Aaz3d HH $ @d HHHHff@  d/ Footnote TableFootnote**.\t.\t/ - :;,.!?9+  `H* `ITOCHeading1Heading2    EquationVariables7m|5mY5nm6ot5pq6q6#r6;s6Dt6Uu6Yv6mwP6rxCm6wyS<6z{6{`'6}U7~/c7 ,77% 7*O7L"7U{@O ) I > )<$lastpagenum> *<$monthname> <$daynum>, <$year>3d +"<$monthnum>/<$daynum>/<$shortyear> ,;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm> -"<$monthnum>/<$daynum>/<$shortyear> .<$monthname> <$daynum>, <$year> /"<$monthnum>/<$daynum>/<$shortyear> 0 <$fullfilename>/ 1 <$filename> 2 <$paratext[Title]> 3 <$paratext[Heading1]>\ 4 <$curpagenum>. 5 <$marker1> 6 <$marker2> 7 (Continued) 8+ (Sheet <$tblsheetnum> of <$tblsheetcount>) 9Heading & Page <$paratext> on page<$pagenum> :Pagepage<$pagenum>ua ;See Heading & Page%See <$paratext> on page<$pagenum>.Y < Table All7Table<$paranumonly>, <$paratext>, on page<$pagenum> =Table Number & Pagev'Table<$paranumonly> on page<$pagenum><  FFA HHA* JJ LL NN PP RA$  x5y  soe>5|  s > 5  ui> 5 t"5 ! u/um5 " ta5 # vn$d5 $ y se5 % w so>/5 & | s$ea5 ' z wfi5 ( z w 5 ) w w 5 * z wele5 + ~ wte5 , w w]5 - z wg5 . ~ wa5 / ~ wa5 0 ~od)5 1 ~+t 5 2 } wu<$6 3 w wn5 4 { wnge5 5 w wx p6 6 w wu6 7  ween6 8 } wHe6 9 we<6 : w e65 ; Y66 < e6G = l$p6< > w wo<6? ? wTa6@ @ w w&'6H A w way>6I B w wp><6J C w6K D w wF6L E } w6Q F w6Z G w w6[ H wJ6^ I w w6h J 6i K { w6n L { w6s M w w6x N w wR6y O w$ 6z P y ws6{ Q w ws6| R } wu6 S z wt6 T w wu6 U wt6 V w wv6 W w wy6 X w ww6 Y w|6 Z y wz6 [ w wz6 \ } ww6 ] wz6 ^ w~6 _ w ww6 ` wz6 a y w~6 b w~6 c w w6 d } w6 e w}6 f w ww6 g ~ w{6 h ~ ~w6 i w ~w6 j w6 k w w}6 l } w7 m  w7 n w w7 o w w7 p w w7 q w ww7 r y w7 s w ww7 t } ww7! u ww7& v w w7+ w w ww7, x { w}7- y w w73 z w ww7/ { w w7? | } ww79 } v 7: ~ w{7E  w7F y ww7G w ww7H w7M w wy7N ww7O w}7P w wz7Q  ww7V w7] x ww7^ { ww7_ w ww7` w w7a wy7b ww7c w}7m 7n 7q w7r 7s w wy56dq5}PF6dq6~wLH6dq7}8C7 Hm3R q897 wHm3R H RH R FootnoteHr@ q98:7 ! Hr@ HzHz  Single LineH'q:9<73 w;;Footnote }  5_;:{     HD q<:=7 7MHD HH  Double LineH q=<@77V>?w Double Line {5c>?=5e?>=7n7q H q@=B7AA Single Line5hA@HZqB@C7q TableFootnoteEGxR qCB7 9EGxR EPwEPw  TableFootnoteod5pDRRHH5xE5 HH 3Fe   HH5zFN5HHlEE DHH5{G6 HH @>?HDoe   HH5}HJ6HHlGG HUV 5~I6 HUV AAJUU`   HUV 5JLH6HUV lII H$ 5K6 FtnH$ ELUU` tn oH$ 5LJ6H$ lKK HUV 5M5 HUV N UUe !  HUV 5NPF5HUV lMM H$ 5O5 H$ P UUe "  H$ 5PN5H$ lOO HH5QD HHAR  ` # Gravitation and Cosmology ;` $ )What is the reason for Hubble Expansion? X` %  u` ' -Hubbles Law says the universe is expanding. ` ( ӸWhat does that mean? ` )  ` * The Cheap, Casual Answer ` + Newtonian Cosmology ` 0 !  