Centrifugal Acceleration in Relativistic Astrophysics

Abstract

Particles moving along prescribed, relativistically rotating trajectories may exhibit quite unexpected, interesting kinematic behavior. Their dynamics may lead to a number of physical processes that could have various important consequences in a wide variety of relativistic astrophysical objects. In this chapter the author gives brief review of the theoretical ideas and results related to Machabeli and Rogava gedankenexperiment (1994) and their astrophysical implications. In particular three astrophysical cases—acceleration of particles by rotating magnetospheres in AGNs, centrifugal acceleration and gamma flares in Crab nebula, and self-trapping as a beaming mechanism for Fast Radio Bursts—are discussed. Conclusions and future research prospects are briefly outlined.

1. Introduction

Crisis is a Hair

Toward which forces creep

Past which forces retrograde….

- Emily Dickinson.

Christiaan Huygens introduced the concept of the centrifugal force in the manuscript entitled “De vi Centrifuga” written in 1659 [1] and published posthumously in Leiden, in 1703. Since then, the concept of the centrifugal force became an integral part of the essential vocabulary of physics and a topic of different interpretations in the philosophy of the physical science. It proved to be a useful notion for studies of a wide variety of problems involving rotation and dynamics of particles/bodies in rotating frames of reference. Rotation is omnipresent in the nature; as long as the humans advanced in the space exploration, it became increasingly evident that the physical processes related to the presence of rotation play an important, often decisive, role in the complex and still incompletely understood texture of the physical universe.

From the very beginning, there was a certain kind of dichotomy in the understanding of the centrifugal force. According to Newton and basic principles of the classic mechanics, the magnitude and the direction of the centrifugal force, acting on matter in a rotating frame of reference, are not connected with motion of the matter, but are entirely and explicitly determined by the rotation of the reference frame. On the basis of this postulate, the centrifugal force is defined in terms of the angular velocity of the instantaneously corotating reference frame and a displacement of the moving body (particle) from the instantaneous rotation axis. Consequently, the direction of the force is always unequivocally determined by the direction of the displacement vector and it always points outward. However, it has to be remembered that the original concept of the centrifugal force, outlined by Huygens [2], emphasized another, drastically different, aspect of the force. In particular, Huygens’ idea was the following: “centrifugal force should not be connected to rotation of a reference frame, but to motion along a curved path in space.” Within the framework of classic, nonrelativistic physics, the difference between these two postulates does not lead to a quantitative difference, because in the Newtonian gravitation theory, based on Euclidean geometry, these two definitions are equivalent. But in Relativity, when the space–time is not Euclidean, they differ from each other [2] and may lead to different results.

In recent decades, the increase of interest in the phenomenon of relativistic rotation was largely due to series of significant results by M. Abramowicz and his collaborators [2, 3, 4, 5] related to physics of relativistic rotation in the vicinity of black holes. In particular, it was found that under certain circumstances, the centrifugal force attracts toward the rotation axis even in the case of a nonrotating Schwarzschild black hole. This outstanding fact led to a number of unusual rotational effects considered and discussed in these publications.

About 30 years ago, Machabeli and Rogava considered [6] a “gedankenexperiment” (viz. an experiment carried out in thought only): centrifugally driven motion of a bead inside a long, straight pipe rotating with a constant angular velocity $\omega =\mathit{\text{const}}$ around an axis normal to its symmetry axis. The motion of the bead was studied in the frame of reference of the rotating pipe. Different classes of exact analytic solutions were found, exhibiting puzzling regimes of motion: it was discovered that centrifugally driven bead could both accelerate and decelerate, and the acceleration could change its sign; actual modes of motion strongly depended on the value of the initial velocity. The purpose of the idealized “gedankenexperiment” was to mimic the motion of centrifugally driven charged particles in various astronomical situations. The possible importance of this result in the context of centrifugally driven dynamics of particles in rapidly rotating pulsar magnetospheres and astrophysical jets was later repeatedly pointed out [7] and extensively studied in a number of subsequent publications [8, 9, 10, 11, 12]. Namely, particle acceleration by rotating magnetospheres in active galactic nuclei was investigated [13]; the role of radiation reaction forces in the dynamics of centrifugally accelerated particles was explored [14]; centrifugal acceleration was studied in isotropic photon fields [15] and in wormhole metrics [16].

