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Extremal Black Holes as Relativistic Systems with Kepler Dynamics

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Abstract

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler.

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Notes

  1. The post-Newtonian expansion is in terms of \(\frac{1}{c^{2}}\), resulting in an expansion in weak gravitational field and low velocity, while the post-Minkowskian is an expansion in gravitational constant \(G\), i. e., a weak gravitational field expansion only [18].

  2. For all bodies orbiting the Sun, the square of the period is proportional to the third power of the semi-major axis of the orbit, with the same proportionality constant [18].

  3. This is a rather interesting technical problem in itself, and we refer the interested reader to [7, 16] for the current state of the art.

  4. We divide out the rest-mass energy \(mc^{2}\) and set \(m=1\) as previously.

  5. This scalar field gives the black hole what is called secondary hair, as the scalar charge is completely determined in terms of mass and charge within a given theory, i. e., for a given value of the scalar coupling constant [24].

  6. Note that this in general will also have an additional \(p_{r}^{2}/r\) term, proportional to the radial momentum only. By means of a constant shift of the radial coordinate, one can set the coefficient of this term to zero, see, e. g., [9]. We will do so in order to facilitate the comparison to Section 2.

  7. This corresponds to the combination \(A+2B+C+D\) in the conventions of [11, 37].

  8. This is because the effective Newton’s constant \(G_{12}\) is eight times larger than the usual gravitational constant. This corresponds to the findings of [11], who also found this in their super-gravity system.

  9. This result has been derived before in [31], though with a different mass function in the sense of Section 3.3, such that the results only coincide for extremal black holes.

  10. This value coincides with the Kaluza – Klein reduction of gravity in \(5\) dimensions [15].

  11. One could further extend our considerations and include magnetic charges as well. We expect the dyonic charges to span a \(U(1)\) charge vector playing a completely analogous role to the \(SU(8)\) charge vector of [8].

  12. In terms of the Schwarzschild radial coordinate, this choice of gauge corresponds to \(A_{0}=-\frac{1}{\sqrt{1+a^{2}}}+\frac{m_{2}Q_{2}}{r}\). After the change \(r\to r+r_{\pm}\), we find the above.

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ACKNOWLEDGMENTS

We are grateful to Andreas Knauf, Tomás Ortín and Cédric Deffayet for stimulating discussions and to the anonymous referees for their useful comments.

Funding

D.N. is supported by the Fundamentals of the Universe research program within the University of Groningen. M.S. is supported by the NWO project 613.009.10.

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Authors and Affiliations

  1. Van Swinderen Institute for Particle Physics and Gravity, University of Groningen, PO Box 72, 9700 AB, Groningen, The Netherlands

    Dijs de Neeling & Diederik Roest

  2. Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands

    Dijs de Neeling, Marcello Seri & Holger Waalkens

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  1. Dijs de Neeling
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Correspondence to Dijs de Neeling, Diederik Roest, Marcello Seri or Holger Waalkens.

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MSC2010

37J06, 70H15, 83C22, 83C57

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Neeling, D.d., Roest, D., Seri, M. et al. Extremal Black Holes as Relativistic Systems with Kepler Dynamics. Regul. Chaot. Dyn. 29, 344–368 (2024). https://doi.org/10.1134/S1560354724020035

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