Transcript 5. Applications in Physics

5. Applications in Physics
A. Thermodynamics
B. Hamiltonian mechanics
C. Electromagnetism
D. Dynamics of a perfect fluid
E. Cosmology
A. Thermodynamics
5.1
5.2
5.3
Simple systems
Maxwell and other mathematical identities
Composite thermodynamic systems: Caratheodory's theorem
5.1.
1-component fluid:
 Q  PdV  dU
 Q  PdV  dU
  
(1st law)
1-forms on 2-D manifold with coordinates (V,U).
d  Q  d PdV  dU

  P 

 P 

dV

dU
 dP  dV

  dV



 U V
 V U

 P 

dU  dV

 U V
  Q  dQ
Simple Systems
0
(δQ is not exact )
δQ is a 1-form in 2-space → its ideal is closed (see Ex 4.31(b))
 Q  T dS
Frobenius' theorem (§4.26)
→
2nd law:  Q  T dS holds for all systems in thermodynamic equilibrium.
5.2. Maxwell and Other Mathematical Identities
T dS  PdV  dU
Switching to (S,V) gives
 T 
 P 






 V  S
  S V
→
Switching to (T,V) gives
→
 S   P 

 

 V T  T V
P
1
dV  dU
T
T
dT  dS  dP  dV
→
 T 
dT  dS  
 dV  dS
 V  S
 P 
 dP  dV  
 dS  dV

S

V
(Maxwell identity)
 S 
 P 

dP

dV
dT  dS  dT  
dV



 dT  dV
 V T
 T V
1
P
1
dP  dV  2 dT  dV  2 dT  dU
T
T
T
 U 
1  P 
P
1
Switching to (T,V) gives
0 
dT

dV

dT

dV

dT



 dV
2
2
T  T V
T
T
 V T
 P 
 U 
→ T

P





T

V

V

T
dS 
→
0
 T 
dT  dS  
 dP  dS
 P  S
 T   S 

 
 dP  dT
 P S  T  P
 T   S   P 

 
   dS  dT
 P  S  T  P  S T
→
 T   S   P 

 
    1
 P  S  T  P  S T
5.3. Composite Thermodynamic Systems:
Caratheodory's Theorem
 Q    PdV
i
i  dU i 
n
i 1
Frobenius’ theorem:
1-forms on 2N-D manifold with coordinates (Vi ,Ui ).
 Q  T dS
d Q   Q  0

n
  Pi 
dQ  Q   
 dU k  dVi   Pj dV j  dU j

U
i, k  1 
j 1
k Vk
n

(See Ex 4.30 )

δQ integrable → equilibrium submanifolds defined by S = constant.
→ Equilibrium states on different submanifolds cannot be bridged adiabatically.
Question: is the converse true?
i.e., Not every state reachable adiabatically → S exisits ?
c.f. 2nd law: Heat can’t be transfer from cold to hot regions in a closed system
without doing work.
Caratheodory: 2nd law → S exisits.
Proof:
If δQ is not integrable, then  V, W , s.t. in the neighborhod of P,
 Q V    Q W   0
but


 Q V , W   0
i.e., the subspace KP of vector fields that annul δQ do not form a hypersurface.
e.g.,
x i  e W e V e  W e  V x i
P
1
1



  1   W   2W 2    1   V   2V 2   
2
2



1
1



  1   W   2W 2    1   V   2V 2   x i
2
2





 1   2 W ,V   O  3  xi
P
P
δQ is not integrable → Any states near P can be reached adiabatically.
Not all states near P can be reached adiabatically → δQ is integrable
B. Hamiltonian Mechanics
4.
5.
6.
7.
8.
9.
Hamiltonian Vector Fields
Canonical Transformation
Map between Vectors & 1-forms Provided by ω
Poisson Bracket
Many Particle System: Symplectic Forms
Linear Dynamical Systems: Symplectic Inner Product &
Conserved Quantities
10. Fibre Bundle Structure of the Hamiltonian Equations
See Frankel
5.4. Hamiltonian Vector Fields
Lagrangian:
Momenta:
L  q, q   L  q , q
i
ppi 
L
 qi
i
d
q
t

i
q 
dt

Hamiltonian:
H  pi qi  L = H  q i , pi 
d L L

0
d t  qi  qi
Euler-Lagrange eq:
Hamilton’s eqs:
i
H
q 
 pi
i
pi  
H
 qi
Let M be the manifold with coordinates ( qi ).
Then the set ( qi , qi,t ) is the tangent bundle T(M).
The set ( qi , pi ) can be taken as the cotangent bundle T*(M),
or a symplectic manifold with the Poincare (symplectic) 2-form
  dqi  dpi
Definition: Symplectic Forms:
A 2-form on M2n is symplectic ( M is then a symplectic manifold ) if
1.
2.
d  0
ω is non-degenerate, i.e., iX ω is non-singular.
Definition: Interior Product:
iv :
s.t.
p  p1
by

iv
iv  0
if α is a 0-form
iv    v
if α is a 1-form
iv  w 2 ,
Properties:
Components:
, w p     v, w 2 ,
i a v  b w  a iv  b iw
iv H  v j j H
,wp 
if α is a p-form


i v      iv       
H   h2 
 hp 
deg 
   iv  
  dqi  dpi
U
d  0
d


H 
H 
 qi i  p i i 

dt
q
p
 pi  qi  qi  pi
 
LU   d  U    d  V   d  U  
Eq(4.67):
If U is tangent to
system trajectory
 i 
 
i
i
 U   dq  dp j  q i  pi i   q ii j dp j  pi  ji dq j  q dpi  pi dq
p 
 q


→
j

H
H
dpi  i dqi  dH
 pi
q
LU   ddH  0
U is called a Hamiltonian vector field.
LU H  U  H  
dH
H H H H
0

 i
i
i
dt
 pi  q  q  p
(Conservative system)
5.5. Canonical Transformation
A coordinate transformation  :  qi , pi   Q j , Pj  is canonical if
 *  dQ j  dPj  dqi  dpi  
This can be achieved through a generating function F.
E.g., given F = F(q,Q ) s.t.
pi 
F
 qi
& Pi  
F
Q i
we have
2 F