In this chapter of the present book, I will try to give a brief but systematic overview of the Machabeli & Rogava gedankenexperiment and its astrophysical implications. The material is divided into two sections: the next one is dedicated to the theoretical ideas and results, while the final one deals with astrophysical applications of the theory. The latter also contains a very short description of unresolved issues and possible directions for further, both physical and astrophysical, research.

2. Theoretical background

In this section, a brief overview of theoretical ideas and results related to the centrifugally driven motion of particles along prescribed trajectories is given. Firstly, main characteristics of the original gedankenexperiment [6] are derived and discussed in detail. Secondly, it is shown how the idea developed and what is the range of related physical results disclosed in the process.

2.1 Original gedankenexperiment: Rotating frame study

In Ref. [6], a straight pipe rotating around an axis normal to the pipe, and a small bead moving inside the pipe without the friction were considered; radii of the bead and the pipe being equal to one other. According to Newtonian mechanics the bead would move with ever-increasing acceleration, if at the moment $t=0$ it was located just above the axis pivot ($r_0=0$) and had an initial velocity $v_0\ne 0$, then the solution of Newton’s second law of motion would imply that the radial distance $r\left(t\right)$ from the axis would increase in time as:

$$r\left(t\right)=\left(v_0/\omega \right)\mathit{sinh}\left(\mathit{\omega t}\right),$$E1

while the bead’s radial velocity $v_r\left(t\right)$ along the pipe would monotonously increase as:

$$v_r\left(t\right)=v_0\mathit{cosh}\left(\mathit{\omega t}\right).$$E2

However, relativity-related common sense tells us that if the pipe is long enough and its walls are absolutely rigid, then the bead’s increasing velocity sooner or later would become relativistic, $v\ll c$ condition sooner or later ceases to hold and the bead’s motion has to be scrutinized by means of special relativity. Intuitively it seems evident that the increase of the velocity would somehow be limited since the total velocity of the bead $v_{\mathit{tot}}\equiv \sqrt{v_r^2+r^2{\omega }^2}$ cannot exceed the speed of light. Therefore, the problem has to be considered within the framework of a relativistic theory. Hereafter, we shall use geometrical units, in which $G=c=1$. If the bead would reach the light cylinder (defined as the radial distance $r^{\ast }\equiv 1/\omega$, that is, $v_{\varphi }\left(t\right)\equiv \omega r^{\ast }=1$) its radial velocity at $r^{\ast }$ should fall to zero value, implying that the radial velocity $v_r\left(t\right)$, being increasing at the initial stage of the motion, must become decreasing as the bead approaches the light cylinder.

In Ref. [6] the problem was considered in a reference frame of the rotating pipe (rotating frame, hereafter referred to as RF), where the dynamics of the bead motion appeared to be the simplest: one-dimensional, radial motion along the straight pipe. If the space–time in the laboratory frame (hereafter referred to as LF) is Minkowskian with the metric:

$${\mathit{ds}}^2\equiv -d{\tau }^2=-{\mathit{dT}}^2+{\mathit{dX}}^2+{\mathit{dY}}^2,$$E3

then applying the transformation of variables: $T=t,X=\mathit{\text{rcos}}\left(\mathit{\omega t}\right),Y=\mathit{\text{rsin}}\left(\mathit{\omega t}\right)$ it can be rewritten in the following form:

$${\mathit{ds}}^2\equiv -d{\tau }^2=-\left(1-{\omega }^2{r}^2\right){\mathit{dt}}^2+{\mathit{dr}}^2,$$E4

where $\tau$ is a proper time of the moving bead, defined in the standard way. The Lagrangian for the derived “1 + 1” space–time metric (4) may be defined as [17]:

$$L\equiv \frac{g_{\mathit{\alpha \beta }}}{2}{\dot{x}}^{\alpha }{\dot{x}}^{\beta }=\frac{1}{2}\left[-\left(1-{\omega }^2{r}^2\right){\left(\frac{\mathit{dt}}{\mathit{d\tau }}\right)}^2+{\left(\frac{\mathit{dr}}{\mathit{d\tau }}\right)}^2\right].$$E5

