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) ≡ (x0 , x1 , x2 , x3 ) ≡ xµ
Event 1
Event 1
S1
x�µ
x�µ + Δx�µ
S
xµ
xµ + Δxµ
2
−Δs2 = −Δx0 +
�
2
Δxi = gµv xv = xµ xv
i
Δs2 = Δsi
2
> 0 timelike separated
Δs2 = = 0 lightlike separated
< 0 spacelike separated
1
2
2
= −Δx�0 +i Δx�i = −Δsi
2
Lecture 1
8.251 Spring 2007
Again, the interval:
−ds2 = ηµv dxµ dv v
ηµv symmetric by definition:
⎛
⎞
−1
1
⎜
ηµv = ⎜
⎝
⎟
⎟
⎠
1
1
aµ → aµ = ηµv av ∀a
Given aµ , bµ , a · b = aµ bµ = ηµv aµ bv
−1
Inverse metric: η µv = (ηµv
)
η vρ ηρµ = δµv (summed over ρ)
η ρµ bµ = η ρµ ηµv bv = δρv bv = bρ
Lorentz Transformation:
β=
v
1
;γ = √
c
1 − βv
x� = (x − βct)γ
y� = y
z� = z
ct� = γ(ct − β x )
�
x0 = γ(x0 − βx1 )
�
x1 = γ(−βx0 + x1 )
�
x2 = x2
�
x3 = x3
2
Lecture 1
8.251 Spring 2007
A linear invertible transform between xµ and x�µ that satisfies Δs2 = Δs�2
x�µ = Lµv xv
L is a Lorentz transfer of LT ηL = η
Light cone coordinates:
x0 , x1 ,
x2 , x3
� �� �
Keep these two
1
x+ = √ (x0 + x1 )
2
1
x− = √ (x0 − x1 )
2
1
x± = √ (x0 ± x1 )
2
3