General Relativity Fall 2018 Midterm exam, 11/13/2018

Transcription

1 Geerl Reliviy Fll 2018 Miderm exm 11/13/2018 NAME seful formuls d coss Chr el symbols µ 1 2 µg g g µ 1 A few coss i cgs u G cm 3 g 1 s 2 c cm s 1 M Su g R Su cm 2 srucios There re 15 quesios bu you will ge mximum grde if you swer 12 correcly whichever oes you prefer f you c swer more you ll ge bous po Do o sped more h few miues per quesio f you do ge swer righ wy move o ex quesio you c lwys come bck ler f quesio seems require leghy clculio i mes you re o hikig bou i righ wy Bewre rick quesios f you do o remember equio eg how my ime derivives re re gi? use dimesiol lys mke sure your swer mkes sese Th oly 20% your grde d grdue course grdes do mer much so ry relx Good luck!

2 desiy rule h h M velociy e reliviy Quesio 1 Mmiclly wh defiig propery Lorez rsformio µ0 µ i erms Mikowski meric µ? Wh physicl priciple does h propery come? 2 Dx m Dx yw D Ds You D D y % A i A! u7u Th origies coscy speed ligh i ieril coordies Quesio 2 Wh re physicl meigs 00 0i i0 d ij compoes sresseergy esor T µ? Too T eergy desiy Tio eergy flux log i i direcio compoe momeum Tii flux i h i direcio j Quesio 3 Se s bes s you c equivlece priciple se i derive grviiol redshif E equio E emier observer where Newi poeil No geodesic equio required jus equivlece bewee wo physicl siuios Wek equivlece priciple cos uiform grviiol field equivle uiformly ccelered referece fre Srog eqeio wihou grviy i Some coordies ll lws physics ke ir specil fom em Siu pho emied e received 51 Jujuy ] L Voss Redski DE o Lf k ssh 2 Los og em E d {

3 3 Quesio Wrie geodesic equio firs i geomeric coordieidepede form secod i erms Chr el symbols Wh ccelerio mssive pricle o geodesic? Wh ccelerio mssive chrged pricle i elecromgeic field? u %M+! ccelerio For i wx M pricle M u chged form geomeric M F i velociy de o field me x pm wih P de d defiiio by works Fu g u lso did geodesic Quesio 5 Give brief explio cocep mifold i jus oe seece o full mmicl defiio Give forml mmicl defiiio ge vecr poi p mifold d give simple exmple ge vecr locl Vecr p ie N p Si se rule Yplf g R chrs clled lieup Leibiz Sfies F mps looks lice lohly Riford h se glp + pril ex smooh oe fucios smooh &µg1pk which plb re h such i e Vplg Glp derivive h 112 operr Quesio 06 Expli wh locllyieril coordie sysem Expli how give some origil oieril coordies xµ oe would fid ew coordies xµ which re loclly ieril some poi p oly skech seps do o do full clculio Are such coordies uiquely deermied? Lies i To coordie Sule G coordies Differeie geodesics w 1 fid deermied g 16 up h ursers 6 h Such sysem dxµ Lorez ge Go you gulp re Smu gulp loclly h srigh B lives o Ymir h sfy formio equios for se equio 0 ox dx oly 0

4 Thig Th Quesio 7 Suppose we re give covri derivive r µ wih coecio µ such h r µ µ V + µ V Wrie explici expressio r µ T i erms coecio coe cie Expli wh fudmel propery covri derivive eeded derive h expressio m o skig you derive i however Bu s Qu s + T Tip TF Fudmel propery f D f for sclr fucio + Leibizmle From 0mV fid X Sy compuig # d mf Tds for courig CT s V Xx Qu fpvbx Quesio 8 Expli where expressio 1 for Chr el symbol comes d skech derivio h expressio co meric compibiliy Tyg o o rg dogs s g Cosrucig lier combiio C gives us T TV Vgrcy Quesio 9 Derive geodesic equios for for 2sphere wih meric ds 2 d 2 +si 2 d 2 wihou explicily compuig Chr el symbols hi you ll eed exremize cio The ifer Chr el symbols geodesic equio Geodesics exkemizefdcyu de Jdc 8 sik Jdc L Euler equios cssock dlfo f Hio d 2 fe ff defsiioie Ed o o fioo de jihl?! ii i20 No zero Chrffel symbols oe T si cos PY ye ce CS zic

5 5 Quesio 10 i Wh defiiio Riem esor i erms covri derivives? f you do remember exc expressio sr wih defiiio i words d ry wrie dow idices ii Expli how Riem esor coecs prllel rspor roud smll closed loops les i words if possible wih equio eve if oly pproxime Pyu Vx urµu G Riem The quifies esor covri commuio o vecr V ⑦ li Vi oe prllel Afer door A Vr wih rsporig A re RM My DV ok DVM V vecr Riem remember oh roud smll fields sig! closed A x dd forged V derivives coefficie remember o ok loop! Quesio 11 Se ll lgebric symmeries Riem esor Give defiiio esor sclr d Eei esor Wrie dow corced Bichi ideiy Rs RBL Rp ymmeric f i los d wo wo iches QDµf±yR pµu AN Rx hose hese Ryu eer Eei Rgs pos 85 symmeric Rso R m bes d firs exchge ri u uder R R R sclr i Bichi Corced gµu R GM ; Quesio 12 i A ordermgiude level wh pproxime vlue sclr Su s ceer? ii Suppose es mss rdilly freefllig wrds Su srig res ifiiy Wh ccelerio whe i reches rdius r 2Rsu? Wh orm velociy h poi? i R Riem sig Poso Resrig Gd Cii Free The fll o equio s d T 10 J k Lie {f idl quifies forces Me Rim } hmt z geodesic squred Riem AM velociy u Do lwys Qur W

6 6 d r2 G derive geerl Quesio 13 Expli how Newi equios d2 x/d2 r relivic equios Jus skech seps wihou doig ll clculios explicily; mke sure se codiios vlidiy Newi pproximio DXT ± F velociies smll geodesic equio comes ddz T D dyig de pio us s D d 2 smll 3 DE 1T meric Qusi Gp gu Ahmd 00 L d evie hmu GT hmu y Eei ssumpios some ddz perurbios comes o siory wih de hou hoo { di field G equio K di e im 8T 8561 Quesio 1 Compue Riem esor followig meric e2 d2 + cosh2 xdx2 + ds2 re C Rge o coordies would wy e T dt h sk X e ll? dx DT if quesio Sih dy 2 + z 2 dz 2 y2 Y dy dds dy v lily rosh d DX died i d7 simple Z dz dy { z Mikowski! swer meric h > fl jus spceime dgue i Riem Quesio 15 Cosider equlmss circulr biry wih l mss M d semimjor x sig qudrupole formul for power rdied i grviiol wves derive ordermgiude esime ime i kes semimjor x biry shrik by fcr 2 s fucio d M Ex rdied M w Power Q hik dff M F Y jo dr M s E hve Q MRL some hs dimesios dimesio ime Ml i E w Ml T ME Ros se correc dimesios