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37

Triad Formulations of Canonical Gravity without a �xedreference frame

Joachim Schirmer

Fakult�at f�ur Physik der Universit�at Freiburg

Hermann-Herder-Str. 3, 79104 Freiburg i.Br. / FRG

e-mail: schirm@phyq1.physik.uni-freiburg.de

FR-THEP 95/8 March 1995

Abstract

One can simplify the triad formulations of canonical gravity by abandon-

ing any relation to a �xed coordinate system. That means in case of the adm

formalism that one can determine the momentum by direct derivation of the

Lagrange-3-form w.r.t the time-derivative of the triad-1-forms, thus the mo-

mentum is most naturally a 2-form. We apply this concept to the Palatini

formulation where we can closely follow Dirac's concept to �nd and eliminate

the second class constraints. Following the same way for the Ashtekar theory it

will turn out to be equivalent to two successive canonical transformations where

the �rst makes explicit use of the spatial dimension being 3 and the second is

usually hidden in the use of densities. At the end we can give a simple version

of the reality constraints.

1 Technical Preliminaries

Avoiding any coordinate system will eliminate all determinants from the theory, but

the associated problems will be contained in the frequently used Hodge operator. Yet

the algebra of this operator is simple because our triads are normalized. To have ane�ective way of handling this operator we �rst introduce some notations and formulaswhere we mainly follow the treatment given in [1]. Let (M;g) be a m-dimensional

pseudo-Riemannian manifold. We �rst de�ne the interior multiplication i of two forms

of di�erent degree.

For q � p let the bilinear mapping i : q(M) � p(M) �! p�q(M); (�; �) 7�! i��

have the following properties:

i�� = g](�; �) = �a�bgab fr p = q = 1 (1.1)

i�(�1 ^ �2) = i��1 ^ �2 + (�1)p1�1 ^ i��2 fr �i 2 pi(M); � 2 1(M) (1.2)

i(�1^�2) = i�2 � i�1 (1.3)

1

These properties de�ne i uniquely. Let faaga=1:::m be a local basis of the tangential

bundle and faag be the dual basis. We will use the following abbreviation for the

basis of p(M):

aa1:::ap := aa1 ^ : : : ^ aap (1.4)

We will not distinguish in the notation between aa 2 �loc(TM) and a[a = gabab 2

1loc(M), so usually we will omit the [-sign. Consequently we do not distinguish

between the interior product with a 1-form and the natural injection of the metric dual

vector of this 1-form. This identi�cation is possible unless variations or derivations

get involved. Then one has to keep in mind if for the de�nition the metric was used.

In a local cobasis the interior product takes the following form:

i�� =1

q!(p� q)!�i1:::iq�i1:::iqj1:::jp�qa

j1:::jp�q (1.5)

which is proved in a simple, but tedious calculation. The scalar product of two p-formsis now given by:

h�j�i := pg](�; �) := i�� = i�� �; � 2 p(M) (1.6)

We notice that the interior product i� is the dual mapping of the exterior multiplica-tion �^ with respect to the scalar product hji:

hi�!j�i = i�i�! = i�^�! = h� ^ �j!i (1.7)

We de�ne the Hodge-dual form of a form as interior product with the canonical volumeform of the pseudo-Riemannian manifold

� : p(M) �! m�p(M) 8p � � := i�� ; (1.8)

so in a local basis we have:

� � =1

p!(m� p)!�i1:::ip�i1:::ipj1:::jm�pa

j1:::jm�p (1.9)

Using the properties of the interior product it is not di�cult to prove the following

identities { here s is the signature of the pseudo-metric:

(i) h��j ��i = (�1)sh�j�i (1.10)

(ii) � �� = (�1)p(m�p)+s� � 2 p(M) (1.11)

(iii) �(� ^ �) = i� �� � i�� = (�1)q(m�q) �� ^ � � 2 q(M) (1.12)

(iv) � ^ �� = h�j�i� = � ^ �� �; � 2 p(M) (1.13)

For an orthonormal co-frame feigi=1:::m the Hodge operator has the following useful

representation:

� ei1:::ip =1

(m� p)!�i1:::ipjp+1:::jme

jp+1:::jm =1

(m� p)!�i1j1 :::�ipjp�j1:::jme

jp+1:::jm (1.14)

2

For the exterior derivative of these forms one can prove the formula:

d �ei1:::ip =1

(m� p)!�i1:::ipjp+1:::jmd e

jp+1:::jm = del ^ �ei1:::ip l (1.15)

Let !ab be the connection forms of an arbitrary linear connection with respect to an

arbitrary basis faag. Then the torsion is given by

T a = Daa = daa + !ab ^ a

b ; (1.16)

so for a torsionfree connection one has

daa = �!ab ^ a

b and (1.17)

daa1:::ap = �!a1b ^ a

ba2:::ap � : : :� !apb ^ a

a1:::ap�1b (1.18)

The connection forms of a metric linear connection with respect to an arbitrary basis

satisfygac!

cb + !c

agcb = dgab ; (1.19)

so in an orthonormal frame holds

!ij + !ji = 0: (1.20)

Using these equations one �nds for an arbitrary connection in an orthonormal basis:

d �ei1:::ip = del ^ �ei1:::ip l = (T l� !l

k ^ ek) ^ �ei1:::ip l

= T l^ �ei1:::ip l + !l

i1 ^ �eli2:::ip + : : :+ !lip ^ �ei1:::ip�1l (1.21)

and hence for an arbitrary basis:

d �aa1:::ap = T b^ �aa1:::apb + (!ba1 + dgba1) ^ �ab

i2:::ip + : : :

+(!bap + dgbap) ^ �aa1:::ap�1b � (!aa �

12gabdgab) ^ �a

a1:::ap (1.22)

For the special case of the Levi-Civit-connection this formula reads:

d �aa1:::ap = �!a1b ^ �a

ba2:::ap � : : :� !apb ^ �a

a1:::ap�1b (1.23)

For the convenience of the reader we give the expression of Christo�el symbols of theLevi-Civit-connection in an arbitrary basis:

!a bc := gad!dc(ab) = 1

2

�gab;c � gbc;a + gca;b + Ca bc �Cb ca + Cc ab

�(1.24)

Ca bc = gadad([ab; ac]) = �gadda

d(ab; ac)

It is possible to represent the connection form of the Levi-Civit connection by the

inner product of an orthonormal basis and its derivatives

!ij = 12

�ijdei � iidej � iijdek � e

k�

= ijdei � iidej �12iij(dek ^ e

k)

= (�1)m+s�h�ej ^ �dei + ei ^ �dej +

12eij ^ �(de

k^ ek)

i(1.25)

3

In the following paragraphs one important point is the functional derivative with

respect to forms. We will illustrate this method with a well known example. Instead

of varying the metric components with respect to a �xed coordinate basis we will vary

the orthonormal frame, i.e. we will vary the four 1-forms which are declared to be

orthonormal. So varying the metric can be represented by varying with respect to

forms. Let us consider the Einstein-Hilbert-action, as usual the variation shall vanish

on the boundary of M :

S[e�] =1

2

ZMR�� ^ �e

�� (1.26)

�S =1

2

ZM[�R�� ^ �e

�� +R�� ^ � �e��]

=1

2

Z@M

�!�� ^ �e�� +

ZMR�� ^ �e

�^ �e���

�S

�e�=

1

2R�� ^ �e

��� =

1

2

�hR�� je

��i �e� + hR�� je

��i �e

� + hR�� je��i �e�

�= � �

�R� �

1

2Re�

�(1.27)

Here R� := i�R�� (1.28)

is the Ricci-form, which is symmetric, i.e. hR�je�i = hR� je�i and

R := i�R� = i�i�R�� (1.29)

is the Ricci-scalar.It will be useful in the last paragraph to use instead of the or-thonormal triads the dual frame of the m (m � 1)�forms �e�. We will show thatone can functionally derive the expression

RM ei ^�i, with respect to �ei and obtain a

well-de�ned 1-form. Let f�igi=1:::m be m (m� 1)�forms independent of e and de�ne

the m 1-forms f�igi=1:::m by:

� ej i ^ �j = �i (1.30)

Then one has ZMei ^ �i =

ZMei ^ �ej i ^ �j = (m� 1)

ZM�ej ^ �j

��i = � �ej i ^ �j + �eji ^ ��j = 0

�

ZMei ^ �i = (m� 1)

ZM� �ej ^ �j +

ZMei ^ �ej i ^ ��j

= (m� 1)ZM��ej ^ �j �

ZMei ^ � �ej i ^ �j

= (m� 1)ZM� �ej ^ �j �

ZMei ^ �ek ^ �ej ik ^ �j

= (m� 1)ZM� �ej ^ �j � (m� 2)

ZM�ek ^ �ejk ^ �j

4

�

ZMei ^ �i =

ZM��ej ^ �j

�

� �ej

ZMei ^ �i = �j

The equation �ej i ^ �j = �i is easily solved:

ek ^ �eji ^ �j = � �ei ^ �k + �ki �e

j ^ �j = ek ^ �i

�h�kjeii+ �kih�jjeji = (�1)shekj ��ii

h�jjeji =

(�1)s

m� 1hej j ��

ji

h�ijeji = (�1)s�

�ij

m� 1hekj ��

ki � heij ��ji

��i = (�1)s

�ei

m� 1hekj ��

ki � heij ��jie

j

�Thus we have proved the formula

�

� �ei

ZMek ^ �k = (�1)s

�ei

m� 1hekj ��

ki � heij ��jie

j

�; (1.31)

which we specialize for the case of dimM = 3; s = 0:

�

� �ei

ZMek ^ �k = ��i

lmil�m �1

2eih�mj �e

mi (1.32)

2 The ADM formulation in triads

The space-time manifold is assumed to be a parametric set of imbedded spacelikehypersurfaces �t, where we call the non-unique parametrization "time"; i.e. there

exists a di�eomorphism i : IR � � �! M , which can be used for identi�cation. Let

(x1; x2; x3) be a coordinate system for �, then (t; x1; x2; x3) is a chart for IR �� and

(�t; �x1; �x2; �x3) := (t; x1; x2; x3) � i�1 is one for M . We will generally distinguish theanalogous quantities on M and � by using a bar for quantities de�ned on M . So ifnot de�ned in another way one obtains the unbarred quantities by pullback with the

map it(x) := i(t; x); it : � �! M: A pseudo-metric �g { signature (-1,+1,+1,+1) {

on M de�nes a normal �e0 to the subspace of the tangential space which is spanned

by f @@�xigi=1;2;3, it is the normal to the f�t = constg-surfaces �t = it � � in M . With

respect to this normal we decompose the time vector �eld @@�t:

@

@�t= �N�e0 +

�~N (2.1)

At this point it seems that the use of a certain identi�cation i leads to a gauge �xingof lapse and shift, because it determines at once the time vector�eld and the normal.

