Michelson-Morley experiment revisited under the light of special relativity

Taking into account the composition of speeds in special relativity, lets suppose that a Michelson-Morley interferometer is moving at a speed of $\vec{v}$: the speed of Earth relative to the local space-time.

Under newtonian mechnics, the conclusion of the Michelson-Morley was of course that either $\vec v = \vec 0$ or that $\vec v$ is a meaningless concept and hence there is no aether.

Under special relativity: $$\forall \vec v, \vec c + \vec v = \vec c$$

Hence the speed of the photons on the (supposed) transversal $t$ and longitudinal $l$ arms have a speed of the same value: $c$, and their time to travel is independtly of $\vec v$ $$T_t = T_l = 2L/c$$.

Consequently in special relativity any (hypothetic) speed $\vec v$ of the earth relative to any form of space-time (I prefer this term to the one of aether) can not be detected with such an interferometer because it is heavily based on the speed of light.

How starting from this basic reasonning can anyone conclude that$$\vec v = \vec 0$$ ?

How can anyone conclude that such a$$\vec v$$ doesn't mean anything?

Were other experiments conducted so as to be less directly dependant of the speed of light?

• Do you understand how the analogous experiment with waves in a medium differs? At the end of the 19th century it was assumed that waves traveled in media, so the expectations for the experiment were based on calculations that made that assumption. – dmckee Jan 5 '15 at 22:38
• "The speed of Earth relative to the local space-time" - there is no such thing as "relative to local space-time" – Señor O Jan 5 '15 at 22:42
• → dmckee: it is rather clear that Michelson worked on the hypothesis the model of the sound (in gaz, liquid or solid) was transposable to light (in an hypothetic aether). – daniel Azuelos Jan 5 '15 at 22:43
• Are you asking what would be predicted in special relativity? If so, velocity "relative to the local space-time" is meaningless. Or are you asking what would be predicted in an aether theory, assuming rulers moving relative to the aether don't contract and clocks moving relative to the aether don't run slow? If so, you should say "relative to the aether", not "relative to the local space-time". If neither of these, please elaborate. – Hypnosifl Jan 5 '15 at 22:58
• → Hypnosifl: "Are you asking what would be predicted in special relativity?": no I didn't write this question. Moreover I used one of its results (special relativity). – daniel Azuelos Jan 5 '15 at 23:11

They expected that light would go the "dotted path", which is longer than the "solid" path thanks to the relative motion $v$, so that it arrives later than the other ray, so that there would be constructive / destructive interference between the two rays which would happen below the mirror, so they expected to see light/dark interference patterns.

But no patterns were visible, so I believe the chain of reasoning was like this:

no interference $\Rightarrow$ no dotted path $\Rightarrow$ $v = 0$

Then the experiment was rotated and they tried again without success.

$\Rightarrow$ $v = 0$ in every direction! $\Rightarrow$ something is wrong

SRT explanation: light doesn't care about $v \ne 0$, it will arrive just at the same time than its other ray, so it is ok to see no interfernce pattern.

• That $+(v_1, v_2) \ne v_1 + v_2$ is indeed more scientific than my sloppy "doesn't care". :-) – mvw Jan 6 '15 at 0:26
• This conclusion was right with the addition of speeds within newtonian mechanics: $\vec c + \vec v = \vec c + \vec v$. But we all know (now) that this hypothesis context was wrong. Within the context of SR, the Michelson-Morley experiment looks like a tautology. – daniel Azuelos Jan 6 '15 at 7:20
• @daniel Azuelos - What do you mean by "tautology"? Of course SR predicts that the same interference pattern will be seen regardless of the interferometer's velocity relative to any arbitrarily-chosen reference frame, but I don't see how this is more "tautologous" then any other prediction, like the prediction that energy will be conserved. And since the aether theory predicts something different, it is a good test of which theory's predictions are correct--experiments aren't just done "within the context of" a particular theory, they can test one theory vs. another. – Hypnosifl Jan 6 '15 at 16:36
• What does the Michelson-Morley experiment demonstrate within the hypothesis frame of the SR and not within the hypothesis frame of an aether and the newtonian mechanics? – daniel Azuelos Jan 6 '15 at 18:01
• @daniel Azuelos - Again, experiments aren't really done "within" a particular hypothesis frame, they just give real-world empirical results which can then be compared to the theoretical predictions of various theories/hypothesis frames, to see which theories correctly predict those results and which ones give predictions that conflict with the real-world results. The fact that the same interference pattern is seen even in multiple experiments where the device has different rest frames is an observation that agrees with the predictions of SR, but not with those of traditional aether theories. – Hypnosifl Jan 9 '15 at 17:45