An elastic Lagrangian for
space-time
Angelo Tartaglia and Ninfa Radicella
Dipartimento di Fisica, Politecnico
di Torino and INFN
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The universe: a dualistic
description
Space-time/Matter-energy
G   T
What is this?
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“Elastic” continua
N+n
N
f X1 , X 2 ,..., XNn   0
ξ
r
u  '
hX1 , X 2 ,..., XNn   0
u  
xμ
Xa
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The strain is described by the differential
change of u

u
u 
 
b
X
 X b
a
a
u a u a x 
 
b
b
X
x X
a  1,...., N  n
  1,...., N
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Metricity
a
b

X

X




dl 2  ab dX a dX b  ab 
d

d



d

d


  


X ' X '  


dl '   ab dX ' dX '   ab 
dx dx



  x x
a
2
a
b
b
 g  dx  dx     2  dx  dx
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Defects
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The “elastic” approach
Stress tensor
Hooke’s law


C




Elastic modulus tensor
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Isotropic medium
C           
Lamé coefficients



    2

Lorentz signature notation
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Correspondence with the usual
way of thinking


1 2



4
S   R    2   Lmatter   gd x
2


Potential term
“Kinetic” term
Geometry
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Fitting the data (307 SnIa)
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10
2

Reduced of the fits
(2 parameters)
CD
2 = 1.017
ΛCDM
2 = 1.019
B =λ+2μ/3=(32) 10-7 Mpc-2
=(32) 10-52 m-2
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Conclusion
The CD theory is a theory of space-time
preserving all general features of GR.
CD introduces the idea of a global symmetry
fixing defect.
Local effects coincide with GR effects
The nature of space-time shows up only at
the cosmic scale, where CD performs at
least as well as other theories, however
providing a compact and consistent picture
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Space time and the ether
…. according to the general theory of
relativity space is endowed with physical
qualities; in this sense, therefore, there
exists an ether. ……. But this ether may
not be thought of as endowed with the
quality characteristic of ponderable
media, as consisting of parts which may
be tracked through time. ……
Albert Einstein, Leiden, 1920
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A. Tartaglia, N. Radicella, Phys. Rev. D, 76, 083501 (2007)
A. Tartaglia, M. Capone, Int. Jour. Mod. Phys. D, 17, 275299 (2008)
A. Tartaglia, M. Capone, V. Cardone, N. Radicella, Int. Jour.
Mod. Phys. D, 18, n. 3, 1-12 (2009)
A. Tartaglia, Geometry, Integrability and Quantization
X., Varna, Bulgaria, 6-11 June 2008, Publisher SOFIA:
Avangard Prima, p. 248-264, 2009
A. Tartaglia, N. Radicella, arXiv:0903.4096
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