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General Relativity and the Einstein Equations$

Yvonne Choquet-Bruhat

Print publication date: 2008

Print ISBN-13: 9780199230723

Published to Oxford Scholarship Online: May 2009

DOI: 10.1093/acprof:oso/9780199230723.001.0001

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(p.653) APPENDIX VII Kaluza–Klein Theories

(p.653) APPENDIX VII Kaluza–Klein Theories

Source:
General Relativity and the Einstein Equations
Publisher:
Oxford University Press

1 Introduction

Already in the 1920s Kaluza and Klein, in search of unification of gravitation with electromagnetism, considered the Einstein equations on a five-dimensional Lorentzian manifold with a 1-parameter isometry group of spatial isometries. They supposed that the Killing vector ξ has a constant length and neglected the equation R(ξ,ξ)=0. They showed that the 14 remaining Einstein equations in vacuo for (V 5, ĝ) split into Maxwell equations for a 2-form F, and Einstein equations for a spacetime (V 4, g) with source the corresponding Maxwell tensor. Later Jordan relaxed the hypothesis of the constancy of the length of ξ, interpreting the 15th potential as a varying gravitational constant. Shortly afterwards, Thiry, a student of Lichnerowicz, gave a geometric formulation of the equations. They interpreted the 15th potential as the induction of vacuum.

In 1968, Kerner indicated how the same technique applied to a d-dimensional spacetime with d-4-dimensional isometry group can lead to the coupled Einstein–Yang–Mills system.

2 Isometries

Recall that an isometry of a pseudo-Riemannian manifold (V̂, ĝ) is a diffeomorphism of V̂ which leaves ĝ invariant; see for instance CB-DM1 III D. Isometries of (V̂, ĝ) constitute a Lie group G r of dimension r with

(2.1)
r 1 2 d ( d + 1 ) if V ^ has dimension d

A one-parameter group of diffeomorphisms of V generated by a vector field X is a group of isometries (subgroup of G r) if and only if the Lie derivative of g with respect to X vanishes:

(2.2)
L X g = 0 , i .e . α X β + β X α = 0.

The orbit O x of a point x under the group G r is the set of all points of V into which x is mapped by all elements of G r. If the group acts simply transitively on O x, this orbit is a submanifold of V of dimension r.

(p.654) 3 Kaluza–Klein metrics

Let (V̂ ĝ) be a pseudo-Riemannian manifold of dimension and let G be a Lie group of dimension d of isometries of (V̂, ĝ), i.e. such that the right action of G on V̂ leaves ĝ invariant. We suppose this action of G on V̂ to be such that it endows V̂ with the structure of a G-principal fibre bundle whose fibres are the orbits of G. These orbits are then all diffeomeorphic to G and G acts simply transitively on them. The base is a n + 1= – d-dimensional manifold V. We denote by π the bundle projection π :V̂ → V. It is proved in CB-DM2 V 13 that, under these hypotheses, the datum of the metric ĝ on V̂ is equivalent to the data of a G-invariant metric Φ on each orbit of G (if these orbits are not null for ĝ}), a G-connection on V̂ whose horizontal spaces are the vector spaces orthogonal to the orbits (hence invariant under the action of G), and a pseudo-Riemannian metric g on the base V.

3.1 Metric in adapted frame

We consider a local trivialization φ of V̂ over the domain U of a chart in V; that is, a mapping φ,

(3.1)
φ : π 1 ( U ) U × G .
We choose local coordinates x̂M in V̂ adapted to the trivialization i.e. such that x̂α = x α, local coordinates in U, for α = 0,1…,n, and x̂m = y m, m=n+1,…,d̂, local coordinates in G. We take a moving frame (êα, êm) such that the êm are Killing vector fields of the right action of G on V̂ they are mapped by φ onto a basis e m of the right Killing vector fields on G. We choose the êα orthogonal1 to the êm and mapped by φ onto a basis e α of the tangent space to UV. Then
(3.2)
φ e ^ m = e m , φ e ^ α = e α A α m e m .
We denote by θ̂A the dual coframe of êA on V̂, by θα and θm the dual coframes respectively of e α on U and e m on G. We find that
(3.3)
θ ^ α = θ α and θ ^ m θ m + A α m θ α ,
One shows that A : = A α m θ α e m is the representant on V, in the trivialization φ, of a G-connection 1-form which takes its values in the Lie algebra G of G.

