Presence of horizon makes particle motion chaotic
We analyze the motion of a massless particle very near to the event horizon. It reveals that the radial motion has exponential growing nature which is the signature of the presence of chaos in the particle motion. This is being confirmed by investigating the Poincar section of the trajectories with the introduction of a harmonic trap to confine the particle’s motion. Two situations are investigated: (a) the black hole is any static, spherically metric and, (b) spacetime represents a stationary, axisymetric black hole (e.g., Kerr metric). In both cases, the largest Lyapunov exponent has upper bound which is the surface gravity of the horizon. We find that the inclusion of rotation in the spacetime introduces more chaotic fluctuations in the system. The possible implications are finally discussed.
The dynamics of a particle around a massive object is one of the central attention in physics. General theory of relativity (GR) successfully explains several phenomena in the motion of astrophysical objects. It explains how the light can bend due to the presence of a mass in the spacetime. Among several compact objects, black holes are one of the fascinating thing in the Universe till now. Theoretically these are the solutions of Einstein’s equations of motion and defined by the region whose boundary is a one way membrane (classically through which nothing can escape), known as event horizon. Recent discovery of LIGO Abbott:2016blz () confirms that black holes are no longer a theoretical concept, rather they indeed have the existence in the spacetime. Till today, researchers are devoting lot of attentions not only to understand the Physics of black holes, but also the kind of phenomena that they induce around themselves both at the astrophysical and the quantum level.
Near horizon Physics is very important in various ways. In this paper, we try to understand how the motion of a particle changes when it approaches very near to the black hole horizon. There have been studies in this direction Bombelli:1991eg (); Sota:1995ms (); Vieira:1996zf (); Suzuki:1996gm (); Cornish:1996ri (); deMoura:1999wf (); Takahashi:2008zh (); Hashimoto:2016dfz (); Li:2018wtz (). These calculations are performed which were mainly based on either Newtonian approximation Sota:1995ms (); Vieira:1996zf () or effective potential technique Bombelli:1991eg (); Suzuki:1996gm (); Cornish:1996ri (); deMoura:1999wf (); Takahashi:2008zh (). Moreover, black hole system is either spinning Takahashi:2008zh () or magnetized Li:2018wtz () with the massive, charged or spining test particles. There is a recent analysis Hashimoto:2016dfz () that considers the effect of Schwarzschild black hole on a massive test particle. To make sure that the particle does not enter into the horizon an extra potential has been added to the system. However, the situation with the massless particles have not been addressed so far. This has been a thriving area of the research since last few years as we know that the interaction of light with gravity is quite non-trivial in nature.
In this paper we have addressed the issue more deeply and in a more general way. Calculations show that the outgoing radial trajectory of the particle grows exponentially with time. It indicates the chaotic motion of the particle. We ascertain this observation numerically using detailed investigation of the Poincare section of the particle trajectories, incorporating a harmonic potential to the system. Both the situations, any static spherically symmetric (SSS) (not necessarily Schwarzschild case) and stationary axisymmetric (Kerr solution), have been investigated. The test particle has been considered to be chargeless and massless. It is observed that a static spherically symmetric horizon induces chaos in a particular energy range, while, the presence of rotation in the spacetime makes the system more chaotic.
In this analysis we have not only found that the chaos is inevitable in presence of horizon, the value of Lyapunov exponent has also an upper bound. Interestingly, the bound can be shown to be given by , where is the surface gravity of the black hole (it is the acceleration of a particle, measured by an asymptotic distance observer, which is very near to the horizon). This has been predicted recently Hashimoto:2016dfz () for a massive particle in a completely different analysis. We found that such a value is compatible with the bound predicted in Maldacena:2015waa () by analyzing the out-of-order correlator of some observables in the Sachdev-Ye-Kitaev (SYK) model. This shows that the bound is very much universal in nature.
Let us now discuss what we can predict by our present analysis. Note that we consider only the massless particles which are following the outgoing trajectories, of which the radial one is the radial null geodesic. It is observed that the same null geodesic is responsible for the Hawking radiation Hawking:1974rv (); Hawking:1974sw () of the particles from the horizon (see Parikh:1999mf (); Banerjee:2008gc (); Banerjee:2008ry () for understanding the Hawking radiation as tunneling). Therefore, it may be worth mentioning that the radiated particles, after escaping from the horizon barrier, exhibit chaotic behavior in their motion due to the influence of the horizon. This implies that the horizon not only radiates (Hawking radiation), it also infuses chaos. Of course, this is not a conclusive statement, rather a suggestive one. In order to get more insight to this one needs to investigate more, may be in the quantum mechanical way. After having this discussions, we now turn our focus to the main analysis.