2 F
2 F
j
k
k
k
dQ  dPj  dQ    k j dQ  k j dq    k j dQ  dq
 q Q
 q Q
 Q Q

j
j
2 F
 2 F
2 F
k
k
dqi  dQ k
dq  dpi  dq   k i dQ  k i dq  
k
i
Q q
 q q
 Q q

i
i
5.6. Map between Vectors & 1-forms Provided by ω
Define
V   V   iV 
or
Vi   ji V j
Note difference in order of indices with eq(5.27)
However, ω is not a metric since
V ,V   V ,V   0
See Ex 5.3
Ex 5.5: A hamiltonian vector field corresponds to an exact 1-form, i.e.
d 
dH   dH    
 dt 
 
5.7. Poisson Bracket

Let

f , g    df , dg
dg  a j
→

ai 
then
 j 
 
 dq  dpi  a
 bj

 q j

p
j 


i
g
 pi
f , g   df , dg

 df , dg



b
j
q j
 pj
  
dg   dg

bi  

g
 qi
→
 ai dpi  bi dqi 
dg 
g
g
dpi  i dqi
 pi
q
g 
g 

 p j q j q j  p j
f
f
g 
g 
i
dq

dp
,

i
 qi
 pi
 p j q j q j  p j
 f g  f g

i
 q  pi  pi  qi
Note sign difference with eq(5.31),
which can be traced to eq(5.27)
5.8. Many Particle System: Symplectic Forms
Symplectic = German for plaiting together
  dqi  dpi
Symplectic Form
Phase space = Symplectic manifold
Ex 5.6 8
5.9. Linear Dynamical Systems: Symplectic Inner
Product & Conserved Quantities
Linear system:
H
1 ij
T pi p j  Vi j q i q j 

2
1
H


Vi j  ki q j  Vi j qi kj
pk   k

2
q
qk 


Tij T
ji
1
Vk j q j  Vi k q i   Vk j q j

2
1
1
kj
H
 T i j  i k p j  T i j pi  jk   T k j p j  T i k pi   T p j
2
2
 pk
 
q
,
p


 


  q    q  ,  p    p  
qi1 , pi1
Linearity: if
i
1
&
2
i
2
1
i
2
i
i
are solutions, so is
2
i
 Solution sub-manifold is also a vector space.
A manifold that is also a vector space is a flat manifold
(M is isomorphic to Rn )
Vi j  V ji
Treating elements of phase space as vectors, we set
Y   q1 ,
, q N , p1 ,
, p N   qi



p
i
 qi
 pi
The (anti-symmetric) symplectic inner product is defined as

 
 Y 1 , Y 2    dqi  dpi  Y 1 , Y 2   qi1 dpi  p1i dqi Y2
 qi1 p2 i  p1 i qi2
The symplectic inner product is time-independent if Y(i) are solution curves.
Proof:



d
d i
 Y 1 , Y 2 
q1 p2 i  p1 i qi2
dt
dt

 qi1 p2 i  qi1 p2 i  p1 i qi2  p1 i qi2
 T i j p1 j p2 i  qi1 Vi j qj2  Vi j qj1 qi2  p1 i T i j p2 j
0
Reminder:
 Y , Y   qi pi  p i qi  0
For time independent T i j and Vi j , we have
pk  Vk j q
qk  T k j p j
j
→
→
i.e., if Y   qi , pi 
d
pk  Vk j q j
dt
d k
q  T k j pj
dt
is a solution, so is
Define the canonical energy by
dY
  qi , pi 
dt

1  dY
EC Y   
,Y 
2  dt

 
EC is conserved on solution curves.
If Y is a solution, then
 
EC Y 
1 i
1 ij
i
q
p

p
q

T p j pi  Vi j q j qi   H



i
i
2
2
LUT i j  0  LUVi j
Let U be a vector field on configuration space s.t.
If
Y   qi , pi 
is a solution, so is
c.f. Ex 5.8
LU Y
The canonical U-momentum is defined by
 

PU Y   LU Y , Y

PU is conserved on solution curves.
For the Klein-Gordon eq.
4-current density =
jμ is conserved:
Setting
we have
j 

tt
  2  m 2   0
i
i
 *        *   
 *   

2m
2m 
 j   0
qi1    x i 
p1 i   t  x i 
qi2    *  x i 
p 2  i   t *  x i 
( DoF =  )
3
0
3
*
*

d
x
j

d
x







     * ,  
t
t

 
5.10. Fibre Bundle Structure of the Hamiltonian Equations
qi defines a configuration space manifold M.
Evolution of system is a curve qi(t) on M.
Lagrangian L( qi , qi, t ) is a function on the tangent bundle T(M).
L
pi  i
is a 1-form field on M, i.e., phase space is a cotangent bundle T*(M)
q , t
Proof:
j'
j'
i
Consider a new set of coordinates Q  Q  q  and new momenta
i
L  q ,t
L
 i
Pj ' 
j'
j'
Q , t  q , t  Q , t
Since qi, t and Q j , t are elements of the (tangent) fibre of T(M), they
transform like contravariant vectors, i.e.,
L i
k
k
j'
P

 j '  pi i j '
q , t   j' Q , t →
( pi & Pj  are 1-forms )
j'
i
q , t
Phase space { qi, pi } is a cotangent bundle T*(M).
H is a function on T*(M).
  d qi  d pi
The symplectic form
is coordinate free.
Proof:

