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Black Holes and Nilpotent Orbits

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Advances in Geometry and Lie Algebras from Supergravity

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Abstract

When on September 14th 2015 the gravitational wave signal emitted 1.5 billion year ago by two coalescing black stars was detected at LIGO I and LIGO II, we not only obtained a new spectacular confirmation of General Relativity but we actually saw the dynamical process of formation of the most intriguing objects populating the Universe, namely black holes.

Deep into that darkness peering, long I stood there, wondering, fearing, doubting, dreaming dreams no mortal ever dared to dream before.

Edgar Allan Poe

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Notes

  1. 1.

    There followed a vast literature some items of which are are quoted in [4,5,6,7,8,9,10].

  2. 2.

    Clarification for mathematicians: General Relativity in \(D=3=1\oplus 2\) dimensions is a rather empty field theory. Einstein equations do not describe the propagation of any particle since there are no solutions of the wave-type and the only degree of freedom is the analogue of the Newton potential. Mathematically this follows from the fact that the Riemann tensor is fully determined by the Ricci tensor and the latter is identified by Einstein equations with the stress-energy tensor of matter fields.

  3. 3.

    The special overall normalization of the Poincaré metric is chosen in order to match the general definitions of special geometry applied to the present case.

  4. 4.

    Clarification for mathematicians: the acronym BPS stands for Bogomolny, Prasad and Sommerfeld. It is a notion occuring in the theory of monopoles where one always derives a bound according to which the energy (or mass) of a quasi-particle corresponding to a localized solution of non linear propagation equations is always larger or equal than some kind of charge carried by the quasi-particle. BPS states are those that saturate the bound and typically correspond to shortened representations of the space-time group. In the case of supergravity black–holes the BPS bound relates the mass of the hole with the modulus of the central charge of the supersymmetry algebra. Because of the scope of this book we omit the original definition of the central charge in terms of superalgebras and we confine to give its expression in terms of special Kähler geometrical items (see Eq. (6.4.4)).

  5. 5.

    As we are going to see later, each orbit of Lax operators always contains representatives such that the Taub-NUT charge is zero. Alternatively from a dynamical system point of view the Taub-NUT charge can be annihilated by setting a constraint which is consistent with the hamiltonian and which reduces the dimension of the system by one unit. The problem of black hole physics is therefore equivalent to the sigma model based on an appropriate codimension one hypersurface in the coset manifold \(\mathrm {G}/\mathrm {H}^\star \).

  6. 6.

    See for instance the lecture notes [11].

  7. 7.

    Clarification for mathematicians: for a short but comprehensive introduction to the theory of Black Holes we refer the interested reader to Chaps. 2 and 3 of Volume II of [35] by the present author.

  8. 8.

    Clarification for mathematicians: Extremal in the GR sense means something different than extremal in the \(\sigma \)-model sense. As we mentioned above the extremal Kerr solution, according to General Relativity is the solution where \(m=\alpha \). In the \(\sigma \)-model sense any extremal solution corresponds to a light-like geodesic of the of the \(\mathrm {U}/\mathrm {H}^\star \) manifold. Light-like geodesics, on their turn are associated with \(\mathrm {H}^\star \) orbits of nilpotent \(\mathrm {U}\) Lie algebra elements. As shown above the extremal Kerr solution is obtained from a \(\mathrm {U}/\mathrm {H}^\star \) geodesic that is not light-like so it is not extremal in the \(\sigma \)-model sense.

  9. 9.

    Such solutions actually correspond to different \(\mathrm {G}_{\mathbb {R}}\)-orbits [36].

  10. 10.

    In the literature, see [36], \(\beta \)-labels are defined as the value of the simple roots \(\beta ^i\) of the complexification \(\mathbb {H}_\mathbb {C}\) of \(\mathbb {H}^\star \) on the non-compact element \(X_c\), viewed as a Cartan element of \(\mathbb {H}_\mathbb {C}\) in the Weyl chamber of \((\beta ^i)\). We find it more practical to work with the equivalent characterization (6.6.17).

  11. 11.

    Note that the action of certain Weyl group elements \(g \in \mathscr {W}\) on specific h.s can be the identity: \(g\cdot h = h\). When such stabilizing group elements g are inside \(\mathscr {W}_H\) the number of different h.s inside each lateral classes is accordingly reduced. If there are stabilizing elements g that are not inside \(\mathscr {W}_H\) than two or more \(\mathscr {W}_H\) orbits coincide.

  12. 12.

    For the rationale of our notation we refer the reader to previous Sect. 5.8.

  13. 13.

    See [32] for details, in particular Eq. (3.13) of that reference for the explicit form of the spin \({\textstyle \frac{3}{2}}\) matrices.

  14. 14.

    Actually even the condition \(\mathscr {H}_1 \, = \, \text{ const }\) suffices to annihilate the Taub-NUT charge allowing for a non trivial real part of the z-field. However in this section we analyze the case \(\mathscr {H}_1 \, = \, 0\) for its remarkable simplicity.

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  1. Pietra Marazzi, Alessandria, Italy

    Pietro Giuseppe Fré

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Fré, P.G. (2018). Black Holes and Nilpotent Orbits. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_6

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