A Lagrangian for the dark
side of spacetime
Angelo Tartaglia
Monica Capone
Dipartimento di Fisica, Politecnico di
Torino
July 20 2005
Paris - Einstein centennial
1
A simple classical problem
Motion of a point massive particle in a
viscous medium
b
S   Ldt
a
1 t  x 2
L e
mx
2
July 20 2005
Paris - Einstein centennial
2
Euler Lagrange equation
x  x 

2
x  0
2
Non-invariance per inversion of x
July 20 2005
Paris - Einstein centennial
3
Covariant formulation of
the same problem
s2
    x
S  m  e
s1
ds
2
1 dx     x
L  m 1  2
e
c dt
July 20 2005
Paris - Einstein centennial
4
Time like vector
   ,  ,  ,  
    3  0
r '   ' , x' ,0,0
2
July 20 2005
2
2
Paris - Einstein centennial
5
Euler-Lagrange equation

x' c 1  v'
2
     c 1  v' v'  0
2 2
2
2
dx'
v'
d
July 20 2005
Paris - Einstein centennial
6
Reference frame of the fluid
   ,0,0,0
r   , x,0,0
 0
 x 2 
x  c1  2  x  0
 c 
July 20 2005
Paris - Einstein centennial
7
Dependence on v
d
 '
dv
   v 
 ' 1  v +
2
x
v
c
v
1 v
2
= Ke
 
χ(v) given
July 20 2005
Paris - Einstein centennial
8
Space time
(4)-spherical symmetry
2

dr
2
2
2
2
2
2
2 
ds  d  a 
 r d  sin d 
2
1  kr


July 20 2005
Paris - Einstein centennial

k  0,1
9
Action integrals
Free
Symm.
2
S   Rd 
1
2


S  6 Vk  a aa  a 2  k d
Simplest “dissip.”
July 20 2005
d   g d 4 x
1
2
S  6 Vk 
1
Vk  4 
r2
r1


r2
1  kr 2

dr
2



e a aa  a  k d
Paris - Einstein centennial
10
Euler-Lagrange
  constant
a  2aa  aa  k  0
2
a  constant
k 0
a  a0 e
July 20 2005
K 2 
e
2
Paris - Einstein centennial
empty and flat
inflating
11
Non-constant χ
Naive example
     given
a  2aa      k  0
2
Non trivial solution
a  a0e
July 20 2005
2 
Paris - Einstein centennial
12
χ depends on a
   a  given
'
d
da


a  2aa  k  a ' aa  a  k  0
2
July 20 2005
2
Paris - Einstein centennial
2
13
Dependence on
a
   (a )

R   ' R  R   a 2  2aa  k  0
a
   ' a  

'
a
R  a aa  a 2  k 
July 20 2005
Paris - Einstein centennial
14
More appropriate action
integral
S  e
July 20 2005
g     
Rd 
Paris - Einstein centennial
15
No pre-defined symmetry
Euler-Lagrange-Einstein equations

G  R     g  
 
2
      
,
2
   
2
July 20 2005
2
,
2
     
,
,  ;
2 ,
2 ,

;
0

Paris - Einstein centennial
16
4-sphere symmetry
Cosmic time
  (  ,0,0,0)
S  6 Vk  e a(aa  a  k )d

July 20 2005
2
Paris - Einstein centennial
2
17
Non trivial if
   a 
   a 
   R 
July 20 2005
Paris - Einstein centennial
18
Let us consider


   (a)
2a ' a  aa  k  a  k  0
July 20 2005
2
2
Paris - Einstein centennial
19
Divergence free vector


 g

,
0

 
 a3
0

K
 3
a
July 20 2005
Paris - Einstein centennial
20
Empty space time field
equations
2


K
2
2
 6 6 a  aa  k  a  k  0
a
Non-trivial implicit solution (k=0)
July 20 2005
a  Hae
Paris - Einstein centennial

a6
36K 2
21
Maximum expansion rate for
aM  6K
6
2
2
6
K
a M  6
H
e
July 20 2005
Paris - Einstein centennial
22
What about γ’s dynamics?
Nothing more than wild guess
An ‘EOS’ needed
July 20 2005
Paris - Einstein centennial
23
Trying an approach similar to
the one of field theory
S   e Rd 

 
  g     A
Assume
July 20 2005


;

;

B

;


;


;

0
Paris - Einstein centennial
24
;

Introducing symmetry
2


a 2
2
2
    B   3 2  
a


K
 3
a
Obtain a complicated equation for a
July 20 2005
Paris - Einstein centennial
25
What could the vector field γ
represent?
Again an analogy:
• point-like defect in space time
•consequent radial displacements field
(one-dimensional strain)
July 20 2005
Paris - Einstein centennial
26
Four-dimensional point defect
July 20 2005
Paris - Einstein centennial
27
Conclusion
• Use the guiding idea of motion through a viscous
medium
• Express the problem covariantly in the Lagrangian
formalism
• Assume basic expansion or contraction as an
intrinsic property of space time
• Adopt, by analogy, a ‘dissipative’ Lagrangian for
empty space time depending on a vector field
• When the vector field depends in turn on the
scale parameter end with an accelerateddecelerated expansion
July 20 2005
Paris - Einstein centennial
28