Components of the corresponding Lagrange equation:

$$\frac{d}{\mathit{d\tau }}\left(\frac{\partial L}{\partial {\dot{x}}^{\alpha }}\right)=\frac{\partial L}{\partial x^{\alpha }},$$E6

may be written in the following way:

$$-\left(1-{\omega }^2{r}^2\right)\left(\frac{\mathit{dt}}{\mathit{d\tau }}\right)=\mathit{\text{const}}\equiv -E,$$E7

$$\frac{d^{2}r}{d{\tau }^2}={\omega }^{2}r{\left(\frac{\mathit{dt}}{\mathit{d\tau }}\right)}^2.$$E8

A simpler equation for $d^{2}r/d{\tau }^2$ may be derived by combining (7) with the algebraic relation between 4-velocities, resulting from their normalization $g_{\mathit{\alpha \beta }}{u}^{\alpha }{u}^{\beta }=-1$ condition:

$$-\left(1-{\omega }^2{r}^2\right){\left(\mathit{dt}/\mathit{d\tau }\right)}^2+{\left(\mathit{dr}/\mathit{d\tau }\right)}^2=-1.$$E9

The result of a simple calculation is:

$${\left(\frac{\mathit{dr}}{\mathit{d\tau }}\right)}^2=-1+\frac{E^2}{1-{\omega }^2{r}^2}.$$E10

On the other hand, rewriting (9) as

$${\left(\frac{\mathit{dt}}{\mathit{d\tau }}\right)}^2={\left[1-{\omega }^2{r}^2-{\left(\frac{\mathit{dr}}{\mathit{dt}}\right)}^2\right]}^{-1},$$E11

and expressing from (7)$\mathit{dt}/\mathit{d\tau }$ by means of $E$_, we can write down the following first-order differential equation for the function $r\left(t\right)$:

$$\mathit{\text{drdt}}=\sqrt{\left(1-{\omega }^2{r}^2\right)\left[1-\frac{\left(1-{\omega }^2{r}^2\right)}{E^2}\right]}.$$E12

A constant parameter $E=-U_t$, coming into view in these equations, may be treated as an energy of the moving bead in the RF and may be determined through initial conditions. For the above-mentioned case, when at the moment $t=0$_, the bead was at the position $r=r_0$ and had a velocity $v_r=v_0$, we can easily find out that

$$E=\frac{\left(1-{\omega }^2{r_0}^2\right)}{\sqrt{1-{\omega }^2{r_0}^2-{v_0}^2}}.$$E13

Obviously, $0<E<\infty$. If, for example, $r_0=0$ and $v_0\ne 0$ then $E={\left(1-v_0^2\right)}^{-1/2}$ and is more than unity ($E\gg 1$, if $v_0\approx 1$). While, if $v_0=0$ and $r_0\ne 0$ then we see that $E={\left(1-{\omega }^2{r}_0^2\right)}^{1/2}$, certainly, is less than unity. Note also that $E\approx 0$ if $\omega r_0\approx 1$.

It must be noted that in the nonrelativistic limit the energy of the moving bead reduces to the following expression:

$$E_{\mathit{nr}}=1+v^2/2-{\omega }^2{r}^2/2,$$E14

where a unity evidently describes rest mass energy (per unit mass) of the bead, while remaining terms have clear “nonrelativistic” physical meaning. In particular, $v^2/2$ is a kinetic energy corresponding to radial motion in the RF, while $-{\omega }^2{r}^2/2$ is a “centrifugal energy” [18], known from classical mechanics.

One can also derive a differential equation for a radial acceleration of the bead $d^{2}r/{\mathit{dt}}^2$. The result is:

$$d^2{\mathit{rdt}}^2=\frac{{\omega }^{2}r}{1-{\omega }^2{r}^2}\left[1-{\omega }^2{r}^2-2{\left(\frac{\mathit{dr}}{\mathit{dt}}\right)}^2\right],$$E15

at its left-hand side appears a radial acceleration of the bead as measured in the RF. Note that in nonrelativistic limit, it reduces to the conventional expression for the centrifugal force $f_{\mathit{cf}}=w^{2}r$.

