Announcements. Review: Lorentz & velocity transformations (relativistic version of Galileo) Transformations (in 1D) Some examples

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1 Announeents Reading for Monda: Chapter.6-. First Mid-ter is in das (Feb. 9 th, 7:30p). It will oer Chapters &. Reiew: Lorentz & eloit transforations (relatiisti ersion of Galileo) Transforations (in D) If S is oing with speed in the positie diretion relatie to S, then the oordinates of the sae eent in the two fraes are related b: Galilean transforation (lassial) Lorentz transforation (relatiisti) t γ( t) t t γ( t ) u u u/ Soe eaples Note: This assues (0,0,0,0) is the sae eent in both fraes. Q Relatiisti transforations γ( t) u u γ( u/ u u/ + u + / t ) Suppose a spaeraft traels at speed 0.5 relatie to the Earth. It launhes a issile at speed 0.5 relatie to the spaeraft in its diretion of otion. How fast is the issile oing relatie to Earth (Hint: Reeber whih oordinates are the pried ones. And: Does our answer ake sense) Veloit transforation: Whih oordinates are pried z S Earth (,,z,t) z' u is what we were looking for! (i.e. eloit easured in S) ' u (',',z',t') Spaeraft ' a) 0.8 b) 0.5 ) d) 0.5 e) 0

2 Q + u + / u u/ Q3 Lu George γ( t) γ( t ) The objet ould be light, too! Suppose a spaeraft traels at speed 0.5 relatie to the Earth. It shoots a bea of light out in its diretion of otion. How fast is the light oing relatie to the Earth (Get our answer using the forula). a).5 b) 0.5 ) d) d e) e George has a set of snhronized loks in referene frae S, as shown. Lu is oing to the right past George, and has (naturall) her own set of snhronized loks. Lu passes George at the eent (0,0) in both fraes. An obserer in George s frae heks the lok arked. Copared to George s loks, this one reads A) a slightl earlier tie B) a slightl later tie C) sae tie Lu γ( t) γ( t ) L Reeber and Lu R Lu George The eent has oordinates ( -3, t 0) for George. In Lu s frae, where the lok is, the tie t is 3γ γ (0 ( 3)), a positie quantit. slightl later tie At 3 o lok in s frae, two firerakers go off to announe the tie. It so happens that these firerakers are at the left and right ends of the train, in s frae. Eent : fireraker eplodes at 3:00 Eent : fireraker eplodes at 3:00 L L L L RRRR Do ou reeber Lu Lu L R Lu Soetie later, the waefronts eet. The eeting point is halfwa between the firerakers in s frae, but is soewhere in the left of the train ar, in Lu s frae. Eent 3: two light pulses eet, shortl after 3:00. Lu

3 L Lu Lu R We found (in an earlier liker question) that aording to Lu: s lok on the right reads a later tie than s lok on the left. Spaetie interal This proble an be soled with the Lorentz transforation (no need to draw trains eer tie ) Reeber this (fro st lass) The distane between the blue and the red ball is: (3) + (4) 5 5 If the two balls are not oing relatie to eah other, we found that the distane between the was inariant under Galileo transforations... but not under Lorentz transforations! (Length ontration.) need new definition for distane Spaetie interal Sa we hae two eents: (,,z,t ) and (,,z,t ). Define the spaetie interal (sort of the "distane") between two eents as: ( s) ( t) ( ) ( ) ( z ) With: t t t z z z Spaetie interal The spaetie interal has the sae alue in all inertial referene fraes! I.e. s is inariant under Lorentz transforations. (Hoework #3!) Reeber Lu Reeber Lu Eent fireraker eplodes Eent light reahes detetor Geoetrial distane between eents is h. Tie between the two eents is t. And we know that: h t or: h 0 ( t) h t Eent fireraker eplodes Eent light reahes detetor Geoetrial distane between eents is t Distane between -oordinates is and: ( t ) ( ) + h And Lu got We an write: 0 t h 0 ( ) ( ) ( h) sine 0 h ( ) ( ) ( ) ( s) ( t) ( ) ( ) ( z), with: h, 0 or: 0 z

4 Spaetie Spaetie t Here is an eent in spaetie. t Here is an eent in spaetie. An light signal that passes through this eent has the dashed world lines. These identif the light one of this eent. The blue area is the future on this eent. The pink is its past. Spaetie Q4 Spaetie t A Here is an eent in spaetie. The ellow area is the elsewhere of the eent. No phsial signal an trael fro the eent to its elsewhere! t A B Now we hae two eents A and B as shown on the left. The spae-tie interal ( s) of these two eents is: a)positie b)negatie )zero If ( s) is negatie in one frae of referene it is also negatie in an other inertial frae! (( s) is inariant under Lorentz transforation). Causalit is fulfilled in SR. t A Spaetie D B C ( s) >0: Tie-like eents (A D) ( s) <0: Spae-like eents (A B) ( s) 0: Light-like eents (A C) ( s) is inariant under Lorentz transforation. Eaple: Waefront of a flash z t0 t>0 Waefront Surfae of a sphere with radius t: (t) z 0 Spaetie interal for light-like eent: ( s) 0 Einstein: '' is the sae in all inertial sstes. Therefore: (t') - ' - ' - z' 0 in all inertial sstes! (Here we assued that the origins of S and oerlapped at t0)

5 Chapter This is the end of our eursion to the relatiisti spaetie. Questions Relatiisti ehanis Preiousl: Ideas of spae and tie Siple algebra Net: Ideas of oentu and energ Slightl ore inoled algebra Moentu The lassial definition of the oentu p of a partile with ass is: pu. In absene of eternal fores the total oentu is onsered (Law of onseration of oentu): n i p i onst. Due to the eloit addition forula, the definition pu is not suitable to obtain onseration of oentu in speial relatiit!! Need new definition for relatiisti oentu! Conseration of Moentu S u u u u If u -u we find: p tot,before 0 p tot,after 0 ' Frae S oes along with u ' Sste is oing to the right with the eloit u. We will use relatiisti eloit transforations here. Classial oentu Galileo (lassial): S u (u,u ) u (-u,-u ) p, before (u, u ) p, before (-u, -u ) p tot, before (0, 0) p, after (u, -u ) p, after (-u, u ) p tot, after (0, 0) p tot, before p tot, after ' ' p, before ( 0, u ) p, before (-u, -u ) p tot, before (-u, 0) p, after ( 0, -u ) p, after (-u, u ) p tot, after (-u, 0) p tot, before p tot, after

6 Veloit transforation (3D) Lorentz transforation Classial: u u z u z Relatiisti: u u / γ z γ u ( u / ) u z ( u / ) ' ' Use: u u/ u γ u / Algebra Part of HW#4 ( ) p tot, before p tot, after Conseration of oentu is etreel useful in lassial phsis. For the new definition of relatiisti oentu we want:. At low eloities the new definition of p should ath the lassial definition of oentu. Q: A Q: C Q3: B Q4: B Answers to liker questions. We want that the total oentu (Σp) of an isolated sste of bodies is onsered in all inertial fraes.