With this choice of frames the metric ĝ V̂ can be written, in π−1(U),

(3.4)
g ^ g ^ A B θ ^ A θ ^ B = g ^ α β θ ^ α θ ^ β + Φ m n ( θ m + A α m θ α ) ( θ n + A β n θ β )
where g g α β θ α θ β is a metric on V and Φ :=(Φmn) a set of functions on V, sometimes called a scalar multiplet. Since the Lie derivatives of the 1-forms θα and θm in the direction of a Killing vector êm vanish, the metric ĝ is invariant (p.655) under the right action of G on V̂ if and only if the coefficients g αβ, Φmn, g α β , Φ m n , A α m depend only on the local coordinates x α on V.

The metric g g α β θ α θ β on V, the connection A and the set of functions Φ:= (Φmn) are geometric elements on the basis2 V. Under a change of local trivialization of the bundle V̂ → V described by a transition mapping3 UG by xh(x) it holds that:

The metric g is invariant.

The connection A transforms as a representant of a G-connection,

(3.5)
A h 1 A h + h 1 d h .
The scalar multiplet transforms as
(3.6)
Φ h h Φ .

3.2 Structure coefficients

The structure coefficients c ^ B C A of a moving frame ∂̂A with dual frame θ̂A are defined by the relations

(3.7)
[ ^ B , ^ C ] = 1 2 C ^ B C A ^ A , d θ ^ A = 1 2 C ^ B C A θ ^ B θ ^ C .
To simplify the computations we choose for θα a natural frame on V, θα = dx α. Then
(3.8)
d θ ^ α = d θ α = 0 , hence c ^ B C α = 0

Since the θm are dual to Killing vectors on G it holds that

(3.9)
d θ m = 1 2 C p q m θ p θ q
where the C p q m are structure constants of the group G. We deduce from the expression of θm
(3.10)
d θ ^ m = 1 2 ( C p q m θ p θ q + ( α A β m β A α m ) d x α d x β ) .
The relation θ m = θ ^ m A α m d x α and the antisymmetry of C p q m imply that
(3.11)
d θ ^ m = 1 2 ( C p q m θ ^ p θ ^ q + 2 C p q m A α p d x α θ ^ q + F α β m d x α d x β )
with
(3.12)
F α β m α A α m β A α m + C p q m A α p A β p .
(p.656) We see from the definitions of the curvature of a connection on a principal fibre bundle (see CB-DM1 Vbis 4) that the F α β m are the components of the representative in the chart of V of the curvature of the connection A, since the curvature of A is the G valued antisymmetric tensor of type Ad4
(3.13)
F : = 1 2 F α β m d x α d x β e m d A + [ A , A ] .
We read on (3.11) that
(3.14)
C ^ n p m = C n p m , C ^ α β m = F α β m , C ^ α q m = C ^ p q m A α p

3.3 Kaluza–Klein connection

Recall that (Chapter 6) the connection coefficients ω ^ B C A of a metric ĝ are given in terms of the metric components ĝAB and the structure coefficients by

ω ^ B C A Γ ^ B C A + ω ˜ B C A ,
where are the Christoffel symbols constructed with the coefficients ĝAB and where
(3.15)
ω ˜ B C A 1 2 ( C ^ B C A g ^ C L g ^ M A C ^ B M L g ^ B L g ^ M A C ^ C M L )
We find easily that
(3.16)
Γ ^ β γ α = Γ β γ α , ω ˜ β γ α = 0 ,
where Γ β γ α are the Christoffel symbols of the metric g on V,
(3.17)
Γ ^ p q m = Γ ^ α β m = Γ ^ β q α = 0 ,
and, with Φmn the inverse matrix of Φmn
(3.18)
Γ ^ α q m = 1 2 Φ m n α Φ n q , Γ ^ p q α = 1 2 g α β β Φ p q
We now compute the ω˜ terms and we find, Greek indices raised with g αβ, small Latin indices lowered with Φmn,
(3.19)
ω ^ α β m = ω ˜ α β m = 1 2 F α β m n , ω ^ m β α = ω ^ β m α = 1 2 F α β , m .
(3.20)
ω ^ p q m = ω ˜ p q m = ω p q m
where ω p q m is the Riemannian connection of the metric Φmn(xmθn induced by ĝ on the orbit G x, i.e. with x = (x α) fixed; therefore
ω ^ p q m 1 2 ( C p q m Φ q r Φ m s C p s r Φ p r Φ m s C q s r ) .
(p.657) To write in compact form other connection coefficients we recall a notation and definition.