In Ref. [6] it was shown that the exact analytic solution of (12) may be expressed by means of elliptical functions. It was derived by introducing two auxiliary variables $\theta \equiv \mathit{arccos}\left(\mathit{\omega r}\right)$ and $\lambda \equiv \mathit{\omega t}$; and a parameter $m\equiv 1/E^2$, reducing (12) to:

$$\frac{\mathit{d\theta }}{\mathit{d\lambda }}=-\sqrt{1-{\mathit{\text{msin}}}^2\theta }.$$E16

The solution of this equation may be written as:

$$\lambda ={\int }_0^{{\phi }_0}\frac{\mathit{d\theta }}{\sqrt{1-{\mathit{\text{msin}}}^2\theta }}-{\int }_0^{\phi }\frac{\mathit{d\theta }}{\sqrt{1-{\mathit{\text{msin}}}^2\theta }},$$E17

with ${\phi }_0=\mathit{arccos}\left(\omega r_0\right)$. It can be rewritten as:

$$\phi =\mathit{am}\left({\lambda }^{\ast }-\lambda \right),$$E18

where $\mathit{am}$ is an amplitude of Jacobian elliptic functions [19] and we have introduced an additional parameter:

$${\lambda }^{\ast }\equiv {\int }_0^{{\phi }_0}\frac{\mathit{d\theta }}{\sqrt{1-{\mathit{\text{msin}}}^2\theta }}.$$E19

Recovering physical variables $t$ and $r$ one comes to the following explicit solution:

$$r\left(t\right)=\frac{1}{\omega }\mathit{cn}\left({\lambda }^{\ast }-\mathit{\omega t}\right)$$E20

where $\mathit{cn}$ is Jckobian elliptic cosine [19].

It is the general solution to the problem. In Ref. [6] a particular case was considered, specified by above-mentioned initial conditions: $r=0$ and $v_0\ne 0$ at $t=0$. For the case, $m=1-v_0^2$ and ${\lambda }^{\ast }=K$_, where $K$ is a complete elliptical integral of the first kind [19]. It allows rewriting the general solution (20) in a different way:

$$r\left(t\right)=\frac{1}{\omega }\mathit{cn}\left(K-\mathit{\omega t}\right)=\left(\frac{v_0}{\omega }\right)\frac{\mathit{sn}\left(\mathit{\omega t}\right)}{\mathit{dn}\left(\mathit{\omega t}\right)},$$E21

where $\mathit{sn}$ and $\mathit{dn}$ are Jacobian elliptical sine and modulus, respectively [19]. The corresponding solution for the radial velocity $v_r\left(t\right)$ is:

$$v_r\left(t\right)=v_0\frac{\mathit{cn}\left(\mathit{\omega t}\right)}{{\mathit{dn}}^2\left(\mathit{\omega t}\right)}.$$E22

Both asymptotic limits—nonrelativistic and ultrarelativistic—of the solutions (21) and (22) are easy to specify. When an initial radial velocity of the bead is nonrelativistic ($v_0\ll 1$), $m\approx 1$ and recalling an asymptotic behavior of Jacobi’s elliptic functions [19] $\left(\mathit{sn}\to \mathit{tanh},\mathit{cn}\to \mathit{sech},\mathit{dn}\to \mathit{sech}\right)$ we can see that the solutions reduce to above-mentioned (1) and (2) Newtonian expressions. Whereas in the ultrarelativistic limit ($v_0\approx 1$), $m\approx 0$, hence $\mathit{sn}\to \mathit{sin},\mathit{cn}\to \mathit{cos},\mathit{dn}\to 1$ and consequently $r\left(t\right)\approx \left(v_0/\omega \right)\mathit{sin}\left(\mathit{\omega t}\right)$, and $v_r\approx v_0\mathit{cos}\left(\mathit{\omega t}\right)$.

