Motion of Spinning Particles around Black Holes

TL;DR

This paper introduces an open-source computational tool for simulating and visualizing the trajectories of spinning particles around black holes, applicable to various spacetime backgrounds.

Contribution

The work provides a modular, open-source code for integrating and visualizing spinning particle motion in spherically symmetric spacetimes, adaptable beyond Schwarzschild.

Findings

The code accurately simulates spinning particle trajectories.

It is adaptable to different spacetime metrics.

Provides visual insights into complex particle dynamics.

Abstract

The motion of spinning particles around compact objects, for example a rotating stellar object moving around a supermassive black hole, is described by differential equations that are, in general, non-integrable. In this work, we present a computational code that integrates the equations of motion of a spinning particle moving in the equatorial plane of a spherically symmetric spacetime and gives a visual representation of its trajectory. This code is open source, freely available and modular, so that users may extend its application not only to the Schwarzschild metric, but also to other backgrounds.

Equations

$$\{\begin{array}{ll}\frac{D{P}^{\mu }}{D\lambda }=& -\frac{1}{2}{R}_{\nu \alpha \beta }^{\mu }{u}^{\nu }{S}^{\alpha \beta }\\ \frac{D{S}^{\mu \nu }}{D\lambda }=& P^{\mu }{u}^{\nu }-u^{\mu }{P}^{\nu },\end{array}$$

$$s^2=\frac{1}{2}{S}^{\mu \nu }{S}_{\mu \nu }.$$

$$m^2=-P_{\alpha }{P}^{\alpha }.$$

$$\mu =-P^{\beta }{u}_{\beta },$$

$$P^{\alpha }=\mu u^{\alpha }-u^{\beta }\frac{D{S}^{\alpha \beta }}{D\lambda },$$

$$P_{\mu }{S}^{\mu \nu }=0.$$

$$\{\begin{array}{ll}{S}^{01}=-S^{10}=& \frac{P_3}{P_0}{S}^{13}\\ S^{03}=-S^{30}=& -\frac{P_1}{P_0}{S}^{13}.\end{array}$$

$$d{s}^2=g_{00}d{t}^2+g_{11}d{r}^2+g_{22}d{\theta }^2+g_{33}d{\varphi }^2.$$

$$s^2=$$

$$g=\det g_{\mu \nu }=g_{00}{g}_{11}{g}_{22}{g}_{33}$$

$$g^{\mu \nu }=\frac{1}{g}\left(\begin{array}{cccc}{g}_{11}{g}_{22}{g}_{33}& 0& 0& 0\\ 0& g_{00}{g}_{22}{g}_{33}& 0& 0\\ 0& 0& g_{00}{g}_{11}{g}_{33}& 0\\ 0& 0& 0& g_{00}{g}_{11}{g}_{22}\end{array}\right).$$

$$s^2=$$

$$\{\begin{array}{ll}{S}^{13}=& -\frac{s{P}_0}{m}\frac{1}{\sqrt{-g_{00}{g}_{11}{g}_{33}}}\\ S^{01}=& -\frac{s{P}_3}{m}\frac{1}{\sqrt{-g_{00}{g}_{11}{g}_{33}}}\\ S^{03}=& -\frac{s{P}_1}{m}\frac{1}{\sqrt{-g_{00}{g}_{11}{g}_{33}}}.\end{array}$$

$$C_k=P^{\mu }{k}_{\mu }+\frac{1}{2}{S}^{\mu \nu }{\nabla }_{\mu }{k}_{\nu }.$$

$$\xi ={\partial }_t\text{ and }\zeta ={\partial }_{\varphi },$$

$$C_t=-E=$$

$$S^{\mu \nu }{\nabla }_{\mu }{\xi }_{\nu }=S^{\mu \nu }{\partial }_{\mu }{\xi }_{\nu }-S^{\mu \nu }{\Gamma }_{\mu \nu }^{\alpha }{\xi }_{\alpha }=S^{\mu \nu }{\partial }_{\mu }{\xi }_{\nu }$$