Definition 3.1 One denotes by D the covariant derivative on sections of vector bundles over V associated with the G-principal fiber bundle V̂ → V. This covariant derivative is taken in both the Riemannian connection of the metric g and the G-connection A. For instance it holds that:

(3.21)
D γ F α β m n γ F α β m ω γ α λ F λ β m ω γ β λ F α λ m + C n p m A γ p F α β n .
and
(3.22)
D α Φ m n = α Φ m n C a n q Φ m q C a m q Φ q n .

Using this definition we find that ω ^ m n α a vector on V with values in the tensor product G * G * , G * dual of G , given by

(3.23)
ω ^ m n α = 1 2 g α β D β Φ m n .
and
(3.24)
ω ^ n α α = 1 2 Φ m q α Φ n q + ω ˜ n α m
with, using the values of the structure coefficients and their antisymmetry
(3.25)
ω ˜ n α m = 1 2 { C ^ n α m g α λ Φ m s C ^ n s λ Φ n Φ p m C ^ α p } = 1 2 Φ m q ( Φ s q C n p s + Φ n C q p ) A α p ) .
Hence
(3.26)
ω ^ n α m = 1 2 Φ m q D α Φ n q
and (we use the relation in Chapter 6 defining a torsionless connection)
(3.27)
ω ^ α n m = ω ^ n α m + C ^ α n m .

4 Curvature tensor

We recall that the curvature tensor tensor of a pseudo-Riemannian metric g ^ g ^ A B θ ^ A θ ^ B in an arbitrary frame θ̂A is given in terms of the connection by

(4.1)
R ^ L M A B L ω ^ M B A M ω ^ L B A ω ^ C B A C ^ L M C + ω ^ L C A ω ^ M B C ω ^ M C A ω ^ L B C

We find by computation, with R λ μ α β the curvature tensor of g,

(4.2)
R ^ λ μ . α β R λ μ α β + 1 2 F m , β α F λ μ m + 1 4 F m , λ α F β μ m 1 4 F m , μ α F β λ m
(p.658) and, with R l m p q the curvature tensor of the metric with constant coefficients (x is fixed) Φ m n ( x ) θ m θ n ,
(4.3)
R ^ l m p q R l m p q 1 4 g γ α Φ r p ( D α Φ m q D γ Φ l r D γ Φ m r D α Φ l q ) .
The definition (4.1) gives
(4.4)
R ^ L M α q λ ω ^ m q α λ ω ^ m q α ω ^ p q α C λ m p + ω ^ λ C α ω ^ m q C ω ^ m C α ω ^ λ q C ;
that is, with D the covariant derivative in the metric g on Greek indices and covariant derivative in the connection A on Latin indices
(4.5)
R ^ L M α q 1 2 D λ D α Φ m q + 1 2 F p , λ α ω m q p + 1 4 F m , γ α F q , λ γ 1 4 Φ p r D α Φ m p D λ Φ q r
On the other hand, with f [ A B ] = f [ A B ] = f A B f B A
(4.6)
R ^ L M α q 1 2 g α β D β Φ p q c l m p + 1 4 F [ m γ α D γ Φ l ] q 1 2 g α β D β Φ p [ l ω m p ] q + 1 4 F [ m γ α D γ Φ l ] q
and
(4.7)
R ^ λ μ α q 1 2 D [ λ F q , μ ] ...... α + 1 2 g α β D β Φ p q F λ μ p + 1 4 F p , [ λ ...... α Φ p n D μ ] Φ q n
The other components of the curvature can be obtained from the symmetries properties of the Riemann tensor and the algebraic Bianchi identity.

