String Theory Notes Page 1 of 20
Chapter 2 + 3 – Relativity & Extra Dimensions
1. Stuff
The UP indices are the real ones. The DOWN indices are missing a minus
(
sign. So dx 0 = -dx 0, dx 1, dx 2, dx 3 )
A three-ball, B3 is surrounded by the two-sphere S2. In general
2pd /2
( ) ( )
vol S d -1(R) = Rd -1vol S d -1 = Rd -1
G ( d2 )
pd /2
( )
vol B d =
G (1 + d2 )
In general
ò Bd
⋅ E dV = òBd
r dV
d -1
Flux of E across S =q
For each extra dimension, we get an extra factor of 1/r.
éG ù = m 3 / kg s 2 , éc ù = ms -1 and é ù = kg m 2 / s . = G / c 3
êë úû êë úû êë úû P
2 (D ) (D )
g = -V and V = 4pG rm . Since the dimensions of the density
change, the dimensions of G must change as well.
G (D ) 2G
(D )
( )
D -2
We get (PD ) =
c3
= ( P) G
G (D )
= VC , where Vc is the product of all the different characteristic
G
lengths of the extra dimensions.
© Daniel Guetta, 2009
, String Theory Notes Page 2 of 20
Chapter 4 – Nonrelativstic Strings
1. Introduction
Small oscillations of a string imply that
¶y
1
¶x
Using (¶y / ¶x ) » tan q , we can derive the fact that the motion of the
string satisfies the wave equation
¶2y m0 ¶ 2y
- =0
¶x 2 T0 ¶t 2
Disturbances along the string therefore move at a velocity
v 0 = T0 / m0
If each point on the string is oscillating sinusoidally and in phase, with
y(t, x ) = y(x )sin (wnt + f ) , we can feed this into our wave equation, and
find solutions for Dirichlet and Neumann boundary conditions
o Dirichlet conditions give
æ n px ö÷ n p T0
yn (x ) = An sin ççç ÷ wn = n = 1,2,
çè a ÷÷ø a m0
o Neumann conditions give
æ n px ö÷ n p T0
yn (x ) = An cos ççç ÷ wn = n = 0,1,
çè a ÷÷ø a m0
This case clearly admits an extra mode of motion (n = 0) which
corresponds to the string translating along the y-axis.
More generally, the most general solution of this equation is given by
y(t, x ) = h+ (x - v0t ) + h- (x + v 0t )
[This is the superposition of a wave travelling towards the left and one
travelling towards the right].
The functions for h+ and h– can be related using initial and/or boundary
conditions – these can be of two forms
o Dirichlet conditions specify the value of y.
© Daniel Guetta, 2009
, String Theory Notes Page 3 of 20
o Neumann conditions specify the value of a derivative of y with
respect to position along the string.
For strings with non-constant densities m(x ) , the wave equation still
applies, since it was derived by considering a small piece of string. The
analysis leading to y(x) involves solving a slightly more complex
differential equation.
2. Lagrangian Mechanics
The lagrangian for a system is defined by
L = T -V
Where T is the kinetic energy and V is the potential energy of the system.
The action for a given path the particle might take is defined as
S= ò
L(t )dt
Hamilton’s Principle states that the path which a system actually takes
is one for which the action S does not change to first order when is
varied infinitesimally.
Usually, the path is parameterised by time as x(t), and the perturbed path
takes the form x (t ) + dx (t ) . The integral then takes the form
tf
S= ò ti
L(x , x )dt
We usually only consider variations to the path that are fixed at the start
and end of the motion, such that dx (ti ) = dx (t f ) = 0 .