Q j'  Q j' q i 
→
dQ j '   j 'i d q i
Pj '  pi i j '
→
d Pj '  i j ' d pi  pi i j ' , k d qk
dQ j '  d Pj '  i j '  j ' k d q k d pi  pi i j ' , k  j ' m d q m  d qk
→
i j '  j 'k   ki

i j ' , m  j 'k  i j '  j 'k , m
dQ j '  d Pj '  d q i d pi  pi i j '  j ' m , k d q m  d qk
where
2 Q j '
 m , k d q  d q  m k d q m  d qk  0
 q q
j'
m
k
 d qi  d pi
since
Reminder: System with constraints leads to non-trivial bundles.
2 Q j '
m k   0
 q q
QED
C. Electromagnetism
11. Rewriting Maxwell’s Equations Using
Differential Forms
12. Charge & Topology
13. The Vector Potential
14. Plane Waves: A Simple Example
5.11. Rewriting Maxwell’s Equations Using Differential Forms
Maxwell’s equations in vacuum with sources, Gaussian units with c = 1:
B 
E
 4 J
t
E
  E  4
B  0
Faraday 2-form:
F
B
0
t
 0
E
 F    E x
y

 Ez
1
F dx   dx
2!
→
 Ex
Ey
0
Bz
 Bz
0
By
 Bx
dF 
 dF 
dF  0

Fi 0  F0i  Ei
Fi j  Fji  i jk Bk
1
F  ,   dx  dx   dx
2!
 

 Ez 
 B y 
Bx 

0 
 
 3F  ,    dF
F ,    0
 
F 0 i , 0   F 00, i   0
F 0 i , j   F i j , 0  
1
F0i , j  Fi j , 0  F j 0, i  Fi 0, j  F ji ,0  F0 j , i 

3!

1
 Ei , j   i j k Bk ,0  E j ,i  Ei , j   ji k Bk ,0  E j ,i 

3!

1
 Ei , j   i j k Bk ,0  E j ,i 

3

1
B 
 i j k    E 

3
 t k

Fi j , k  

1
Fi j , k  Fj k ,i  Fk i , j  Fji ,k  Fk j ,i  Fi k , j 

3!
1
Fi j , k  F j k ,i  Fk i , j 

3
 dF  0

i,j,k cyclic.
1
 i j l Bl ,k   j k l Bl ,i   k i l Bl , j   1   B

3
3
corresponds to the homogeneous eqs.
 0
 E
 F      F     E x
y

  Ez
F
F
0
i
,
F
,
F
0i
,i
ij
,j
 E i ,i
F
i0
,0
Ex
Ey
0
Bz
 Bz
0
By
 Bx
Ez 
 B y 
Bx 

0 
F i 0  F 0i  Ei
F i j  F ji  i jk Bk
0
   E  4  4 J

E 
 4  J i  4 J i
 i jk Bk , j  Ei ,0     B 

 t i

 Inhomogeneous eqs:
Metric volume form =
F  ,  4 J 

det  dt  dx  dy  dz
åF 
1
 F 
2!
åF 
11
 



F
dx

dx
 


2!  2!

→
 å F   
1
    F  
2!
 dt  dx  dy  dz
å F 

1
  0 i F  
2!

1
1
 k k i l   k l i k  Bl   3Bi  Bi   Bi

2!
2!
 å F i j 
1
1
  i j F     0 k i j F 0 k   k 0 i j F k 0 
2!
2!
0i
 0
 B
x
åF  
  By

  Bz
d å F  dF

dåF

i j0
d å F 

Bx
By
0
Ez
 Ez
0
Ey
 Ex
1
 jk 0 i F jk
2!
Bz 
 E y 
Ex 

0 
E  B , B  E

B 
 i jk    E 

 t k

i jk
 B
E  B, B  E
F

1
i jk F jk
2!
1
 i j k  j k l Bl
2!
 0 k i j F 0 k   k i j Ek   i j k Ek
E  B , B  E
d å F 
 
E  B , B  E
 E


 dåF

 
 3F  ,  
E  B , B  E

E 
  i j k    B 


t

k
i,j,k cyclic.
J  J
å J 

i j0
å J 
i jk
  , J
 å J          J 
å J  J 
k
  i j 0 J   
  0 k i j J k   k i j J k
ki j0 J
0
  i j k J   
0i j k J
 i jk J0
 Inhomogeneous eqs are given by
Magnetic monopole:
dF  4 å J m
d å F  4 å J
Alternative Approach
F  Ei dx i  dt 
See §7.2, Frankel
§4.6, Flanders
1
 i j k Bk dx i  dx j
2!


 E x dx  E y dy  E z dz  dt  Bx dy  dz  B y dz  dx  Bz dx  dy

d  dx  
x


 d  dt 
t
dF  Ei , j dx j  dx i  dt 

d  dx i 

 xi
1
1
 i j k Bk ,l dx k  dx i  dx j   i j k Bk ,t dt  dx i  dx j
2!
2!
1
1
E j , i  Ei , j   i j k Bk ,t  dt  dx i  dx j   i j k   B dx k  dx i  dx j

2!
3!
  