Remark 4.1 The transformation law ( 3.6 ) of Φ under a change of trivialization implies the identity

(4.8)
D α D β Φ m q D β D α Φ m q + ρ α β p m Φ p q + ρ α β p q Φ m p ,
where ρ α β m n is the representative of the curvature of the gauge connection in the vector bundle associated to the given G-principal bundle by the adjoint representation; that is,
(4.9)
ρ α β m n C p n m F α β p .
Therefore
(4.10)
Φ m q D α D β Φ m q Φ m q D β D α Φ m q + 2 F α β r . c r p p
In particular if the group is compact (up to the product by an Abelian group) we have the symmetry property
(4.11)
Φ m q D α D β Φ m q Φ m q D β D α Φ m q

(p.659) 5 Ricci tensor and K–K equations

The components of the Ricci tensor of (V̂,ĝ) result immediately from those of the curvature tensor. They simplify to

(5.1)
R ^ M B : = R ^ M B A A B A ω ^ M B A M ω ^ A B A ω ^ C B A ω ^ A B C + ω ^ A C A ω ^ M B C
After straightforward computation, using Remark 4.1 to make apparent the symmetry and simplify the expression of R̂αβ. We find
(5.2)
R ^ α β = R α β 1 2 F m , β μ F α μ m 1 4 { Φ m q ( D α D β + D β D α ) Φ m q + D α Φ m q D β Φ m q } .
Definitions and computation also give
(5.3)
R ^ α l 1 2 D λ F l , α λ + 1 4 F l , α λ Φ n q + 1 2 c l q p Φ n q D α Φ n q + 1 2 c p n p Φ n q D α Φ l q
(5.4)
R ^ m n R m n + 1 4 { F m , α β F n α β + 2 Φ p q D α Φ m p D α Φ n p Φ p q D α Φ p q D α Φ m n 2 D α D α Φ m n }
with R mn the Ricci tensor of the metric on G with constant coefficients (x fixed) Φmnθmθn, that is,
(5.5)
R m n ω q n p ω p m q + ω p q p ω m n q ,
where ω p q p = 0 if the group is compact up to the product by an Abelian group.

Example. Let GSU(2). The structure constants are then totally antisymmetric with

(5.6)
C 12 3 = 1
Suppose the metric Φ is diagonal, given by, with a m some constants
(5.7)
Φ m = 1 , 2 , 3 a m ( θ m ) 2
We find that the only non-zero connection coefficients of the metric ϕ are those with three different indices and
(5.8)
ω 12 3 = 1 2 ( a 1 a 3 1 + a 2 a 3 1 1 ) = ω 21 3
We find that R mn = 0 if mn. Let us compute R 11; R 22 and R 33 are then obtained by circular permutation of indices. We find a result already obtained in Chapter 5 for Bianchi IX spacetimes.
(5.9)
R 11 = 1 2 a 2 a 3 { a 1 2 ( a 2 a 3 ) 2 }

(p.660) 6 Equations in conformal spacetime metric

As was already known from the five-dimensional theory, the Kaluza–Klein d̂-dimensional vacuum equations Ricci(ĝ) = 0 take a form more familiar to physics if they are expressed with a metric g˜ on the basis V conformal to the original metric g. We recall that if

(6.1)
g ˜ α β exp ( 2 h ) g α β
then (see Appendix VI),
(6.2)
R α β = R ˜ α β + ( n 1 ) ( α β h α h β h ) + g α β { λ λ h + ( n 1 ) λ h λ h } .

We suppose that the group G is compact (up to product by an Abelian group). Then c m p q is totally antisymmetric, in particular

(6.3)
C m p p = 0.
We have then the following lemma.

Lemma 6.1 If the group G is compact (up to product by an Abelian group), then Φ m n D β Φ m n is a covariant vector on V given by:

(6.4)
Φ m n D β Φ m n Φ m n β Φ m n = β f ,
with
(6.5)
f : = log det Φ
where det Φ is the determinant of the matrixmn).

