Gravity in general relativity and Einstein-Cartan-Sciama-Kibble theory is summarized. In general relativity, gravity is described using a symmetric connection and Riemannian geometry. The Einstein field equations relate the Ricci tensor and scalar to the stress-energy tensor. In ECSK theory, the connection is non-symmetric due to the inclusion of torsion. This leads to modifications in the covariant derivative, curvature, and Einstein field equations compared to general relativity. ECSK theory aims to extend general relativity to combine macroscopic and microscopic scales.

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
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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.