$$E=$$

$$E=-P_0-\frac{1}{2}{S}^{10}{\partial }_r{g}_{00}$$

$$E=-P_0-\frac{1}{2}\frac{s{P}_3}{m}\frac{{\partial }_r{g}_{00}}{\gamma },$$

$$C_{\varphi }=J=$$

$$J=$$

$$J=P_3+\frac{1}{2}{S}^{13}{\partial }_r{g}_{33}$$

$$J=P_3-\frac{1}{2}\frac{s{P}_0}{m}\frac{{\partial }_r{g}_{33}}{\gamma }.$$

$$\{\begin{array}{ll}{P}_0=& -\frac{2m\gamma \left(2m\gamma E+sJ{\partial }_r{g}_{00}\right)}{4m^2{\gamma }^2+s^2{\partial }_r{g}_{00}{\partial }_r{g}_{33}}\\ P_3=& \frac{2m\gamma \left(2m\gamma J+sE{\partial }_r{g}_{33}\right)}{4m^2{\gamma }^2+s^2{\partial }_r{g}_{00}{\partial }_r{g}_{33}}.\end{array}$$

$$(P_1{)}^2=-\frac{m^2+g^{00}(P_0{)}^2+g^{33}(P_3{)}^2}{g^{11}}.$$

$$\frac{D{S}^{01}}{D\lambda }=P^0{u}^1-P^1.$$

$$-\frac{s{P}_3}{m}\frac{D}{D\lambda }\left(\frac{1}{\gamma }\right)-\frac{s}{m\gamma }\frac{D{P}_3}{D\lambda }=P^0\dot{r}-P^1.$$

$$\frac{D}{D\lambda }\left(\frac{1}{\gamma }\right)=\frac{d{x}^{\mu }}{d\lambda }{\nabla }_{\mu }\left(\frac{1}{\gamma }\right)=\frac{dr}{d\lambda }{\partial }_r\left(\frac{1}{\gamma }\right)=-\frac{\dot{r}}{{\gamma }^2}{\partial }_r\gamma .$$

$$\dot{r}\left[P^0-\frac{s{P}_3}{m{\gamma }^2}{\partial }_r\gamma \right]=P^1-\frac{s}{m\gamma }\frac{D{P}_3}{D\lambda }.$$

Full text

Keywords: Black Holes. Equations of Motion. Spinning Particles.

1 Introduction

Macroscopic astrophysical objects classically have a spin angular momentum associated with their rotation. When these objects, such as stars or compact objects (black holes or neutron stars), belong to physical systems (e.g. binaries), it is well known that the existence of spin has consequences on the behavior of the dynamics of the system. A particular case of interest is that of spinning bodies orbiting around supermassive black holes and its relation with the production of gravitational waves.

In order to introduce the classical spin to the description of the dynamics of extended bodies in curved spacetimes, Mathisson demonstrated in 1937 that the equations of motion show an interaction between the Riemann curvature tensor and the spin of the test particle [1]. Later, in 1951, Papapetrou made important contributions in the covariant treatment of the equations of motion, showing once again, that a spinning object moves in orbits that will differ from the geodesics [2]. Another significant input to this framework was made by Dixon in 1970, when he reformulated the equations of motion in a generalized form to hold exactly for an extended body [3]. Most of these approaches are restricted to the pole-dipole approximation, reducing the problem by considering that the body is not itself contributing to the gravitational field, and where just monopole and dipole or the mass and spin terms, respectively, are taken into account [4]. Finally, it has been studied the use of the so-called spin supplementary conditions, which fixes the center of the mass of the spinning body [5, 6, 7, 8, 9]. Following this logical path of results, the formalism that will be used in this work to describe the dynamics of a spinning particle in a curved spacetime is described by the Mathisson-Papapetrou-Dixon (MPD) equations.