Ii Static spherically symmetric black hole
Consider a static, spherically symmetric black hole background, given by
where the horizon is determined by and . It has a coordinate singularity at this position. To remove this let us adopt the Painleve coordinate transformation:
Under this transformation, the above metric takes the following form:
It has a timelike Killing vector and the energy of a particle, moving under this background, is given by , where is the four momentum vector. Our aim in the following is to find this energy in terms of other components of momentum. To find it, we take help of the covariant form of the dispersion relation , with , mass of the particle. Expanding this under the background (3), we obtain
where, only the radial and directions motion have been considered, i.e., the particle is moving only along the radial and the directions. For a massless particle, solution of the above equation (4) with gives the energy. It is found that it has two values:
Positive sign denotes the energy for the outgoing particle, while, the other sign is for ingoing particle. In this paper we will be mainly interested to investigate the dynamics of the outgoing particles, therefore, throughout the discussions only the positive sign will be considered.
Next aim is to find the trajectory of the particle. It will be computed from the Hamilton’s equations of motion. The equations of motions for the energy (5) can be written as
Before, proceeding further, let us now discuss what happens near the horizon if the particle has only radial motion. For this we consider and make an expansion of near the horizon as
where, we have retained only the first order term. Here is the surface gravity of the black hole. Substitution of this in (6) leads to a very simple equation
The solution is
where, is the integration constant and is parameter with respect to which the derivative is taken, i.e., . The above solution implies the exponential growth of radial coordinate which can be attributed to the appearance of chaos. We shall show explicitly that this indeed the case by keeping this particle in a harmonic potential and allowing it to move along the direction as well. In this situation, it may be worth to point out that the Lyapunov coefficient () is bounded by the following equation:
Now if one consider the quantum nature of the black hole, then it has a temperature, given by the Hawking expression . In that case the above bound reduces to a very well known form . This was first mentioned in Maldacena:2015waa () for the SYK model.
There is a very interesting connection with the above radial null geodesic (11) with the Hawking effect in the context of tunneling mechanism Parikh:1999mf (); Banerjee:2008gc (); Banerjee:2008ry (). It must be noted that precisely this path has been used to find the tunneling probability from the horizon for the outgoing particles. One finds that this is non-zero and leads to Hawking radiation with the temperature, mentioned above. Therefore, after escaping from the horizon the radiated particles will exhibit chaotic motion. Hence the current analysis implies that the horizon not only radiates, it also makes the radiation chaotic.
With this let us now confirm if the presence of horizon really creates chaos in the system. For that we consider two harmonic potentials and along and directions, respectively. Here , and are spring constants while and are the equilibrium positions of these two harmonic systems. Also we choose the form of as (10). Now if the particle moves under the influence of these potentials then the total energy of this particle is
and correspondingly, the equations of motion will have the form as
We will use these equations to study the motion of the particle numerically.
Iii Kerr black hole
After having the discussion on the effect of the SSS balck hole on the particle in this section we investigate the effect of rotation of the black hole on the over all dynamics the system. For this we can proceed in the similar way like we did for SSS. The Kerr metric in dragging Painleve coordinate is given in Jiang:2005ba (). Due to the large size of the equations, the metric and the trajectories for this case have been provided in supplementary material. It may be noted that here also if one concentrates only on the radial trajectories, it is given by (setting and in eq. (30) of supplementary materials)
where . Note that in this case also we are getting the same radial equation in the near horizon limit. So again the solution is (12). As a result of this the Lyapunov coefficient bound would be given by (13). Therefore we can see that the rotating black hole also appears to infuse chaotic fluctuations in the motion of particles. In the next section we will ascertain this observation by numerically solving the full set of dynamical equations and will also investigate the detailed influence of the rotation on the chaotic dynamics of the particles.
Iv Numerical Analysis
In the last section we analytically showed that the presence of black hole horizon induces the exponential growth of the radial trajectory of the particles which indicated the chaotic behaviour for both SSS and Kerr black holes. In this section we ascertain the claim by analyzing the Poincar section of the system obtained by solving the dynamical equations of motion for SSS (Eqs. 15-18) and Kerr (Eqs. 30-33) black holes. First we present numerical results for the SSS black hole and then those for the Kerr black hole in which we systematically analyze the effects of the rotation parameter on the chaotic fluctuations.
For SSS black hole model, the dynamical equations (Eqs. 15-18) are numerically solved using the fourth order Runge-Kutta scheme with fixed . For present study we have considered , , , and . The variables , , and are initialized with the random numbers and is obtained from Eq. 5 for a fixed energy .