5

But we use this concept only to �nd the 3+1-decomposed Lagrangian which corre-

sponds to the covariant action principle. When we have obtained it, we will turn

our point of view and try to reconstruct the space-time metric from the decomposed

data. If we do not know the space-time-metric, the normal on the right-hand-side of

the equation will be unknown. By varying lapse and shift, we will vary the normal,

and thus it is no surprise that the constraints associated with lapse and shift are

equivalent to Einstein's equations restricted to the normal, i.e. i�e0�G = 0. Now let

f�e�g�=0:::3 be adapted orthonormal tetrades onM , such that f�eigi=1;2;3 are always par-

allel to the hypersurfaces �t. The metric �g will be represented by the dual cotetrades

f�e�g�=0:::3; �g = ����e� �e�. Using the imbedding it we can pull back the tetrades

e� = i�t �e�; i�t �e

0 = Ni�td�t = 0 We �rst decompose the Einstein-Hilbert Lagrange form1:

�L =1

2

��R�� ^ ��e

���=

1

2

�2 �R0i ^ ��e

0i + �Rij ^ ��eij�

=1

2

h2 d�!0i^ ��e

0i + 2 �!0k^ �!ki ^ ��e

0i + 3�Rij ^��eij + �!i0 ^ �!0

j^��eiji

(1:23)=

1

2

h2 d

��!0i ^ ��e

0i�� 2�!0i ^ �!0

j ^ ��eji + 3�Rij ^ ��e

ij + �!0i ^ �!0j ^ ��eiji

=1

2

h2 d

��!0i ^ ��e

0i�+ (3�Rij � �!0i ^ �!0j) ^ ��e

iji

(2.2)

Neglecting the exact form which turns into a surface integral after integration, we can

write the action in the 3+1-decomposed form:

S(�e) =ZM

�L =1

2

ZM

�3�Rij � �!0i ^ �!0j

�^ ��eij

=1

2

Zdt

Z�i�t i@=@�t

�3�Rij � �!0i ^ �!0j

�^ ��eij (2.3)

Here i@=@�t is the natural injection of a vector �eld in a form. Using �e0 = �N d�t andi�td�t = 0, this reduces to

S(�e) =1

2

ZIRdt

Z�i�t (

3�Rij � �!0i ^ �!0j) ^ i�t i@=@�t

4� �eij

=1

2

ZIRdt

Z�N(Rij � !0i ^ !0j)^

3� eij ; (2.4)

where we have identi�ed i�t3�Rij and Rij, which is de�ned on � as Levi-Civit curvature

form to the metric de�ned by the triads feig. Using the relation !0i = �iiK, whereK

is the (0; 2)-tensor of the extrinsic curvature, one can easily recognize the form givenabove as the adm action-integral:

S(e; _e) =1

2

Zdt

Z�NhRij � !0i ^ !0jje

iji�

=1

2

Zdt

Z�N(R�K i

iKjj +K i

jKji)� (2.5)

1Greek letters are summed from 0 to 3, Latin letters from 1 to 3

6

We now de�ne the Lagrange function as follows { the index 3 on the Hodge-operator

will be understood for the rest of the paragraph.

L(ei; _ei; N; ~N ) =Z�

N

2(Rij � !0i ^ !0j) ^ �e

ij (2.6)

To make the dependance of _ei obvious it is necessary to use the relation between the

time-derivative of the triads and the extrinsic curvature:

_ei =d

dtjs=t

(i�s�ei) =

d

dsjs=0

�i�t�

@=@�ts

��ei�= i�tL@=@�t�e

i (2.7)

because it holds it+s = �@=@�ts � it, where �

@=@�ts denotes the ow of the time-vector�eld

@@�t. So we take the Lie-derivative and pull back the result:

L@=@�t�ei = L �~N

�ei + �Ni�e0d�ei

= L �~N�ei � �N �!i

�(�e0)�e� + �N �!i

0

_ei = i�tL@=@�t�ei = L ~Ne

i� aije

j +N!i0 (2.8)

aij := Ni�t �!ij(�e0) (2.9)

Here aij is the rotational parameter which is characteristic for a triad theory. This

relation corresponds to the equation

_qab = L ~Nqab + 2NKab (2.10)

in the usual adm theory, which could be derived from (2.8). One is tempted to

treat the rotational parameter a like shift and lapse as an additional variable, whichturns out to be a gauge parameter, since its time derivative does not appear in theLagrangian. But this will lead to redundancies: If in (2.10) _q; N and ~N are given,

one can determine Kab, and if in (2.8) _e; N and ~N are given, one can determine !and a. So one can not arbitrarily choose a to determine !0i from _e. The reason is

the symmetry of the extrinsic curvature, which is a consequence of the fact that ouroriginal connection on M was torsionfree.

�T 0 = 0 =) i�t�T 0 = i�t (d�e

0 + �!0� ^ �e�) = !0

i ^ ei = 0

=) h!0ijeji = h!0j jeii (2.11)

We de�ne the symmetric and antisymmetric part of the 1-forms _ei and L ~Nei in the

following way

_eiS=A := 12( _ei � h _ejjeiiej) (2.12)

L ~NeiS=A := 1

2(L ~Ne

i� hL ~Ne

jjeiiej) (2.13)

and can split the equation (2.8) in these parts:

_eiS = L ~NeiS�N!0

i (2.14)

_eiA = L ~NeiA� aije

j (2.15)

7

This split will not be possible in the Palatini-case, where the connection is not assumed

to be torsionfree and thus the extrinsic curvature is not symmetric. In the Palatini-

case the rotational parameter a will become a variable comparable with lapse N and

shift ~N .

We notice that only _eiS appears in the Lagrange function. We can now either introduce

a �xed basis faaga=1:::3 and �nd a variable, such that _eiS is a proper time derivative

{ _eiS is not the time derivative of eiS = ei. Then we end up with the metric adm

formulation

qab = �ijei(aa)e

j(ab) (2.16)

_qab = 2h _eiSjejiei(aa)e

j(ab)2 (2.17)

and of course we can write the Lagrangian in terms of qab and _qab. But we can alsodisregard the fact, that only the symmetric part of _ei appears in the Lagrangian andexpect a primary constraint. The momentum form is easily found:

pi :=�L

� _ei= !0j ^ �ei

j = h!0j jeji � ei � h!0jjeii �e

j (2.18)

The momentum is naturally a vector-valued 2-form, generally in a space of dimension

n it is a n � 1-form. This is analogous to the usual adm formulation, where themomentum-form is a (0,2)-tensorvalued 3-form, where one usually splits the 3-forminto a density and d3x. We de�ne the contracted momentum

p := hpij �eii = 2h!0ije

ii (2.19)

and would like to reexpress the extrinsic curvature !0i by a simple calculation asfollows

!0i = �hpj j �eiiej + 1

2pei; (2.20)

but this would be inconsistent, if hpij �eji 6= hpj j �eii, since we know that the extrinsic

curvature is symmetric. Thus it is only possible to obtain !0i from pi, if pi ^ ej =

pj ^ ei.This is a consequence of the nonsolvability of the equation

pi ��L

� _ei(e; _ei) = pi +

1

N

�_ekS � L ~Ne

kS

�^ �eik = 0 ; (2.21)

which has to be regarded as a constraint equation. Multiplying this constraint by ejand taking the antisymmetric part yields the necessary consistency condition

pi ^ ej � pj ^ ei � 0 (2.22)

which is an equivalent formulation of the constraint implied by the solvability of(2.21). So this is a primary constraint implied by the symmetry of the extrinsic

2Here _ej denotes �ij _ei, not ( _ei)

[, they di�er by a sign.

8

curvature, which is a consequence of the fact that the fourdimensional connection

to be constructed from ei and _ei is torsionfree.We can now perform the Legendre-

transform.

H =Z�pi ^ _ei � L

=Z�pi ^ _eiS +

Z�pi ^ _eiA �

Z�

N2(Rij � !0i ^ !0j) ^ �e

ij

=Z�pi ^ L ~Ne

i�

Z�aijpi ^ e

j +Z�

N2

hhpij �e

jihpj j �e

ii � 1

2p2 �R

i� (2.23)

The signi�cance of a is here at �rst that of a Lagrange multiplier of the rotationalconstraint. But if one want to reconstruct the metric space-time one has to relatea to a space-time quantity as given in equation (2.9), in order that the Hamiltonianequation for _e and the geometric equation (2.8) agree. It seems that the Hamiltoniandescription does not determine the quantity bi := Ni�t �!0i(�e0), but we will show in the

following paragraph, that one has bi = �hdN jeii(3.7). Since the time dervatives of

lapse N and shift ~N do not appear in the Lagrangian one �nds the following primary

constraints:

pN =�L

� _N� 0 p _~N

=�L

�_~N� 0 ; (2.24)

which give rise to the following secondary constraints:

CH = fH; pNg = �hhpij �e

jihpj j �eii � 1

2p2 �R

i� 0

CDi = fH; pN ig = �dpi + pj ^ iidej = �Dpi � !j

k(ei)pj ^ ek � 0

(2.25)

where in the case of the di�eomorphism constraint a surface term had to be neglected.Using the rotational primary constraint

CRji =

12(pj ^ ei � pi ^ e

j) : (2.26)

we could rede�ne the di�eomorphism constraint in a concise way Dpi � 0, but this

will not be possible in the Palatini case, and since we prefer to work with integrated

constraints where our di�eomorphismconstraint has the simple meaning of the mo-

mentum mapping of the action of the di�eomorphism group of �, we stick to the

original version. But this is a matter of taste and we will have the choice also in theAshtekar formulation. We can represent the Hamiltonian as sum of the integrated

constraints

H(e; p;N; ~N; a) =R�N

iCDi(e; p) +R@�N

jpj +R� a

ijCR

ji +

R�NCH

:= HD(e; p; ~N) + HR(e; p; a) + HH(e; p;N)

(2.27)

and we will call the integrated versions of the constraints like the constraints itself,di�eomorphism, rotational and Hamiltonian constraint. We can now determine the

9

equations of motion:

_ei =�H

�pi= L ~Ne

i� aije

j +N(hpj j �eiiej � 1

2pei) (2.28)

_pi = ��H

�ei= L ~Npi + aj ipj �N �

�Ri �

12Rei

�+ �(ridN � ei�N)

�N�hpij �e

jipj �

12ppi

�+N

2

�hpj j �e

kihpkj �e

ji � 1

2p2��ei (2.29)

The Ricci-form Ri is de�ned as in (1.29) and �N denotes as usual the Laplacian

of the lapse function. In the equation for the momentum surface terms had to beneglected in order to derive the �rst and the last term. We will show the contributionof the Hamiltonian part for the equation of motion.

