Torsion: what it is and how does it work
Paweł Laskoś-Grabowski
Institute for Theoretical Physics, University of Wrocław
January 10, 2009
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 1 / 25
Introduction Conventions
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 2 / 25
Introduction Conventions
Basic symbols & conventions
D = 4
c = = 1
� = diag(-, +, +, +)
Greek indices correspond to the curved space. Latin indices
correspond to the tangent space.
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 3 / 25
Introduction Einstein Cartan theory
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 4 / 25
Introduction Einstein Cartan theory
Poincar� group
Initially devised to be the symmetry group of Maxwell equations
Consists of (local) translations, rotations and boosts
Algebra consists of 10 generators: momenta and angular momenta
Commutation relations:
[P�, P�] = 0
[J��, P�] = i(���P� - ���P�)
[J��, J��] = i(���J�� - ���J�� - ���J�� + ���J��)
We now choose it to be group of fundamental symmetries of our
theory
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 5 / 25
Introduction Einstein Cartan theory
Differential forms
Coordinate-independent approach
ą a" ąi1i2...ik dxi1 '" dxi2 '" . . . '" dxik
Wedge product: ą '" � is a form of greater rank
Exterior derivative dą:
"ąi1i2...ik
dą = dxi1 '" dxi2 '" . . . '" dxik '" dxj
"xj
j
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 6 / 25
Introduction Einstein Cartan theory
Einstein Cartan theory
Gauge fields for Poincar� group
a
Translations Pa field of frames ea a" e� dx�
ab
Lorentz rotations &!ab = -&!ba field of connection �ab a" �� dx�
Transformations under Lorentz group element �
� = �����T - (d�)��T
e = ��e
Transformations under translation �a
� = �
e = e + d� + ���
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 7 / 25
Introduction Einstein Cartan theory
Curvature and torsion
Two important two-forms
Rab = d�ab + �a '" �cb
c
a
T = dea + �a '" eb
b
Bianchi identity (curvature is covariant constant)
dR = dd� + d(� '" �) = d� '" � - � '" d�
= d� '" � + � '" � '" � - � '" d� - � '" � '" �
= (d� + � '" �) '" � - � '" (d� + � '" �) = R '" � - � '" R
Similar identity for torsion
dT = dde + d(� '" e) = d� '" e - � '" de
= d� '" e + � '" � '" e - � '" de - � '" � '" e
= (d� + � '" �) '" e - � '" (de + � '" e) = R '" e - � '" T
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 8 / 25
Introduction Einstein Cartan theory
Relation to  classical general relativity
a b
e is invertible g�� = �abe�e�
May be proven that
1 a a a
Ra - �bR + �b = �T
b 2 b
a a � d a � d a
Tbc = �Sbc + Sdb�c - Sdc�b
D-2 D-2
Classical GR assumes also T = 0 and e = const. Everything, incl. R
may then be expressed in terms of g.
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 9 / 25
Mathisson Papapetrou equation Free spinning particle
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 10 / 25
Mathisson Papapetrou equation Free spinning particle
Representation
Particles represented as elements of connected Poincar� group
ę!
P+ = R4 � Lę! = {(z, �)}
+
z  coordinates of particle location
� relates to momentum and spin:
pa = m�a0, m > 0
1
Sab�ab = ��12�-1 a" -iS
2
a b a b
where (�ab)cd = -i(�c �d - �d�c ),  = const
1
It follows that SabSab = 2, paSab = 0
2
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 11 / 25
Mathisson Papapetrou equation Free spinning particle
Free lagrangian
1
Classical spinning particle: L0 = m �2 + i Tr(�3s-1a)
2

Ł
Relativistic analogue: Lp = paża + i Tr(�12�-1�)
2
Variation w.r.t. z obviously Wa = 0
General variation of � is of such form: �� = i � � �
Then follows ��-1 = -i�-1 � �  because
0 = �(��-1) = �� �-1 + � ��-1 = i � � ��-1 + � ��-1
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 12 / 25
Mathisson Papapetrou equation Free spinning particle
Free lagrangian (cont.)
Variation w.r.t. �
If kab = żapb = (zapb) , then
�(paża) = mża��a0 = mi( � �)ab�bża
0
= -i( � �)bakab = -i Tr( � � k)
The other term yields
 
Ł Ł
i �(Tr(�12�-1�)) = i Tr(�12 � -i�-1( � �)�)
2 2

+ i Tr(�12�-1i( � �) �)
2

Ł
+ i Tr(�12�-1i( � �)�)
2
i
= Tr(i��12�-1( � �) )
2
S
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 13 / 25
Mathisson Papapetrou equation Free spinning particle
Free lagrangian (cont.)
Angular momentum conservation
i
Full variation now reads �Lp = -i Tr(k � �) + Tr(S( � �) )
2
1
Integrating by parts gives �Lp = -i Tr((zapb + Sab) ( � �))
2
Conserved charge  total angular momentum:
Mab = zapb - zbpa + Sab
If we expand @ = 0, multiply by pa and recall that pa`ab = 0, we ll
obtain p ż
"
Furthermore, then pa = mża/ -ż2 and `ab = 0
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 14 / 25
Mathisson Papapetrou equation Particle in gravitational field
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 15 / 25
Mathisson Papapetrou equation Particle in gravitational field
Lagrangian with fields
Analogous design of lagrangian