   *
  dt  dx  dy  dz

 
1
i
p! 1
ip

i1 i p
dt  dx  dy  dz  
    *
dt  dx  dy  dz  
 dx  dt
i
    *
dx dx   dx dt 

dx  dt  
 dt dx   dt dt 
i
i
i
i

dt  dx  dy  dz  dx  dt  * dx  dt


i

* dx i  dt 

→

→
0
0 1

 1

* dx  dt  dy  dz
1
 i j k dx j  dx k
2!

dt  dx  dy  dz  dx  dy  * dx  dy

1


* dx  dy  dt  dz

* dx i  dx j   i j k dx k  dt
1
 i j k Bk dx i  dx j
2!
1
1
*F   i j k Ei dx j  dx k   i j k Bk  i j l dx l  dt
2!
2!
F  Ei dx i  dt 
where

1
 i j k Ei dx j  dx k  Bl dx l  dt
2!
 i j k Bk  i j l   j j k l   j l k j  Bk  2 Bl
*F 
1
 i j k Ei dx j  dx k  Bl dx l  dt
2!
d *F 


1
 i j k Ei , l dx l  Ei ,t dt  dx j  dx k  Bi , j dx j  dx i  dt
2!
1
1

 i j k Ei , l dx l  dx j  dx k    i j k Ei ,t  Bk , j  dt  dx j  dx k
2!
 2!

1
1
    E  i j k dx i  dx j  dx k   i j k  Ei ,t   i l m Bl , m  dt  dx j  dx k
3!
2!

J  dt  J i dxi
dt  dx  dy  dz  dt  *dt
→
*dt  dx  dy  dz
dt  dx  dy  dz  dx  *dx
→
*dx  dt  dy  dz
*dx i  
1
 i j k dt  dx j  dx k
2!
*J   dx  dy  dz 
1
J i i j k dt  dx j  dx k
2!
 Inhomogeneous eqs are given by
d * F  4 * J
Ex 5.14
12. Charge & Topology
Charge = Topology
1. Wheeler: Wormhole (handle) → Pair of  charges.
Objections:
a. Origin of wormhole unknown.
b. Linkage of distant pair of charge unacceptable.
2. Sorkin: Wormhole creating pair of nearby charges of same sign.
5.13. The Vector Potential
dF  0
←
F  dA
A  Vector potential
F is invariant under a gauge transformation: A  A  A  d f
F  F   dA  dA  F
A cannot be defined in region with magnetic monopole.
Ex 5.16
5.14. Plane Waves: A Simple Example
Let
F  F u
→
ut x
1 d F
1

dF  d  F  u  dx   dx  
dt  dx  dx   dx
2 du
2

d F
1  d Fi j
i
j

 
 
dt  dx  dx 
dx  dx  dx 
2  du
du




dF12
dF
dF
dt  dx  dy  23 dt  dy  dz  13 dt  dx  dz
du
du
du
dF02
dF
dF
dx  dt  dy  03 dx  dt  dz  23 dx  dy  dz
du
du
du
 dBz dE y 
dBx


dt

dx

dy

dt  dy  dz

du 
du
 du

dF  0
→
 dBy dEz 
dBx


dt

dx

dz

dx  dy  dz

du
du
du


By  Ez
Bx  0
Bz  Ey
Static fields
ignored
D. Dynamics of a Perfect Fluid
15.
16.
17.
18.
Role of Lie Derivatives
The Comoving Time-Derivative
Equation of Motion
Conservation of Vorticity
5.15. Role of Lie Derivatives
Perfect fluid:
• No viscosity.
• No heat conduction (adiabatic).
Quantities conserved in any fluid element ( local conservation laws ) :
• Mass.
• Entropy.
• Vorticity.
Conservation laws are more transparent within the framework of Lie derivatives.
5.16. The Comoving Time-Derivative
Equation of continuity:



L
    0
 t
V 



 div  V   0
t
where τ is the volume 3-form:

t
 LV 
  dx  dy  dz
= time-derivative operator in a frame travelling with the fluid element.
Proof :
Let ( x, y, z, t ) be the coordinates of a fluid particle in the
Galilean space-time.
The tangent U to the “world-line” of the fluid particle is
 d x d y d z dt 
U 
,
,
,

d
t
d
t
d
t
d
t


 V x , V y , V z , 1 
( parameter of
world-line = t )
The time-derivative operator in a frame travelling with the fluid element is U.
L U W  U , W    V i i  t , W 
  V , W    t , W 
  t  W i i

W
 L V W  L t W
 t , W    t W     W    t    t W   
   t W i  i
i = x, y, z.
   t W t  t    t W i  i
if W is purely spatial, i.e., W t = 0

L U W    t  LV  W
 t W
if W is purely spatial
This holds if W is replaced by any purely spatial (n0) tensor.
Reminder: The Galilean space-time is a fibre bundle with t as base.
Ex 5.19
5.17. Equation of Motion
Adiabatic flow: specific entropy S conserved →


  t  LV  S  0


Euler’s equation of motion ( see Landau & Lifshitz, “Fluid Mechanics”, §2 ) :
v
1
  v    v   p  
t

( p = pressure,
Φ = gravitational potential )
In Cartesian coordinates:
 i
1
V  V j  jV i    i p   i
t

Equation valid only in Cartesian coordinates because :
• Index mismatch (allowable only in orthonormal bases).
• j V i is a tensor only for transformations with coordinate independent Λi’j .
 i
1
V  V j  jV i    i p   i
t

is not a tensor equation in general coordinates.
Usual remedy is to introduce a covariant derivative (see Chap 6).
An alternative approach via Lie derivative is as follows.
Index mismatch can be resolved using V i = Vi for Cartesian coordinates :

1
Vi  V j  jVi    i p   i
t

Non-tensorial transformation behavior is resolved using
L V   V  V  V
j
V
i
j
i
j
iV
j
 V j  j Vi 
( d involves only spatial derivatives )
→

