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 = dab + 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 = abee
May be proven that
1 a a a
Ra - bR + b = T
b 2 b
a a d a d a
Tbc = Sbc + Sdbc - Sdcb
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 = ma0, m > 0
1
Sabab = 12-1 a" -iS
2
a b a b
where (ab)cd = -i(c d - dc ), = 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 W a = 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) = mżaa0 = mi( )abbża
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(i12-1( ) )
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 = paeż + 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 = eżpb
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 - żS )
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 = paeż + pażz"e + i Tr(12-1(ż + żz"))
2
1
a a
= paeż + pażz"e + Tr(S(ż + żz"))
2
1
a a
a" -(pae) z + pażz"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) = W ae + pa = (D pa - żab pb)e + paż"e
After insertion we see
L 1
a
= -eD pa + ż Tr(SR)
z 2
a a a ab
+ żab 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
Wyszukiwarka
Podobne podstrony:
SHSpec 06 6402C25 What Auditing Is and What It Isn t04 How The Heart Approaches What It YearnsAnimals 1 What animal is it worksheetBeginning Dialogues with Multiple Choice Questions What time is itEverything Is Not What It SeemsSHSpec 38 6409C03 Clearing What It IsKelis What It Is (Part 2)Kelis What It IsMadonna What It?els Like For A GirlWhat s It Worth A General Manager s Guide to Valuation (Harvard Business Review HBR)Become a Computer Game Developer Do You Have What it Takeseverlast what it s likeBon Jovi If That s What It TakesEverlast What It s Likei want to know what love isWhat Child is This chór SAB a capellawięcej podobnych podstron