� [hpij �ejihpj j �e

ji�] = � [(pi ^ ej) � �(pj ^ e

i)]

= (�pi ^ ej + �ej ^ pi) � � (pj ^ e

i) + (pi ^ ej) � � �(pj ^ e

i)

eklm � � �(pj ^ ei)

(1:13)= (�pj ^ e

i + �ei ^ pj) � �eklm � �eklm � � (pj ^ e

i)

Multiplication by 13!hpi ^ e

jjeklmi leads to�pi ^ e

j��� �

�pj ^ e

i�=��pj ^ e

i + �ei ^ pj�� ��pi ^ e

j���ek^ ik

�pi ^ e

j�� ��pj ^ e

i�

and �nally one obtains:

�hhpij �e

jihpj j �e

ii�i= �pi ^ 2e

jhpj j �e

ii+ �ei ^

h2pjhpij �e

ji � �eihpj j �e

kihpk j �e

jii

Analogously one �nds

�(p2�) = �pi ^ 2pei + �ei(2ppi � p2 �ei)

and all momentum terms in the equation of motion are proved. The derivation of the

Ricci-scalar term is more di�cult:

��NRij ^ �e

ij�= N

h�Rij ^ �e

ij + �ek ^�Rij ^ �e

ijk

�iThe second term is responsible for the Einstein-force term as shown in (1.27). It isleft to discuss the variation of the curvature-form:

N�Rij ^ �eij = Nd(�!ij ^ �e

ij)

= d(N�!ij ^ �eij)� dN ^ �!ij ^ �e

ij

modulo exact forms = 2h!ij jeiiej ^ �dN

10

We neglect the exact form which, leads to a surface integral, and substitute the

variation of the connection form by the variation of the triads:

dei = �!ij ^ e

j

�dei = ��!ij ^ e

j� !i

j ^ �ej

ii�dei = �h�!i

jjeiiej � ii

�!i

j ^ �ej�

Hence we have:

N2�Rij ^ �e

ij = h�!ij jeiiej ^ �dN

= �ii(d�ei + !i

j ^ �ej) ^ �dN

= (d�ei + !ij ^ �e

j) ^ �(dN ^ ei)

= d(�ei ^ �dN ^ ei) + �ei ^D �(dN ^ ei)

= �ei ^ � [ridN � ei�N ] (2.30)

Of this derivation we keep in mind the last transformation for later use:

D �(dN ^ ei) = �(ridN � ei�N) (2.31)

Using the equations of motion for each part of the Hamiltonian it is no di�culty todetermine the constraint algebra. All surface integrals are consequently neglected.

fHD( ~N);HD( ~M)g = HD([ ~N; ~M ])

fHD( ~N);HR(a)g = HR(L ~Na)

fHD( ~N);HH(N)g = HH(L ~NN)fHR(a1);HR(a2)g = �HR([a1; a2])fHR(a);HH(N)g = 0fHH(N);HH(M)g = HD(NdM ] �MdN ])

�HR (hdN ^ dM jeiji+ hNdM �MdN j!i

ji)

(2.32)

We calculate only three of these brackets:

fHD( ~N);HD( ~M )g =Z�

24�HD( ~N)

�ei^�HD( ~M )

�pi��HD( ~N)

�pi^�HD( ~M)

�ei

35=

Z�

h�L ~Np

i^ L ~Mei + L ~Ne

i^ L ~Mpi

i=

Z�pi ^ L[ ~N; ~M] = HD([ ~N; ~M ])

This result reassures that the di�emorphism constraintHD is the momentummapping

of the action of the di�eomorphism group.

fHR(a);HH(N)g =Z�

(aj ipj ^N

�hpk j � e

iiek � 1

2pei

�+ aije

j^

11

��N(hpij �e

jipj �

12ppi) +

N

2(hplj �e

mihpmj �e

li � 1

2p2) �ei

N � (Ri �12Rei) + �(ridN � ei�N

�)

=Z�aijrir

jN� = 0 ;

since rirjN is symmetric because the connection is torsionfree.

fHH(N);HH(M)g =Z�

�� �

�ridN � ei�N

�^M

�hpj j � e

iiej � 1

2pei

�+ �

�ridM � ei�M

�^N

�hpj j � e

iiej � 1

2pej

��=

Z�

�Nrir

jM �MrirjN

�pj ^ e

i

=Z�

hri(Nr

jM �MrjN)� (riNrjM �riMr

jN)ipj ^ e

i

(2:34) =Z�

(pj ^ L(NdM ]�MdN ])e

j

+

�!j

i(NdM ]�MdN ]) + hdN ^ dM jej iie

i^ pj

�)= HD(NdM ]

�MdN ])

�HR

�hdN ^ dM jeiji+ hNdM �MdN j!i

ji�

(2.33)

Here we used the identity

DY i = (rjYi)ej = L~Y e

i + !ij(~Y )e

j (2.34)

As expected the rotational constraint or more precisely the generated vector�elds on

the in�nite dimensional manifold of triads and triad-momenta form an ideal. So we

can go through the canonical analysis without introducing a �xed coordinate system.One can also de�ne the Poisson bracket between the 1-form ei and the 2-form pj :

fei(r); pj(s)g = �ij�(r; s) r; s 2 � (2.35)

Here � has to be regarded as a (0,3)-tensor�eld on � � �; �(r; s) is a 1-form withrespect to r with values in �2(Ts�) or a 2-form with respect to s with values in

�1(Tr�). In a local chart (U; fxaga=1;2;3 �(r; s) has the following form:

�(r; s) = 12�abcdx

a(s) ^ dxb(s) dxc(r)�(x1(r)� x1(s))�(x2(r)� x2(s))�(x3(r)� x3(s))

(2.36)Despite the use of the �-symbol this is a tensor since in coordinate transformations

the product of �-functions transforms with a determinant. Finally we give the repre-

sentation of the usual adm momentum �, as canonical conjugate to the (0,2)-tensorq a (2,0)-tensorvalued 3-form in terms of the momentum 2-form and the triads

� = �k(iei ej pk ^ ej) = ei ejp

(i^ ej); (2.37)

12

where the symmetrisation is not really necessary, since one can de�ne the metric

momentum only on the constraint surface where pi^ej is symmetric. This eqution can

easily be veri�ed using the well known relation between � and the extrinsic curvature

K = �!0i ei

� =�K � qtr (q]K)

� �; (2.38)

and equation (2.18).

3 The Hilbert-Paltini-action

The Hilbert-Palatini-action was suggested by Palatini [2] mainly in order to avoidsecond time derivatives of the relevant �elds in the action integral. This can also beachieved by neglecting a surface integral

1

2

ZM

�R�� =

ZM

��d(�e� ^ �d�e�)| {z }boundary term

�1

2(d�e� ^ �e�) ^ �(d�e� ^ �e�) + 1

4(d�e� ^ �e�) ^ �(d�e� ^ �e�)| {z }

�rst order Lagrangian

�(3.1)

but then the action is no longer gauge invariant under SO(1,3) rotations. FollowingPalatini's suggestion one regards the action not only as dependant on the metric, but

also on the connection. So the connection is now no longer torsionfree and metric,since if these conditions hold, the connection is uniquely determined by the metric.Unfortunately varying w.r.t. the connection does not yield both necessary conditions,i.e. if one considers a general linear connection the Euler-Lagrange equation for theconnection only gives a relation between torsion and non-metricity [3]. In the vacuum

case this relation states that the connection is torsionfree, if it is metric, and vice versa.So one usually assumes that one of the conditions for the Levi-Civit connection issatis�ed and only varies in the class of either metric or torsionfree connections. Since

in case of a non-metric connection the compatibility of the gauge group of the tetradesand the connection form is destroyed { the connection form for a metric connection

is soIR(1; 3)-valued { we prefer to drop the torsion condition here. Since our canonicaldescription shall be equivalent to the covariant formulation, which is of course easier,

we �rst revise the covariant derivation of the equations. We show that assuming thatour connection satis�es the condition of metricity

�!�� + �!�� = 0 (3.2)

the Euler-Lagrange equation for the connection form yields in the vacuum case that

our connection is torsionfree.

S(�e; �!) =1

2

ZM

�R��(�!) ^ ��e��

��!S =1

2

ZM[d��!�� ^ ��e

�� + ��!�� ^ �!�� ^ ��e

�� + ��!�� ^ �!�� ^ ��e

��]

13

(1:23)=

1

2

ZM��!�� ^ �T �

^ ��e��� + d(��!�� ^ ��e��) (3.3)

Assuming that the variation vanishs on the boundary one obtains

�S

��!��= �T �

^ ��e��� = 0: (3.4)

This turns out to be a zero torsion condition. We show this abandoning the summation

convention for a short while.X�6=��

�T �^ ��e��� =

X�6=��

��e�h �T�j�e��i + ��e�

X�6=��

h �T �j�e��i + ��e

�X�6=��

h �T �j�e�

�i = 0

1: h �T �j�e��i = 0 8�; � 6= �

2:X�6=�

h �T �j�e��i = 0 8�; �

We write (h �T �j�e��i)�=1:::m;� 6=� =: ~x and summarize the second equation for �xed � forall � in the matrix equation 0BB@

0 1 : : : 1...

. . ....

