Torsion in Gravity Theory
Paweł Laskoś-Grabowski
Institute for Theoretical Physics, University of Wrocław
November 14, 2009
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 1 / 21
1
General relativity
Introduction
Mathematical framework
Prospects
2
Einstein Cartan theory
Introduction
Main part
Relation to the general relativity
3
Mathisson Papapetrou equation
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 2 / 21
General relativity Introduction
History
Special relativity (1905): measurements differ between inertial frames
Aim: incorporate gravity in SR
General relativity (1915): influence of gravitational field on time and
space
Schwarzschild solution (1916)
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 3 / 21
General relativity Introduction
Tests
-1/2
dĆ
L L2
Bending of light: = E2 - (r - 2M)
dr r2 r3
Perihelion precession of Mercury: contributes 43 of 5600 per century
-1/2
" 2GM GM
Gravitational redshift: z = = 1 - - 1 H"
 r r
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 4 / 21
General relativity Mathematical framework
Generalizing the SR formalism
Minkowski spacetime replaced by a manifold
Curvilinear coordinates �! non-constant transition matrix
"x�
�� a" �� (x�) =
� �
"x�
Metric encodes (local) gravitational field
��� g�� a" g��(x�)
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 5 / 21
General relativity Mathematical framework
Mathematical entities
Christoffel symbol a.k.a. the affine connection
1
� = g� ("�g�� + "�g�� - "�g��)
��
2
Covariant derivative
"�f = "�f "�A� = "�A� + �� A�
��
"�A� = "�A� - �� A� etc.
��
They do not commute
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General relativity Mathematical framework
Curvature tensor
Mathematical entities, cont.
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 7 / 21
General relativity Mathematical framework
Curvature tensor
Mathematical entities, cont.
Commutator of covariant derivatives: R����A� = ["�, "�]A�
�

. . . or transports. W � - W� = V R� �� �
��
Coordinate expression
R� = "��� - "��� + �� �� - �� ��
��� �� �� �� �� �� ��
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 7 / 21
General relativity Mathematical framework
Einstein Field Equations
Ricci tensor: R�� = R����
Ricci scalar: R = R��
1
Einstein tensor: G�� = R�� - Rg��
2
G�� = 8ĄT��
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 8 / 21
General relativity Prospects
Future
Further testing
Gravitational waves
Development
Predicted by GR
The need for quantum gravity
Not observed directly
Indirect proofs exist
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 9 / 21
Einstein Cartan theory Introduction
Do we need more?
GRT is older than spin, doesn t take it into account
More general theory arises from field-theoretical approach
History of the ECT
1922 designed by French mathematician �lie Cartan
1950s  rediscovered by Sciama & Kibble
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 10 / 21
Einstein Cartan theory Introduction
Mathematical foundations
Poincar� group: rotations, boosts & translations
Tangent space at a point of a manifold
Differential forms
0-forms  smooth functions on open subset of a manifold
1 2 k
k-form � a" �� �2����k dx� '" dx� '" � � � '" dx�
1
Wedge product � '" � = (-1)rank(�)rank(�) � '" �
"��1�2����k 1 2
k
Exterior derivative d� = dx� '" dx� '" � � � '" dx� '" dx�
"x�
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 11 / 21
Einstein Cartan theory Main part
Tetrad, connection, curvature, torsion
Global Poincar� symmetry changed to local one-form gauge fields
a
Translations Pa tetrad field ea a" e� dx�
ab
Lorentz rotations &!ab = -&!ba connection field �ab a" �� dx�
Important two-forms
Curvature Rab = dRab + �a '" �cb
c
a
Torsion T = dea + �a '" eb
b
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 12 / 21
Einstein Cartan theory Main part
Tetrad, connection, curvature, torsion
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 12 / 21
Einstein Cartan theory Main part
Transformation laws and identities
Under local Lorentz transformation �ab
� = �����T - (d�)��T
e = ��e
R = ��R��T
Q = ��Q
Bianchi identity dRab + �a '" Rcb - Ra '" �cb = 0
c c
dQa + �a '" Qb = Ra '" eb
b b
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Einstein Cartan theory Relation to the general relativity
GRT as a special case of ECT
a b
g�� = �abe�e�
ECT reduces to GRT when
a
1
The tetrad is invertible (as a matrix e�)
2
The tetrad is covariantly constant (w.r.t. spin connection)
3
Torsion vanishes
The first assumption is quite useful in any case!
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 14 / 21
Einstein Cartan theory Relation to the general relativity
Is ECT more  true than GRT?
Yes as a generalisation and for the aforementioned reasons
But the effects are extremely weak
No because it still isn t a quantum theory. . .
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 15 / 21
Mathisson Papapetrou equation
Mathisson Papapetrou equation
Equation of motion in ECT
Particles represented as elements of the connected Poincar� group
ę!
P+ = R4 � Lę! = {(z, �)}
+
z is the particle location, � encodes momentum and spin:
pa = m�a0, m > 0
1
a b a b
Sab�ab = ��12�-1 a" -iS where (�ab)cd = -i(�c �d - �d �c )
2
Lagrangians:

Ł
Lfree = paża + i Tr(�12�-1�)
2
a 
L = pae�ż� + i Tr(�12�-1D� �) + field part
2
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Mathisson Papapetrou equation
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 17 / 21
Mathisson Papapetrou equation
Mathisson Papapetrou equation, cont.
Free solution: Wa = 0, @ab = 0 where Mab = zapb - zbpa + Sab
"
paSab = 0 pa = mża/ -ż2, `ab = 0
Nonfree equations of motion:
a
Spin precession eq.: Jab - Jba + (D� S)ab = 0 where Jab = e�ż�pb
a a 1
MPE: (D� pa)e� = pażT� + ż Tr SR��
2
Solving this equation is nontrivial, even in super-simplified cases
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 18 / 21
Mathisson Papapetrou equation
Graphical results
0 i i
For a nonspinning particle, � = 0 and e0 = 1 + ąz1, ei0 = e0 = 0, eji = �j
10 10
8 8
6 6
z0 z0
4 4
2 2
0 0
0 2 4 6 8 10 0 2 4 6 8 10
z1 z1
ą = 0.0005 ą = 0.001
10
10
8
8
6
6
z0 z0
4
4
2 2
0 0
0 2 4 6 8 10 0 2 4 6 8 10
z1 z1
ą = 0.005 ą = 0.01
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 19 / 21
Bibliography
Sources & further reading
Robert M Wald, General Relativity, 1984
Krzysztof A Meissner, Klasyczna teoria pola, 2002
A P Balachandran, G Marmo, B-S Skagerstam, A Stern, Gauge
Symmetries and Fibre Bundles, 1983
A Trautman, Einstein Cartan theory
Paweł Laskoś-Grabowski, The Einstein Cartan theory: the meaning and
consequences of torsion, 2009
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 20 / 21
Myron Mathisson
Warsaw 1897  Cambridge 1940
Paweł Laskoś-Grabowski (IFT UWr) Torsion in Gravity Theory November 14, 2009 21 / 21