1
1
 i V j V j   V j  j Vi  dV 2
2
2


1
1 2


L
V


dp

d


V 


V 

2 

 t


i
V 2  V jV j  V V 
v
1
  v    v   p  
t

v  (  v )   v    v 
1 2
v
2
v
1
1
 v     v    p    v 2
t

2


1
1 2


L
V


dp

d


V 


V 

2 

 t

5.18. Conservation of Vorticity
Vorticity   V
~ dV


1
1 2


L
V


dp

d


V 


V 

2 

 t

( d involves only spatial derivatives )


Since  , d   0  L V , d 


t 
d both sides →


1
  L V  dV  2 d   dp

 t

Case I : p = p(ρ )
→
d   dp  d  
dp
d  0
d



L
dV  0
 t
V 


( Helmholtz circulation theorem )
Case II : p = p(ρ, S )
p
p
dp  
d



  S  dS



S


Since



L
S 0
 t
V 


d S  both sides of

dS  d   dp  0



L
dS  0

V 
 t

→


1

L
dV

d   dp

2
V 

 t



dS    LV  dV  0
 t



   LV  dS  dV
 t


gives

( Ertel’s Theorem )
Since any two 3-forms are proportional in our 3-D space, we can write
α = some scalar function, τ = volume 3-form
dS  dV   
  
    *
→
  *  dS  dV 


  LV       0
 t






L


0


L



V 
V 
 t

 t

Ex 5.21 :


Ex.5.22
1

1




L
 0

V 
 t

→
S     V 
S , i  i j k  j Vk

1

 i j k S , i Vk , j
E. Cosmology
19.
20.
21.
22.
23.
The Cosmological Principle
Lie Algebra of Maximal Symmetry
The Metric of a Spherically Symmetric 3-Space
Construction of the Killing Vectors
Open, Closed, & Flat Universes
5.19. The Cosmological Principle
General relativity → Cosmology
Assuming universe to be homogeneous & isotropic in the large scale,
D.G. → only 3 cosmology models (different initial metrics) are possible:
Flat,
Open,
Closed.
This result can be derived without using general relativity or Riemannian geometry.
Mass distribution of the universe:
• Small scale [ 1015 m (nuclear) ~ 1017 m (interstellar) ] : lumpy.
• Star cluster = Galaxy : lumpy
• Cluster of galaxies ( 101 – 103 galaxies ) : lumpy
• Cluster of galaxy clusters = Supercluster : lumpy
• Beyond superclusters : homogeneous & isotropic
Since the universe is evolving, the “observed”
homogeneity is an interpolation to the “present time”.
Spacetime is thus treated as a foliation with leaves of
constant time hypersurfaces.
A hypersurface is space-like is g is positive-definite
on all vectors tangent to it.
Definition of homogeneity
Let G be the isometry Lie group of manifold S with metric tensor field g.
The Lie algebra G of G is that of the Killing vector fields of g.
Elements of G are diffeomorphisms of S onto itself.
The action of G on S is transitive if  P, Q S,  gG s.t. g(P) = Q.
A manifold S is homogeneous if its isometry group acts transitively on it,
i.e., the geometry is the same everywhere on S.
Elements of G which leaves a point P on S fixed form a subgroup HP of G.
HP is called the isotropy group of P.
The isotropy group HP of P maps any curve through P to another curve through it.
 H P : TP → TP
(c.f. adjoint representation of a Lie group)
A manifold S is isotropic about P if its HP = SO(m).
If S is isotropic about all P, it is isotropic.
A cosmology model M is a homogeneous cosmology if it has a foliation of
homogeneous space-like hypersurfaces.
Similarly for isotropic cosmology.
The universe is observed to be homogeneous on the large scale about us.
Cosomological principle: likewise for all observers in the universe.
Ex 5.23
5.20. Lie Algebra of Maximal Symmetry
Let S be a 3-D manifold & ξ a Killing vector field on it, i.e.,
L g

ij
  k g i j, k   k , i g k j   k , j g i k  0
 k , i g k j   g k mm  , i gk j   g k m , i  m  g k m m , i  g k j
  gk j g k m   gk j , i g k m  m  g m j m , i
,i


  g m j m   g k j , i g k m m
,i
→
L g

ij
 gk j g k m , i m  g m j m , i
  g m j , i  g k j , i g k m  m  g m j  m , i
  j , i  gk j , i g k m m
 g k mm g i j , k   j , i  gk j , i g k m m  i , j  gk i , j g k m m
  j , i  i , j   m g k m  g i j , k  g k j , i  g k i , j 
  j , i  i , j  2 m Gimj  0
where
Gimj 
1 km
g  g k j , i  g k i , j  g i j , k  = Christoffel symbol
2
 j , i  i , j  2 m Gimj  0 is symmetric in i & j → ½ n(n+1) eqs for n variables ξj .
 ξ is over-determined for n > 1.
→ A general g may have no Killing vector fields.
Task: Find criteria for g to have the maximal set of Killing vectors.
k eq. gives
 j , ik  i , jk  2 m , k Gimj  2 m Gimj , k  0
(1)
i → j → k →i :
k , ji   j , ki  2 m , i G mjk  2 m G mjk , i  0
(2)
i , k j  k , i j  2 m , j Gkim  2 m Gkim , j  0
(3)
(1)+(3)(2) :
i , j k  m , k Gimj  m , i G mj k  m , j Gkmi   m  Gimj , k  G mj k , i  Gkmi , j 
 K i jk mlm , l  H i jk m m
where
K i jk ml  Gimj kl  G mj k i l  Gkmi  jl
H i jk m  Gimj , k  G mj k , i  Gkmi , j
 i j , k  K i jk mlm , l  H i jk m m
 For a given g, if ξi and ξi , j at point P are known,
then all higher derivatives of ξ at P are known.
→ ξ is known in any neighborhood of P where ξ is analytic.
Given ξi , the symmetric part of ξi , j is given by
1
 j , i  i , j   m Gimj

