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The attractor of evaporating Reissner–Nordström black holes

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Abstract

Hiscock and Weems showed that there is an attractor behavior in the evolution of asymptotically flat Reissner–Nordström black hole under Hawking evaporation. If the initial charge-to-mass ratio Q/M of the black hole is relatively small, then the ratio first increases until the black hole hits the attractor and then starts to discharge toward the Schwarzschild limit. Sufficiently charged black holes, on the other hand, simply discharge steadily toward the Schwarzschild limit. In this work, we further investigate the nature of the attractor and found that it is characterized by the mass loss rate being equal to the charge loss rate. The attractor is not necessarily related to the specific heat in a general evaporating black hole spacetime, but for the Reissner–Nordström case part of the attractor lies very close to the boundary of the region where specific heat changes sign.

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Notes

  1. We follow the convention of [4] in which \(\hbar \ne 1\), but rather it is the Planck area. Consequently temperature has dimension of length.

  2. We used the word ”hit” colloquially here. The curves can come very close together near the attractor, but none of the curves ever intersect. After all, solutions of ODE is unique.

  3. We emphasize that, as per Footnote 2, the attractor is actually not a single curve but a small region.

  4. Here, one employs the chain rule

    $$\begin{aligned} \frac{{\text {d}}\!{T}}{{\text {d}}\!{t}}=\frac{\partial T}{\partial M}\frac{{\text {d}}\!{M}}{{\text {d}}\!{t}} + \frac{\partial T}{\partial Q}\frac{{\text {d}}\!{Q}}{{\text {d}}\!{t}}. \end{aligned}$$
    (7)

    The explicit expression can then be obtained from Eq. 3, 4, and 5. Note that [4] contains a typo in the second term in the denominator: the square is missing in \(3840 \pi ^2\).

  5. Despite the change of sign of specific heat, there is no change to the geometry, and hence, there is arguably no real ”phase transition”. See, e.g., [8, 9].

  6. In view of Footnotes 2, 3, it should be clear that this attractor curve is an idealization: in actual fact, there is a thin ”attractor region” formed by many curves. It is possible that these curves are in a tubular neighborhood of the idealized curve studied here.

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Acknowledgements

YCO thanks the National Natural Science Foundation of China (No.11705162) and the Natural Science Foundation of Jiangsu Province (No.BK20170479) for funding support.

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Authors and Affiliations

  1. Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou City, 225002, Jiangsu Province, China

    Yen Chin Ong

  2. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yen Chin Ong

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Ong, Y.C. The attractor of evaporating Reissner–Nordström black holes. Eur. Phys. J. Plus 136, 61 (2021). https://doi.org/10.1140/epjp/s13360-020-00995-4

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