1 : : : 1 0

1CCA ~x = 0 (3.5)

Since this (m � 1) � (m � 1)-matrix has determinant (�1)m(m � 2); ~x = 0 is theonly solution if m 6= 2 and thus it follows �T = 0. Despite this result we keep in mindthat the Palatini-theory is another theory than Einstein's relativity, since spin �eldscould now couple to torsion [4]. As long as there are no experimental data there isonly Hehl's argument, that a theory where only the translatory part of the Poincar

group couples to geometry seems to be inconsequent. If future experiments really

con�rm a spin-torsion-coupling it may well be that one has to modify the Einstein-Lagrangian, because it does not give rise to derivative terms for the connection inthe �eld equations. For our canonical description this means that the there is no

time-development of the connection, it is fully determined by constraints.

The Hamiltonian description we are seeking for should yield the same result: The

connection to be constructed on the fourdimensional manifold IR � � � M shouldprove torsionfree. On � one can only observe the pullback of the equation �T = 0, soit is useful to realise �rst the meaning of the pullback of the di�erent components of�T on �. We can di�er four equations:

i�t�T 0 = 0 i�t

�T i = 0 i�t i0�T 0 = 0 i�t i0

�T i = 0

The �rst equation assures the symmetry of the extrinsic curvature (2.11). The secondimplies that the induced connection on � is torsionfree:

0 = i�t�T i = i�t (d�e

i + �!i� ^ �e

�) = dei + !ij ^ e

j = 0 (3.6)

14

The third equation yields an important relation between lapse N and i�t i0�!0i.

0 = i�t i0�T 0 == i�t

hi0d( �Nd�t) + �eii0�!

0i

i= �

1

NdN + eii�t i0�!

0i

De�ning bi := Ni�t �!0i(�e0) we write this equation as follows:

dN + biei = 0 (3.7)

The last equation is the well known relation between the extrinsic curvature, the

rotational term and the Lie-derivative along the normal:

0 = i�t i0�T i = i�t (i0d�e

i + �!i�(�e0)�e

� � �!i0) = i�tL�e0�e

i + eji�t �!ij(�e0)� i�t �!

i0

Ni�tL�e0�ei = �aije

j �N!0i (3.8)

We will now decompose the Lagrange 4-form in another way than in the last para-graph, since there is a priori no dependance of the connection on the tetrades.

�R�� ^ ��e�� = 2 �R0i ^ ��e

0i + �Rij ^ ��eij

= 2d(�!0i ^ ��e0i) + 2�!0i ^ d��e

0i + 2�!0j ^ �!ji ^ ��e

0i

+( 3�Rij + �!i0 ^ �!0j) ^ ��e

ij

mod.ex.forms

(1:15)= 2�!0i ^ d�ej ^ ��e

0ij + 2�!0j ^ �!ji ^ ��e

0i + ( 3�Rij + �!0i ^ �!0j) ^ ��eij(3.9)

The pullback i�t after insertion of the time vector �eld @=@�t will be simple, if we �rsteliminate �e0 from the last factor �e:::. Then the dualised form ��e::: will contain a factor�e0 = �Nd�t in any case and only if the time vector �eld is inserted there the term will

not vanish after the pull back to �. Technically spoken i�t i@=@�t is performed then by

changing4

� to3

� and multiplying with the lapse function.

�L =1

2

h2�!0i ^ d�ej ^ i

0��eij � 2�!0j ^ �!j

i ^ i0��ei + ( 3�Rij + �!0i ^ �!0j) ^ ��e

iji

= ��!0i(�e0)d�ej ^ ��eij + �!0i ^ i0d�ej ^ ��e

ij� �!0i(�e0)�!

ij ^ ��e

j� �!i

j(�e0)�!0i ^ ��ej

+12( 3�Rij + �!0i ^ �!0j) ^ ��e

ij (3.10)

Using the de�nitions bi := Ni�t �!0i(�e0) and aij := Ni�t �!ij(�e0) we obtain:

L = i�t i@=@�t�L

=hN2(Rij + !0i ^ !0j)� bidej + !0i ^ Ni�t i0d�ej

i^

3�eij �

�bi!

ij � aij!0i

�^

3�ej

=hN2(Rij + !0i ^ !0j)� bi(dej + !jk ^ e

k) + akjek ^ !0i + !0i ^Ni�t i0d�eji^ �eij

In order to introduce _ei we again have to use the space-time picture:

_ei = i�tL@=@�t�ei = L ~Ne

i +Ni�tL�e0�ei = L ~Ne

i +Ni�t i0d�ei (3.11)

15

Then we obtain the following Lagrangian:

L(ei; _ei; !0i; !ij ; N; ~N; aij; bi)

=Z�

�_ei ^ !0j ^ �ei

j�h�

N2(Rij + !0i ^ !0j)

+bi(dej + !jk ^ ek) + akiek ^ !0j + L ~Nei ^ !0j

i^ �eij

�(3.12)

We notice that the time derivatives of the connection form components !ij; !0i; bi and

aij do not appear in the Lagrangian like those of lapse and shift. This is not surprising,

since in the covariant equations �S�!��

= 0 did not contain an exterior derivative of the

connection form. We can reduce the number of constraints that we get, if we identify

at this point�L

� _ei= pei = !0j ^ �ei

j : (3.13)

Formally spoken this identi�cation solves the pair of second class constraints

pei � !0j ^ �eij� 0 p!0i � 0 : (3.14)

The spatial part of the connection form �!0i is then already eliminated as a redun-dant degree of freedom. Before this identi�cation one would obtain the equationLNne

i + aij � N!i0 = 0 as nondynamical Euler-Lagrange-equation for !0i. After

the identi�cation we obtain this equation in the Hamiltonian picture as equation of

motion for _ei. The Hamiltonian is now easily obtained (pei � pi):

H(ei; pi; !ij ; N; ~N; aij; bi) =

=Z�pi ^ L ~Ne

i� aije

j^ pi + bi(dej + !jk ^ e

k) ^ �eij

+N2

�hpij �e

jihpj j �e

ii �

12p2 �R

�� (3.15)

Compared to the adm Hamiltonian there is one additional term. Since dej +!jk ^ e

k

is the torsion of the connection we have an explicit torsion potential. Apart from thetwo constraints which were already discussed and solved the Lagrangian gives rise to

�ve other primary constraints:

pN � 0 p ~N � 0 paij � 0 pbi � 0 p!ij � 0 (3.16)

These constraints give rise to the following secondary constraints:

CH = fH; pNg = 12

�hpij �e

jihpj j �eii �

12p2 �R

�� Hamiltonian

CDi = fH; pN ig = �dpi + pj ^ iidej di�eomorphism

CRij = fH; paijg = �

12(pi ^ ej � pj ^ ei) rotational

CTi = fH; pbig = (dej + !j

k ^ ek) ^ �eij = T j ^ �eij torsion

CCij = fH; p!ijg = �(dN + bke

k) ^ �eij �NT k ^ �eijk connection constraint(3.17)

16

The Hamilton function is again a sum of integrated secondary constraints. But only

four of the �ve secondary constraints appear in the Hamiltonian.

H(ei; pi; N; ~N; !ij; aij; bi)

=R�N

iCDi(e; p) +R� aijCR

ij(e; p) +R�CH(e; p; !) +

R� biCT

i(e; !)

= HD( ~N ; e; p) + HR(B; e; p) + HH(N ; e; p; !) + HT (Ai; e; !)

We observe that the rotational, torsion and connection constraint imply the remaining

equations which assure that the connection to be constructed on IR�� is torsionfree:

Because of the rotational constraint the extrinsic curvature is symmetric. Note that

this is a secondary constraint here whereas it is a primary constraint in the admdescription. The other equations are found in the following way:

0 � CCij^ ej = 2(dN + bk ^ e

k) ^ �ei +NT j ^ �eij � 2(dN + bkek) ^ �ei

=) dN + bkek � 0 (3.18)

because of the torsion constraint and thus both terms in this sum for CC vanishseparately, hence

T k^ �eijk � 0 (3.19)

and this equation implies T k � 0 as shown above (3.4). Note that these argumentsare independent of the space dimension. In order to simplify the constraint analysiswe substitute torsion and connection constraint by the following equivalents:

C i1 := dei + !i

j ^ ej� 0 (3.20)

C2 := dN + bidei� 0 (3.21)

We do not expect any tertiary constraints since there is no equivalent on the co-

variant level, but it is not obvious that there exists an extension of the Hamiltonianby a sum of integrated primary constraints which does conserve all constraints. Tocheck the absence of tertiary constraint it is necessary to �nd integration functions

K1;K2;K3;K4; �5 such that

fH +Z�K1pN +

Z�K2

ipN i +Z�K3ijpaij +

Z�K4ipbi +

Z��5ij ^ p!ij ; CH=D=R=1=2g � 0

(3.22)

We will �rst analyze the constraints because then it will become obvious how to choose

these integration variables. For de�nition and a survey we summarize once again all

constraints we have found. One should note the di�erence of test integration forms

and canonical variables:

17

primary secondary

H3(�ij ; p!ij) := 1

2

R��ij ^ p!ij

H1(�i; ei; !ij) :=

R��i ^ (dei + !ij ^ e

j)

second �ij = ��ji 2 1(�) �i 2 1(�)

class H4(Ri; pbi) :=R�Ripbi H2(�; e

i; N; bi) :=R�� ^ (dN + bie

i)

Ri 2 C1(�) � 2 2(�)

�rst HN (U ; pN ) :=R�UpN HH(X; ei; pi; !ij) =

R�

X2[hpij �e

jihpjj �eii � 1

2p2 �R]�

class H ~N(V i; p ~N ) :=

R�V ipNi HD(~Y ; ei; pi) =

R�pi ^ L ~N

ei

Ha(Wij ; paij ) :=R�Wijpaij HR(Zij ; e

i; pi) := �R�Zijp

i ^ ej Zij = �Zji

Theorem:

The constraints H1; : : : ;H4 form second class pairs. The other constraints HN ; H ~N ;

Ha;HH; HD; HR can be substituted by equivalent constraints HN ; H ~N ; HA; HH;

HD;HR which are �rst class. The Poisson bracket relations of those substitutedconstraints correspond to the relations in the adm formulation and since they are�rst class, they also equal the Dirac brackets of the original constraints which one

could have calculated directly as well. All surface terms are consequently neglected.

