Agenda 2/12/2017. Modern Physics for Frommies V Gravitation Lecture 6. Special Relativity Einstein s Postulates. Einstein s Postulates

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1 /1/17 Fromm Institute for Lifelong Learning Uniersit of San Franiso Modern Phsis for Frommies V Graitation Leture 6 Agenda Speial Relatiit Einstein s Postulates 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 (1) Relatiit Priniple: Einstein s Postulates The laws of phsis hae the same form in all inertial referene frames. Inertial frame one in whih Newton s laws are alid. i.e. one in whih an objet subjet to no eternal fore moes in a straight line with onstant eloit. () Constan of the speed of light: Obserers in all inertial frames measure the same alue for the speed of light in a auum. Light propagates through empt spae with a definite speed,, independent of the eloit of soure or obserer. 15 Februar 17 Modern Phsis V Leture 6 3 Postulate (1) is the same as Galilean relatiit etended to inlude not onl the laws of mehanis but those of the rest of phsis (in partiular eletriit and magnetism). Postulate () iolates our ommonsense notions. No ether, just E and B feeding off eah other and propagating through empt spae. Not so bad. After all the ether proed impossible to detets so we hae no proof of its eistene. Fails to meet the testabilit requirement. Speed of light in a auum is alwas measured to be regardless of the relatie speeds of the soure and the obserer. 15 Februar 17 Modern Phsis V Leture 6 4

2 /1/17 Our ommonsense is based on a lifetime of eperiene in whih we deal with eloities that are er small in omparison to. 8 = 3 1 m/se = 3,, m/se Walking.9 m/se Running 6.7 m/se Automobile 6.8 m/se Airliner 68. m/se Earth esape eloit 11,7. m/se Light traels approimatel 1 foot in 1 nse or se. Simultaneit Time an no longer be regarded as an absolute quantit. The time interal between eents and een whether eents are simultaneous depends on the obserer s referene frame. Theorem: If eents are simultaneous and oloated in frame F the are simultaneous and oloated in F. If eents are widel separated in spae we must take into aount the time it takes for light from them to reah us. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 6 O and both A and B are fied in the same referene frame as is the obserer If eents are simultaneous to an obserer in one referene frame, are the simultaneous to another obserer moing w.r.t. the first? If obserer O is ½ wa between A and B and sees the light flashes at the same time then the eents are simultaneous. equialent points of iew 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 8

3 /1/17 Another Eample: Now onsider just one moing train ar with a light flasher mounted at the enter. t = t = 1 When the light from A and B arries at O he obseres them to be simultaneous. In O 1 s frame the light from B 1 has alread arried and that from A 1 has et to arrie so he obseres them not to be simultaneous. Simultaneit is not an absolute onept but is relatie. Obserer on train. Light flashes arrie front and bak simultaneousl. t = Obserer on platform. Arrial of flashes is not simultaneous. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 1 Eents, World lines, et. Eents: Eent what + loation + time 4-D etors using olumn matri notation z t To keep same units, t t. z t Instead of a 3-D spatial loation + a 1-D temporal loation, write it as a 4-D loation in spaetime. The oordinates (,,t) hange as we hange referene frames. For ease of drawing let s look at a -D etor, V, in two frames rotated in the plane of V. V V V V V θ Notes: As promised, the omponents are frame dependent. In fat, V = V osθ + V sinθ Aside: V = V sinθ + V osθ osθ = sinθ sinθ osθ (1) The transformation mies oordinates = f (, ) and = g (, ) () Though omponents of V hange, its length...does not. V + V = V + V inariant 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 1

4 /1/17 Spaetime Diagrams t ends t light flashes 45º 45º 45º 45º Obserer on the Train Obserer on the Platform 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 14 The Lorentz Transformation Transformations between stationar (w.r.t. eah other) frames Eent = displaement from origin loation + time A 4-D etor using olumn matri notation Change frames ia a transform(ation). In Galilean relatiit we need onl deal with the 3-D etor z Time is an absolute 15 Februar 17 Modern Phsis V Leture 6 15 Translation onl Transform = = z = z ± z (not shown) 15 Februar 17 Modern Phsis V Leture 6 16

