� Lecture 1 8.251 Spring 2007 Lecture Topics • Announcements, introduction Lorentz transformations • • Light-cone coordinates Reading: Zwiebach, Sections 2.1-2.3 Strings at Many Scales Classical strings, cosmic strings QCD strings, gluons or flux tubes hold quarks together as qq+ Flux tube - String of 0.2 fm AdS/CFT Anti-desiltem/conformal field theory Fundamental Strings Standard model of particle physics, cosmology, inflation Zero thickness, mass, measure Relativistic Strings Intervals: (ct, x, y, z) ≡ (x 0 , x 1 , x 2 , x 3) ≡ xµ S S1 Event 1 xµ x�µ Event 1 xµ + Δxµ x�µ + Δx�µ −Δs 2 = −Δx 02 + Δx i2 = gµvx v = xµx v = −Δx�02 +i Δx�i2 = −Δs i2 i Δs 2 = Δs i2 > 0 timelike separated Δs 2 = = 0 lightlike separated < 0 spacelike separated 1Lecture 1 8.251 Spring 2007 Again, the interval: −ds2 = ηµvdxµdvv ηµv symmetric by definition: ⎞⎛ −1 1 ηµv = ⎜⎜⎝ ⎟⎟⎠1 1 aµ → aµ = ηµva v∀a Given aµ, bµ, a b = aµbµ = ηµv aµbv · Inverse metric: ηµv = (η−1)µv ηvρηρµ = δµv (summed over ρ) ηρµbµ = ηρµηµvbv = δρvbv = bρ Lorentz Transformation: v 1 β = ; γ = c √1 − βv x� = (x − βct)γ y� = y z� = z ct� = γ(ct − βx) x 0� = γ(x 0 − βx1) x 1� = γ(−βx0 + x 1) 2� 2 x = x 3� 3 x = x 2� �� � Lecture 1 8.251 Spring 2007 A linear invertible transform between xµ and x�µ that satisfies Δs2 = Δs�2 x�µ = Lµv x v L is a Lorentz transfer of LT ηL = η Light cone coordinates: 0 1 2 3 x , x , x , x Keep these two 1 x + = √2(x 0 + x 1) x− = √12(x 0 − x 1) 1 x± = √2(x 0 ± x 1)
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