3. Lagrangian Mechanics for a nonrelativistic string
For a string, the kinetic energy is the sum of the kinetic energies of all the
infinitesimal segments that comprise the string:
2
a 1 æç ¶y ö÷
T = ò0
m ç ÷ dx
2 0 ççè ¶t ÷÷ø
The potential energy comes from the work that must be done to stretch
each individual segment of the string. If each small part of the string is
stretched by an amount d , then the work done stretching it is T0 d and
© Daniel Guetta, 2009
Chapter 2 + 3 – Relativity & Extra Dimensions
1. Stuff
The UP indices are the real ones. The DOWN indices are missing a minus
(
sign. So dx 0 = -dx 0, dx 1, dx 2, dx 3 )
A three-ball, B3 is surrounded by the two-sphere S2. In general
2pd /2
( ) ( )
vol S d -1(R) = Rd -1vol S d -1 = Rd -1
G ( d2 )
pd /2
( )
vol B d =
G (1 + d2 )
In general
ò Bd
⋅ E dV = òBd
r dV
d -1
Flux of E across S =q
For each extra dimension, we get an extra factor of 1/r.
éG ù = m 3 / kg s 2 , éc ù = ms -1 and é ù = kg m 2 / s . = G / c 3
êë úû êë úû êë úû P
2 (D ) (D )
g = -V and V = 4pG rm . Since the dimensions of the density
change, the dimensions of G must change as well.
G (D ) 2G
(D )
( )
D -2
We get (PD ) =
c3
= ( P) G
G (D )
= VC , where Vc is the product of all the different characteristic
G
lengths of the extra dimensions.
© Daniel Guetta, 2009
, String Theory Notes Page 2 of 20
Chapter 4 – Nonrelativstic Strings
1. Introduction
Small oscillations of a string imply that
¶y
1
¶x
Using (¶y / ¶x ) » tan q , we can derive the fact that the motion of the
string satisfies the wave equation
¶2y m0 ¶ 2y
- =0
¶x 2 T0 ¶t 2
Disturbances along the string therefore move at a velocity
v 0 = T0 / m0
If each point on the string is oscillating sinusoidally and in phase, with
y(t, x ) = y(x )sin (wnt + f ) , we can feed this into our wave equation, and
find solutions for Dirichlet and Neumann boundary conditions
o Dirichlet conditions give
æ n px ö÷ n p T0
yn (x ) = An sin ççç ÷ wn = n = 1,2,
çè a ÷÷ø a m0
o Neumann conditions give
æ n px ö÷ n p T0
yn (x ) = An cos ççç ÷ wn = n = 0,1,
çè a ÷÷ø a m0
This case clearly admits an extra mode of motion (n = 0) which
corresponds to the string translating along the y-axis.
More generally, the most general solution of this equation is given by
y(t, x ) = h+ (x - v0t ) + h- (x + v 0t )
[This is the superposition of a wave travelling towards the left and one
travelling towards the right].
The functions for h+ and h– can be related using initial and/or boundary
conditions – these can be of two forms
o Dirichlet conditions specify the value of y.
© Daniel Guetta, 2009
, String Theory Notes Page 3 of 20
o Neumann conditions specify the value of a derivative of y with
respect to position along the string.
For strings with non-constant densities m(x ) , the wave equation still
applies, since it was derived by considering a small piece of string. The
analysis leading to y(x) involves solving a slightly more complex
differential equation.
2. Lagrangian Mechanics
The lagrangian for a system is defined by
L = T -V
Where T is the kinetic energy and V is the potential energy of the system.
The action for a given path the particle might take is defined as
S= ò
L(t )dt
Hamilton’s Principle states that the path which a system actually takes
is one for which the action S does not change to first order when is
varied infinitesimally.
Usually, the path is parameterised by time as x(t), and the perturbed path
takes the form x (t ) + dx (t ) . The integral then takes the form
tf
S= ò ti
L(x , x )dt
We usually only consider variations to the path that are fixed at the start
and end of the motion, such that dx (ti ) = dx (t f ) = 0 .
3. Lagrangian Mechanics for a nonrelativistic string
For a string, the kinetic energy is the sum of the kinetic energies of all the
infinitesimal segments that comprise the string:
2
a 1 æç ¶y ö÷
T = ò0
m ç ÷ dx
2 0 ççè ¶t ÷÷ø
The potential energy comes from the work that must be done to stretch
each individual segment of the string. If each small part of the string is
stretched by an amount d , then the work done stretching it is T0 d and
© Daniel Guetta, 2009