5 /1/17 Stationar frames rotation onl Continuous series of eents trajetor V θ = osθ + sinθ = sinθ + osθ In 3-D this beomes rather mess 1 so I won t show it. If the eents are loations the trajetor is alled a world line. a lim = t t lim = t t Similarl for and z Rotation and Translation: Composed of two suessie transforms. 1. H. Goldstein, Classial Mehanis, Addison-Wesle (195) p. 19 Transformations between stationar frames do not hange or a, the are inariants. Now, onsider two inertial frames moing relatie to one another. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture is eloit of frame w.r.t. frame 1 (-diretion). = onst. boost 1 = z t 1 1 z 1 t z 1 z 1 = t = + t z 1 1 = 1 = z 1 = 1 = + 1 Now, appl Einstein s postulates 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6

6 /1/17 Einstein s Postulates (1) Relatiit Priniple: The laws of phsis hae the same form in all inertial referene frames. Inertial frame one in whih Newton s laws are alid. i.e. one in whih an objet subjet to no eternal fore moes in a straight line with onstant eloit. z F z F At t = t = the frames are oinident. A soure at the origin emits a pulse of light. Einsteins postulate(1) Phsial laws must be phrased identiall in the sstems () Constan of the speed of light: Obserers in all inertial frames measure the same alue for the speed of light in a auum. Light propagates through empt spae with a definite speed,, independent of the eloit of soure or obserer. (1) A wae equation of the form 1 E E = t desribes the propagation of light in both frames () will be tne same in both frames 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 1 γ = 1 is alled the Lorentz fator. is alled β = the speed Returning to the transforms for = γ ( - t) = γ ( + t ) Substituting for = γ ( t) +γ t And soling for t t = ( 1 γ ) + γ t γ 15 Februar 17 Modern Phsis V Leture 6 3 In summar, the Lorentz transforms (frame boost in ) are: = γ ( - t) = z =z t = γ ( t - / ) Note that for << these redue to the familiar Galilean transforms. = t = + t z 1 1 = 1 = z 1 And the inerses = γ ( + t) = z=z t = γ ( t + / ) = 1 = Februar 17 Modern Phsis V Leture 6 4

7 /1/17 γ γ = 1 1 Consequenes of the Lorentz Transforms limγ = 1 lim γ = The world turned upside down -An English ballad, supposedl plaed b the British band at Lord Cornwallis surrender at the siege of Yorktown (1781) 3 Januar 13 Modern Phsis V Leture Januar 13 Modern Phsis I Leture 3 6 Time an no longer be regarded as an absolute quantit. The time interal between eents and een whether eents are simultaneous depends on the obserer s referene frame. Simultaneous t = t 1 t = But t = γ ( t / ) Simultaneit Theorem: If eents are simultaneous and oloated in frame F the are simultaneous and oloated in F. 15 Februar 17 Modern Phsis V Leture 6 7 Thus simultaneit is not an absolute onept but is relatie. Can we Lorentz transform to a frame in whih the order of eents is reersed? Consider two eents t = t 1 - t t = t 1 - t t = γ ( t / ) Let t > then, t < if / > t > > = t NOT ALLOWED > 15 Februar 17 Modern Phsis V Leture 6 8

8 /1/17 There are metaphsial arguments inoling ausalit, predestination, free will et. The superluminal murder trial: F F u Let u and u be the eloities of an objet in F and F respetiel. F moes to the left with eloit w.r.t. F z In Galilean relatiit we simpl hae = - u and = + u This is not in agreement with the Lorentz transforms 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 3 = γ ( - ut) And the inerses = γ ( + ut) = = z =z z=z t = γ ( t - u/ ) t = γ ( t + u / ) = = t t Using differential alulus or else some mess algebra u = u 1 = u γ 1 z z = u γ 1 + u = u 1+ = u γ 1+ z z = u γ 1+ Note the effets transerse to the boost due to the Lorentz transformation of the time oordinate. Again, for and u << these redue to the Galilean eloit transforms 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 3

9 /1/17 Eample: Veloit Addition Find speed of w.r.t. Earth Earth is frame F. is frame F., are along flight Eample: Constan of Replae with light from 1 s headlight in preious eample w.r.t. 1 Find speed of light w.r.t. Earth Earth is frame F. 1 is frame F., are along flight + u = = = = u (.6 ) (.6 1+ ) Galilean transform = 1. There is no addition of eloities that will result in > TRY IT Februar 17 Modern Phsis V Leture u = = = = u (.6 1 ) u = = = = u (.99 1 ) Februar 17 Modern Phsis V Leture 6 34 Eperiments and obserations Classiall, a partile moing at.98 oers m in se. m µ mountain Radioatie dea law: N where t 1/ = seonds µ =.98 µ t (ln ) t1 / = N e Count for some period of time Top = 1 ± 31.6 µ Bottom = 54 ± 3. µ Dea law onl 45 µ should surie the trip Relatiistiall, we realize that the quoted half-life of se. is that of a µ at rest. At β =.98, time dilation is signifiant. An obserer in the lab will pereie that a lok moing with the µ to be slowed b a fator of γ. β =.98 γ = 5 Dea law orreted 538 µ should surie the trip. Agreement with eperiment. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 36

