Lecture 4 8 251 Spring 2007 Lecture 4 Topics Gravitational potentials Compacti cation Large extra dimensions Reading Zwiebach Sections 3 7 3 10 Gravitation Newtonian Gravity Fm GM m r2 r When working in regime with high velocities or strong gravity elds instead work with Einstein s Gravity When working in a regime with small distances or extremely strong gravity elds instead work with String Theory GR ds2 dx dx ds2 g x dx dx g x g x Still symmetric still invertible Signature 1 negative eigenvalue 3 positive eigenvalues 1 Lecture 4 8 251 Spring 2007 If gravity weak approximate g x h x small uctuation In E M used A x A A Could de ne new coordinates x x x h h GT h h h There s a dramatic change of language when moving from Newtonian gravity to Einstein s gravity but we ll use the same language for string theory as GR May eventually need to change language for string theory as we understand it better Planckian Units 3 Fundamental Constants m3 kg s2 m c 3 108 s G 6 67 10 11 kg m2 s Suppose want fundamental length mass and time lp mp and tp 1 06 10 34 h G 1 lp3 mp t2p c 1 1 h lp tp mp lp2 tp Gh 1 6 10 33 cm c3 h c mp 2 17 10 5 g G lp tp 5 4 10 44 s c Note lp small tp extremely small mp fairly large lp 2 Lecture 4 mp 10 5 g 8 251 Spring 2007 mp c2 1019 GeV melectron c2 0 5 MeV melectron 10 27 g mproton melectron c2 1 GeV New accelerator LHC will accelerate protons to maximum of 7000 GeV which is much smaller than 1019 GeV Cosmological Constant g Energy density mass density 0 7 10 29 cm 3 Note critical mass density of universe to make it at is 10 29 g cm3 g In Planckian units mp lp3 5 3 1093 cm 3 Planckiang vacuum energy 5 3 1093 7 6 10122 True vacuum energy 0 7 10 29 That s quite an error Odd because in an equation with order 1 expect solution to also be order 1 unless equation very peculiar E M F qE E E 2 Gravity F m g g Vg g gravitational eld energy charge Vg gravitational potential energy mass in all dimensions like E In 4D Q 4 r Vg 4 GM r 2 Vg 4 4 G m D In D dim 2 Vg 4 G D m Note D number of space time dimensions d number of space dimensions D d 1 Note not keeping G the same LHS has units the same in all dimensions 3 Lecture 4 8 251 Spring 2007 RHS s m has units that changes with dimension so G must too g 4 G m G 5 lp M M G 3 G 5 L G 4 L L 3 3 lp2 c3 Gh 2c 5 3 c G G L G L c3 h h h G 5 lp 5 3 c3 h 5 G 5 lp 3 G lp 2 In general D G D lp D 2 G lp 2 Imagine a 5D world with a G 5 Compactify 1 dimension subject to e ectively 4D x1 x2 x3 x4 5 2 Vg 5 4 G 5 m 1 2 3 5 m m x x x 5 m above has correct units but correct value Check by integrating over volume 2 R dx1 dxx dx3 dx4 m M 0 For a 4D observer 1 2 3 5 4 m M x x x 2 m 4 Lecture 4 8 251 Spring 2007 2x1 x2 x3 Vg x1 x2 x3 4 G 5 4 2 R m G G 5 2 R So the extra length is 2 R length of compact dimension lc If had more compact dimestions all circles would get G D lc D 4 volume of compact space G Experimental Implications With current particle accelerators have explored l 10 16 cm Since E h c l an l of 10 18 cm correlates to E 20TeV D Say lp 10 18 cm lp 10 33 cm for an acc that could detect Find lc D G D lp D 2 lc D 4 G lp2 Take D 5 lc lp 5 3 lp2 10 18 3 10 33 2 1012 cm 107 km If this were the case we would have seen this dimension already Take D 6 lc2 lp 6 4 lp2 lc lp 6 2 lp 10 3 cm At least not laughable But people don t talk in 10 3 cm They talk in microns 1 m 10 6 cm 10 3 mm So D 6 gets us lc 10 m Note light s 0 5 m 5 Lecture 4 8 251 Spring 2007 String theory says we live in a D brane We and photons and electrons and all particles are open string attached to the D brane Can t see out Graviton is closed strings hanging out outisde the D brane 5 years ago asked experimenters how far had tested gravity Answer cm 2006 hep ph 0611184 Kapner et al R 44 m So possible that D 6 6