APPENDIX VII Kaluza–Klein Theories
APPENDIX VII Kaluza–Klein Theories
- General Relativity and the Einstein Equations
- Oxford University Press
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.
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
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:
The orbit 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.
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,
We choose local coordinates x
̂ adapted to the trivialization i.e. such that x
= x α
, local coordinates in U
, for α = 0,1…,n
, and x
= y m
̂, local coordinates in G
. We take a moving frame (êα
) such that the êm
are Killing vector fields of the right action of G
̂ they are mapped by φ onto a basis e m
of the right Killing vector fields on G
. We choose the êα
to the êm
and mapped by φ onto a basis e α
of the tangent space to U
We denote by θ̂A
the dual coframe of êA
̂, by θα
the dual coframes respectively of e α
and e m
. We find that
One shows that
is the representant on V
, in the trivialization φ, of a G
-connection 1-form which takes its values in the Lie algebra
With this choice of frames the metric ĝ V̂ can be written, in π−1(U),
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
in the direction of a Killing vector êm
vanish, the metric ĝ is invariant
under the right action of G
̂ if and only if the coefficients g αβ
depend only on the local coordinates x α
The metric 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 U → G by x ↦ h(x) it holds that:
The metric g is invariant.
The connection A transforms as a representant of a G-connection,
The scalar multiplet transforms as
3.2 Structure coefficients
The structure coefficients of a moving frame ∂̂A with dual frame θ̂A are defined by the relations
To simplify the computations we choose for θα
a natural frame on V
= dx α
Since the θm are dual to Killing vectors on G it holds that
are structure constants of the group G
. We deduce from the expression of θm
and the antisymmetry of
We see from the definitions of the curvature of a connection on a principal fibre bundle (see CB-DM1 Vbis 4) that the
are the components of the representative in the chart of V
of the curvature
of the connection A
, since the curvature of A
valued antisymmetric tensor of type Ad4
We read on (3.11
3.3 Kaluza–Klein connection
Recall that (Chapter 6) the connection coefficients of a metric ĝ are given in terms of the metric components ĝAB and the structure coefficients by
where are the Christoffel symbols constructed with the coefficients ĝAB
We find easily that
are the Christoffel symbols of the metric g
and, with Φmn
the inverse matrix of Φmn
We now compute the ω˜ terms and we find, Greek indices raised with g αβ
, small Latin indices lowered with Φmn
is the Riemannian connection of the metric Φmn
induced by ĝ on the orbit G x
, i.e. with x
= (x α
) fixed; therefore
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:
Using this definition we find that a vector on V with values in the tensor product dual of , given by
with, using the values of the structure coefficients and their antisymmetry
and (we use the relation in Chapter 6
defining a torsionless connection)
4 Curvature tensor
We recall that the curvature tensor tensor of a pseudo-Riemannian metric in an arbitrary frame θ̂A is given in terms of the connection by
We find by computation, with the curvature tensor of g,
the curvature tensor of the metric with constant coefficients (x
The definition (4.1
that is, with D
the covariant derivative in the metric g
on Greek indices and covariant derivative in the connection A
on Latin indices
On the other hand, with
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
where 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,
In particular if the group is compact (up to the product by an Abelian group) we have the symmetry property
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
After straightforward computation, using Remark 4.1 to make apparent the symmetry and simplify the expression of R̂αβ
. We find
Definitions and computation also give
with R mn
the Ricci tensor of the metric on G
with constant coefficients (x
, that is,
if the group is compact up to the product by an Abelian group.
Example. Let G ≡ SU(2). The structure constants are then totally antisymmetric with
Suppose the metric Φ is diagonal, given by, with a m
We find that the only non-zero connection coefficients of the metric ϕ are those with three different indices and
We find that R mn
= 0 if m
. 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.
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
then (see Appendix VI
We suppose that the group G is compact (up to product by an Abelian group). Then is totally antisymmetric, in particular
We have then the following lemma.
Lemma 6.1 If the group G is compact (up to product by an Abelian group), then is a covariant vector on V given by:
det Φ is the determinant of the matrix
Proof By the definition of (Φmn) as the inverse matrix of (̂Phi;mn) and the definition of D it holds that
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
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:
where ρ is a tensor field on V given by
Proof Using the formulae (6.2) and (5.2) gives
The relation (6.7
) between h
implies the cancelling of the second derivatives
We now compute
, using (5.4
) and the identity
We see that, when f
, the second derivatives of h
cancel in the combination
, as well as the terms in ∂λ f
and ∂λ 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
which are equations for a weighted Yang–Mills field.
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.
The vectors êα are a basis of the horizontal space of the G-connection.
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.
For notations and proofs see CB-DM1 IIIB 2.
i.e. transforms by a change of trivialization of the bundle by a law of the type F → h −1 Fh (see CB-DM1 Vbis).