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Conlusions from the Algebrai Poinare Lemma
Norbert Dragon
Maximilian Kreuzer
Memorial Conferene
Wien
26.06.2011
Variational Derivatives and Noether Currents
Gravitational Energy Momentum Complex
Cruial (Kreuzer) Lemma
Variational Derivatives and Currents
ÆW = jm
m
=j
ÆAm
1
mn
mn
= - T EnergyMomentum = j
2
matter
eletromagneti
ÆW
Ægmn
matter
Noether, U(1)
Noether, Translation
Why?
Why does a Chern-Simons mass in three dimensions not ontribute to the
energy momentum tensor?
Fields
Jet-Spae
2 F are dierentiable maps
:
R4 ! Rn
7! (
x
x)
and dene a orresponding map , the prolongation to the jet-spae
J1 = R 4+n+4n
:
R 4 ! R 4+n+4n
7! ( ) = (
x
x
x; (x); (x))
Higher Jet-spae Jk = fBase, Target, derivatives up to kthorderg = kJ1
Loal Ation
The Lagrangian density L is a funtion of jet-spae
L:
R 4+n+4n
x; ;
(
)
! R
7
!
L(
x; ;
)
whih gives rise to the loal ation S
S:
F
! R
7! [
S
℄=
R
d4x (L Æ )(x) :
The equations of motion state that the Euler-derivative of the Lagrangian
vanishes for physial elds
L = L - m L ; L Æ m phys
=
0:
Noether Theorem
Innitesimal transformations Æ are maps of the jet-spae to R n
Æ
:
R 4+n+4n
x; ;
(
)
! Rn
7
!
(
Æ x; ;
)
They are innitesimal symmetries of the ation S , 9Km :
L
L
Æ + (mÆ ) m + mKm = 0 ,
L
m
Æ + mj = 0
L
m
j = Æ m + Km + nBmn ; Bmn = -Bnm :
To eah innitesimal symmetry of the ation there orresponds a onserved
urrent and vie versa.
Seond Noether Theorem
Innitesimal gauge symmetry, if Æ is linear in an arbitrary funtion Æ
=
R + (m)Rm :
To eah innitesimal gauge symmetry of the ation there orresponds an identity
among the Euler derivatives of the Lagrangian and vie versa.
Æ L
!
L
L
m
=R
- m R
=0 ;
e.g. eletromagnetism m(nFmn) = 0 , gravity DmGmn = 0 .
R
R L
Rm
L
m
= m
+( m )
Frozen Gauge Theories
L
m(j R ) = 0
L
L
Æi mi + ÆAk mAk + Km + nBnm
L
L
m
m
+ RA
) :
= -(R
i
k
A
= 0 and for
If the matter elds satisfy their equations of motion L Æ bakground gauge elds with innitesimal symmetries, ÆAk = 0 ,
L
L
jm = Æi mi + Km + nBmn = -RmA A
:
k
m+
matter
m
matter
matter
matter
i
k
matter
matter
rigid
phys
matter
k
In frozen gauge theories the Noether urrent is, up to improvement terms, the
variational derivative with respet to the gauge eld.
Example
In gravitational theories, the gauge eld is the metri gmn and transforms as
Ægkl = nngkl + kngnl + lngkn = nngkl + Rmng mn ;
Rmng = Æmkgnl + Æmlgnk :
The urrent, orresponding to innitesimal isometries of gmn = mn, is the
kl
kl
energy-momentum tensor ontrated with the Killing eld
j
m
L
1
mn
mn = = T n ; T
:
2 gmn
matter
phys; gmn
=mn
Charges from Boundary Values
Q=
Z
x0=onst
L
m(jm + Rm ) = 0 ;
Z
Z
dD-1xj0 =
dD-1xiB0i =
x0=onst
V
dD-2xniB0i :
In gauge theories, harge is determined by boundary values.
B0i unique up to kC0ik.
Gravitational Energy Momentum Complex
Gravitational energy density annot be ovariant, beause gravitational eets
an be made to vanish along eah hosen geodesi.