Proof:

The constraintsH1; : : : ;H4 are obviously second class, one obtains for (fHi;Hjg)ij=1:::4

a skewsymmetric matrix of functionals:

(fHi; Hjg)ij=1:::4 =

0BBB@0 0 �

R��ij ^ �i ^ ej 0

0 0 0R�Ri� ^ eiR

��ij ^ �i ^ ej 0 0 0

0 �R�Ri� ^ ei 0 0

1CCCA =: (Sij)ij=1:::4

(3.23)

This matrix is invertible and thus the matrix of Poisson brackets of all constraints hasat least rank 4 and because the constraints H1; : : : ;H4 yield this simple symplectic

structure we regard these constraints as the fundamental second class pairs. We know

that on the surface described by these second class constraints the induced connection

is torsionfree, but the extrinsic curvature is not necessarily symmetric. We suppose

that all other constraints are �rst class, since there should not be less �rst classconstraints than in the adm case. Starting with the primary constraints we can set

H ~N := H ~N HA := Ha (3.24)

since these constraints are �rst class. HN does not commute with H2 because of thelapse dependance. Thus we substitute HN in the following way:

HN (U ; pN ; pbi) := HN (U ; ei; pi)�H4(hdU jeii; pbi)

=Z�

�UpN � hdU jeiipbi

�(3.25)

18

One can easily check that this constraint commutes also with H2. Regarding the

secondary constraints we notice that none of them commutes with H1 and H2 because

of their momentum dependance. But the Hamiltonian constraint HH does not even

commute with the primary constraint H3. And indeed calculating for example the

Poisson bracket

fHH(N);HH(M)g = 0 (3.26)

we obtain a result which di�ers from the adm result, because here the Ricci-scalar-

term remains underived since the connection form ! does not depend on the triads e

before restriction to the constraint surface and thus �H(N)�ei

and �H(N)�pi

are both linear

in N . In order to make the Hamiltonian constraint HH commute with H3 we have toadd a term H1 with a certain argument which we now determine:

~HH(X) := HH(X) �H1(�i(X))

f ~HH(X);H3(�ij)g '3 0

fHH(X);H3(�ij)g =Z�

h�d(X �eij)�X!j

k ^ �eik�X!i

k ^ �ekji^

12�ij

(1:23)' �1

2

Z��ij ^ dX ^ �e

ij

fH1(�i);H3(�ij)g = �

Z��ij ^ �

i^ ej

Thus we obtain �i ^ ej � �j ^ ei = +dX ^ �eij or equivalently:

�i ^ �eki = +dX ^ ek

In the same way as we transformed pi to !0i in (2.20) we obtain

�i = �hdX ^ ekj �eiiek + 1

2hdX ^ ejj �e

jiei = �(dX ^ ei)

and our corrected Hamiltonian constraint ~H reads:

~H(X) := HH(X)�H1(�(dX ^ ei))4 (3.27)

~H; HD and HR do now commute with all primary constraints. Forming the Poisson

brackets among themselves we are already back to the adm algebra if one restricts

3' denotes: after restriction to the second class surface4The correction term seems to be a pure surface integral

H1(�(dX ^ ei)) =R��(dX ^ ei) ^ (dei + !ij ^ e

j)

=R�D � (dX ^ ei) ^ e

i �R�d��(dX ^ ei) ^ e

i�

(2:31)=

R��(ridX � ei�X) ^ ei �

R@�

: : :

= �2R�rir

iX�;

but Gauss's law does not hold for connections with torsion:

ririX� 6= L(ei�riX)� = d(riX � ^ � ei)

19

to the second class constraint surface after calculation of the brackets. We will show

this for the two most di�cult examples:

f ~HH(X); ~HH(Y )g

= �fH1(�(dX ^ ei));HH(Y )g � fHH(X);H1(�(dY ^ ei))g

= �

Z�

�

�ei

h�(dX ^ ek) ^ (de

k + !kj ^ e

j)i^ Y

�hpj j �e

iiej � 1

2pei

�+ (X ! Y )

= �

Z�

hd � (dX ^ ei)� !k

i ^ �(dX ^ ek)i^ Y

�hpj j �e

iiej � 1

2pei

�+ (X ! Y )

�

Z�

"�

�ei� (dX ^ ek)

# �dek + !k

j ^ ej�^ Y

�hpj j �e

iiej � 1

2pei

�+ (X ! Y )

'

Z�� � (ridX � ei�X) ^ Y

�hpj j �e

iiej � 1

2pei

�+ (X ! Y )

(2.33)= HD(XdY ]

� Y dX])�HR

�hdX ^ dY jeiji+ hXdY � Y dXj!i

j

�(3.28)

In order to calculate the bracket fHD(~Y ); ~HH(X)g we split HH(X) in the parts

HH kin(X; e; p) := 12

Z�X�hpij �e

jihpj j �e

ii � 1

2p2�

(3.29)

HH pot(X; e; p; !) := �12

Z�XRij(!) ^ �e

ij (3.30)

and obtain:

fHD(~Y );HH kin(X)g = HH kin(L~YX)

fHD(~Y );HH pot(X)g = 12

Z�L~Y e

i^XRjk(!) �e

jki

'12

Z�L~Y �e

jk^XRLC

jk (e)

= 12

Z�

�L~Y (X �e

jk^Rjk)

�(L~YX)(Rjk ^ �ejk)�X � ejk ^ L~YRjk(e)

�mod.bound.terms.

(2:30)= HH pot(L~YX) �

Z�L~Y e

j^ �(rjdX � ej�X)

' HH pot(L~YX) + fHD(~Y );H1(�(dX ^ ei))g

Thus we have

fHD(~Y ); ~H(X)g ' HH(L~YX) ' ~HH(L~YX) (3.31)

since the correction term vanishes on the second class constraint surface.

In the last mainly technical step we must change ~HH ; HD and HR in such a waythat they also commute with H1 and H2, the secondary second class constraints. It

20

is obvious that one has to add H3- and H4-terms with certain integration functions.

We just show this for the case of the rotational constraint:

fHR(Zij);H1(�i)g =

Z�Zije

j^D�i

m:e:f:'

Z�DZij ^ �

i^ ej

fHR(Zij);H2(�)g =Z�Zijb

i� ^ ej

Thus we substitute HR in the following way:

HR(Zij) := HR(Zij) +KR(Zij;!ij ; bi; p!ij ; pbi) := HR(Zij)�H3(DZij)�H4(Zijbj)

(3.32)

In the same way one can �nd correction terms for the di�eomorphism and the Hamil-tonian constraint HD and ~HH :

HD(~Y ) := HD(~Y ) +KD(~Y ) KD(~Y ) = KD(~Y ; e; !; b;N; p!; pb)

KD(~Y ) := H3(L~Y !ij) +H4(bjhL~Y ejjeii) (3.33)

HH(X) := ~H(X) +KH(X) KH(X) = KH (X; e; p; !; b;N; p!; pb)

KH(X) := �H3

�12[ii�j � ij�i + ekiij�k]

��H4

�bi(Xhpj j �e

ii �12p�ij)

�(3.34)

�i := DhXhpk j �eiie

k �12pei

iThe Hamiltonian constraint is quadratic in the momentum p, and thus the correctionterm depends on p. One should convince oneself that despite of the dependance of

the integration functions on the canonical variables the Poisson brackets restricted tothe second class constraint surface are unchanged. Using Leibniz's rule for derivingthe correction terms the derivation of the integration function yields in any case theconstraint functional which vanishs after restriction to the constraint surface.

We will show now at an example that having proved that there are exactly two pairs

of second class constraints one could also have calculated the Dirac brackets of the

original constraints in order to �nd the Poisson bracket relations of the corrected �rst

class constraints:

fHH(X);HH(Y )gD = fHH(X);HH(Y )g � fHH(X);Hi(�)gSij(�; �)fHj(�);HH(Y )g

= �fHH(X);H1(�)gS13(�; �)fH3(�);HH(Y )g

�fHH(X);H3(�0)gS31(�0; �0)fH1(�

0);HH(Y )g

where Sij is the inverse of the matrix of second class constraints. This expression

should be independent of the chosen integration forms � and �, and so we choose� depending on Y resp. �0 depending on X, such that the product S13(�; �) �

fH3(�);HH(Y )g and fHH(X);H3(�0)gS31(�0; �0) is 1.The calculation will be equiv-

alent to the determination of the correction term for HH to ~HH :

S13(�; �) =�Z

��ij ^ �

i^ ej

��1

21

fH3(�);HH(Y )g = 12

Z��ij ^ dY ^ �e

ij =Z�h�ij je

iihdY jeji�

=Z��ij ^ �(dY ^ e

i) ^ ej

So one has to choose �i = �(dY ^ei) and �0i = �(dX^ei) and the calculation proceeds

as shown in (2.33):

fHH(X);HH(Y )gD = �fHH(X);H1

��(dY ^ ei)

�g+ fHH(Y );H1

��(dX ^ ei)

�g

' HD(XdY ]�Y dX])�HR

�hdX^dY jeiji+ hXdY �Y dXj!i

ji

�One could as well have chosen � depending on X and �0 depending on Y , suchthat fHH(X);H1(�)gS

13(�; �) and S31(�0; �0)fHH(�0);H(Y )g is 1. This calculation

is equivalent to the determination of the correction term for ~HH to HH.

Finally we have to show that there is an extension of our Hamiltonian by primary

constraints which conserves all constraints. Since our Hamiltonian is a sum of con-straints:

H(e; p;N; ~N; !; a; b) = HH(N) +H1(�(bjeji)) +HD( ~N ) +HR(a)

' ~HH(N) +HD( ~N) +HR(a)

and we know how to correct these constraints in order to get �rst class constraints we

suppose that

H := H(e; p;N; ~N; !; a; b) +KH(N) +KD( ~N) +KR(a)| {z }special �rst class Hamiltonian

+ HN (U) +H ~N (~V ) +HA(W )| {z }

arbitrary primary �rst class term

(3.35)

is a �rst class Hamiltonian. This is actually the most general �rst class Hamiltonian

for our in�nite-dimensional problem ([5](1.32)). To prove that the �rst part is �rstclass, we rewrite it in the following way:

H = H(e; p;N; ~N; !; a; b) +KH(N) +KD( ~N ) +KR(a)

= HH(N) +H1(�(bjeji)) +KH(N) +HD( ~N) +HR(a) (3.36)

Forming Poisson brackets with primary constraints yield secondary constraints, form-

ing Poisson brackets with secondary constraints yield secondary constraints, since oneobtains the same result as if one had taken the bracket with

HH(N) +HD( ~N) +HR(a) = HH(N)�H1(�(dN ^ ei)) +KH(N) +HD( ~N ) +HR(a)

apart from a term H1(�) which is a result of the di�erent derivations of the integrandfunctions of the H1 term and which vanishes on the surface described by the second

class constraints.