Figure 1 is taken from Ref. [6]. It shows the temporal evolution of the radial velocity of the bead $v_r$ (dashed line) and the quantity $v_{\phi }\equiv \mathit{\omega r}$ (solid line), which is equal to the azimuthal velocity of the bead in the laboratory frame. The curves correspond to four different values of $v_0$. Namely, Figure 1(a) is drawn for the case $v_0=0.001$. Initially, both velocities ($v_r$ and $v_{\phi }$) grow almost synchronously, but later radial velocity slows its increasing pace, it reaches its maximum value ($v_{\mathit{max}}\approx 0.5$) and begins to decrease, while $v_{\phi }$ continues increasing and reaches $v_{\phi }=1$ value at the moment $t^{\ast }=K/\omega$, when the bead is at the distance $r=r^{\ast }$ from the rotation axis and its radial velocity becomes zero. This is a “turning point” since here $v_r\left(t\right)$ changes its sign, and the bead begins to move toward the rotation axis with increasing speed. A modulus of the radial velocity in the time interval $t^{\ast }<t<2t^{\ast }$ varies exactly in the same way as in the previous interval $0<t<t^{\ast }$. At the moment $t=2t^{\ast }$ the bead is at its starting point $r=0$ just above the rotation axis and it has the same velocity $v_0$ but is directed, this time, in the opposite direction. Therefore, the bead passes over the rotation axis and, in the interval, $2t^{\ast }<t<4t^{\ast }$ repeats in the left half of the pipe the same kind of motion. Figure 1(b) is drawn for the case $v_0=0.3$. Qualitatively it resembles the previous case. However, a maximum value of $v_r$ is slightly larger and an average radial velocity of the motion is apparently higher.

Figure 1.

Dependence of the bead’s radial velocity v_r (dashed line) and azimuthal velocity v_ϕ≡ω r (solid line) for four different values of initial velocity: (a) v₀ = 0.001, (b) v₀ = 0.3, (c) $v_0=\sqrt{2}/2$, and (d) v₀ = 0.99.

An interesting “threshold” case $v_0=\sqrt{2}/2$ is represented in Figure 1(c). From (15) we can see that for this particular case, initial radial acceleration is equal to zero. Therefore, originally, the bead moves almost uniformly, but further on it continues to move with a decreasing radial speed. Other qualitative features of the motion remain the same as in above-specified cases. Finally, Figure 1(d) represents the case of a strongly relativistic initial speed $v_0=0.99$. It corresponds to the asymptotic case $m\approx 0$ discussed above and consequently the curves for $v_r$ and $v_{\phi }$ would be well represented by usual trigonometric cosine and sine, respectively.

Thus, we see that the character of the bead motion is “oscillatory.” The period of the “oscillations” $P\equiv 4t^{\ast }=4K/\omega$. The period tends to infinity when $v_0\to 0$ (as it should be), while when $v_0\to 1$_, the period naturally tends to: $P\to 2\pi /\omega$.

In Ref. [6] the authors introduced the concept of an “effective potential” $U\left(r\right)$, which turns out to be quite useful for shedding more light on the qualitative characteristics of derived solutions. Substituting $\mathit{dr}/\mathit{dt}$(15) from (12) it is possible to rewrite the equation of motion as:

$$\frac{d^{2}r}{{\mathit{dt}}^2}={\omega }^{2}r\left[\left(2m-1\right)-2m{\omega }^2{r}^2\right]\equiv -\frac{\mathit{dU}\left(r\right)}{\mathit{dr}},$$E23

and for our particular case ($r_0=0,v_0\ne 0$), taking into account (13), we can rewrite it in the following way:

$$\frac{d^{2}r}{{\mathit{dt}}^2}={\omega }^{2}r\left[\left(1-2v_0^2\right)-2\left(1-v_0^2\right){\omega }^2{r}^2\right],$$E24

while the explicit expression for $U\left(r\right)$ is:

$$U\left(r\right)=\frac{{\omega }^2{r}^2}{2}\left[\left(1-2m\right)+m{\omega }^2{r}^2\right].$$E25

In Figure 2, which is also taken from [6], the function $U\left(x\right)\left(x\equiv \mathit{\omega r}\right)$ is shown for four different values of $v_0$, corresponding to the cases shown in Figure 1(a–d). It provides graphic illustration of the bead’s motion in terms of its motion in the specific kind of potential “well.” Namely, in (a) and (b) cases, the curves have “secondary” minima, ascertaining that the bead beginning motion from the point $x=0$ accelerates while “rolling down” to the “secondary” minimum point, then hampers until it reaches the point $x=1$, and stops and begins to move toward the rotation axis. Case (c) is represented by a “potential well” with an almost flat bottom, where initially, as we have seen earlier, motion happens to be almost uniform. For the fourth case, the form of the “well” naturally implies the motion with the negative radial acceleration during the whole course of the motion.