Although the motion of spinning particles in black holes backgrounds is, in general, a non-integrable problem, it has been analyzed using the MPD equations under different approaches. One of the first methods considers only the equatorial motion of a spinning particle in Schwarzschild and Kerr spacetimes, and obtains the behavior of some of the properties of the innermost stable circular orbit [10]. Generalizations of this analysis, including the characteristics of black hole solutions from other gravitational theories, are presented in papers such as [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. The motion of spinning particles in static and spherically symmetric backgrounds have been also studied in literature, including the Schwarzschild–de Sitter spacetime [26, 27, 28], the Reissner-Nordström-de Sitter black hole [29] and in general Schwarzschild-like space–times [30]. Motion in backgrounds from alternative gravity theories was studied too. For example, in the background of a Schwarzschild black hole surrounded by quintessential matter field [31], in four-dimensional Einstein–Gauss–Bonnet space–time [32], around a charged Hayward black hole [33] or in a cosmological and general static spherically symmetric background [34]. All of these approaches prove that the spin of the test particle produce a non geodesic orbit, due to the existence of an additional force.

There are currently various astrophysical systems of interest that have emerged in recent years, including the collision processes between spinning particles [35, 36, 37, 20, 38, 39, 40, 41], the study of black hole backgrounds as accelerators for spinning particles [31, 21, 15, 42, 23] or the emission of gravitational radiation from spinning particles orbiting around or plunging into a black hole [43, 44, 45, 25, 14].

In this work, we provide a numerical code, which is publicly accessible and modifiable, to solve the equations of motion of a spinning particle orbiting around a spherically symmetric and static spacetime. Furthermore, it will give us the possibility of visualizing the trajectories based on the initial values of the system. This chapter is organized as follows: in Sec. 2, we present the MPD equations and the effects on the spin tensor when the motion is restricted to the equatorial plane. In Sec. 3, we introduce the metric that describe a spherically symmetric and static background to determine the components of the spin tensor. In Sec. 4, we obtain the components of the momentum vector from the associated conserved quantities of the motion. Then, the expressions of the radial and angular velocities are found in Sec. 5. Later, we show a brief summary of the step by step to solve the equations of motion computationally in Sec. 6. and finally, in Sec. 7, the code is commented.

2 Equations of Motion for a Spinning Particle

2.1 The Mathisson-Papapetrou-Dixon Equations

The MPD equations describe the motion of a spinning particle in a curved background. They state that [1, 2, 3]

[TABLE]

where $P^{\mu}$ and $u^{\mu}=\frac{dx^{\mu}}{d\lambda}$ are the momentum and velocity vectors of the test particle, $R^{\mu}_{\,\,\nu\alpha\beta}$ is the Riemann tensor and $S^{\mu\nu}$ represents the spin tensor that defines the spin angular momentum of the test particle through the relation

[TABLE]

As is well known, the proper mass of the test particle is given by the norm of the momentum vector,

[TABLE]

but, introducing the quantity

[TABLE]

equations (1) gives the relation

[TABLE]

indicating that the momentum and the velocity vectors are no longer parallel for spinning particles. This fact gives the freedom to introduce an additional condition which may produce different trajectories [12, 5]. There are many equations that we can introduce, but we will specifically use the Tulczyjew spin-supplementary condition [46, 5], which restricts the spin tensor to generate only rotations through the relation

[TABLE]

Due to the spherical symmetry of the background, it is possible to consider, without loosing generality, that the orbital motion is restricted to the equatorial plane, $\theta=\frac{\pi}{2}$, which implies $P_{2}=0$. For simplicity, we will only consider the spin aligned or anti-aligned orbits. Thus, we impose the condition $S^{2\mu}=0$ in equation (6), to obtain the non-vanishing components of the spin tensor as

[TABLE]

3 Spherically Symmetric Static Spacetimes

In this work, we will consider a spherically symmetric and static spacetime, with the following general line element

[TABLE]