22

Thus we have shown that three of the four di�erent equations which represent in

3+1 dimensions that the connection on the fourdimensional manifold is torsionfree

are realised by second class constraints, whereas the fourth is obtained by a �rst class

constraint. If one restricts to the second class constraint surface from the beginning the

only di�erence to the adm formulation is that there is one more secondary constraint

and that the part of the phase space where the rotational constraint does not equal

zero can be interpreted as a region, where the zero component of the torsion does

not vanish. As a by-product we have seen how Dirac's framework for second class

constraints works for a �eld theory when the canonical variables are forms rather than

functions.

4 The Ashtekar formulation

It is well known the main idea of the Ashtekar formulation is the restriction to fourdimensions and the use of a complex selfdual connection [6]. We will show that the

e�ect is a canonical transformation in the complexi�ed phase space. In order toobtain a simpli�cation it is necessary to perform a second transformation which isnormally done by the use of densities. We are free to decide if we want to regard theselfdual connection as an independent variable or if we prefer to work with the uniqueselfdual connection given by the selfdual projection of the Levi-Civit-connection. Sincewe know that the Palatini theory reduces to the adm theory in the vacuum case

{ otherwise spin could couple with torsion [4] { we attempt a formulation wherethe four-dimensional connection is the complex selfdual projection of the Levi-Civit-connection. It is no contradiction that the connection form of the spatial restrictionof the connection turns out to be one of the canonical variables, as in the adm theoryit is no contradiction that the momentum consists mainly of one component !0i of

the connection. Before describing the canonical formulation we shortly summarizesome selfdual notation. Our �rst step is the transition to the complexi�ed tangential

bundle over the real space-time manifold M . We consider the complexi�ed frame

bundle LC(M) which carries a certain real structure, because we still regard themanifold as real. We will regard the complexi�ed bundle of tetrades where the gauge

group is now SOC(1; 3) as a subbundle of the tangential bundle. As usual there is aunique metric torsionfree connection whose components

�!�� =12

h�g��; � �g� ;� + �g �;� + �C�� �

�C� � + �C ��

i(4.1)

are real only if they belong to a real basis, for example a coordinate basis.

In the Lie-algebra soIR(1; 3) one can de�ne the dualization by

A�� :=12���

��A�� (4.2)

23

Because of the signature of the metric one �nds

e~A = �A (4.3)

so the eigenvalues of the dualisation operator are �i and hence a decomposition in

eigenvectors is only possible in the complexi�ed algebra soC(1; 3). One calls elements

of soC(1; 3) with the property~A = iA selfdual

and ~A = �iA antiselfdual.

The subspaces so+C(1; 3) := fA 2 soC(1; 3)j ~A = iAg and

so�C(1; 3) := fA 2 soC(1; 3)j ~A = �iAg

form ideals and it holds

soC(1; 3) = so+C(1; 3) � so�C(1; 3) :

For a proof one shows g[A;B] = [A; ~B] = [ ~A;B] (4.4)

One can introduce the projectors to the selfdual and antiselfdual parts

P�A := 12(A� i ~A) P�A =: A� (4.5)

and with help of (4.4) one can directly show

P�[A;B] = [P�A;B] = [A;P�B]; (4.6)

which proves that so�C(1; 3) form ideals of soC(1; 3). The connection form with respectto a tetrad base is soC(1; 3)-valued, and the associated curvature form is a soC(1; 3)-

valued 2-form. Since

�R+ + �R� = �R = d�! + �! ^ �! = d�!+ + �!+^ �!+| {z }

selfdual

+ d�!� + �!� ^ �!�| {z }anti-selfdual

+ [�!+ ;̂ �!�]| {z }0

(4.7)

the (anti-) selfdual part of the curvature is the curvature to the (anti-) selfdual part

of the connection form. We now return to the action principle. Using the relation

�R�� ^ �e� = 0 (4.8)

we can consider the selfdual Einstein-Hilbert action instead of the usual one.

S =Z�

�R+

�� ^ ��e�� =

1

2

Z�( �R�� ^ ��e

��� i �R�� ^ �e

��) =1

2

Z�

�R�� ^ ��e�� (4.9)

Varying with respect to the tetrades yields (the complexi�ed version of) Einstein's

equations. In order to obtain the Hamiltonian formulation we decompose the Lagrangeform (�0ijk � �ijk):

24

�L = �R+

�� ^ ��e��

= 2 �R+

0i ^ ��e0i + �R+

ij ^ ��eij

= �i�ijk �R+ jk^ ��e0i + �R+

ij ^ ��eij

= +i�ijki0 �R+ jk^ ��ei + �R+

ij ^ ��eij

= d�t ^ [�ijk �Ni0 �R+ jk^

3��ei + �N �R+

ij^3��eij] (4.10)

Now we use the following equation

�Ni0 �R+ jk = �Ni0

�d�!+jk + �!+j

� ^ �!+�k�= i �N�e0d�!

+jk + di �N�e0 �!+jk�

�!+D(i �N�e0 �!+jk)

= L �N�e0d�!+jk�

�!+Di �N�e0 �!+jk; (4.11)

where L denotes the Lie-derivative and obtain

�L = d�t ^

�iL@=@�t�!

+jk ^ �e

jk� iL �~N

�!+jk ^ �e

jk� i �!

+

Di �N �e0 �!+jk ^ �e

jk +N �R+

ij^3� �eij

�:

(4.12)

Since �R+ij = d�!+

ij + �!+ik ^ �!+k

j + �!+0i ^ �!+

0j

= d�!+ij + 2�!+

ik ^ �!+kj (4.13)

one de�nes a connection form on � as follows

Aij := 2i�t �!+ij = !ij + i�ij

k!0k (4.14)

{ here !ij and !0i denote the Levi-Civit-connection and the extrinsic curvature on �as usual { because for the curvature F de�ned by A holds

Fij = 2i�t

�d�!+

ij + 2�!+ik ^ ��!+k

j

�= 2i�t

�R+ij: (4.15)

Using the de�nition one obtains

i�tL@=@�t�!+jk = 1

2_Ajk (4.16)

i�t�!+Di �N�e0 �!

+jk = ADi�t i �N�e0 �!+jk =: 1

2ADZjk (4.17)

Zjk = �Zkj := 2Ni�t i0�!+jk = ajk + i�jk

lbl: (4.18)

As usual we pass over to the spatial Lagrange form:

L = i�t i@=@�t�L

= 12

hi _Ajk ^ e

jk� iL ~NAjk ^ e

jk� i ADZjk ^ e

jk +NFij ^ �ejki

(4.19)

At this point one could recognize 12i�i

jk _A as the canonical momentum of �12�ijke

jk.

But this would be obvious only in a Palatini-like theory where the connection is not�xed to be the selfdual projection of the Levi-Civit-connection. On the other hand

25

usually one does not vary w.r.t. A in order to obtain the Levi-Civit-connection. We

are more careful and keep in mind that the de�nition of A implies a dependance on the

extrinsic curvature, which is closely linked to the time derivative of the triads. Thus

the connection form A itself depends on _e, so every term of the sum in the Lagrange

form contributes to the calculation of the momentum. To determine the momentum

we write the Lagrangian in the following way

L(e; _e;N; ~N) =d

dt

Z�

i

2Ajk ^ e

jk�

i

2

Z�L ~N (Ajk ^ e

jk)�i

2

Z�d(Zjke

jk)

+Z�

�i _ej ^Ajk ^ e

k� iL ~Ne

j^Ajk ^ e

k + iZjkADej ^ ek +

N

2Fij ^ �e

ij

�:(4.20)

We ignore the boundary terms and notice that one of the terms vanishs in the admlike Ashtekar theory:

ZjkDej^ ek = Zjk

!Dej ^ ek + iZjk�mj

l!0mel^ ek

= Zjk!0m ^ �emj^ ek

= Zjk(h!0kjeji � �jkh!0mje

mi)� = 0 (4.21)

Considering this result we are not surprised since an explicit dependance on Zij wouldmean an explicit dependance on the rotational parameter aij and bi de�ned in (2.9)

and (3.7), which does not appear in the adm Lagrangian. But there is a dependanceon _ei

A(2.15) contrary to the adm theory in the �rst term:

i _ej ^Ajk ^ ek = i _ejA ^ Ajk ^ e

k + i _ejS ^Ajk ^ ek

= i _ejA ^ !jk ^ ek + i _ejS ^Ajk(e

i; _eiS) ^ ek

= �i _ejA ^ dej + i _ejS ^Ajk(ei; _eiS) ^ e

k and (4.22)

Ajk = !jk(e) + i�jkl1N(L ~Ne

lS � _elS) (4.23)

So the Lagrangian depends linearly on _ejA, but is quadratic in _ejS . This means a

second class constraint for _ejA, so that one can disregard this degree of freedom asone knows from the adm theory. We can guess at this point that the canonical

momentum which we derive now has an additional contribution �idej and hence

hpij �eji is no longer symmetric. We now derive the momentum-2-form.

