2.2 The Michelson-Morley Experiment

1 The Michelson-Morley The Michelson-Morley waves need a medium through which to travel. In 1864 James ClerkMaxwell showed that light is an electromagnetic wave. Therefore it was assumedthat there is anetherwhich propagates light waves. This ether was assumed to beeverywhere and unaffected by matter. This ether could be usedto determine anabsolute reference frame (with the help of observing how light propagates throughthe ether). Michelson-Morley Experiment (circa 1885) was performed to detect theEarth s motion through the ether as follows:The viewer will see the two beams of light which have traveledalong The Michelson-Morley Experiment2arms display some interference pattern. If the system is rotated, then the influenceof the ether wind shouldchange the time the beams of light take to travel alongthe arms and therefore should change the interference pattern.

2 The experimentwas performed at different times of the day and of the year. NO CHANGE INTHE INTERFERENCE PATTERN WAS OBSERVED!Example (Exercise ).SupposeL1is the length of arm #1 andL2is thelength of arm#2. The speed of a photon (relative to the source) on the trip over to the mirror isc vand so takes a time ofL1/(c v). On the return trip, thephoton has speed ofc+vand so takes a time ofL1/(c+v). Therefore the roundtrip time ist1=L1c v+L1c+v=L1(c+v) +L1(c v)c2 v2=2cL1c2 v2=2L1c11 v2/c2=2L1c(1 v2c2) photon traveling along arm #2 must follow a path (relative to a stationary observer) ofWe need the time the photon travels (t2) and the angle at which the The Michelson-Morley Experiment3leaves the mirror to be such thatL22+(vt22)2=(ct22)2(the photon must travel with a component of velocity upstream to compensatefor the wind).

3 ThenL22=(ct22)2 (vt22)2L22=t22(c24 v24)t2=2L2 c2 v2=2L2c1 1 v2/c2 Now11 x= n=0xnfor|x|<1, sot1 2L1c(1 +v2c2).Also,(1 +x)m= 1 +mx+m(m 1)2!x2+m(m 1)(m 2)3!x3+ and withm= 1/2 andx= v2/c2,t2 2L2c(1 +( 12) ( v2c2))=2L2c(1 +v22c2).For the Earth s orbit around the sun,v/c 10 4so the approximation is appro-priate. Now the rays recombine at the viewer separated by t=t1 t2 2c(L1 L2+L1v2c2 L2v22c2).Now suppose the apparatus is rotated 90 so that arm #1 is now transverse to theether wind. Lett 1andt 2denote the new round trip light travel times. Then ( The Michelson-Morley Experiment4above, replacingL1withL2int1to determinet 2and replacingL2withL1int2todeterminet 1):t 1 2L1c(1 +v22c2)andt 2 2L2c(1 +v2c2).

4 Then t =t 1 t 2 2c(L1 L2) +v2c3(L1 2L2)and t t 2c(L1 L2) +2v2c3(L1 L22) {2c(L1 L2) +v2c3(L1 2L2)}=v2c3(L1+L2).This is the time change produced by rotating the 1892, Fitzgerald proposed that an object moving through the etherwind with velocityvexperiences a contraction in the direction of the ether windof 1 v2/c2. That is, in the diagram above,L1is contracted toL1 1 v2/c2and then we gett1=t2whenL1=L2, potentially explaining the results of theMichelson- morley Experiment . This is called the Lorentz-Fitzgerald under this assumption, it turns out (see the following example) that theMichelson- morley apparatus with unequal arms will exhibita pattern shift over a 6month period as the Earth changes direction in its orbit around the Sun.

5 In 1932,Kennedy and Thorndike performed such an Experiment and detected no such The Michelson-Morley Experiment5 Example (Exercise ).Suppose in the Michelson-Morley apparatus that L=L1 L26= 0 and that there is a contraction by a factor of 1 v2/c2in thedirection of the ether wind. Then show t=2c L(1 +v22c2). in Exercise , we havet1=2L1c1 1 v2/c2t2=2L2c1 1 v2/c2and so t=t1 t2=2L1c1 1 v2/c2 2L2c1 1 v2/c2=2c(L1 L2)1 1 v2/c2 2c(L1 L2)(1 +( 12) ( v2c2))=2c(L1 L2)(1 +v22c2). the equation in the above exercise expresses tas a function ofvonly(c,L1, andL2being constant; although they may be contracted, but this istakencare of in the computations), we would see tvary with a period of 6 months (asmentioned above).

6 Suggestion to explain the negative result of the michelson -Morleyexperiment was the idea that the Earth drags the ether alongwith it as it orbitsthe sun, galactic center, etc. This idea is rejected becauseofstellar aberration,discovered by James Bradley in The Michelson-Morley Experiment6 Example (Exercise ).Light rays from a star directly overhead enter atelescope. Suppose the Earth, in its orbit around the Sun, ismoving at a rightangle to the incoming rays, in the direction indicated in thefigure below. In thetime it takes a ray to travel down the barrel to the eyepiece, the telescope will havemoved slightly to the right. Therefore, in order to prevent the light rays from fallingon the side of the barrel rather than on the eyepiece lens, we must tilt the telescopeslightly from the vertical, if we are to see the star.

7 Consequently, theapparentposition of the star is displaced forward somewhat from the actual position. Showthat the angle of displacement, , is (in radians) = tan 1(v/c) v/cwherevis the Earth s orbital velocity. Verify that (roughly the anglesubtended by an object mm in diameter held at arm s length). This is alsoabout 1% of the angular size of the moon in the night The Michelson-Morley = tan 1(v tc t)= tan 1(vc).Nowtan 1x= n=0( 1)nx2n+12n+ 1for|x|<1and so tan 1(vc) have the Earth s orbital velocity implyingv/c= 10 4(see Exercise ). So = 10 4= 10 4(360 2 ) (60 1 ) (60 1 ) . the Earth revolves around the Sun in its nearly circular annual orbit, theapparent position of the star will trace a circle with angular radius.

8 This isindeed observed. If the Earth dragged a layer of ether along with it, the light rays,upon entering this layer, would acquire a horizontal velocity component matchingthe forward velocityvof the telescope. There would then be no aberration speed of light is constant and the same in all directions and inall inertial : 6/24/2019