Using this metric, together with equations (6) and (7), the spin angular momentum of the test particle is written as

[TABLE]

The determinant of the metric tensor is given by

[TABLE]

while the inverse tensor is calculated as

[TABLE]

Then, using these results we obtain

[TABLE]

Solving for the spin tensor component $S^{13}$ and using this result in equations (7) gives

[TABLE]

4 Conserved Quantities in the Motion of a Spinning Test Particle

Given the existence of a Killing vector $k$, we associate the existence of a conserved quantity in the motion of the spinning particle through the relation

[TABLE]

Assuming that the metric tensor components do not depend explicitly on the coordinates $t$ and $\phi$, there are two Killing vectors,

[TABLE]

corresponding to the conservation of energy and angular momentum, respectively. For the first of these vectors we have

[TABLE]

The covariant derivative in the second term gives

[TABLE]

and because $\xi_{\mu}=g_{\mu\nu}\xi^{\nu}=g_{00}\delta^{0}_{\mu}$, we obtain

[TABLE]

Considering the dependence $g_{00}=g_{00}(r)$, we get

[TABLE]

and using equations (13) this becomes

[TABLE]

where we introduce $\gamma=\sqrt{-g_{00}g_{11}g_{33}}$. A similar procedure for the Killing vector $\zeta$ gives

[TABLE]

This time, the Killing vector satisfies $\zeta_{\mu}=g_{\mu\nu}\zeta^{\nu}=g_{33}\delta^{3}_{\mu}$, and therefore

[TABLE]

Considering now the dependence $g_{33}=g_{33}(r)$ ( because considering that the motion occurs in the equatorial plane implies that there is no dependence with the coordinate $\theta$ ), we obtain

[TABLE]

and using equations (13) this becomes

[TABLE]

From equations (20) and (24) we solve to obtain the components $P_{0}$ and $P_{3}$ of the momentum vector as

[TABLE]

The component $P_{1}$ is obtained from the normalization condition (3), giving

[TABLE]

5 The Equations of Motion revisited

Now, we are able to write explicitly the equations of motion given in (1). Using a parameter $\lambda$ for which $\dot{u}^{0}=1$, we have that the equation for the component $S^{01}$ of the spin tensor is

[TABLE]

Using equations (7) and (13) in the left hand side gives

[TABLE]

Since the factor $\frac{1}{\gamma}=\frac{1}{\sqrt{-g_{00}g_{11}g_{33}}}$ depends only on the coordinate $r$, we have that

[TABLE]

Replacing in the equation of motion gives

[TABLE]

The last term in the right hand side can be evaluated using the equation of motion for $P_{3}$ arising from (1),

[TABLE]

This gives the final result

[TABLE]

A similar procedure using the equation of motion for the component $S^{03}$ of the spin tensor, gives the following relation,

[TABLE]

Note that using the anti-symmetry properties (in the first two indices) of the Riemann tensor and the fact that the particle is moving on the equatorial plane $\dot{\theta}=0$, we have that

[TABLE]

and

[TABLE]

Therefore, the equations of motion are now

[TABLE]

From the first of these relations, we obtain an expression for the radial velocity, $\dot{r}$,

[TABLE]

while the second relation gives an equation for the angular velocity $\dot{\phi}$,

[TABLE]

6 Summary of the Equations and Procedure to Solve Them Computationally

In order to solve the equations of motion for the particular case of the spinning particle in the equatorial plane of a spherically symmetric and static spacetime, we will impose initial values for the following quantities:

• •

Proper mass of the particle: $m$ (conserved)

• •

Initial position: $t_{0}$, $r_{0}$ and $\phi_{0}$

• •

Total Energy of the particle: $E$ (conserved)

• •

Orbital Angular Momentum : $j$ (conserved)

• •

Spin Angular Momentum : $s$ (conserved)

The total angular momentum is determined by the values of $j$ and $s$ by

[TABLE]

Given these initial values, we proceed with the following steps:

Obtain the initial values for the components of the momentum using the equations

[TABLE] 2. 2.