2
Hence, a Killing vector field on S is determined given some appropriate values
i  i  P 
Ai j   i , j   P 
at a single point P  S.
Number of independent choices of ηi is n.
That of Ai j is ½ n(n1).
→ Maximal number of Killing vector fields is ½ n(n+1).
In which case, M is maximally symmetric.
A maximally symmetric connected manifold is homogeneous.
Proof :
S is maximally symmetric →
 a Killing field whose tangent at P = any desired value.
The 1-par Lie group associated with the Killing field
maps P to any point Q in some coord patch U of P.
By extending the map across different coord patches, P can be
mapped to any Q in S.
Thus, the isometry group G maps P to any Q in S.
→ G acts transitively on S (S is homogeneous ).
Let G be the isotropy group of P.
→ P is fixed under any action of G.
→ The Killing fields associated with G vanish at P.
i
V , W   V i , j W j  W i , j V j
V , W  i  gi k V k , j W j  W k , j V j 
  gik V k   gik , j V k
,j

 W j   g W k   g W k
ik , j

 ik
,j
V j

 Vi , j W j  Wi , j V j  gi k , j V kW j  W kV j 
If V and W are any 2 Killing fields of G, then [V , W] = 0 at P.
Hence, the Lie algebra of G is a subalgebra of that of the isometry group.
Ex.5.24: The isotropy group of a space-like S is SO(m).
I.e., a maximally symmetric space-like manifold is isotropic.
5.21. The Metric of a Spherically Symmetric 3-Space
Let S be a space-like 3-manifold.
If the isotropy group of S is SO(3), then S is spherically symmetric everywhere.
The Killing vectors of SO(3) define spheres S2 by their integral curves.
→ they foliate S.
Spherical coordinates: r labels different leaves; (θ, φ) = coord on each leaf.
Metric of S induces metric on each S2 → volume 2-form & its integral (total area).
Intrinsic definition of r :
area  4 r 2
→
r
area
4
Caution: r defined this way need not be monotonically increasing everywhere.
E.g., 2-manifold S2 (leaves are circles) :
r 1st increases, then decreases when moving
away from P towards P.
At every point Q on a leaf S2,  a unit normal vector n s.t.
g  n ,V   0
g n, n   1
 V  TQ  S 2 
( n is orthogonal to S2 )
( n is normalized )
The unit normal vector field is C everywhere except at the poles where θ= 0 or π.
The poles (θ= 0 ) on different spheres can be related by demanding that they lie on
the integral curve of n through the pole of an arbitrarily chosen sphere.
Example: 2-manifold S2
n is orthogonal to the leaves S1.
Poles (φ = 0 ) on different leaves lie on
integral curve of n.
θ & φ are constant on any integral curve of the unit normal vector field.
→ Integral curves of n are coordinate lines of r.
Since θ & φ are tangent to S2 :
g r   g   r ,    0
g r  g r ,    0
area  4 r 2
i.e.,
→
g   r 2
 f r

g
r2
 0

g   0
g   r 2 sin2 



2
2
r sin  
0
where f(r) is to be determined by the rest of the isometries of S.
Ex 5.25
5.22. Construction of the Killing Vectors
Any vector field on S can be written as
V   l m  r  Yl m r  l m  r  Ylm   l m  r  Ylm
sum over repeated
indices implied
where (see §4.29)
ij
kn
Y lm  å dY l m  g  jk g Yl m , n
Ylm  Yl m  g i j Yl m , j
With
g i j  diag  1 , sin 2 


g d  d  sin  d  d
 i j  i j sin
we have
Ylm   g  j Yl m , j  Yl m , 
Ylm   g  j Yl m , j 


Ylm   g     g Yl m , 

Y 
 
lm
 g    g  Y l m , 
1
Yl m ,
sin 2 
1
1
Y

Yl m , 
lm ,
sin 2 
sin 
1
1


sin

Y


Yl m , 


lm ,
sin 2 
sin 
 sin 
V   l m  r  Yl m r  l m  r  Ylm   l m  r  Ylm

1


  l m  r  Y l m r   l m  r  Y l m ,    2 Y l m ,   
sin 


1
 1

  lm r 
Y l m ,   
Y l m ,   
sin 
 sin 

V    l m  r  Y l m,    l m  r 
V   l m  r  Yl m
r
V   lm r
1
Y l m, 
sin 
1
1
Y


r
Yl m, 
l m, 
lm  
2
sin 
sin 
1
1
1




V   l m  r  Y l m r   l m  r  Y l m,    l m  r 
Y l m ,      l m  r  2 Y l m ,    l m  r 
Y l m,   
sin 
sin 
sin 




If V is to be a Killing vector, it components must satisfy the Killing eq
i.e.,
K i j  V k g i j , k  V k , i g k j  V k , j g ik  0
where g is the 3-D metric tensor
g i j  diag  f  r  , r 2 , r 2 sin 2 

LV g  0
K i j  V k g i j , k  V k , i g k j  V k , j g ik  0
→
K ij  g i m V k g m j , k  g i m V k , m g k j  g i m V k , j g mk  0
g diagonal →
K ji   ji g i i V k g ii , k  g ii V j , i g j j  V i , j  0
K  g  V k g  , k  g  V  ,  g   V  , 
( no summation on i & j )
 g  V k g  , k  2V  , 