Figure 2.

Effective potential as a function of the quantity v_ϕ≡ω r for four different values of the initial velocity: (a) v₀ = 0.001, (b) v₀ = 0.3, (c) $v_0=\sqrt{2}/2$, and (d) v₀ = 0.99.

Recently the gedankenexperiment in the RF was reconsidered by Khomeriki and Rogava [20] and the exact solutions for the bead motion were found and analyzed for most general sets of initial conditions. The qualitative nature of the motion is largely the same, but there are interesting quantitatively new classes of solutions. We address the interested reader to this publication for more details.

It must be noted that in Ref. [6], Gia Machabeli and I have interpreted the surprising behavior of the bead in terms of reversal in direction of a centrifugal force. Criticizing that interpretation, Miller and Abramowicz [21] pointed out the connection of the bead’s behavior with relativistic dependence of its mass on velocity. They recommended defining the relativistic centrifugal force in the way that excludes the “reversal” of the force and argued that the deceleration of the bead has to be ascribed to the relativistic mass variation. In Ref. [22], F. de Felice considered relativistic dynamics on a rotating disk and for the case of radially constrained motion, he interpreted the vanishing of the bead’s radial velocity at the light cylinder as the result of an infinite Doppler shift experienced by the inertial observer.

If we follow the standard definition and assume that the centrifugal force is an “apparent” force, appearing only in non-inertial frames of reference then one may wonder: is it possible to derive above-described dynamics of the bead inside the rotating pipe in the laboratory inertial frame (LF), relative to which the pipe rotates? Below it will be shown that all quantitative results of Ref. [6] are not related at all to peculiarities of the frame in which the problem is considered. We shall discuss also two other interesting aspects of the problem, viz. surprising and remarkable analogy of the “pipe+bead” gedankenexperiment with: (a) motion of classic simple gravity pendulum, and (b) the radial geodesic infall motion of a test particle in Schwarzschild geometry.

2.2 The original solution: LF study

In this section, we reconsider the gedankenexperiment in the LF, where the space–time is Minkowskian. Our purpose is to show that the peculiar dynamics of the system disclosed in the RF are not related at all to the arguable concept of a reversible or irreversible centrifugal force; on the contrary, it can be easily derived within the LF where no inertial forces are present.

Owing to the symmetry of the system, it is convenient to write the spatial part of the metric in polar coordinates:

$${\mathit{ds}}^2\equiv -d{\tau }^2=-{\mathit{dt}}^2+r^{2}d{\phi }^2+{\mathit{dr}}^2,$$E26

The metric has three nonzero Christoffel symbols:

$${\mathrm{\Gamma }}_{\mathit{\varphi \varphi }}^r=-r,{\mathrm{\Gamma }}_{\mathit{r\varphi }}^{\varphi }={\mathrm{\Gamma }}_{\mathit{\varphi r}}^{\varphi }=1/r.$$E27

The radial velocity and angular velocity of a bead/particle are defined as:

$$v\equiv \mathit{dr}/\mathit{dt},\omega \equiv \mathit{d\varphi }/\mathit{dt},$$E28

while the Lorentz factor is:

$$\gamma \equiv \mathit{dt}/\mathit{d\tau }={\left(1-r^2{\omega }^2-v^2\right)}^{-1/2},$$E29

The motion of the bead is free in the radial direction, while the pipe wall acts on the bead with tangential force of reaction. The radial component of the equation of motion

$$\frac{d^{2}r}{d{\tau }^2}+{\mathrm{\Gamma }}_{\mathit{\alpha \beta }}^r\left(\frac{{\mathit{dx}}^{\alpha }}{\mathit{d\tau }}\frac{{\mathit{dx}}^{\beta }}{\mathit{d\tau }}\right)=0,$$E30

is reduced to:

$$\frac{d^{2}r}{d{\tau }^2}=r{\left(\frac{\mathit{d\varphi }}{\mathit{d\tau }}\right)}^2=r{\omega }^2{\gamma }^2.$$E31