In Fig. 1 we show the Poincar section of the particle trajectory (for SSS type black hole) projected over the (,) plane for different energies. The section is defined by the condition and . We have considered the energies , and as indicated in the plots. For low energy the Poincar section exhibits the regular KAM (Kolmogorov-Arnold-Moser) tori nonlinear:02 () and the corresponding orbit is mainly confined near the center of the harmonic potential (). Different colors in the figures indicate the trajectory of the particles for different initial conditions. As the total energy of the system is increased the trajectory approaches near to the black hole horizon () as a consequence of this KAM tori starts getting distorted and appeared to be pinched as shown in the figure for E=76.8 and E=77. Further increase in the energy () results the complete breaking of regular Tori and appearance of scattered points in the plane. This feature of the Poincar section supports the chaotic nature of the particle trajectory near the horizon as showed in the last section. Due to presence of horizon we have some upper bound on the energy. At present increasing the energy further above we find that during time evolution becomes less than the position of the horizon () that brings numerical instability in our calculation.
In order to see the effect of the rotation of the black hole on the particle dynamics next we consider the Kerr black hole model. The corresponding dynamical equations (Eq. 30-33) are solved using the Runge-Kutta fourth order. The initial conditions are chosen in the similar line as discussed for the SSS model. In Fig. 2 we show the Poincar sections for different energies with fixed rotation parameter . All the other parameters are considered same as the SSS model. We observe the similar feature of the KAM tori upon increase of energy for Kerr black hole as we observed for the SSS model. The regular KAM tori appeared at low energy gets squeezed along with appearance of some region filled with the scattered points as the energy is increased to . At high energy () as the surface of the trajectory approaches near the horizon there is complete breaking of these tori which is quite evident with the filling of the region with the random points. Interestingly here we obtain that the chaotic nature appears at relatively lower energy than those obtained with the SSS black hole. This particular feature suggests that the rotation of the black hole may introduce more chaotic fluctuations in the trajectories of the particles approaching towards the horizon. We confirm this by analyzing the nature of the particle trajectory by changing the rotation parameter for fixed energy . In Fig. 3 we plot the Poincar section in the plane for different rotation parameters , and at fixed energy . We clearly find that the increase in introduces chaotic fluctuations in the trajectories and at very high rotation the trajectory becomes fully chaotic. The nature of the appearance of chaos in the system upon increase in the for fixed energy appears to be same as those obtained while increasing the energy for fixed (see Fig. 2).
The trajectories of a massless and chargeless particle in a very near horizon region have been studied. The theoretical analysis showed that the radial motion is exponentially increasing function of time parameter. This indicates the chaotic motion of the particle. Carefully investigating the situation in presence of harmonic potential, it has been observed that this is indeed the case. Here both SSS and Kerr black holes cases have been investigated. For SSS case the chaos occurs at the particular energy range. Note that our SSS metric is not necessarily restricted to Schwarzschild spacetime, rather it incorporates all candidates of this type. Therefore even the SSS analysis is much more general than the earlier ones. Interestingly, we observed that the rotation parameter induces more chaos to the particle motion.
In this calculation, the outgoing path is taken to be the geodesics for a zero mass particle for which the radial direction is the null geodesic. This particular path plays important role in investigating the Hawking radiation as tunnelling phenomenonParikh:1999mf (); Banerjee:2008gc (). It reveals that horizon not only radiates quantum mechanically, but also can influence the trajectories of the particles when they are very near to the horizon. If this is the case, then it may be possible that the radiated particles may follow chaotic motion after being escaped from the horizon barrier. Note that due to complexity of the problem here we have not been able to provide any rigorous analytical calculations to support the claim. To be more sure about the detailed nature of the effect of the horizon on the chaotic fluctuation we need to investigate more on this. Finally it must be mentioned that our analysis is completely classical in nature. Probably a systematic calculation based on the quantum mechanics may unravel more information about the chaotic behavior of the particles in black hole spacetime.
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Trajectories in Kerr spacetime
In the Painleve coordinate transformation the Kerr metric can be written in this form Jiang:2005ba ():
and . is the mass and is the angular momentum per unit mass. We call it as rotation parameter. The event horizon is given by . It leads to the location of horizon as
Using and identifying the energy of the massless particle as , we find the energy of the particle which is moving through the two harmonic potentials in the above background can be calculated as:
Now since we are interested near to the horizon, expanding upto the first order one obtains
Using this and replacing all the variables in Eq. (26) the energy turns out to be,
Now the equation of motions are
These equations have been used to study the motion of the particle numerically.