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 <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.