Equations we can work with #` ,  @` - The Meaningful, Lasting Answer ]` / !Einsteins theory of gravitation z` . UVGeneral Relativity ` 1 (  Some insight into whats going on HH5RDHHlUQQ e "d5SUUHH5TS HHcYqU h & Escape Velocity m  h 4 it2Conservation of energy  E  implies thatn >` 5 e  a[ h 3 $If  v   = 0  then p leHH5USHHlRXTT d5VXXanosHH5WV nweHHs` -tXin ` 2 Newtonian Cosmology te;` 6 vi iX 7 .CStarting point:  The  universe  has mass  M  and is intu H 7 *contained in a sphere of radius  Ro ,` 9 Connection to reality: I h : "5 Density, not mass    Constant   r f` ;   h <  Hubble Constant s ` =  ` >   h ?  t ` @  s` A m0 : The universe is  open  and expands forever li1` B 6h ?D65`_aY> 6[_H-5a`bYH-: T uHHH-5bacY CH-e uveH?HH?l5cbdYwiH?l anfaHH?fgO5dceY fgO RH%fgO5edfY H%fgOH5H5 M#SSO5fegY [#SSO#SdZ#SdZ mZ! O5gfhY Z! OZ{Z{ vS"6Xj5hgiY S"6XjS2DS2D v H5ihkYjj5ji 6p{E6eE](|5'equal[char[E],times[over[num[1.00000000,"1"],num[2.00000000,"2"]],char[m],power[indexes[0,1,char[v],char[infty]],num[2.00000000,"2"]]]][_Q5kiYll5lkj*6y1'equal[char[E],plus[times[over[num[1.00000000,"1"],num[2.00000000,"2"]],char[m],power[char[v],num[2.00000000,"2"]]],minus[over[times[char[G],char[M],char[m]],char[R]]]]]N5mYqS nnTn5nmgo-N$exXHdZ'natop[(*q"Magenta"q*)equal[(*q"Magenta"q*)times[(*q"Magenta"q*)over[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)m],power[(*q"Magenta"q*)indexes[(*q"Magenta"q*)0,1,char[(*q"Magenta"q*)v],char[(*q"Magenta"q*)infty]],num[(*q"Magenta"q*)2.00000000,"2"]]],plus[(*q"Magenta"q*)times[(*q"Magenta"q*)over[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)m],power[(*q"Magenta"q*)char[(*q"Magenta"q*)v],num[(*q"Magenta"q*)2.00000000,"2"]]],minus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M],char[(*q"Magenta"q*)m]],char[(*q"Magenta"q*)R]]]]],equal[(*q"Magenta"q*)power[(*q"Magenta"q*)indexes[(*q"Magenta"q*)0,1,char[(*q"Magenta"q*)v],char[(*q"Magenta"q*)infty]],num[(*q"Magenta"q*)2.00000000,"2"]],plus[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)v],num[(*q"Magenta"q*)2.00000000,"2"]],minus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M]],char[(*q"Magenta"q*)R]]]]]]d6ocr[ssq"ge ca>5pqMen9X > ca>% *q'equal[char[v],indexes[0,1,char[v],string["ESCAPE"]],power[id[over[times[num[2.00000000,"2"],char[G],char[M]],char[R]]],over[num[1.00000000,"1"],num[2.00000000,"2"]]]]}wca@5qmSMenUppTp*)00HH6ro "geHH|ta  G],char[s)M  ` 8 chThe Flat Universe ;` D en qX h F a"u  or  v )0,u` G "q ]` H q*"Note:  M  is constant, but ` J Ma3R ,  r  and  H  change with time t num` I 00 0 h K *Integrate the first equation and findw  h L en/which means that the universe grows likex [(* h M ha@Also, if  t  is the present age of the universe, theny cW` N $for H 0  = 75  km/secMpc. 5t` O M4  Getting hard to accomodate globular clusters! eqHH6soA"]HHlXrr ar[H6tV02" u~Wo5 ll6uvt*) ll? 6vuwt? ?? H! O6wvxt 8chH! OH{H{ vn" cO6xwyt v" cO" sdZ" sdZ MBV-O6yxzt BV-OBV=dZBV=dZ R ?66zy{t I?6 K?Q?~$Q?6{z}t L|| uve6|{aAlS6N76e))'] u'Bequal[char[v],over[times[char[d],char[R]],times[char[d],char[t]]]];fgV6}{~t rd;fgVKK so ~6~}t]6~RI"˭n>i)I?SW'equal[power[id[over[times[char[d],char[R]],times[char[d],char[t]]]],num[2.00000000,"2"]],plus[over[times[num[2.00000000,"2"],char[G],char[M]],char[R]],times[num[2.00000000,"2"],char[k]]]]w6e6!