Proof By the definition of (Φmn) as the inverse matrix of (̂Phi;mn) and the definition of D it holds that

(6.6)
Φ m n D β Φ m n Φ m n β Φ m n 2 c p m m A β p .
The result of the lemma is then the consequence of the identity (6.3) satisfied by the structure constants of the considered groups, together with the formula giving the derivative of a determinant.

We choose as conformal factor the function

(6.7)
exp ( 2 h ) ( d e t Φ ) 1 n 1 i .e . h = 1 2 ( n 1 ) f
and we prove the following lemma.

Lemma 6.2 If g˜αβ ≡ exp(2h)gαβ with h given by (6.7), then the following identity holds:

(6.8)
R ^ α β + g α β n 1 Φ m n R ^ m n R ˜ α β ρ α β .
(p.661) where ρ is a tensor field on V given by
(6.9)
ρ α β 1 2 ( F m , β μ F α μ m g α β 2 ( n 1 ) F m , λ μ F m , λ μ )
(6.10)
1 4 D α Φ m q D β Φ m q + 1 4 ( n 1 ) α f β f g α β ( n 1 ) Φ m n R m n .

Proof Using the formulae (6.2) and (5.2) gives

(6.11)
R ^ α β R ˜ α β + ( n 1 ) ( α β h α h β h ) + g α β { λ λ h ( n 1 ) λ h λ h } . 1 2 F m , β μ F α μ m 1 2 α β f + 1 4 D α Φ m q D β Φ m q .
The relation (6.7) between h and f implies the cancelling of the second derivatives α β h and α β f , therefore
(6.12)
R ^ α β R ˜ α β + ( n 1 ) ( α β h α h β h ) + g α β { λ λ h + ( n 1 ) λ h λ h } 1 2 β α f 1 2 F m , β μ F α μ m + 1 4 D α Φ m q D β Φ m q .
We now compute Φ m n R ^ m n , using (5.4) and the identity
(6.13)
Φ m q D α D β Φ m q D α ( Φ m q D β Φ m q ) D α Φ m q D β Φ m q = α β f D α Φ m q D β Φ m q .
We find
(6.14)
Φ m q R ^ m n Φ m q R m n + 1 4 F m , α β F m , α β 1 4 λ f λ f 1 2 λ λ f .
We see that, when f = 2(n − 1)h, the second derivatives of h and f cancel in the combination R ^ α β + 1 n 1 g α β Φ m n R ^ m n , as well as the terms in ∂λ fλ f and ∂λ hλ h. We thus obtain the formulae (6.8) and (6.9).

Theorem 6.3 If the isometry group G is compact (up to the product by an Abelian group), the d˜-dimensional Einstein Kaluza–Klein equations in vacuo, Ricci(ĝ) =0, can be written as an Einstein–Yang–Mills wave map type system on the n + 1-manifold V.

Proof Equations (5.4), R̂mn = 0, are wave equations in the metric g with quadratic non-linear terms in the first derivatives of Φ and in the curvature F.

Under the hypothesis made on the group G, Equations (5.3), R̂αl = 0, reduce to

(6.15)
R ^ α l 1 2 D λ F l , α λ + 1 4 F l , α λ λ f 1 2 | det Φ | 1 : 2 D λ ( F l , α λ | det Φ | 1 : 2 ) = 0
which are equations for a weighted Yang–Mills field.

(p.662) The previous lemma gives Einstein equations for the metric g̃ with sources quadratic in the first derivatives of Φ and a Yang–Mills type field. □

These results justify the name given to the system obtained for the unknowns (g̃,Φ,A). We will not pursue further here this general study. Applications are found in Chapter 14 and in the section “S 1 invariant Einstein universes” in Chapter 16.

Notes:

(1) The vectors êα are a basis of the horizontal space of the G-connection.

(2) The basis V is the quotient of V̂ by the isometry group. It can be identified with a submanifold of V only if the family of horizontal spaces is completely integrable, i.e. tangent to an n + 1-dimensional submanifold of V̂.

(3) For notations and proofs see CB-DM1 IIIB 2.

(4) i.e. transforms by a change of trivialization of the bundle by a law of the type Fh −1 Fh (see CB-DM1 Vbis).