisation used in $S$ and they are therefore spurious: only $a$, $b$, and $d$ can be probed experimentally. This is done by expanding them in series of $v$ (one can prove that $a$, $b$ and $d$ must be even, so the first-order term has to be zero):

\begin{aligned} a &= 1 + \alpha v^2+\cdots\\ b &= 1 + \beta v^2 +\cdots\\ d &= 1 + \delta v^2+\cdots \end{aligned}

Then any experimental test of special relativity can be analysed in this framework, and this results on upper bounds on combinations of $\alpha$, $\beta$ and $\delta$, as SR corresponds to $\alpha=\beta=+1/2$ and $\delta=0$.

For Michelson-Morley experiments, the difference $\delta\tau$ in the optical path between the two arms reads

$$\delta\tau = (l_1 + l_2)(2\beta + 2\delta -1)v^2\cos2\theta,$$

where $l_1$ and $l_2$ are the lengths of the arms, and $\theta$ is the orientation of one of the arm. For example, for the original Michelson-Morley experiment, taking $\delta\tau < 0.005\lambda$,

$$\beta + \delta = 0.5\pm10^{-3}$$

whereas for Joos experiment

$$\beta + \delta = 0.5\pm3\times10^{-5},$$

a two-order of magnitude improvement.

[1] Reza Mansouri and Roman U. Sexl. A test theory of special relativity: I. simultaneity and clock synchronization. General Relativity and Gravitation, 8(7):497–513, 1977.

[2] Reza Mansouri and Roman U. Sexl. A test theory of special relativity: II. first order tests. General Relativity and Gravitation, 8(7):515–524, 1977.

[3] Reza Mansouri and Roman U. Sexl. A test theory of special relativity: III. Second-order tests. General Relativity and Gravitation, 8(10):809–814, 1977.

• Is it necessary to invoke SR here? How do $\beta+\delta$ and fringe displacement relate? – Geremia Aug 22 '17 at 15:20
• Please see my edit. – user154997 Aug 22 '17 at 15:35
• ok, sorry, I kept editing after my last comment. Done now! – user154997 Aug 22 '17 at 15:42

You can never say that anything is zero, you can only put an upper bound on it, because the effect can always just be made to be smaller and smaller until it is indistinguishable from zero to your apparatus. Saying "The expected result was $x$, but it was no more than $x/20$" is basically saying "I couldn't measure the effect.", but in more precise language.

• Yes, which is why I asked: "Did the following later experiments constrain Michelson and Morley's upper bound on the displacement of the fringes?" – Geremia Aug 21 '17 at 19:48
• But it does address the actual question asked in the title and the start of the question, and the misconception of "How is a 0.01 or 0.02 displacement in the fringes a null-result?" – PhillS Aug 21 '17 at 19:51
• Your answer is only a partial answer. If you can show whether the subsequent M-M experiments tighten the upper bound, that would be great. thanks – Geremia Aug 21 '17 at 19:58
• Perhaps I am really asking about how subsquent M-M experiments improved upon M&M's 1887 result. – Geremia Aug 21 '17 at 19:59
• @Geremia if that's what you want, then you should edit your question. – Jerry Schirmer Aug 21 '17 at 21:10