L =Z�i _ej ^Ajk ^ e

k� iL ~Ne

j^Ajk ^ e

k +N

2Fij ^ �e

ij (4.24)

� _ei

Z�i _ej ^Ajk ^ e

k = i

Z�� _ei ^Aik ^ e

k + i

Z�_ej ^ � _eiAjk(e; _e) ^ e

k

i

Z��j ^ � _eiAjk(e; _e) ^ e

k = �1N

Z�� _ei

�_elS ^ �

j^ �elj

�= �

1N

Z�� _ei

�_el ^ �jS ^ �elj

�= �

1N

Z�� _ei ^ �jS ^ �eij ;

26

where �i 2 1(�) and �iS = 12(�i + h�j jeiiei) is de�ned as usual. Hence we obtain

�

� _ei

Z�i _ej ^Ajk ^ e

k = iAij ^ ej�

1

N_ejS ^ �eij

�

� _ei

Z�iL ~Ne

j^Ajk ^ e

k =1

NL ~Ne

jS ^ �eij

�

� _ei

Z�

N

2Fjk ^ �e

jk =�

�ei

Z�

N

2(i�t

�Rjk + i�jkli�t

�R0l) ^ �ejk

=�

� _ei

Z�

�N

2(Rjk + !0j ^ !0k) ^ �e

jk + iNi�t (�R0l ^ �el)| {z }

0

�

= �!0j ^ �eij

pi =�L

� _ei= iAjk ^ e

k�

1

N( _ejS � L ~Ne

jS +N!0

j) ^ �eij = iAjk ^ ek : (4.25)

We compare this with the momentum 2-form of the adm theory

pi = iAij ^ ej = i!ij ^ e

j� �kij!0k ^ e

j = �idei + pADMi (4.26)

and notice that the new momentumdoes not transform homogeneous under rotations.As in the adm theory the equation (4.25) can only be solved on a submanifold of thephasespace on which the following condition holds:

0 = iAij ^ ej � pi = �idei + !0l ^ �ei

l � pi

=) (pi + idei) ^ ej � (pj + idej) ^ ei

= pi ^ ej � pj ^ ei + id(eij) = 0 (4.27)

Our Hamiltonian is only determined up to this constraint, so we obtain:

H(ei; pi; N; ~N;Z) =Z�_ei ^ pi � L�

Z�Zij(p

i^ ej + idei ^ ej)

=Z�

hpi ^ L ~Ne

i� Zij(p

i + idei) ^ ej � N2Fij(p) ^ �e

iji(4.28)

We call the arbitrary integration function of the rotational constraint Z, because if we

derive the equations of motion and want to compare the Hamiltonian equation withthe geometrical equation obtained by i�tL@=@�t we have again to identify the integration

parameter with a space-time quantity by equation (4.18). We will see in a moment

that this Hamiltonian is not suitable for further consideration, since the substitutionof the curvature form Fij by momentum terms ends up in a lengthy expression. The

substitution of A by p on the constraint manifold yields

Aij = �i(hpkjeijiek�

12p �eij) p := hpij �e

ii: (4.29)

For convenience we de�ne also

Ai :=i2�ijkAjk Ajk = �i�ij

kAk (4.30)

27

and analogously Fi and Zi. Then the relation between Ai and pi reads

Ai = hpkj �eiiek� 1

2pei () pi = Ak ^ �e

ki (4.31)

and �nally we obtain for the last term of the Hamiltonian

�N2Fij ^ �e

ij = iNFk ^ ek modulo exact forms

= N2

�hpij �e

jihpj j �e

ii �

12p2 + 2ihpij �ejihde

jj �eii � iphdeij � e

ii��

�ihdN jeiipj ^ �eij: (4.32)

This is quite an awkward expression for deriving the Hamiltonian equations, buthaving in mind that

N2R� = N

2

h�d(ei ^ �dei)�

12(dei ^ e

j) ^ �(dej ^ ei) + 1

4(dei ^ e

i) ^ �(dej ^ ej)i

(4.33)one can easily see that one could have obtained this Hamiltonian by a canonicaltransformation (ei; pADMi ) 7�! (ei; pADMi � idei) =: (ei; pi) of the adm Hamiltonian(2.23). So it is not surprising that the Hamiltonian equations one could derive from(4.28) are really equivalent to the adm equations of motion, but they contain even

more terms than in the adm case. One notices that the canonical transformation isonly possible for spatial dimension 3, since otherwise the 2-form dei can not be addedto the n � 1-form pi. The gauge group SO(3) acts on the con�guration space whichis invariant under the canonical transformation. But wheras the action of the groupto the momentum can in the adm case be obtained by the obvious lift to the phase

space, one has to transform also the group action to the new phase space in order toobtain the correct action of the group to the new momentum

ei 7�! Sije

j pADMi 7�! S�1jip

ADM

j = SijpADMj

pi 7�! Sijpj � idSi

j ^ ej S 2 �� SOC(3) (4.34)

In order to simplify the expression one considers again the Lagrange form and notices:

L =Z�

h�

i2Ajk ^ (e

jk)� + i2Ajk ^ L ~Ne

jk + N2Fij ^ �e

iji

(4.35)

One would like to perform another canonical transformation which turns the conection

form A itself into the momentum. We now de�ne and would like to determine thecanonical conjugate qi 2 2(�) such that� Z

�qi ^ �i;

Z�Aj ^ �

j

�=

Z�

"��

�ek

Z�qi ^ �i

�^

��

�pk

Z�Aj ^ �

j

��

��

�pk

Z�qi ^ �i

�^

��

�ek

Z�Aj ^ �

j

�#

=Z��i ^ �

i

28

where �i 2 1(�) and �j 2 2(�). Using equation (4.31) one can calculate the

derivatives of the second term.

�

�pk

Z�Aj ^ �

j = ejh�jj �eki �

12ekh�jj �e

ji

�

�ek

Z�Aj ^ �

j = �jhpkj �eji � h�jj �eiiikpi ^ ej �

12�kp+

12ikpj ^ e

jh�ij �e

ii

One can now guess ��pk

R� q

i ^ �i = 0, because otherwise one can hardly get rid of

the momentum dependence. Consequently one supposes for the canonical coordinate

qi = c �ei and obtains

�

�ek

Z�qi ^ �i = c�ikle

l^ �i = c � eik ^ �i and

c � eik ^ �i ^�ejh�

jj � eki � 1

2ekh�jj � e

ji

�= �c�i ^ �

i

so our new con�guration variable isqi = � � ei: (4.36)

Introducing coordinates fxaga=1;2;3 for a moment we can link these varibles with thedensities used in Ashtekar's formulation. Since in integrals one has expressions likeZ

�qi ^ �i =

12

Z�dxa ^ dxb ^ dxcqiab�ic =

12

Z�d3x�abcqiab�ic

we consider

12�abcqiab = �

14�il�abc�jkle

jkab = �

12�il�abc�jkle

jae

kb = ��

ilecl det eia =

ecldet eai

= ��ilEcl

(4.37)

where eia = ei(@=@xa); eai = dxa(ei) and thus eiaeaj = �ij . So the densities Ec

l are justthe coordinate expressions of the 2-forms qi { up to a sign and an �-symbol usually

hidden in the volume d3x. The densities Eai determine directly the coordinates of the

dual triads

eai =Eai

det1=2Eai

(4.38)

and these of course determine the triads itself

eia =1

2det eai�ijk�abce

bje

ck =

1

2det1=2Eai

�ijk�abcEbjE

ck (4.39)

what insures that the metric can be reconstructed from the new variables.

Substituting (ei; pi) in the Hamiltonian by (qi; Ai) one obtains

H(qi; Ai) =Z�Ak ^ �e

ki ^ L ~Ne

i� Zij(Ak ^ �e

ki + idei) ^ ej � N

2�ijkFij ^ e

k

=Z�Ak ^ L ~Nq

k + ZkDqk� iNFk(A) ^ �q

k (4.40)

29

where Dqk = dqk +Akj ^ q

j = dqk � i�kjlAl^ qj

D~q = d~q + i ~A�^ ~q (4.41)

and Fk = dAk +i2�k

ijAi ^ Aj

~F = d ~A+ i2~A�^ ~A (4.42)

Using the vector notation we obtain what we will call the Ashtekar Hamiltonian:

H(~q; ~A) =Z�

�~A

�^ L ~N~q �

~Z�^ D~q � iN ~F

�^ �~q

�(4.43)

One easily recognizes the three terms as di�eomorphism, rotational and Hamiltonian

part. The derivation of the equations of motion is simple up to a single point: We have

to derive a termR� ~�

�^ �~q =

R� �i ^ �q

i with respect to the 2-form qj. Reexpressedby the triads the problem reads:

�

� �ei

Z�em ^ �m �i 2 2(�)

Since there are as many independent dual triads as triads itself for any space dimensionand one can reexpress the metric in terms of coordinates of the dual forms �ei as well

as in terms of the triads the problem has a simple solution which we described in(1.32). Then we �nd the following equations of motion:

_~q =�H

� ~A= L ~N

~q � i ~Z � ~q � iD(N �~q) (4.44)

_~A = ��H

�~q= L ~N

~A+D~Z �N ~RicF � N4F �~q (4.45)

Here ijFji =: ( ~RicF )i is the Ricci-form and F = hFij je

iji is the Ricci-scalar of F .These equations of motion are the complex extensions of the adm equations as can

be checked using the following relations

Ai = i2�ijkAjk =

i2�ijk!jk � !0i (4.46)

Zi = i2�ijkZjk =

i2�ijkajk � bi =

i2�ijkajk + hdN jeii (4.47)

In order to check the equations one needs the equation of motion for !0i which caneither be obtained from the adm equations of motion or by use of the equation of

motion for the extrinsic curvatureKab, where the indices are w.r.t. a �xed (coordinate)

basis fdxaga=1;2;3[3]:

_Kab = L ~NKab + 2NKacKcb �NKKab

�NRicRab +rarbN + N4(R �KcdK

cd +K2)hab

30

hab = �ijei(@a)e

j(@b) !0i = �Kabdxbeai eai = dxa(ei)

Using _eai = dxa( _ei) = �dxa(ej) _e

j(ei)

_ej = L ~Nej � ajke

k�N!0

j

one �nally obtains

_!0i = �(Kabeai)�dxb

= L ~N!0i �DhdN jeii + aki!0k +

N(RicRij �KikKkj +KKij)e

j�

N4(R�KklK

kl +K2)�ijej: (4.48)

It is nearly obvious that this equation is the real part of equation (4.45), if one

regards all di�erential geometric objects as real, i.e. ei; ~N;N; aij; !0i. Regarding the

equations of motion and having in mind the relation hdN jeii+ bi = 0 and (4.47) onecan easily understand why in the equation of motion for _q there appears a derivativeterm of the lapse contrary to the adm theory and why there is no derivative term forthe lapse in the equation for A. If one decomposed Z into real and imaginary partanother derivative term for the lapse would appear. We notice a di�erence to the

usual Ashtekar formulation. There is no need to densitize our lapse function. If wehad done so, the last term of equation (4.45) containing the Ricci-Scalar of F wouldvanish. This is not dramatic since we will see in a moment that the Hamiltonianconstraint in the Ashtekar formulation is just F = 0 and since our constraint algebrawill prove �rst class we will never leave the constraint surface. The Ricci-scalar-term

corresponds to the last term in equation (4.48) which we recognize as the the theEinstein-tensor G(n; n) or the Hamiltonian constraint of the adm formulation hereexpressed by the extrinsic curvature instead of the momentum. But since one doesusually not omit the terms with �ei in the adm equation (2.29) whose sum vanishson the constraint surface and in order to perform a correct constraint analysis, we

keep the Ricci-scalar term here. Finally we note that of course A does not transformhomogeneous under rotations as well and that the rotational part in the equation of

motion for A could have been derived easily from the canonical transformation of the

SO(3)-action, using equations (4.31) and (4.34).

