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Third type of spacetime with the coexistence of integrability and non-integrability.

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Title: Third type of spacetime with the coexistence of integrability and non-integrability.
Authors: Lu, Junjie1 (AUTHOR), Wu, Xin1 (AUTHOR) wuxin_1134@sina.com
Source: European Physical Journal C -- Particles & Fields. Mar2026, Vol. 86 Issue 3, p1-17. 17p.
Subjects: Conformal mapping, Geodesic motion, Black holes, Kerr black holes, Hamilton-Jacobi equations, Spacetime, Schwarzschild metric
Abstract: The integrability or non-integrability of a spacetime usually refers to whether the motion of massive or massless particles in the spacetime is integrable or not. The standard black hole spacetimes such as the Schwarzschild and Kerr metrics are always integrable for both timelike and null geodesics. They belong to a first type of spacetime. However, the Melvin type spacetimes as a second type of spacetime are non-integrable, regardless of whether they are for massive or massless particle motion. In this paper, we discover the possibility of a third type of spacetime with non-integrable dynamics of timelike geodesics and integrable dynamics of null geodesics. In fact, conformal transformations may transform type one solutions into type three. This is due to the conformal factors preventing the separation of variables from the Hamilton–Jacobi equation and leading to the absence of a fourth constant of motion for the massive particle dynamics. Nevertheless, the massless particle motion still remains integrable in these metrics for any conformal factors because the conformal factors have no effect on the null geodesics whatsoever. The conformal Kerr metric is an example of the third type of spacetime. In addition to the conformal transformation method, other paths may yield the third type of spacetime. The Kerr–Bertotti–Robinson black hole metric and the accelerating Schwarzschild spacetime are two examples of non-conformal solutions that are also of type three. [ABSTRACT FROM AUTHOR]
Copyright of European Physical Journal C -- Particles & Fields is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Third type of spacetime with the coexistence of integrability and non-integrability.
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  Data: <searchLink fieldCode="AR" term="%22Lu%2C+Junjie%22">Lu, Junjie</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Wu%2C+Xin%22">Wu, Xin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> wuxin_1134@sina.com</i>
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  Data: <searchLink fieldCode="JN" term="%22European+Physical+Journal+C+--+Particles+%26+Fields%22">European Physical Journal C -- Particles & Fields</searchLink>. Mar2026, Vol. 86 Issue 3, p1-17. 17p.
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  Data: <searchLink fieldCode="DE" term="%22Conformal+mapping%22">Conformal mapping</searchLink><br /><searchLink fieldCode="DE" term="%22Geodesic+motion%22">Geodesic motion</searchLink><br /><searchLink fieldCode="DE" term="%22Black+holes%22">Black holes</searchLink><br /><searchLink fieldCode="DE" term="%22Kerr+black+holes%22">Kerr black holes</searchLink><br /><searchLink fieldCode="DE" term="%22Hamilton-Jacobi+equations%22">Hamilton-Jacobi equations</searchLink><br /><searchLink fieldCode="DE" term="%22Spacetime%22">Spacetime</searchLink><br /><searchLink fieldCode="DE" term="%22Schwarzschild+metric%22">Schwarzschild metric</searchLink>
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  Data: The integrability or non-integrability of a spacetime usually refers to whether the motion of massive or massless particles in the spacetime is integrable or not. The standard black hole spacetimes such as the Schwarzschild and Kerr metrics are always integrable for both timelike and null geodesics. They belong to a first type of spacetime. However, the Melvin type spacetimes as a second type of spacetime are non-integrable, regardless of whether they are for massive or massless particle motion. In this paper, we discover the possibility of a third type of spacetime with non-integrable dynamics of timelike geodesics and integrable dynamics of null geodesics. In fact, conformal transformations may transform type one solutions into type three. This is due to the conformal factors preventing the separation of variables from the Hamilton–Jacobi equation and leading to the absence of a fourth constant of motion for the massive particle dynamics. Nevertheless, the massless particle motion still remains integrable in these metrics for any conformal factors because the conformal factors have no effect on the null geodesics whatsoever. The conformal Kerr metric is an example of the third type of spacetime. In addition to the conformal transformation method, other paths may yield the third type of spacetime. The Kerr–Bertotti–Robinson black hole metric and the accelerating Schwarzschild spacetime are two examples of non-conformal solutions that are also of type three. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of European Physical Journal C -- Particles & Fields is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1140/epjc/s10052-026-15482-w
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      – Code: eng
        Text: English
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        PageCount: 17
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      – SubjectFull: Conformal mapping
        Type: general
      – SubjectFull: Geodesic motion
        Type: general
      – SubjectFull: Black holes
        Type: general
      – SubjectFull: Kerr black holes
        Type: general
      – SubjectFull: Hamilton-Jacobi equations
        Type: general
      – SubjectFull: Spacetime
        Type: general
      – SubjectFull: Schwarzschild metric
        Type: general
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      – TitleFull: Third type of spacetime with the coexistence of integrability and non-integrability.
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            NameFull: Lu, Junjie
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            NameFull: Wu, Xin
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            – D: 01
              M: 03
              Text: Mar2026
              Type: published
              Y: 2026
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            – TitleFull: European Physical Journal C -- Particles & Fields
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