Lecture 4 8.251 Spring 2007 Lecture 4 - Topics • Gravitational potentials • Compactification • Large extra dimensions Reading: Zwiebach, Sections: 3.7 - 3.10 Gravitation Newtonian Gravity Fm = −(GMm/r2) rˆ· When working in regime with high velocities or strong gravity fields, instead work with: Einstein’s Gravity When working in a regime with small distances or extremely strong gravity fields, instead work with: String Theory −ds2 = ηµν dxµdxν GR = gµν (x)dxµdxν ⇒ −ds2 gµν (x) = gνµ(x) Still symmetric, still invertible. Signature (-,+,+,+) (1 negative eigenvalue, 3 positive eigenvalues) 1� �� � � � � Lecture 4 8.251 Spring 2007 If gravity weak, approximate gµν (x) = ηµν + hµν (x) small fluctuation In E & M, used Aµ(x): Aµ → Aµ + ∂µ Could define new coordinates: x�µ = xµ +µ (x) hµν = ηµαηνβ hαβ hµν GT h�µν = hµν + ∂ν + ∂µ → µ ν 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: 3 G = 6.67 × 10−11 kg ms2 · c = 3 × 108 m s ¯kg · m2 h = 1.06 × 10−34 s Suppose want fundamental length, mass, and time: lp, mp, and tp l3 G = 1 p · mptp 2 c = 1 lp · tp ¯mplp 2 h = 1 · tp G¯h lp = ≈ 1.6 × 10−33cm c3 ¯hc mp = G ≈ 2.17 × 10−5g tp = lp ≈ 5.4 × 10−44s c Note: lp small, tp extremely small, mp fairly large. 2�Lecture 4 8.251 Spring 2007 2 2 2mp mpc melectron melectronc mproton melectronc10−5g 1019 GeV 10−27 g 0.5 MeV 1 GeV New accelerator LHC will accelerate protons to maximum of 7000 GeV which is much smaller than 1019 GeV. Cosmological Constant Energy density (mass density) ≈ 0.7 × 10−29 g 3cmNote critical mass density of universe (to make it flat) is ≈ 10−29g/cm3 In Planckian units, mp/l3 = 5.3 × 1093 g p cm3 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 = ρ� · �E = −�Φ �2Φ = −ρ Gravity: F�= m�g g = −�Vg �g: gravitational field Vg: gravitational potential (energy/mass) in all dimensions like E�= (energy/charge) In 4D: QΦ = 4πr V (4) = −GM/r g �2V (4) = 4πGρmg In D-dim: �2Vg (D) = 4πG(D)ρm (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. 3Lecture 4 8.251 Spring 2007 RHS’s ρm has units that changes with dimension so G must too. � · �g = −4πGρm [G(5)] M L4 = [G] M L3 ⇒ [G(5)] = L · [G] � lp = G¯h c3 ⇒ G = l2 pc3 ¯h ⇒ [G] = L2 c3 ¯h ⇒ [G(5)] = L3 [c3] [¯h] G(5) = (l(5) p )3 c 3/¯h G(5) (l(5) p )3 = G (lp)2 In general: G(D) (lp (D))D−2 = G (lp)2 Imagine a 5D world with a G(5). Compactify 1 dimension subject to effectively 4D x1, x2, x3, x4 �2V (5) = 4πG(5)ρ(5) g m ρ(5) = m δ(x 1)δ(x 2)δ(x 3)m · (5)ρm above has correct units, but correct value? Check by integrating over volume: � � 2πR dx1dxxdx3 dx4ρm = M 0 For a 4D observer: ρ(4) = Mδ(x 1)δ(x 2)δ(x 3) = 2πρ(5) m m 4� � Lecture 4 8.251 Spring 2007 G(5) �2 Vg (x1, x2, x3) = 4π 2πR ρ(4) x1,x2 ,x3 m G = G(5)/2πR So the extra length is 2πR =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 = ¯hc/l, an l of ≈ 10−18 cm correlates to E = 20TeV. Say lp (D) = 10−18cm (lp = 10−33cm) for an acc. that could detect Find lc: G(D) (lp (D))D−2 = = (lc)D−4 G l2 p Take D = 5: lc = (lp (5))3/lp 2 = (10−18)3/(10−33)2 = 1012cm = 107km If this were the case, we would have seen this dimension already! Take D = 6: lc 2 = (lp (6))4/lp 2 lc = (lp (6))2/lp = 10−3cm At least not laughable. But people don’t talk in 10−3 cm. They talk in microns. 1 µm = 10−6 cm = 10−3 mm So D = 6 gets us lc = 10µm. Note: light’s λ ≈ 0.5µm. 5Lecture 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 ≥ 44µm. So possible that D = 6!