Chaos in a Magnetized Modified Gravity Schwarzschild Spacetime

Abstract

Based on the scalar–tensor–vector modified gravitational theory, a modified gravity Schwarzschild black hole solution has been given in the existing literature. Such a black hole spacetime is obtained through the inclusion of a modified gravity coupling parameter, which corresponds to the modified gravitational constant and the black hole charge. In this sense, the modified gravity parameter acts as not only an enhanced gravitational effect but also a gravitational repulsive force contribution to a test particle moving around the black hole. Because the modified Schwarzschild spacetime is static spherical symmetric, it is integrable. However, the spherical symmetry and the integrability are destroyed when the black hole is immersed in an external asymptotic uniform magnetic field and the particle is charged. Although the magnetized modified Schwarzschild spacetime is nonintegrable and inseparable, it allows for the application of explicit symplectic integrators when its Hamiltonian is split into five explicitly integrable parts. Taking one of the proposed explicit symplectic integrators and the techniques of Poincaré sections and fast Lyapunov indicators as numerical tools, we show that the charged particle can have chaotic motions under some circumstances. Chaos is strengthened with an increase of the modified gravity parameter from the global phase space structures. There are similar results when the magnetic field parameter and the particle energy increase. However, an increase of the particle angular momentum weakens the strength of chaos.

Chaos in a Magnetized Brane-World Spacetime Using Explicit Symplectic Integrators

“…From the above discussion, we can see that chaos should become strong as magnetic field parameter b increases. The result fits well with that in the previous references [59][60][61][67][68][69][70][71]75,76]. These facts show that the external electromagnetic forces lead to an absence of the fourth constant (i.e., the azimuthal motion), and they are an important source for inducing chaos.…”

“…However, they are integrable because enough conserved quantities of motion are present. When these black holes suffer from some perturbations, such as external magnetic field [57][58][59][60] and the discs or rings around black holes [61,62], the spacetimes may be non-integrable. In this case, chaos may occur.…”

Explicit Symplectic Methods in Black Hole Spacetimes

“…It also accepts the number of splitting terms k = 3 (Zhou et al 2022). The splitting number is k = 5 for the Reissner-Nordström black hole (Wang et al 2021b) and a magnetized modified gravity Schwarzschild spacetime (Yang et al 2022). The splitting number is k = 6 for the Reissner-Nordström-(anti-)de Sitter black hole (Wang et al 2021c).…”

“…The black hole spacetimes that allow such splitting methods in the previous studies (Wang et al 2021a(Wang et al , 2021b(Wang et al , 2021c are several examples of the metric family of Equation (8). A modified gravity Schwarzschild black hole solution based on the scalar-tensor-vector modified gravitational theory (Yang et al 2022) resembles one of the metric family of Equation (8). Brane-world black holes (Deng 2020a; Hu & Huang 2022) are also an example of the same metric family.…”

Measurement of the Central Galactic Black Hole by Extremely Large Mass-Ratio Inspirals

“…In the framework of GR, the Schwarzschild or Kerr metric describes the space-time of an uncharged BH. However, in modified and alternative theories of gravity, there are other possible solutions for the description of the space-time of BHs [26][27][28][29][30][31]. The predictions of different theories of gravity are different, so a universal and reasonable theory about the GWs of an X-MRI should be model-independent.…”