9.2.2 The Oxidation Rules

Similarly to the case of the Kaluza-Klein vector, the combinations of derivatives and fields under-braced in the above formula are constants of motion of the dynamical system defined by the Lagrangian (9.2.5) and have the interpretation of magnetic charges. Indeed the magnetic charges are just the upper nv components of the full 2nv vector of magnetic and electric charges. This latter is defined as follows:

and all of its components are constants of motion.

In view of this the final form of the D = 4 field-strengths is the following one:

This concludes the review of the oxidation formulae that allow to write all the fields of D = 4 supergravity corresponding to a black-hole solution in terms of the fields parameterizing the σ -model defined by (9.2.5).

One very important point to be stressed is that the metric (9.2.3) admits a typically large group of isometries. Certainly it admits all the isometries of the original scalar manifold Mscalar enlarged with additional ones related to the new fields that have been introduced {U, a, ZM }. In the case when the D = 4 scalar manifold is a homogeneous symmetric space:

One can show [4, 5, 15], that the manifold Q with the metric (9.2.3) is a new homogeneous symmetric space

whose structure is universal and can be described in general terms.

General Structure of the Uσ Lie Algebra The Lie algebra Uσ of the numerator group always contains, as subalgebra, the duality algebra UD = 4 of the parent supergravity theory in D = 4 and a universal sl(2, R)E algebra which is associated with the gravitational degrees of freedom {U, a}. Furthermore, with respect to this subalgebra Uσ admits the following universal decomposition, holding for all N - extended supergravities:

where W is the symplectic representation of UD = 4 to which the electric and magnetic field strengths are assigned. Indeed the scalar fields associated with the generators of W(2,W) are just those coming from the vectors in D = 4. Denoting the generators of UD = 4 by T a , the generators of SL(2, R)E by Lx and denoting by

W iM the generators in W(2,W), the commutation relations that correspond to the decomposition (9.2.17) have the following general form:

where the 2 × 2 matrices (λx )ij , are the canonical generators of SL(2, R) in the fundamental, defining representation:

we denote the antisymmetric symplectic metric in 2n dimensions, n = nv being the number of vector fields in D = 4, as we have already stressed. The symplectic character of the representation W is asserted by the identity:

The fundamental doublet representation of SL(2, R) is also symplectic and by εij =

0 1 we have denoted the 2-dimensional symplectic metric, so that:

−1 0

In (9.2.18) we have used the standard convention according to which symplectic indices are raised and lowered with the appropriate symplectic metric, while adjoint representation indices are raised and lowered with the Cartan-Killing metric.

Orbit of Solutions Using the transformations of the isometry group Uσ every solution of the σ -model generates an entire Uσ orbit of solutions which reflects in a similar Uσ orbit of supergravity solutions. Consequently the black-hole solutions are conveniently organized into Uσ orbits.

9.2.3 General Properties of the d = 4 Metric

It is convenient to summarize some general properties of the d = 4 metric in (9.2.7). First we consider the case of non-extremal black-holes v2 > 0 and in particular the Schwarzschild solution which, as it was shown in [3, 18] is the unique representative of the whole Uσ orbit of regular black-hole solutions.

The Schwarzschild Case Consider the case where the function exp[−U (τ )] and the extremality parameter are the following ones:

the reader can immediately verify that the metric (9.2.7) at AKK = 0 is turned into the standard Schwarzschild metric:

by means of elementary algebra the reader can verify that the metric (9.2.7) at AKK = 0 is turned into the extremal Reissner Nordström metric:

which follows from the non-extremal one:

when the mass is equal to the charge: m = q.

It follows from the discussion of this simple example that the extremal blackhole metrics (9.2.7) are all suitable deformations of the extremal Reissner Nordström metric, just as the regular black-hole metrics are suitable deformations of the Schwarzschild one.