10 /1/17 Alternatiel, eamine the problem from the point of iew of an obserer traeling with the µ. Čerenko Radiation This obserer sees the m flight path as length ontrated b a fator of 1/γ to 4 m. The time to trael this ontrated differene is thus redued b a fator of 1/5. Dea law orreted 538 suriing µs. > /n Idential result, in agreement with eperiment, is obtained b using either time dilation or spae ontration. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 38 Apparent Paradoes The twin parado: Idential twins, Pat and Mike, born at the same moment At age : Pat traels to α-centauri at a speed of.9 and then returns to Earth at the same speed. Mike remains on Earth and works the famil farm. Pat returns to Earth to find that 8.4 ears hae passed. i.e Mike is 8.4 ears of age. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 4

11 /1/17 OK, that s the piture in Mike s frame. What happens in Pat s frame? Pat will see himself at rest and Mike traeling at.9 Doesn t this mean that at the reunion their ages will be reersed? PARADOX Proper appliation of Lorentz transforms Pat is indeed ounger One an also inoke General Relatiit sine aelerations are inoled. Smmetr in what the twins obsere is onl apparent. Smmetr is broken b the out and bak nature of Pat s journe. Onl Mike is alwas in a single inertial frame. Pat is in seeral 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture D Minkowski Spaetime Reall the matri formulation of the Lorentz transform for a boost along the (+) -ais t γ 1 = 1 z t γ γ t z An eent loation is noted as a 4-D etor with the 4 th dimension being time. To keep the same units for all aes replae t with t. 15 Februar 17 Modern Phsis V Leture 6 43 t Hermann Minkowski The iews of spae and time whih I wish to la before ou hae sprung from the soil of eperimental phsis, and therein lies their strength. The are radial. Heneforth spae b itself, and time b itself, are doomed to fade awa into mere shadows, and onl a kind of union of the two will presere an independent realit. Hermann Minkowski in his talk at the 8th Assembl of German Natural Sientists and Phsiians, September 1, Februar 17 Modern Phsis V Leture 6 44

12 /1/17 Minkowski spae and Minkowski diagrams A t B Eents A and B are onneted b a trajetor alled a worldline spaelike timelike t = lightlike Points or loations are eents. Eents moe on paths alled world lines t -Spaeship leaes origin with eloit. Slope=/ - Light signal eloit Onl timelike world lines are allowed Cannot see outside light one 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 46 Lorentz Inariants and 4-dimensional distane: interal (s) In 3-D: is the square of the distane between points. s = onl if the two points are oloated. Galilean transforms d and hene d are the same in an inertial frame and are thus said to be inariant. Is there a quantit, similar to d, whih is inariant under the Lorentz transforms in 4-D? Consider two inertial frames, F and F, and the quantities ( ) ( ) s = t s t = 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 48

13 /1/17 Appling the Lorentz transforms we find that s = s, so the quantit s is an inariant. Inluding the other spatial oordinates, and z, ( ) we hae s = + + z t as an inariant. In analog to the distane between two points in 3-D we an determine the 4-D separation of two eents in Minkowski spae. ( s) = ( ) + ( ) + ( z) ( t ) s is alled the spaetime interal between eents Appling the Lorentz transforms we find that s = s, so the quantit s is an inariant. Inluding the other spatial oordinates, and, ( ) we hae s = + + z t as an inariant. s is the time interal eperiened b a lok moing between eents. (proper time) s = for an two points ling on the light one Meanwhile, bak at the ranh, atuall the farm, Pat and Mike are tring to figure things out. 15 Februar 17 Modern Phsis V Leture Februar 17 Modern Phsis V Leture 6 5 Pat and Mike isit 4-spae: t B C 4. l A A-Centauri In 3-D we hae the triangle inequalit A + B C In our 4-D spae this Inequalit is reersed due to our definition of the interal. So A + B C Ireland The traeling twin returns the ounger man. Poor Mike! 15 Februar 17 Modern Phsis V Leture 6 51