Finally we will perform the full constraint analysis. We have already encountered

the �rst primary constraint, the rotational constraint. The secondary constraints

associated with lapse and shift read { as usual we disregard all boundary terms:

CDi = fH; pN ig =�

�N i

Z�Aj ^ L ~Nq

j

= Aj ^ iidqj + dAj ^ iiq

j

= iiAjDqj + Fj ^ iiq

j (4.49)

CH = fH; pNg = �i ~F�^ �~q = �1

2F� (4.50)

31

Using the identity iiqj = ��ki

j �qk and the relation:

(~F�^ �~q)i = �ijkhF

jjeki� = � i

4�ij

k�jlm�krshFlmjersi�

= i( ~RicF�^ ~q)i (4.51)

we can write the di�eomorphism constraint as follows

CDi = iiAjDqj� i( ~RicF

�^ ~q)i (4.52)

and the integrated version then reads

HD( ~N) =Z�

~A�^ L ~N~q =

Z�

�~A( ~N )D~q � i ~N( ~RicF

�^ ~q)

�(4.53)

We reformulate also the rotational constraint in our variables

CRi = Dqi (4.54)

and notice that the di�erential form of the di�eomorphism constraint can be simpli�edusing the rotational constraint can be simpli�ed using the rotational constraint

~CD i = �ii ~F�^ ~q = �i( ~RicF

�^ ~q)i (4.55)

Usually this version of the di�eomorphism constraint is used in the Ashtekar formu-

lation, its integrated form reads

~HD( ~N) = �Z�i ~N~F

�^ ~q = �i

Z�

~N( ~RicF�^ ~q); (4.56)

but we prefer HD =R�~A

�^ L ~N~q, because it is just the momentummapping of action

of the di�eomorphism group on the phase space. Thus it is not even necessary tocalculate the Poisson-bracket between two di�eomorphism constraints in this form.

That the di�erential part of the di�eomorphism constraint has a rotational part is nota special feature of the Ashtekar formalism as we have seen in (2.25). We can write

the Hamiltonian as a sum of integrated constraints, neglecting all boundary terms:

H =Z�N iCDi +

Z�ZiCR

i +Z�NCH

= HD( ~N) + HR(~Z) + HH(N)(4.57)

We determine again the constraint algebra, proving that the constraints are �rst class:

fHD( ~N);HD( ~M)g = HD([ ~N; ~M ])

fHD( ~N);HR(~Z)g = HR(L ~N~Z)

fHD( ~N);HH(N)g = HH(L ~NN)

fHR(~Z1);HR(~Z2)g = �iHR(~Z1 � ~Z2)

fHR(~Z);HH(N)g = 0

fHH(N);HH (M)g = HD(NdM ] �MdN ])�HR(h ~AjNdM �MdNi)

(4.58)

32

The �rst and the third equation are proved as in the previous sections.

fHD( ~N);HR(~Z)g =Z�

�iL ~N

~A�^ (~Z � ~q) + L ~N~q

�^ D~Z

�=

Z�

�iL ~N

~Z( ~A� ~q)� L ~N~Z

�^ d~q

�= +

Z�L ~N

~Z�^ D~q = HR(L ~N

~Z)

fHR(~Z1);HR(~Z2)g =Z�

�iD ~Z1

�^ (~Z2 � ~q)� iD ~Z2

�^ (~Z1 � ~q)

�= �i

Z�

~(~Z1 �~Z2)

�^ D~q = �iHR(~Z1 �

~Z2)

fHR(Z);HH(N)g =Z�

�iD ~Z

�^ D(N �~q) + iN(~Z � ~q)( ~RicF + 1

4F �~q)

�=

�(~F � ~Z)

�^ �~q � i( ~RicF

�^ ~q) � ~Z

�(4:51)= 0

fHH(N);HH(M)g =Z�

�� iN( ~RicF + 1

4F �~q)

�^ D(M �~q)

+iM( ~RicF + 14F �~q)

�^ D(N �~q)

�=

Z�i( ~RicF

�^ �~q) ^ (NdM �MdN)

=Z�i ~RicF

�^ ~q � hNdM �MdN j �~qi

(4:53)= HD(NdM ]

�MdN ])�Z�h ~AjNdM �MdNi

�^ D~q

= HD(NdM ]�MdN ])�HR(h ~AjNdM �MdNi)

So these variables permit a simple derivation of the constraint algebra. At this pointwe have managed to give an "adm like" Ashtekar formalismwhere only the metric was

regarded as dynamical variable whereas the connection was �xed as the Levi-Civit-connection or its selfdual projection. In a Palatini-like Ashtekar theory, where also

the connection is regarded as a variable, the solution of the constraints should leadto equation (4.46) by which the arbitrary connection form is linked to the Levi-Civit-

connection of the triads and the extrinsic curvature. Our analysis also gives a simpleinterpretation of the reality constraints. We know that the Ashtekar Hamiltonian is

just a canonical transformation of the complexi�ed adm Hamiltonian, so we consider

�rst the complexi�ed adm theory. Since the di�erential equation is real for a realrotational term it is clear that a real initial condition has a real time development, so

a pair of real triads and momentum forms (ei; pADMi ) as initial condition will producea real development of the metric, even in the case that the rotational parameter is

not real. Let us now suppose that the spatial metric represented by triads �ijei ej

is real for a certain time. Then there is a SOC(3)-valued function S on IR � � suchthat ~ei := Si

jej are real triads, having real components with respect to a coordinate

33

basis. One can easily see that (~ei; ~pADMi ); ~pADMi := SijpADMj satisfy the equations of

motion (2.28,2.29) for another rotational term. Considering the split equation for the

triads (2.14) one notices that provided lapse and shift are real the extrinsic curvature

~!0i is real and consequently the momentum form ~pADMi is real. Thus the most general

initial condition which leads to a real metric is a pair of real forms (ei; pADMi ) modulo

of course a possible complex rotation. One could write the condition in the form

�ijei ej = h = real (4.59)

�k(iei ej pADMk ^ ej)(2:37)= � = real; (4.60)

but one could as well simplify the second condition by

pi ei = real (4.61)

and replace the �rst one by

�ijpADMi pADMj = real (4.62)

Now we have only to transform this condition from the complexi�ed adm theory tothe Ashtekar variables. Formally one can restate the reality conditions as (qi; Ai �i2�jki !jk(q) = �!0i is real modulo a complex rotation or

�ijqi qj = real (4.63)

qi !0i(A; q) = real; (4.64)

but it is quite di�cult to obtain !ij from qi, so the second condition is di�cult tocheck given the coordinates of qi and Ai. The �rst condition is obviously equivalent

to the condition that the spatial metric h is real, so one can reexpress equation (4.60)in terms of Ashtekar variables using

pADMi = i�ijDej = �i�ijD �q

j; (4.65)

which is a simple consequence of equations (4.25) and (4.26). If one introduces coor-

dinates and uses the expressions (4.38) and (4.39) for the coordinates of triads anddual triads in terms of the densities one �nds

�ab123 = �cde(ei ej p(i ^ ej))(dxa; dxb; @c; @d; @e)

= �i1

detEai

( ~Ef�Df

~E(a) � ~Eb) = real

and this is up to the factor �1=detEia, which is clearly real, the coordinate expres-

sion for the second reality constraint, which is usually derived by requiring that theHamiltonian ow leaves the metric real [7]. So the reality constraints given in the

Ashtekar theory just state that the reconstructed adm metric and momentum is real.

34

Our analysis has shown that it might be useful to perform calculations in canonical

gravity as far as possible without reference to a �xed reference frame. We have seen

in which way the coordinate free version of Ashtekar's formulation is related to the

complexi�edadm theory and how the rotational parameters of the complexi�ed theory

are linked to the real theory. One might ask if triads have a real physical signi�cance

or if they are only useful tools for deriving equations for the metric. In Minkowski

space one usually derives the energy momentum tensor by considering the variation

of the Lagrangian as a consequence of a translation in space-time which could be

written as a derivation with respect to the 1-form dx�. The obvious generalisation

for a curved spacetime is the variation w.r.t. the four orthogonal 1-forms e� what

really yields the energy-momentum-tensor. This could be seen as an argument thatthe tetrades really are physically signi�cant.

Acknowledgment: I would like to thank P. Gl�o�ner for help and motivation.

References

[1] W. Thirring Lehrbuch der mathematischen Physik Bd. 2 Springer Wien, NewYork 1990

[2] A. Palatini, Rend. Circ. Math. Palermo 43, 203 (1919)

[3] J. SchirmerDas Wirkungsprinzip in der allgemeinen Relativit�atstheorie diplomathesis Freiburg 1994

[4] F. Hehl, P. von der Heyde, G. D. Kerlick, J. M. Nester, General Relativitywith Spin and Torsion: Foundations and Prospects Rev. Mod. Phys., Vol 48, 3,

393-416, 1976

[5] P. A. M. Dirac Lectures of Quantum Mechanics Yeshiva University Press, New

York 1964

[6] A. Ashtekar Lectures on Non-Pertubative Canonical Gravity, World Scienti�c,

Singapur 1991

[7] D. Guilini Ashtekar Variables in Classical General Relativity in Canonical Grav-

ity: From Classical to Quantum Ed. J. Ehlers, H. Friedrich, Springer, BerlinHeidelberg 1994

[8] R. Abraham, J. E. Marsden Foundations of Mechanics, Benjamin/Cummings,Reading Massachusetts 1978

[9] R. Arnowitt, S. Deser, C. W. Misner The Dynamics of General Relativity in:

Gravitation An Introduction to Current Research ed. L. Witten, Wiley, New

York 1962

35