 Yl m ,   
1 r
 2 V 2r  2  l m  r  Y l m,    l m  r  
 
r
sin


 ,  

1
 Y l m ,  Y l m , 

 2   l m  r  Y l m   l m  r  Y l m ,    l m  r  
 2 cos 
 sin  sin 

r
K  g  V k g  , k  2V  , 
Yl m ,  
Yl m ,   

1
r
2

2
 2 2 V 2r sin   V 2r sin  cos    2  l m  r  2   l m  r 

r sin 
sin

sin



Y
Y


2
 V r  2cot  V   2  l m  r  l m 2,     l m  r  l m ,   
r
sin 
sin  

Y
Y


2
K  V r  2cot  V   2  l m  r  l m 2,     l m  r  l m ,   
r
sin 
sin  

Y
Y
Y
1




 2   l m  r  Y l m   l m  r   cot  Y l m ,   l m 2,      l m  r   cot  l m ,   l m ,    
sin  
sin  sin   


r
Y
1
Y

K  2   l m  r  Y l m   l m  r  Y l m,    l m  r   l m ,   l m2,  cos  
 sin  sin 

r
Yl m ,    
2

K  K  2   l m  r  Y l m   l m  r   Y l m ,   cot  Y l m ,   2  
sin   

r




Y l m ,   cot  Y l m , 
→
 Yl m
1  
 2 
 sin 
sin  sin   

Yl m ,  
2

1  Yl m
 2
2
 sin  
4
K  K   l m  r  Y l m  2 l m  r  L2 Y l m  0
r
2
 l m  r    l m  r  l  l  1  0
r
 L2 Yl m
Y
Y
Y
1




K  2   l m  r  Y l m   l m  r   cot  Y l m ,   l m 2,      l m  r   cot  l m ,   l m ,    
sin  
sin  sin   


r
1
 Y l m ,  Y l m , 

K  2   l m  r  Y l m   l m  r  Y l m,    l m  r  
 2 cos  
 sin  sin 

r


Yl m ,   
Yl m ,  

 Yl m ,  
1 

K

K


r
Y

cot

Y


2

r

cot

    l m    l m , 
lm ,
lm   

2
2
sin

sin  


 sin 
1 
K  K    l m  r  Fl m   l m  r  Gl m

2
where
Fl m  Y l m ,   cot  Y l m ,  
Yl m ,  
sin 2 
Y
Y

Gl m  2  l m ,    cot  l m ,  
sin  
 sin 
K i j  V k g i j , k  V k , i g k j  V k , j g ik  0
K  V  ,  g   V  ,  g

 Yl m ,  
 Yl m ,    2
 r sin   l m  r   2    l m  r  
r

 sin   , 
 sin   ,  

2
2

 Y l m ,   
 l m  r  Y l m ,     l m  r  

sin




Yl m ,  
Y l m ,  Y l m ,   

 2Y l m , 
 Yl m ,  
 r sin    l m  r  
 2cos  3    l m  r   
 cos  2  3  
2
sin  
sin  sin   
 sin 
 sin 

2
2
Fl m  Y l m ,   cot  Y l m ,  
sin 2 
K   r 2 sin    l m  r  Gl m   l m  r  Fl m
l m  r  Gl m   l m  r  Fl m  0
Non-trivial solutions requires
Yl m ,  
 Yl m ,  
Gl m  2 
 cot 

sin

sin



Yl m ,  

l m  r  Fl m   l m  r  Gl m  0
Fl m
det
Gl m

Gl m
 0   Fl m
 Fl m
  G  
2
2
lm
0   Fl m    Gl m 
2
2
Fl m  Y l m ,   cot  Y l m ,  
→
Yl m ,  
sin 
2
0
Yl m ,  
 Yl m ,  
Gl m  2 
 cot 
0

sin  
 sin 
0  Yl m ,    cot  Yl m , 
G:
→
F:
m0
 i m Yl m ,   cot  Yl m 
Yl m ,   cot  Yl m ,  
or
Y l m ,   cot  Y l m ,   m
 If m  0, then
If m = 0, then
Y00  1
&
m2  1
Yl m
2
sin 2 
i.e.,
Yl 0 ,   cot  Yl 0 , 
Y10 
1
Yl m
2
sin 
m  1
, which can only be satisfied by
3
cos 
4
Hence, the only solutions are l = 0, or 1.
1
1
1




V   l m  r  Y l m r   l m  r  Y l m,    l m  r 
Y l m ,      l m  r  2 Y l m ,    l m  r 
Y l m,   
sin 
sin 
sin 




2
 l m  r    l m  r  l  l  1  0
r
For l = 0 :
 00  r   0
→
V   00  r  r  0
For l  2 :
l m   l m  0
→
For l = 1:
2
 1m  r   2 1m  r   0
r
l m  0
→
&
V 0
1m  r   r1m  r 
1
1
1




V  r 1m  r  Y1m r  1m  r  Y1m,    1m  r 
Y1m,     1m  r  2 Y1m,    1m  r 
Y1m,   
sin 
sin 
sin 




η1m & ζ1m are determined by the rest of the Killing eqs, i.e.,
Krr = 0
K rθ = 0
Krφ = 0
1
1
1




V   l m  r  Y l m r   l m  r  Y l m,    l m  r 
Y l m ,      l m  r  2 Y l m ,    l m  r 
Y l m,   
sin 
sin 
sin 




K i j  V k g i j , k  V k , i g k j  V k , j g ik  0
K rr  V r g rr , r  2V r, r g rr  l m Yl m f, r  2l m , r Yl m f  0
l m f, r  2 l m , r f  0
→
The divergence (see §4.16) of a vector is given by
For a vector on S2,

d * a  d  ji a j dx i

  sin  d  d
  ji a j 
,k
dx k  dx i
   a     a   d  d
,
, 

d å a  d * a    a  
so that
  j a j    j a j 
,
,

 d  d

 sin  a   sin  a   d  d
,
, 

Treating K = (K rθ , K rφ ) as a 1-form on S2, we have
d * K  sin  Kr   sin  Kr   d  d
,
, 