{6e6eAQ y'yequal[char[M],times[over[num[4.00000000,"4"],num[3.00000000,"3"]],char[pi],power[char[R],num[3.00000000,"3"]],char[rho]]]ZfXp8e6"tV{Wrg6e696g6eg6e4 'kequal[char[H],times[over[num[1.00000000,"1"],char[R]],over[times[char[d],char[R]],times[char[d],char[t]]]]]al[$i8e6:Vh[dWs96B99Z S'6'equal[power[char[H],num[2.00000000,"2"]],plus[over[times[num[8.00000000,"8"],char[pi],char[G],char[rho]],num[3.00000000,"3"]],over[times[num[2.00000000,"2"],char[k]],power[char[R],num[2.00000000,"2"]]]]]],c;6CVm[nWt,cr[d6M[]]6!HH6N HHequm[4.000[3 ` E ow Gravitation [3;` P ar(Einsteins General Theory of Relativity X` Q ˃ u` S )Space and time are intimately connected. ` T e  U -The geometry of spacetime is curved by the 0@ U erpresence of mass and energy. h` V l[  W 6:,The equations that determine this curvature # W ,have contained within them the conservation @@ W Jof energy, including mass  as  energy in  E = mc 2 . ]` X ,c [z Y ho0For distances the size of a galaxy, differences "2 Y ch*between this theory and Newtonian gravity @ Y )are  nearly  impossible to detect. cHH6P[HHls 9>6Sq9X8>9>M՜% [3'^equal[(*q"Blue"q*)power[(*q"Blue"q*)id[(*q"Blue"q*)over[(*q"Blue"q*)times[(*q"Blue"q*)char[(*q"Blue"q*)d],char[(*q"Blue"q*)R]],times[(*q"Blue"q*)char[(*q"Blue"q*)d],char[(*q"Blue"q*)t]]]],num[(*q"Blue"q*)2.00000000,"2"]],over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)2.00000000,"2"],char[(*q"Blue"q*)G],char[(*q"Blue"q*)M]],char[(*q"Blue"q*)R]]]|I"9@6To atru56e6Wmhe56e56eA S'ne'equal[(*q"Blue"q*)power[(*q"Blue"q*)char[(*q"Blue"q*)H],num[(*q"Blue"q*)2.00000000,"2"]],over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)8.00000000,"8"],char[(*q"Blue"q*)pi],char[(*q"Blue"q*)G],char[(*q"Blue"q*)rho]],num[(*q"Blue"q*)3.00000000,"3"]]]eT 58e6Xorv[6kYݱDu F'atop[(*q"Magenta"q*)equal[(*q"Magenta"q*)int[(*i2iq"Magenta"q*)times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)R],over[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],num[(*q"Magenta"q*)2.00000000,"2"]]],diff[(*q"Magenta"q*)char[(*q"Magenta"q*)R]]],num[(*q"Magenta"q*)0.00000000,"0"],char[(*q"Magenta"q*)R]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)id[(*q"Magenta"q*)times[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M]]],over[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],num[(*q"Magenta"q*)2.00000000,"2"]]],int[(*i2iq"Magenta"q*)diff[(*q"Magenta"q*)char[(*q"Magenta"q*)t]],num[(*q"Magenta"q*)0.00000000,"0"],char[(*q"Magenta"q*)t]]]],equal[(*q"Magenta"q*)times[(*q"Magenta"q*)over[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],num[(*q"Magenta"q*)3.00000000,"3"]],power[(*q"Magenta"q*)char[(*q"Magenta"q*)R],over[(*q"Magenta"q*)num[(*q"Magenta"q*)3.00000000,"3"],num[(*q"Magenta