As far as $\mathit{dr}/\mathit{d\tau }=\mathit{\gamma v}$ and

$$\frac{\mathit{d\gamma }}{\mathit{dt}}={\gamma }^{3}v\left({\omega }^{2}r+\frac{\mathit{dv}}{\mathit{dt}}\right),$$E32

it is quite easy to calculate that:

$$\frac{d^{2}r}{d{\tau }^2}={\gamma }^2\left[\left(1+{\gamma }^2{v}^2\right)\frac{d^{2}r}{{\mathit{dt}}^2}+{\omega }^{2}r{\gamma }^2{v}^2\right],$$E33

and it is equally straightforward to show that:

$$1-{\gamma }^2{v}^2={\gamma }^2\left(1-r^2{\omega }^2-2v^2\right),$$E34

$$1+{\gamma }^2{v}^2={\gamma }^2\left(1-r^2{\omega }^2\right),$$E35

which, in turn, after taking into account (31) and (33), allows us to derive the following equation:

$$d^2{\mathit{rdt}}^2=\frac{{\omega }^{2}r}{1-{\omega }^2{r}^2}\left(1-{\omega }^2{r}^2-2v^2\right).$$E36

One can easily see that it is exactly the same Eq. (15), which has been derived in the previous subsection within the RF analyses of the same problem!

It is worthwhile to note that taking into account (34) and (35), we can rewrite this equation in the following, surprisingly beautiful, form:

$$\frac{d^{2}r}{{\mathit{dt}}^2}=\left(\frac{1-{\gamma }^2{v}^2}{1+{\gamma }^2{v}^2}\right){\omega }^{2}r.$$E37

Apart from its elegant form, this equation turns out to be quite informative because it distinctly shows asymptotic peculiarities of the bead dynamics. Namely, if the motion is nonrelativistic ($\mathit{\gamma v}\ll 1$), (37) reduces to the usual classical equation for centrifugal acceleration: $d^{2}r/{\mathit{dt}}^2={\omega }^{2}r$; while in the ultrarelativistic limit ($\mathit{\gamma v}\gg 1$), the sign of the right-hand side of (37) changes (!) to the opposite: $d^{2}r/{\mathit{dt}}^2=-{\omega }^{2}r$. Note, furthermore, that when ${\gamma }_0{v}_0=1$ (viz. $v_0=1/\sqrt{2}$) above-mentioned remarkable acceleration sign reversal occurs from the very beginning of the motion!

Another mathematically elegant feature of the problem is that (36) can be rewritten as a differential equation for the function $y\equiv v^2\left(r\right)$. The resulting equation is:

$$\frac{d^{2}y}{{\mathit{dt}}^2}+\frac{4{\omega }^{2}r}{1-{\omega }^2{r}^2}y=2{\omega }^{2}r,$$E38

with the solution, which can be written as:

$$y\left(r\right)=\left(1-{\omega }^2{r}^2\right)+\mathit{\text{const}}\cdot {\left(1-{\omega }^2{r}^2\right)}^2.$$E39

In particular, for the “conventional” initial condition when initially ($t=0$) the bead is situated on top of the pivot $r_0=0$ and has a velocity $v_0\ne 0$, the latter equation implies that $\mathit{\text{const}}=v_0^2-1$. Therefore, for the function $v\left(r\right)$ we obtain the result that had been derived in Ref. [6]:

$$v\left(r\right)=\sqrt{\left(1-{\omega }^2{r}^2\right)\left[1-\frac{\left(1-{\omega }^2{r}^2\right)}{{\gamma }_0^2}\right]}.$$E40

2.3 An interesting pendulum analogy

The gedankenexperiment reveals” surprising dynamics of relativistically rotating particles. But it happens to be remarkable also due to a couple of nontrivial and noteworthy analogies.

One of them shows when we introduce new set of variables:

$$\varphi \equiv 2\mathit{arccos}\left(\mathit{\omega r}\right)$$E41

$$\lambda \equiv \mathit{\omega t}$$E42

$${\mathrm{\Omega }}^2\equiv 1-v_0^2$$E43

and after, some uncomplicated calculations, find out that our equation of motion reduces to the following, remarkably familiar equation:

$$\frac{d^2\varphi }{d{\lambda }^2}+{\mathrm{\Omega }}^2\mathit{\text{sinϕ}}=0,$$E44

we arrive at the well-known pendulum equation!

