Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory
There are two free coupling parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>13</mn></msub></semantics></math></inline-formula> and &...
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MDPI AG
2023-08-01
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| Online Access: | https://www.mdpi.com/2218-1997/9/8/365 |
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| author | Caiyu Liu Xin Wu |
| author_facet | Caiyu Liu Xin Wu |
| author_sort | Caiyu Liu |
| collection | DOAJ |
| description | There are two free coupling parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>13</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>14</mn></msub></semantics></math></inline-formula> in the Einstein–Æther metric describing a non-rotating black hole. This metric is the Reissner–Nordström black hole solution when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, but it is not for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. When the black hole is immersed in an external asymptotically uniform magnetic field, the Hamiltonian system describing the motion of charged particles around the black hole is not integrable; however, the Hamiltonian allows for the construction of explicit symplectic integrators. The proposed fourth-order explicit symplectic scheme is used to investigate the dynamics of charged particles because it exhibits excellent long-term performance in conserving the Hamiltonian. No universal rule can be given to the dependence of regular and chaotic dynamics on varying one or two parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>13</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>14</mn></msub></semantics></math></inline-formula> in the two cases of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The distributions of order and chaos in the binary parameter space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>c</mi><mn>13</mn></msub><mo>,</mo><msub><mi>c</mi><mn>14</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> rely on different combinations of the other parameters and the initial conditions. |
| first_indexed | 2024-03-10T23:31:16Z |
| format | Article |
| id | doaj.art-053e8d6deb974d10b771b4237d8c918f |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T23:31:16Z |
| publishDate | 2023-08-01 |
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| series | Universe |
| spelling | doaj.art-053e8d6deb974d10b771b4237d8c918f2023-11-19T03:16:38ZengMDPI AGUniverse2218-19972023-08-019836510.3390/universe9080365Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther TheoryCaiyu Liu0Xin Wu1School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, ChinaSchool of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, ChinaThere are two free coupling parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>13</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>14</mn></msub></semantics></math></inline-formula> in the Einstein–Æther metric describing a non-rotating black hole. This metric is the Reissner–Nordström black hole solution when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, but it is not for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. When the black hole is immersed in an external asymptotically uniform magnetic field, the Hamiltonian system describing the motion of charged particles around the black hole is not integrable; however, the Hamiltonian allows for the construction of explicit symplectic integrators. The proposed fourth-order explicit symplectic scheme is used to investigate the dynamics of charged particles because it exhibits excellent long-term performance in conserving the Hamiltonian. No universal rule can be given to the dependence of regular and chaotic dynamics on varying one or two parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>13</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>c</mi><mn>14</mn></msub></semantics></math></inline-formula> in the two cases of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msub><mi>c</mi><mn>14</mn></msub><mo><</mo><mn>2</mn><msub><mi>c</mi><mn>13</mn></msub><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The distributions of order and chaos in the binary parameter space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>c</mi><mn>13</mn></msub><mo>,</mo><msub><mi>c</mi><mn>14</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> rely on different combinations of the other parameters and the initial conditions.https://www.mdpi.com/2218-1997/9/8/365modified theory of gravityblack holesmagnetic fieldchaossymplectic integrator |
| spellingShingle | Caiyu Liu Xin Wu Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory Universe modified theory of gravity black holes magnetic field chaos symplectic integrator |
| title | Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory |
| title_full | Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory |
| title_fullStr | Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory |
| title_full_unstemmed | Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory |
| title_short | Effects of Coupling Constants on Chaos of Charged Particles in the Einstein–Æther Theory |
| title_sort | effects of coupling constants on chaos of charged particles in the einstein aether theory |
| topic | modified theory of gravity black holes magnetic field chaos symplectic integrator |
| url | https://www.mdpi.com/2218-1997/9/8/365 |
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