"q*)2.00000000,"2"]]]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)id[(*q"Magenta"q*)times[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M]]],over[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],num[(*q"Magenta"q*)2.00000000,"2"]]],char[(*q"Magenta"q*)t]]]]N96lon"qrw"Men6pgtaL4$)n'equal[(*q"Magenta"q*)char[(*q"Magenta"q*)R],cross[(*q"Magenta"q*)string[(*q"Magenta"q*)"constant"],power[(*q"Magenta"q*)char[(*q"Magenta"q*)t],over[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],num[(*q"Magenta"q*)3.00000000,"3"]]]]]m[NH6qoea"rx"qti6u("M`D<$q*'equal[char[t],times[over[num[2.00000000,"2"],num[3.00000000,"3"]],over[num[1.00000000,"1"],indexes[0,1,char[H],num[0.00000000,"0"]]]],times[sn[num[8.70000000,"8.7"],num[9.00000000,"9"]],string[" years"]]]a"NBH6vo200rya")Gd6}")n001"HH6~ [q"HHo ` R Fundamental Geometry ;` Z taTwo kinds of vectors X` [ )n u` ] a"$Vectors:  Coordinates  A m ge h ^ Ma-aka Contravariant vectors, Stick vectorsq (P` _ ve *m` b *q&Covectors:  Coordinates  A m  h ` "3+aka Covariant vectors, Lasagna vectorsz HH6HHl d6eov[300HH6 s,1HHj0, 0000000,00  ` \ "]Fundamental Geometry ;` a %Inner Products and the Metric Tensor GX` c  }u e "2Inner products (i.e. dot products) are possible  H e .only  between a vector and a covector!{ } f 1Problem:  Angles are set by the inner product m@ f between stick vectors! ds ` g )  Need a way to convert vectors into r` h + covectors, or covectors into vectors ari` i v o h j .Answer:  The metric tensor  g mn} HH63akHHl H6qQZc6QZcQu)fg- 6 s)fg- 99  Have a directionMfg6 \"]Mfg]]  Magnitude = lengthtsH6 }]zr odZ??6Z??y  b?u?Z~~l$-u6eby$-uu-?6 !s u-? --lul?Q?6r o?Q?vecrslQZQ?~$c?6 $c?Qc?c$H$6H$HHHHHHZ$-6$-ZQ??$zfg6 fgzfg  Have a  sense , fg6  fg  not a direction g6  g x x fgC6  fgC  Magnitude = densityd66HH6 HH6r ` d rsThe Relativistic Line Element ;` k  X m +This distance between two nearby points in ?u H m spacetime is given by~ ` n   h o Special Relativity:  ` p   h q 7General Relativity: Spherically symmetric mass  HH6HHl H6>{ ;u6 re;u;uu_Hl6_HlC_;c6;c = ns;cc;l6;l;ll;Z6;Z;ZZ;H6hRe;Hlemt ;HH;Q6 tw;Qintin;QQ;?6;?;??;-6 ;-Ge;--;66;6;66  H66 u7e$'equal[(*q"Green"q*)string[(*q"Green"q*)"Inner Product"],sum[(*q"Green"q*)times[(*q"Green"q*)indexes[(*q"Green"q*)1,0,char[(*q"Green"q*)A],char[(*q"Green"q*)mu]],indexes[(*q"Green"q*)0,1,char[(*q"Green"q*)B],char[(*q"Green"q*)mu]]],char[(*q"Green"q*)mu]]]c[~6 c[~clcl =  Number ofcsK6 csKcc   noodles  pierced bych6 chcc  the  stick6+YGqMr:0'tequal[(*q"Red"q*)times[(*q"Red"q*)abs[(*q"Red"q*)char[(*q"Red"q*)A]],abs[(*q"Red"q*)char[(*q"Red"q*)B]],cos[(*q"Red"q*)indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)theta],times[(*q"Red"q*)char[(*q"Red"q*)A],char[(*q"Red"q*)B]]]]],cdot[(*q"Red"q*)diacritical[(*q"Red"q*)4,0,0,0,0,char[(*q"Red"q*)A]],diacritical[(*q"Red"q*)4,0,0,0,0,char[(*q"Red"q*)B]]],sum[(*q"Red"q*)times[(*q"Red"q*)indexes[(*q"Red"q*)0,1,char[(*q"Red"q*)g],times[(*q"Red"q*)char[(*q"Red"q*)mu],char[(*q"Red"q*)nu]]],indexes[(*q"Red"q*)1,0,char[(*q"Red"q*)A],char[(*q"Red"q*)mu]],indexes[(*q"Red"q*)1,0,char[(*q"Red"q*)B],char[(*q"Red"q*)nu]]],char[(*q"Red"q*)mu]]]N^T6}d7HH7 