DmTmn = 0
Violation of the onservation of energy and momentum of matter by
gravitational eets (by exhange?).
Do the Einstein equations ontain exat onservation laws?
Cmn = f(g) Gmn - tmn ; mCmn = 0 ; g = j det g::j :
Cmn = Cnm and mCmn = 0 , Cmn = klXkmln ;
Xklmn = -Xlkmn = -Xklnm ; Xklmn + Xlmkn + Xmkln = 0:
Nonovariant identity whih holds in all oordinate systems.
Proof of the Symmetri Identity
Cmn = kBkmn for all n, with Bkmn = -Bmkn. Cmn = Cnm implies
k(Bkmn - Bknm) = 0 ;
so for eah pair mn
Bkmn - Bknm = -2lAlkmn ; Alkmn = -Aklmn = -Alknm :
Solve for Bkmn ,
Bkmn = -l(Alkmn + Almnk - Alnkm) :
For Cmn = kBkmn this means
Cmn = kl(Almkn + Aknlm) = klXlmkn :
rmsn
Determination of X
f(g) Gmn - tmn = rsXrmsn :
l.h.s.: two derivatives ! Xrmsn no derivatives
rmsn
rs
mn
rn
ms
X = h(g) g g - g g :
Comparison of g-terms: h = -f=2 and g ddhg = h,
p mn
mn
i.e. f and h homogeneous of degree 1. Using = gg , one has
1
mn
rs
mn
rn
ms
mn ;
gG
= - r s - + t
2
rs
mn
rn
ms
mn
mn
rs( - ) = 2 g T + t :
Landau Lifshitz Energy Momentum Complex
2tmn = 2 g Gmn + rs rsmn - rnms) =
1
mn
kl
km
ln
mn
ks
lr
= k l - k l + g gklr s 2
mr ks
ln
nr ks
lm
rs mk
nl
- gklg r s - gklg r s + gklg r s +
1
pq rs :
ml
nk
mn
kl
+
2g
g
-g g
2g
g
g
g
l
pr qs
pq rs k
8
t00 not denite. Stati gravitational energy density is negative (Newton).
Gravitational Waves in at spae-time to be investigated.
Transformation of the Energy Momentum Complex
2t mn(x (x)) =
2 x m x n
x
tu(x)+
2t
det
x xt xu m n 2 x x
x
rstu - ruts +
+ rs det
t xu
x
x
2 x m x n x
rs
tu
ru
ts
su
tr
+ s det
r 2 - - +
t
u
x
x
x
2 x m x n xl 2x k
x
rs
tu
ru
ts
su
tr
2
- - +
+ s det
t
u
k
r
l
x
x
x
x
x
x
k 2 x m x n 2x l
x
x
rs
tu
ru
ts
+
- :
k det
l
t
u
s
r
x
x x x x x
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Conservation Laws
rs( mn + mn
2 g T
m (g Tmn + tmn) = 0 :
rs mn - rn ms) =
t
;
Conserved harges (ADM-mass)
P
Z
m = d3
x gT
0m + 0m
t
are determined by boundary values
Z
Z
1
1
m
rs
m0
r0
ms
3
P (V) = 2 d xrs - = 2
V
V
d2
ri m0 - r0 mi
fi r Corresponding symmetries: Ængkl = Lvngkl, vn = nll .
:
Cruial (Kreuzer) Lemma
Let and denote invariant dierential forms in jet-spaes with star-shaped
base and target manifold, whih depend on tensors only
d
= 0 , = LdDx + P(F) + d :
P invariant polynomial in F = 12dxmdxn(mAn - nAm - [Am;An℄) ,
L = 0 , L = P(F) + d ;
P(F) + d = 0 , P(F) = 0 = d :
Proof to long to be given here: Part of a series of letures (available).
Enumerates all topologial densities, e.g. D = 4, gravity,
p
RklmnRrstumnrsklst , RmnrsRklsrmnkl .
g
Conerstone to determination of all anomaly andidates.
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