Abstract
On one hand the Geroch group allows one to associate spacetime independent matrices with gravitational configurations that effectively only depend on two coordinates. This class includes stationary axisymmetric four- and five-dimensional black holes. On the other hand, a recently developed inverse scattering method allows one to factorize these matrices to explicitly construct the corresponding spacetime configurations. In this work we demonstrate the construction as well as the factorization of Geroch group matrices for a wide class of black hole examples. In particular, we obtain the Geroch group SL(3, ℝ) matrices for the five-dimensional Myers-Perry and Kaluza-Klein black holes and the Geroch group SU(2, 1) matrix for the four-dimensional Kerr-Newman black hole. We also present certain non-trivial relations between the Geroch group matrices and charge matrices for these black holes.
Highlights
Gravity in four-dimensions.1 The corresponding Lie algebras are the affine-extensions of the Lie algebras of the hidden symmetry groups in three dimensions
We present certain non-trivial relations between the Geroch group matrices and charge matrices for these black holes
Our present study brings in two new elements: (i) we extend the factorization algorithm developed there to incorporate fivedimensional asymptotically flat boundary conditions, (ii) we present a fairly non-trivial example involving the group SU(2, 1) of the general factorization algorithm presented there
Summary
We present a brief review of dimensional reduction to two dimensions. (i) we use the notation xm = (ρ, z) and onwards use x to collectively denote the two-dimensional coordinates, (ii) the Lax equations require us to consider the generalization V (x) → V(t, x), a quantity that depends on the spectral parameter t with the property V(0, x) = V (x), (iii) Pm and Qm are respectively the symmetric and anti-symmetric parts of the Lie algebra element ∂mV V −1 = Pm + Qm, Pm♯ = Pm and Q♯m = −Qm. The integrability condition for equations (2.5) is equivalent to the equations of the motion of the two-dimensional gravitational system if and only if the spectral parameter satisfies certain spacetime dependent differential equation. This is the recipe we use in the later sections to study Geroch group description of black holes
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