HH  c   ` l The Schwarzschild Radius ;` r Mr+Consistent results with Newton or Einstein *q"X` s ed )u h u r[5Set escape velocity equal to the speed of light: * h v *q(This gives the Schwarzschild radius *q[` w ed )x x )BDThis is  also  the value of the radial coordinate  r  )A@ x "R'which makes the line element singular! (*q` y Re * { *).Many tests of General Relativity are based on @ { *qthis Schwarzschild Metric. *HH7ndHHl cha7_+| / Lv5'equal[(*q"Blue"q*)times[(*q"Blue"q*)char[(*q"Blue"q*)d],power[(*q"Blue"q*)char[(*q"Blue"q*)s],num[(*q"Blue"q*)2.00000000,"2"]]],sum[(*q"Blue"q*)sum[(*q"Blue"q*)times[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)g],times[(*q"Blue"q*)char[(*q"Blue"q*)mu],char[(*q"Blue"q*)nu]]],indexes[(*q"Blue"q*)1,0,times[(*q"Blue"q*)char[(*q"Blue"q*)d],char[(*q"Blue"q*)x]],char[(*q"Blue"q*)mu]],indexes[(*q"Blue"q*)1,0,times[(*q"Blue"q*)char[(*q"Blue"q*)d],char[(*q"Blue"q*)x]],char[(*q"Blue"q*)nu]]],equal[(*q"Blue"q*)char[(*q"Blue"q*)nu],num[(*q"Blue"q*)1.00000000,"1"]],num[(*q"Blue"q*)4.00000000,"4"]],equal[(*q"Blue"q*)char[(*q"Blue"q*)mu],num[(*q"Blue"q*)1.00000000,"1"]],num[(*q"Blue"q*)4.00000000,"4"]]]Nd7a b ~*qth7  nkBd1'atop[(*j4jq"Magenta"q*)equal[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)s],num[(*q"Magenta"q*)2.00000000,"2"]]],plus[(*q"Magenta"q*)times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)c],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)t],num[(*q"Magenta"q*)2.00000000,"2"]]],minus[(*q"Magenta"q*)id[(*q"Magenta"q*)plus[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)x],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)y],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)z],num[(*q"Magenta"q*)2.00000000,"2"]]]]]]]],uequal[(*q"Magenta"q*)plus[(*nq"Magenta"q*)times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)c],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)t],num[(*q"Magenta"q*)2.00000000,"2"]]],minus[(*q"Magenta"q*)id[(*q"Magenta"q*)plus[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)theta],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]],power[(*q"Magenta"q*)string[(*q"Magenta"q*)"sin"],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)theta],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)phi],num[(*q"Magenta"q*)2.00000000,"2"]]]]]]]]]q"MNcU7 "]]q*lu7r*q-?rlq"o"M' atop[(*q"Magenta"q*)equal[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)s],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)id[(*q"Magenta"q*)plus[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],minus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)c],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)r]]]]]],power[(*q"Magenta"q*)char[(*q"Magenta"q*)c],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)t],num[(*q"Magenta"q*)2.00000000,"2"]]]],minus[(*q"Magenta"q*)id[(*i1iq"Magenta"q*)plus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]]],id[(*q"Magenta"q*)plus[(*q"Magenta"q*)num[(*q"Magenta"q*)1.00000000,"1"],minus[(*q"Magenta"q*)over[(*q"Magenta"q*)times[(*q"Magenta"q*)num[(*q"Magenta"q*)2.00000000,"2"],char[(*q"Magenta"q*)G],char[(*q"Magenta"q*)M]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)c],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)r]]]]]]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)theta],num[(*q"Magenta"q*)2.00000000,"2"]]],times[(*q"Magenta"q*)power[(*q"Magenta"q*)char[(*q"Magenta"q*)r],num[(*q"Magenta"q*)2.00000000,"2"]],power[(*q"Magenta"q*)string[(*q"Magenta"q*)"sin"],num[(*q"Magenta"q*)2.00000000,"2"]],char[(*q"Magenta"q*)theta],char[(*q"Magenta"q*)d],power[(*q"Magenta"q*)char[(*q"Magenta"q*)phi],num[(*q"Magenta"q*)2.00000000,"2"]]]]]]]ovN7[q"Bgea"d7*.0ntq*HH7 000HHnt  7   ` t 0,Cosmology and C ` } *qGeneral Relativity [(*f` z [( M` ~ "M*Another solution of Einsteins equations: `  nt&The Universe is Isotropic and Uniform ` arCosmological Dust ` 2. 0 h q"&  The Robertson-Walker Metric ` ta u` .0r  is a radial coordinate q*)` q*'R  is a time-dependent scale factor *)2HH7 [q"HHl q"M7#e(*w>,(*')equal[(*q"Green"q*)char[(*q"Green"q*)c],power[(*q"Green"q*)id[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)num[(*q"Green"q*)2.00000000,"2"],char[(*q"Green"q*)G],char[(*q"Green"q*)M]],char[(*q"Green"q*)R]]],over[(*q"Green"q*)num[(*q"Green"q*)1.00000000,"1"],num[(*q"Green"q*)2.00000000,"2"]]]]Nd7$ool*qGe7(M75k9$te'bequal[char[R],over[times[num[2.00000000,"2"],char[G],char[M]],power[char[c],num[2.00000000,"2"]]]]NFH7)soWad7AraalHH7B aHHQfa7   ` | Is the Universe Flat? ;` "M X 7#4The universe is flat if the density of the universe u H eq#equals the critical density: ,po` d[BSince  50  km/secMpc   H  100  km/secMpc,  ` r[:r CRIT   = (5 - 20 )10 -27   kg/m 3 '` )n (D` 00(So what is the density of the universe? ]a`  $~` o1For the local group, isolated from Virgo cluster ` 6M    M GALAXY  + M M31  +  ... ` ],9 10 12   M SUN  = 210 42   kg c` 2"/D  5 M pc =  1.510 23   m `  ` A r  M / D 3  = 0. 610 -27  kg/m 3 l,`  BI`  7Close!   