Determine the initial values of the spin tensor components

[TABLE] 3. 3.

Calculate the Riemann tensor components $R_{\mu\nu\alpha\beta}$. 4. 4.

Numerically solve the equations

[TABLE]

and

[TABLE]

to obtain the new values of the coordinates $r$ and $\phi$. 5. 5.

Begin the process again at step 1.

7 The MSPBH Code

The MSPBH (Motion of Spinning Particles around Black Holes) Code111The MSPBH Code is available at https://github.com/cdelv/MSPBH is a Python implementation that solves the equations of motion described above to obtain the trajectory of a spinning particle around a spherically symmetric compact object described by a metric tensor. The program uses symbolic algebra packages SymPy and EinsteinPy to calculate the required quantities, such as partial derivatives of the metric and the Riemann tensor, and the integration of the equations of motion is performed using odeint from the SciPy package. The code reads all the necessary variables, constants, initial conditions, and information about the metric needed to calculate the trajectories from a configuration file. When executing the code without configuration, a file called template.txt is created with a usage example corresponding to the last stable orbit in the Schwarzschild metric.

Some of the parameters and options in the configuration file include the definition of the metric itself. For example, if the options User_tensor and User_metric are set to FALSE, the program will use the incorporated Schwarzschild metric from the library Einsteinpy, and compute the Riemann tensor. When User_tensor is TRUE, the program will use the components of the tensors given by the user on the configuration file (this option has precedence over User_metric. In fact, when User_metric is TRUE and User_tensor is FALSE, the program will use the function Create_User_Metric_Tensor to create a custom metric tensor with the library Einsteinpy. If User_metric is False, you don’t have to give the Riemann and metric tensor components on the template. In this case, once the metric tensor is determined, the code will compute the Riemann tensor using the EinsteinPy library and proceeds to calculate the orbit.

A successful execution of the code will provide a series of plots (is the user sets options Plot_trajectoryy and/or Plot_energy to TRUE), including the trajectory of the particle and the behavior of the coordinates $r(t)$ and $\phi(t)$, and two output files: ’log.out’ and ’data.csv’. The first one has information about all the variables read from the configuration file, while the second is a CSV file where you will find the information about time, $r$ and $\phi$ coordinates, energy, and angular momentum of the particle at each step in the integration. In Figure 1, the trajectory and the evolution of the coordinates of a non spinning particle can be seen.

For a spinning particle there are two possible configurations: the spin aligns with the angular momentum of the orbit or not. On Figure 2 we show the trajectory of the same particle described in Figure 1 but this time with a non-zero spin aligned with the angular momentum.

8 Conclusion

We have considered the equations of motion of a spinning particle moving in the equatorial plane of a spherically symmetric and static background to write the expressions for the radial and angular velocities in terms of the components of the momentum vector, the spin tensor and the Riemann tensor. After given a brief summary of the step by step algorithm to solve these equations of motion numerically, we have presented a Python implementation named the MSPBH Code, which solves numerically the equations of motion and provides a visualization of the trajectory. The code also gives the coordinates of the spinning particle together with the corresponding energy and angular momentum of the spinning particle.

We have illustrated the implementation by considering two examples (non spinning particle and spinning particle moving around a Schwarzschild black hole) and showing the obtained trajectory. The results confirm that the spin of the particle causes a change in the trajectory, producing a fall into the black hole.

The MSPBH Code provides a practical and visual tool to demonstrate how the spin of the particle affects its own dynamics in a spherically symmetric and static spacetime. It is open source, freely available and modular, so that users may extend its application not only to the default schwarzschild metric, but also to other backgrounds included in the EinsteinPy library or even by defining its own metric.

Acknowledgements. This work was supported by the Universidad Nacional de Colombia. Research Incubator No. 64-Computational Astrophysics and Hermes Grant Code 41673.