122 |
5 Cosmology and General Relativity |
The most important properties of the Lie derivative, which immediately follow from its definition are the following ones:
[ v, d] = 0
(5.2.81)
[ v, w] = [v,w]
The first of the above equations states that the Lie derivative commutes with the exterior derivative. This is just a consequence of the invariance of the exterior algebra of k-forms with respect to diffeomorphisms. The second equation states on the other hand that the Lie derivative provides an explicit representation of the Lie algebra of vector fields on tensors.
The Lie derivatives along the Killing vectors of the frames V a and of the H- connection ωi introduced in the previous subsection are:
vA V a = WAi Caib V b |
(5.2.82) |
vA ωi = − dWAi + Cikj WAk ωj |
(5.2.83) |
This result can be interpreted by saying that, associated with every Killing vector kA there is a an infinitesimal H-gauge transformation:
WA = WAi (y)Ti |
(5.2.84) |
and that the Lie derivative of both V a and ωi along the Killing vectors is just such local gauge transformation pertaining to their respective geometrical type. The frame V a is a section of an H-vector bundle and transforms as such, while ωi is a connection and it transforms as a connection should do.
5.2.3.4 Invariant Metrics on Coset Manifolds
The result (5.2.82), (5.2.83) has a very important consequence which constitutes the fundamental motivation to consider coset manifolds. Indeed this result instructs us to construct G-invariant metrics on G/H, namely metrics that admit all the above discussed Killing vectors as generators of true isometries.
The argument is quite simple. We saw that the one-forms V a transform as a linear representation DH of the isotropy subalgebra H (and group H). Hence if τab is a symmetric H-invariant constant two-tensor, by setting:
ds2 = τabV a V b = τabVαa (y)Vβb(y) dyα dyβ |
(5.2.85) |
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gαβ |
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we obtain a metric for which all the above constructed Killing vectors are indeed Killing vectors, namely:
5.2 Mathematical Interlude: Isometries and the Geometry of Coset Manifolds |
123 |
kA ds2 = τab |
kA V a V b + V a kA V b |
(5.2.86) |
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= τab DH(WA) ac δdb + DH(WA) bc δda V c V d |
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= 0 by |
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invariance |
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= 0 |
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(5.2.87) |
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The key point, in order to utilize the above construction, is the decomposition of the representation DH into irreducible representations. Typically, for most common cosets, DH is already irreducible. In this case there is just one invariant H-tensor τ and the only free parameter in the definition of the metric (5.2.85) is an overall scale constant. Indeed if τab is an invariant tensor, any multiple thereof τab = λτab is also invariant. In the case DH splits into r irreducible representations:
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we have r irreducible invariant tensors τ |
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in correspondence of such irreducible |
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blocks and we can introduce r independent scale factors: |
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λ1τ (1) |
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Correspondingly we arrive at a continuous family of G-invariant metrics on G/H depending on r-parameters or, as it is customary to say in this context, of r moduli. The number r defined by (5.2.88) is named the rank of the coset manifold G/H.
In this section we confine ourself to the most common case of rank one cosets (r = 1), assuming, furthermore, that the algebras G and H are both semi-simple. By an appropriate choice of basis for the coset generators T a , the invariant tensor τab can always be reduced to the form:
τab = ηab = diag(+, +, . . . , +, −, −, . . . , −) |
(5.2.90) |
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where the two numbers n+ and n− sum up to the dimension of the coset:
G |
= dim K |
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n+ + n− = dim H |
(5.2.91) |
124 |
5 Cosmology and General Relativity |
and provide the dimensions of the two eigenspaces, K± K, respectively corresponding to real and pure imaginary eigenvalues of the matrix DH(W ) which represents a generic element W of the isotropy subalgebra H.
Focusing on our example (5.2.43), that encompasses both the spheres and the pseudo-spheres, depending on the sign of κ, we find that:
n+ = 0; n− = n |
(5.2.92) |
so that in both cases (κ = ±1) the invariant tensor is proportional to a Kronecker delta:
ηab = δab |
(5.2.93) |
The reason is that the subalgebra H is the compact so(n), hence the matrix DH(W ) is antisymmetric and all of its eigenvalues are purely imaginary.
If we consider cosets with non-compact isotropy groups, then the invariant tensor τab develops a non-trivial Lorentzian signature ηab . In any case, if we restrict ourselves to rank one cosets, the general form of the metric is:
ds2 = λ2ηabV a V b |
(5.2.94) |
where λ is a scale factor. This allows us to introduce the vielbein
Ea = λV a |
(5.2.95) |
and calculate the spin connection from the vanishing torsion equation:
0 = dEa − ωab Ecηbc |
(5.2.96) |
Using the Maurer Cartan equations (5.2.61)–(5.2.62), (5.2.96) can be immediately solved by:
ωabηbc ≡ ωac = |
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2λ Cacd Ed + Caci ωi |
(5.2.97) |
Inserting this in the definition of the curvature two-form
Rab = dωab − ωac ωcb |
(5.2.98) |
allows to calculate the Riemann tensor defined by: |
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Rab = Rabcd Ec Ed |
(5.2.99) |
Using once again the Maurer Cartan equations (5.2.61)–(5.2.62), we obtain:
Rabcd |
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2λ Cabe Cecd − |
8 Caec Cebd + |
8 Caed Cebc − |
2 Cabi Cicd |
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(5.2.100) which, as previously announced provides the expression of the Riemann tensor in terms of structure constants.