Does  dark matter  make up the rest? 7 HH7DIsHHl 7#7JeniE)uehbKcr'`atop[(*j1jq"Green"q*)equal[(*x-524288xl65536ll65536ll65536lq"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],power[(*q"Green"q*)char[(*q"Green"q*)s],num[(*q"Green"q*)2.00000000,"2"]]],times[(*q"Green"q*)power[(*q"Green"q*)char[(*q"Green"q*)c],num[(*q"Green"q*)2.00000000,"2"]],char[(*q"Green"q*)d],power[(*q"Green"q*)char[(*q"Green"q*)t],num[(*q"Green"q*)2.00000000,"2"]]]],minus[(*q"Green"q*)times[(*q"Green"q*)power[(*q"Green"q*)char[(*q"Green"q*)R],num[(*q"Green"q*)2.00000000,"2"]],id[(*q"Green"q*)char[(*q"Green"q*)t]],id[(*i1iq"Green"q*)plus[(*q"Green"q*)over[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)d],power[(*q"Green"q*)char[(*q"Green"q*)r],num[(*q"Green"q*)2.00000000,"2"]]],plus[(*q"Green"q*)num[(*q"Green"q*)1.00000000,"1"],minus[(*q"Green"q*)times[(*q"Green"q*)char[(*q"Green"q*)k],power[(*q"Green"q*)char[(*q"Green"q*)r],num[(*q"Green"q*)2.00000000,"2"]]]]]],times[(*q"Green"q*)power[(*q"Green"q*)char[(*q"Green"q*)r],num[(*q"Green"q*)2.00000000,"2"]],char[(*q"Green"q*)d],power[(*q"Green"q*)char[(*q"Green"q*)theta],num[(*q"Green"q*)2.00000000,"2"]]],times[(*q"Green"q*)power[(*q"Green"q*)char[(*q"Green"q*)r],num[(*q"Green"q*)2.00000000,"2"]],power[(*q"Green"q*)string[(*q"Green"q*)"sin"],num[(*q"Green"q*)2.00000000,"2"]],char[(*q"Green"q*)theta],char[(*q"Green"q*)d],power[(*q"Green"q*)char[(*q"Green"q*)phi],num[(*q"Green"q*)2.00000000,"2"]]]]]]]]"qNG7Ke(*"qch7S0," ~9,,i'Dequal[(*q"Blue"q*)indexes[(*q"Blue"q*)0,1,char[(*q"Blue"q*)rho],string[(*q"Blue"q*)"CRIT"]],over[(*q"Blue"q*)times[(*q"Blue"q*)num[(*q"Blue"q*)3.00000000,"3"],power[(*q"Blue"q*)char[(*q"Blue"q*)H],num[(*q"Blue"q*)2.00000000,"2"]]],times[(*q"Blue"q*)num[(*q"Blue"q*)8.00000000,"8"],char[(*q"Blue"q*)pi],char[(*q"Blue"q*)G]]]]enN[7T)we eem[d5"]Left"Gd6"qRightqd7en Reference0dDow"GdSr]*qdV,p*qdo"q"]d00,cdarred*)*qden00dd ed ch a 1f a Body. f b 002 ]Bulleted\t(. e"f c CellBody. f d  CellHeading. f e  rFootnote. f fT Heading1Body. f gT  Heading2Body. f hT   HeadingRunInBody. f i l yIndented. f j g Numbered.\t. f kE  Numbered1.\tNumbered. f l f TableFootnote. f mT   TableTitleT:Table : .  f nP TitleBody. f o T   TableTitleT:Table : . f p   CellHeading. f q  CellBody. f r   CellFooting. f s  Body. @ t  lHeader. @ u  blaFooter. f v $ Body. f w Body. f x  Body. f y  Body. f z   Body. f {  Body. f | $ Body. f } $ Body. $f ~ Body. f   Body. Hf  Body. f Body. f  Body. f  l.Body. f Body. f Body. Hf Body. f Body. f  Body. f l.Body. f  Body. f H.Body. f  Body. f $ Body. f   Body. f  Body. Hf Body. $f H.Body. % -  Emphasis  EquationVariables )   ڝ  ڝ tu ڝ w  [  $)  w w $)  tu )  w ڝ )  )  ڝ w ڝ  ڝ w ڝ ڝ tu tu tu w  ڝ  ڝ ڝ ڝ ڝ w  ڝ ڝ w w uo&  )   )  uo&  ڝ ! Z  Z  Z  Z  Z  Z Z Z  Z  Z Z  Z  Z  Z Z& Z Z  Z  Z  Z  Z  Z! Z  Thin Medium Double Thick@  Very Thin     oH p q rH p q rH p q rH p q rH p q rFormat A   oH p q rH p q rH p q rH p q rH p q rFormat B U e V UComment  d BlackT!WhiteddA&Reddd Greendd  Blue d Cyan d Magentad  Yellow  Times-Roman Times-Bold Times-ItalicSymbolSymbolHelvetica-BoldTimes-BoldItalicTimes HelveticaeSymbol RegularRegular BoldRegularItalicr|P\ɼ5qJJCm7+ S< OGu{Wb2`'iʘjU}XM/c}sSq ,3">+D5E@ UEʲO?|r񦤛3Ԣ//_HaLz#Dcd7{U7`'ȘFL{H\J