A seminar about Gravity in both General Relativity and ECSK theories.

1. Torsion and Gravity
(Gravity in ECSK theory: the extension of
General Relativity)
by
Halo A. Abdulkhalaq
Sep, 2015

2. Overview
• Introduction
• Gravity in GR
- Line element
- GR Metric
- Curvature
- Einstein’s Field equation
• Gravity in ECSK theory
- Geometric connection

3. - Curvature in ECSK
- Einstein Field equations in ECSK
• Summary

4. Introduction
• To describe spacetime geometry and
gravitation, Einstein proposed General
relativity (GR) in 1915.
• He wrote his theory in the language of
advanced mathematics (differential geometry)
• Despite the difficulty of this theory, it changed
scientists’ understanding to the universe.
• The main principle of GR is equivalence
principle.

5. • There are Weak (WEP), Einstein (EEP) and
Strong (SEP) equivalence principles
• Gravitational waves, expansion of the
Universe, black holes and Structure formation
are product of GR
• The perihelion precession of Mercury, The
bending of light and Gravitational redshift are
successful tests of GR
• Einstein assumed the geometry of spacetime
to be symmetric
• Hence, GR is working with Riemannian
geometry ( )4V

6. • And the antisymmetric part of connection is
vanishes
• While, ECSK theory inserts the antisymmetric
part, i.e. ECSK theory assumes geometry to
be asymmetric
• Hence, ECSK (Einstein-Cartan-Sciama-Kibble)
theory works with Riemann-Cartan geometry
( )
• In ECSK the antisymmetric part of connection
is Contorsion which is written in terms of
Torsion.
4U

7. • GR works in macroscopical scale
• ECSK theory is composed to extend GR to
combine both macroscopic and microscopic
scales.

8. Gravity in GR
• In Newtonian physics gravity is attraction force
between two objects
• For a particle physicists gravity comes from
exchange of virtual particle ‘’graviton’’
• A string theorist assumes graviton not to be a
particle but a closed string with finite size
• For Einstein gravity is geometry
• All theories has their evidence for supporting
their idea

9. Line element:
• In three dimensional (3-dim) space the line
element is the distance between two points:
• In GR, 4-dim space spacetime
and point event, hence the line element
is:
(c=1)
2222
dzdydxds 
22222
dzdydxdtds 

10. • Then generally,
= diagonal(-1,1,1,1), is Minkowski metric
• In GR, , this new metric is
introduced to describe curved spacetime,
while the Minkowski one used for flat
spacetime only.

 dxdxds 2

 g

11. Curvature
• We need to introduce parallel transport,
connection and covariant derivative
• Parallel transport:
1. In flat space time:

12. 2. In curved spacetime:

13. Connection:
• Vectors parallely transported in curved
spacetime need a connection to meet
• In GR the spacetime connection is called
Christoffel Symbol or Levi-Civita connection:
• since
  )(
2
1




gggg 
  




 )( 0][ 


14. Covariant derivative:
• In curved spacetime we replace partial
derivatives to covariant derivative:
  




 VVV 

15. Curvature
• The commutation of a covariant derivative of
a vector is curvature
• is curvature, it is given by:
  


 VRV  ,

R
         












 R

16. Einstein’s Field equation
• Einstein’s equation :
• is matter energy-momentum tensor
• is Einstein’s tensor which
describes geometry
• R is Ricci scalar and
 GTG 8
T
 RgRG
2
1

R

17. Gravity in ECSK theory
• Geometric connection:
here,
i.e. This is contorsion, its given by:
, where is Torsion tensor
0][ 

  








 K ][)(



 K ][
)(
2
1 






 SSSK  
S

18. Curvature in ECSK
• The result of commutation of covariant
derivative is:
• Notice here an extra component appears to
the connection, which is Torsion
  






 VSVRV  ,

19. Field equations in ECSK
• The field equation becomes:
Where,
And
   GG 8


  )2( S
 







  )))()( (( SSSST 

20. Summary
• GR published in 1915
• In the language of differential geometry
• Based on equivalence principle
• Different theories of gravity
• Importance of connection
• Covariant derivative and curvature in both GR
and ECSK
• Field equation in GR and ECSK

21. References
• F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M Nester. General relativity
with spin and torsion: Foundation and prospects. Rev. Mod. Phys., 48:393,
1976.
• R.M. Wald. General Relativity. University of Chicago, 1983.
• T. Clifton, P.G. Ferreira, A. Padilla, and C. Skordis. Modied gravity and
cosmology. Phys.Rept., 513:1, 2012.
• S.M.Carroll. Spacetime and Geometry: An Introduction to General Rel-
ativity. Chicago, Addison Wesley, 2004.
• C.W. Misner, K.S. Throne, and J.A. Wheeler. Gravitation. Freeman,1973.
• H. Stephani. Relativity: An Introduction to Special and General Relativity.
3id edition, 2004.
• J.B.Hartle. An Introduction to Einstein's General Relativity. Pearson
Education, Inc., 2003.
• B.F.Schutz. A First Course in General Relativity. Cambridge University Press,
2nd edition, 2009.

22. • S. Capozziello, G. Lambiase, and C. Stornaiolo. Gemetric classication of
torsion tensor of space-time. Ann.Phys., 10:713, 2001.