1 
 K 
sin  K r    sin  K r  

,
, 
sin  
1
1
1




V   l m  r  Y l m r   l m  r  Y l m,    l m  r 
Y l m ,      l m  r  2 Y l m ,    l m  r 
Y l m,   
sin 
sin 
sin 




Kr  Kr  0
sin  K 

→
r
,
  sin  K r 
,
0
K ji   ji g i i V k g ii , k  g ii V j , i g j j  V i , j
→

Kr  g

V
r
,
g rr  V

,r

1
1

Y
f


Y


Yl m, 
l m , r l m, 
lm , r
2 l m l m, 
r
sin 
1
 f  sin  Yl m ,   ,    l m , r  sin  Y l m ,  ,    l m , rY l m , 
2 lm
,
r
1
1
1


r



Y
f


Y


Yl m, 
Kr  g V ,  g rr  V , r
lm
l m, 
lm , r
l m, 
lm , r
2
2
2
r sin 
sin 
sin 
sin  Kr 

sin  K 


r
sin  K 

r
,
,
1
1

Y
f


Y l m ,    l m , rY l m ,  
lm
l m , 
lm , r
r 2 sin 
sin 
  sin  Kr 
,
1
1


  2 l m f  l m , r   sin  Y l m,   
Yl m ,   
,
sin 
r


1

  2 l m f  l m , r  sin  L2Yl m  0
r

1


f


→
l m , r   sin   l  l  1 Yl m  0
 2 lm
r



dK  d K r d  K r  d 

→
→
→
Kr ,   Kr ,   0
1
1

Y
f


Y


Y l m, 
l m , r l m, 
lm , r
2 l m l m, 
r
sin 


K r  g Kr  r Kr
2
 l m Yl m ,  f  r 2 l m , rY l m ,   r 2 l m , r
K r ,   l m Yl m ,   f  r 2 l m , r Y l m ,    r 2 l m , r
K r 
l>0
 Kr ,  d  d  Kr ,  d  d
  K r  ,   K r  ,   d  d   0
K r 
1
 f  l m , r  0
2 lm
r
1
Y l m, 
sin 
1
Y l m , 
sin 
1
1
1

Y
f


Y


Y l m, 
lm
l m, 
lm , r
l m, 
lm , r
2
2
2
r sin 
sin 
sin 
K r  g Kr  r2 sin2  Kr   l m Yl m,  f  r2l m , rYl m,   r2 sin  l m , rYl m, 
Kr ,    l m Y l m,  f  r 2 l m , r Y l m ,   r 2 l m , r  sin  Y l m ,  
K r  ,   K r , 
,
2
1

  r2
sin

L
Yl m  0
 r  l m , r  sin  Y l m ,   
Ylm ,   
lm, r
,
sin 


2
r 2 l m , r sin  l l  1 Yl m  0
→
 lm, r  0
 l m  const
l>0
Summary: For a non-trivial solution, l = 1 and
2
 l m  r    l m  r  l  l  1  0
r
1
 f  l m , r  0
2 lm
r
→
→
1m  r 1m
1
 f  1m , r  0
2 1m
r
1m f  r 1m , r  0
l m f, r  2 l m , r f  0
→
r 1m f , r  2 1m  r 1m , r  f  0
 1m  r f , r  2 1  f  f 

df
dr
2
r
 f  1 f
→
f 
→
1
1  C r2
f 1
ln
 2ln r  ln C
f
→
f 1
 C r2
f
f 
1
1  C r2
1m f  r 1m , r  0

→
d1m
1m
1  Cr 2  1 
ln1m  ln 
  ln Vm
2
2  r

Cr 2  1
1m 
Vm
r
dr
f
r

dr
r Cr 2  1
Vm = const
1m  r 1m  Vm Cr 2  1
5.23. Open, Closed, & Flat Universes
Robertson-Walker model :
(homogeneous, isotropic 3-space)
See I.D.Lawrie, “A Unified Grand Tour
of Theoretical Physics”, 2nd ed., Chap 14.

1
2
2
2
2
2 
d 2  dt 2  a 2  t  
dr

r
d


sin

d



2
1

C
r




1
2
2
2
2
2
i.e., g   diag  1 ,  a 2  t 
,

a
t
r
,

a
t
r
sin






2
1

C
r


t = proper time. Obervers with fixed r, θ & φ (comoving) are in free fall.
r
If C  0, then
r
C
a t   a t 
C

1
2
2
2
2
2 
d  dt  a  t  
dr

r
d


sin

d



2
1

k
r


2
2
2
gives
where
k
C
 1
C
Case C = 0 is the same as k = 0 with
d 2  dt 2  a 2  t  dr 2  r 2  d 2  sin 2  d 2 
[ Spatial section is
Euclidean (flat) ]
Surface of fixed r coordinate is a sphere with physical radius
r
dr
  r, t   a  t  
1 k r
0
2
 a  t  sin 1 r

  a t  r
for
 a  t  sinh 1 r

k 1
k 0
k  1
Circumference of the equator ( θ = π/2 ) of the sphere is
c  r, t   a  t 
2

r d
 2 ra t 
0
r

2


 2 
1

sin r


2 
for


 2 a sinh  2 
a

Hubble’s law:
a t 
H t  
a t 

k 1
Closed
k 0
Flat
k  1
Open
velocity between galaxies
distance between galaxies
0r<1