G B Tupper
UCT



Zwicky (1933)
1970’s: Galactic Rotation Curves
+
Standard Model of Particle Physics
Beyond SMPP : Dark Matter
or
Modified Newtonian Dynamics (MOND) ?
o
Milgrom (1983)[1]
Modify inertia, minimum acceleration
 a

F  a 
a
0


x
 x  
1 x
a0





a0  H 0   DE
o


Bekenstein & Milgrom (1984)[2]
Keep a  
Modify Poisson equation
  
 
 
 
   
   4GN 

a0  
 
 
 

S MOND
 x    x   x  x
2






  d 3 x   
 2
8GN  a0



2
2





2
2

 
x
2
2
 x  
  x  1  2 ln 1  x   x 
1 x
x
2
 x, x  1
3






Bekenstein (2004)[3] : TeVeS
Bimetric theory g   e2 g~  2 sinh 2 A A
Matter frame Tensor g  
~
g
Einstein-Hilbert 

~
Constrained Vector A A g  1
Scalar 
2
Zlosnik et al (2006): constraint  e

 A A




Zlosnik et al (2006)[4]: Nonlinear Aether MOND
One metric g 
Timelike vector A
Action c1  1  c3 , c2  0 S  S EH  S A  Smatter
 

 

S A   d 4 x  g 4ao2   ln 1     A A  1
 
F F 
2a
2
0
Newtonian  E   , B  0
  1  A   2   3
32

Milgrom Bimetric (2009)[5]

Bekenstein & Milgrom (1984)[2]
2
2





  


S MOND   d 3 x   
  2 
8GN  a0 





“like nonlinear electrodynamics and QCD.”

Pagels & Tomboulis (1978)[6]: quantum field
theory, renormalization breaks conformal
invariance ; effective action





1  F   F  g3 dg 3
F  F 
t  ln

QCDeff  
4
2
4  QCD  g3  g 3 
4 g 3 t 


Asymptotic freedom  g 3   bg 33 small g
Infrared slavery, confinement
Linear potential
QED
QCD flux tube
  g3    g3

3
3
bg
Interpolate   g 3   
2
1  bg 3


1  F   F
ln
2  4QCD

 

  1  ln g 2
 bg 2

Large field  weak coupling
small field  strong coupling
IR
QCDeff
4QCD 
 F   F

4 
4QCD






3
2
QCD




Gluons self interact
Renormalizable
Asymptotically free
Local gauge invariance
GRAVITY






Broken scale invariance,
effective action
Gravitons self interact
Nonrenormalizable,
effeftive field theory [7]
Asymptotically safe [8]
General coordinate
invariance
??
“Dielectric Infrared
Modified Nonlinear
Dynamics”
Require
o Pure gravity
o Reduce to General Relativity for ‘strong fields’
o Natural MONDian limit for weak fields
2
not f  as Ricci    
 Note GR has General Coordinate Invariance
and
Local Lorentz Invariance

m
Fermions (matter) require tetrad field e
g   e e mn
m


n
GR has hidden LLI   
Christoffel connection torsionless



T      0

 
Weitzenbock connection 
 ea ea,


nonzero torsion T  0
zero curvature 
     0
Weitzenbock spacetime: absolute parallelism





K    

Contorsion tensor

Denote

Up to a “surface term”
S EH



 




K  K  g (K K  K K )
1
1
4
4

d
x

g


d
xeK  K


16GN
16GN
“Teleparallel equivalent of GR”

K K
Is GCI scalar but
 LLT K  K  4  divergence  0

Hypothesis: (infrared ) effects in quantum gravity
break LLI (like conformal invariance in QCD) but
preserve GCI (like gauge invariance in QCD)

Propose effective action:
K K 
1
4

S DIMOND 
d xeK  K 
2

16GN
 2a0 


GR limit automatic
Newtonian MOND limit automatic



Is DIMOND ‘beautiful’ ?
Can one derive breaking of LLI and so effective
action?
What about cosmology – especially ‘dark
energy’ ? [10]
[1]M Milgrom, Astrophys J 270(1983)365,371,384.
[2]J Bekenstein and M Milgrom, Astrophys J 286(1984)7.
[3]J D Bekenstein, Phys Rev D70(2004)083509.
[4]T G Zlosnik, P G Ferreira and G D Starkman, Phys Rev
D75(2007)044017.
[5]M Milgrom, Phys Rev D80(2009)123536.
[6]H Pagels and E Tomboulis, Nucl Phys 143(1978)485.
[7] J F Donoghue. Phys Rev D50(1994)3874.
[8] M Reuter, Phys Rev D57(1998)971.
[9]H I Arcos and J G Pereira, Int J Mod Phys D13(2004)2193.
[10]G R Bengochea and R Ferraro, Phys Rev D79(2009)124019.