a
L = pae�ż� + i Tr(�12�-1D� �) + field part
2
ab
Ł
Covariant derivative  (D� �)a = �a + ż��� �cb
b b
Varying w.r.t. � is formally identical
i
�L = -i Tr(J � �) + Tr(SD� ( � �))
2
a
where Jab = e�ż�pb
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 16 / 25
Mathisson Papapetrou equation Particle in gravitational field
Lagrangian with fields (cont.)
Spin precession equation
Carrying on. . .
i
�L = -i Tr(J � �) + Tr(S( � �) + S[ż���, � �])
2
i
a" -i Tr(J � �) + Tr(-` � � + Sż��� � � - ż���S � �)
2
1
= -i Tr((J + D� S) � �)
2
EOM: Jab - Jba + (D� S)ab = 0
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 17 / 25
Mathisson Papapetrou equation Particle in gravitational field
Lagrangian with fields (cont.)
Varying w.r.t. z
This will hurt only for a while. . .

a a
�L = pae��ż� + paż�z�"�e + i Tr(�12�-1(�ż��� + ż�z�"��)�)
2
1
a a
= pae��ż� + paż�z�"�e + Tr(S(�ż��� + ż�z�"��))
2
1
a a
a" -(pae�) �z� + paż�z�"�e + Tr(-`�� - S�� + Sż"��)�z�
Ł
2
� � �
If Jab = Jab - Jba then 0 = J + D� S = J + ` + [ż�, S], so we expand
�
` = -J - [ż�, S] into the following
�L 1
a a
�
= -(pae�) + paż"�e + Tr(J�� + [ż�, S]�� - S�� + Sż"��)
Ł
�z� 2
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 18 / 25
Mathisson Papapetrou equation Particle in gravitational field
Lagrangian with fields (cont.)
Varying w.r.t. z (cont.)
We will now apply three simplifications at once:
"�� dz
�� = = "��ż
Ł
d�
"z
Tr([ż�, S]��) = Tr(żS��� - Sż���) = Tr(żS[��, �])
ba ab  ab
�
Tr(J��) = (Jab - Jba)�� = -2Jab�� = -2ea żpb��
�L
a a  ab
= -(pae�) + paż"�e - ea żpb��
�z�
1
+ Tr(żS[��, �] - żS"�� + żS"��)
2
. . . and "�� - "�� + [��, �] = R�!
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 19 / 25
Mathisson Papapetrou equation Particle in gravitational field
And we re done!
Now we expand the first term
a a a a a
(pae�) = Wae� + pa
� = (D� pa - ż�ab pb)e� + paż"e�

After insertion we see
�L 1
a
= -e�D� pa + ż Tr(SR�)
�z� 2
a a a  ab
+ ż�ab pbe� - paż"e� + paż"�e - ea żpb��

a a a b a b
paż ("�e - "e� + ��be - �be�)
a
T�
Mathisson Papapetrou equation (with torsion)
1
a a
(D� pa)e� = pażT� + ż Tr SR��
2
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 20 / 25
Finish Motivations and conclusions
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 21 / 25
Finish Motivations and conclusions
What s all this for?!
E C theory allows us to consider spinning particles
No inherent reason for T = 0
What is torsion itself, anyway?
Simpliest physical realisation of translation  movement of a test
particle by a fixed amount of affine parameter
MP equation leads to EOMs for particles in contorted spacetime
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 22 / 25
Finish Motivations and conclusions
What s all this for?!
Curvature
Vector is rotated, when transported along a closed path
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 22 / 25
Finish Motivations and conclusions
What s all this for?!
Torsion
Closed paths  don t close , i.e. translation addition doesn t commute
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 22 / 25
Finish Motivations and conclusions
What s all this for?!
E C theory allows us to consider spinning particles
No inherent reason for T = 0
What is torsion itself, anyway?
Simpliest physical realisation of translation  movement of a test
particle by a fixed amount of affine parameter
MP equation leads to EOMs for particles in contorted spacetime
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 22 / 25
Finish Bibliography
Outline
1
Introduction
Conventions
Einstein Cartan theory
2
Mathisson Papapetrou equation
Free spinning particle
Particle in gravitational field
3
Finish
Motivations and conclusions
Bibliography
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 23 / 25
Finish Bibliography
Selected references & further reading
Meissner KA, Klasyczna teoria pola, PWN, Warsaw 2002
Balachandran AP, Marmo G, Skagerstam B-S, Stern A, Gauge
Symmetries and Fibre Bundles. Applications to Particle Dynamics,
Springer 1983
Nakahara M, Geometry, Topology and Physics, IOP 2003
Mathisson M, Neue Mechanik materieller Systeme, Acta Phys Polon 6
(1937) 163 200
Freidel L, Kowalski-Glikman J, Starodubtsev A, Particles as Wilson
lines of gravitational field, 2008,gr-qc/0607014v2
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 24 / 25
Myron Mathisson
Warsaw 1897  Cambridge 1940
Paweł Laskoś-Grabowski (IFT UWr) Torsion January 10, 2009 25 / 25