General Relativity 05/12/2008. Lecture 15 1
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1 So Fr, We Hve Generl Reltivity Einstein Upsets the Applecrt Decided tht constnt velocity is the nturl stte of things Devised nturl philosophy in which ccelertion is the result of forces Unified terrestril nd celestil mechnics & brought order to the Universe Spring Frmes of Reference Science is Frught with Assumptions This is ll fine, but ccelerting with respect to wht?? Why the Erth, of course! The Erth is t the center of the universe... The Erth is t the center of the solr system... The world is flt... The geometry of the Universe is flt... The surfce of the Erth is the nturl reference frme... Time nd spce re independent concepts These ssumptions cn hve drmtic impct on our views of Nture Spring Spring Lecture 15 1
2 Recll the Rotting Drum Exmple An ccelerting frme of reference feels lot like grvity In fct, it feels exctly like grvity The essence of Generl Reltivity is the recognition tht grvittionl force is n rtifct of doing physics in prticulr reference frme! Imgine Being in Cr Windows re pinted blck Move the cr to outer spce Now imgine plcing few objects on the dshbord of this blcked-out cr, still in outer spce. If the cr ccelertes forwrd, wht hppens to these objects on the dshbord? (Why?) If you didn t know the cr ws ccelerting, wht would you infer bout force cting on the objects? How would tht force depend on the msses of the objects? Spring Spring Grvity vs. Accelertion Cn you tell the difference between forwrd ccelertion nd grvity from str being brought up behind the cr? Cn you tell the difference between grvity nd ccelertion? Spring Spring Lecture 15 2
3 Constnt Velocity Elevtor An Exercise Chnging Your Perspective Close your eyes nd imgine you re being ccelerted upwrds by the room round us Accelerting Elevtor The nturl (i.e. inertil) coordinte systems re flling pst you t 9.8 m/s 2! You re being ccelerted upwrds t 9.8 m/s 2 by the norml force of the set you re in. Spring Spring A Conclusion: Doing Newtonin mechnics in non-inertil frme of reference cn force you to invoke fictitious-forces, relly just unexpected forces, i.e., rtifcts from doing physics in tht frme. Since these fictitious-forces re invoked to explin wht is ctully n ccelertion of the entire reference frme, they re necessrily proportionl to mss. Do you understnd why? Exmples: Centrifugl force in rotting systems Actully n exmple of grvity in the theory of Generl Reltivity Grvity! Spring Einstein in n Elevtor Film clip from Nov/PBS Spring Lecture 15 3
4 Wht s left?! If we blme grvity on our doing physics in the wrong reference frme, is ll of grvittionl physics wiped out? No! There is still n interction there, just more subtle thn Newton thought... Newton couldn t explin wht grvity ws Thought of it s instntneous ction t distnce Wht s wrong with ction t distnce? Mttress Exmple Imgine 2 bowling blls on mttress, on erth As they roll round on the mttress, they mke dimples in its surfce If they get close to ech other, they roll into ech other s dimples they re ttrcted to ech other Spring Spring Bowling Blls on Mttress Grvity = Geometricl Distortions Film clip from Nov/PBS Spring Spring Lecture 15 4
5 A Geometricl Approch Mss tells spce-time how to curve Spce tells mss how to move This nturlly explins the Universlity of Free Fll Accelertion All objects move long the sme geometricl distortions Grvity is property of the geometry of spcetime How fst re you moving? If time is dimension like spce, then how mny meters long is one second: Answer: 300,000,000 meters: c = 300,000,000 m/s Then in ech second, we re trveling 300,000 km through the time dimension! We re ll going t the speed of light! If we ccelerte ourselves to substntil speed through spce, we give up some of our speed in the time dimension: our clock runs slower compred to sttionry observer our totl speed through spcetime is lwys c Spring Spring Mss Tells Spce-Time How to Curve The illustrtions you ve seen re wht would occur if the world were 2-dimensionl. This llows us to show the curvture in the 3 rd dimension. In relity, grvity cuses 3 sptil nd 1 time dimensions to curve, which is tough to visulize! Spce Tells Mss How to Move Objects trvel long stright lines in curved spcetime. They don t ccelerte due to grvity Spring Curvture in this room! Spce (spcetime for tht mtter) seems flt to us Curvture is smll Strength of reltivity in this room is given by 2GM/(Rc 2 ), roughly Ner sun, this is bout 10-6 Actul rdius of curvture on erth is bout one lightyer Is there some wy to mesure curved spce? Yes! Orbiting stellites with gyroscopes Spring Lecture 15 5
6 The Curvture of Spce is Rel It hs been mesured: GR is right! How? Curved Spce is Rel (Prt Deux) Orbiting gyroscope (Grvity probe B, nd others) Tips becuse spce is curved, just like the rrow before Northern ltitude line Equtor L gyroscope xis rottes Spring Spring Quntifying curvture Let s tke projectile trveling stright up initil speed v (up) mens hng time is t=2v/g height cquired is h = g( t/2) 2 = v 2 /g in this time, we trvel c t = 2vc/g meters through the time dimension If we drew n rc of height h nd length 2vc/g, we would find tht its rdius ws R = c 2 /g 1 light yer note this is independent of initil velocity (could be bullet or superbll) spce time Spring A Contemporry View Curved Spcetime forms stge on which other physics hppens Generl Reltivity (GR) is very successful description of the interction between spcetime nd objects GPS wouldn t work without it Einstein s Field Eqution: (just for fun ) Einstein Tensor describing how spcetime is curved G = 8 T + Cosmologicl Constnt: Einstein s Biggest Blunder (resurrected s drk energy) Stress-Energy Tensor describing distribution of mss nd energy Spring Lecture 15 6
7 A Genuine Scientific Revolution The Generl Reltivity view Relegted grvity to the interction between mss nd spcetime Abolished the notion tht the geometry of spcetime is everywhere flt Mixed the concepts of spce nd time GR does not men everything is reltive! The bsic concept is tht the equtions/lws tht describe physicl systems should not depend on your reference frme. Coordinte Invrince would be better term... Einstein wnted to cll it The Theory of Invrints References nd Assignments References dmoz.org/science/physics/reltivity/overviews/ especilly links titled Generl Reltivity Reltivity Visulized, by L. C. Epstein gret intuitive development, low mth (low PC lso) Assignments Red Hewitt Chpters 35 nd 36 HW5: 9.R.13, 9.E.9, 9.E.14, 9.E.43, 9.P.7, 10.E.16, 35.R.27, 35.E.6, 35.E.19, 35.E.20, 35.E.37, 35.P.3, 35.P.10, 36.R.7, 36.E.2, 36.E.6 Spring Spring Lecture 15 7
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