Note that one could easily derive (44) from (43) by rewriting it in above-introduced notation as $\mathit{d\varphi }/\mathit{d\lambda }=-2\sqrt{1-{\mathrm{\Omega }}^2{\mathit{sin}}^2\left(\varphi /2\right)}$, and taking one more derivative by $\lambda$. It is common knowledge that (44) describes nonlinear oscillations of a free “simple gravity” pendulum. In fact, it is closely related to another big achievement for Christiaan Huygens, who studied in his “Horologium Oscillatorium” in 1673. With this analogy, now being so apparent, the striking resemblance of Ref. [6] solutions with a pendulum motion becomes more transparent and understandable. If we introduce the concept of an “analogous pendulum”, governed by (44), then the initial conditions ($r_0=0$, ${\left(\mathit{dr}/\mathit{dt}\right)}_{t=0}=v_0$) are replaced by ${\varphi }_0=\pi$ and ${\left(\mathit{d\varphi }/\mathit{d\lambda }\right)}_{\lambda =0}=-2v_0$. Therefore, it turns out that the analogous pendulum rotates in a vertical plane with the effective frequency $\mathrm{\Omega }$. Remarkably enough, the time interval, needed by the bead to reach $\mathit{\omega r}=1$ (“light cylinder”), corresponds to the time needed by the analogous pendulum to reach its stable equilibrium ($\varphi =0$) point. This time interval is finite, as it, certainly should be.

2.4 A captivating black hole analogy

Another striking analogy turns out to be even more unexpected. One can show that the “bead-pipe” gedankenexperiment features an unusual analogy with a specific kind of geodesic motion in Schwarzschild geometry!

Let us consider a radial geodesic “fall” of a test particle onto a Schwarzschild black hole with $M$ mass. Let a radial velocity of the particle at infinity be $V_{\infty }$ pointed inwards. If one denotes by $E={\gamma }_{\infty }\equiv {\left(1-{V_{\infty }}^2\right)}^{-1/2}>1$ the specific energy of the particle per its rest mass, then for its radial velocity relative to the observer at infinity $V_{\widehat{r}}\equiv \sqrt{g_{\mathit{rr}}}\mathit{dr}/\mathit{dt}$ one gets

$${V_{\widehat{r}}}^2=\frac{E^2}{4}-{\left[\frac{E}{2}-\frac{1}{E}\left(1-\frac{2M}{r}\right)\right]}^2,$$E45

(See, e.g., Ref. [23], where this equation is written in different notation). For the quantity ${\mathit{dV}}_{\widehat{r}}/\mathit{dt}$ (called by McVittie “nontenzor radial acceleration of the particle”) one gets:

$$\frac{{\mathit{dV}}_{\widehat{r}}}{\mathit{dt}}=\frac{2M}{{\mathit{Er}}^2}\left[\frac{E}{2}-\frac{1}{E}\left(1-\frac{2M}{r}\right)\right]\sqrt{1-\frac{2M}{r}}.$$E46

Inspecting the expression it is easy to see that the acceleration of the particle is negative (that is, the modulus of the particle’s infall velocity is increasing), until the particle reaches the radial distance $r_1=4M/\left(2-{{\gamma }_{\infty }}^2\right)$, where the acceleration changes its sign, and $V_{\widehat{r}}$ reaches its maximum velocity $V_{\mathit{max}}={\gamma }_{\infty }/2$.

More precisely, we have the following regimes of the motion:

• $V_{\infty }\ll 1$ (${\gamma }_{\infty }\approx 1$): the particle begins to move with increasing speed, at $r_1\approx =4M$ the speed reaches its maximum value ($V_{\mathit{max}}\approx 1/2$) and further on it decelerates, down to zero radial velocity, when the particle approaches the black hole horizon $r\to 2M$;

• $V_{\infty }=\sqrt{2}/2$, (${\gamma }_{\infty }=\sqrt{2}$): the “threshold” case; with ${\mathit{dV}}_{\widehat{r}}/\mathit{dt}=0$ ($V_{\mathit{max}}=V_{\infty }$, $r_1\to \infty$), and the particle decelerating smoothly, all along, down to the horizon;

• $V_{\infty }>\sqrt{2}/2$: the acceleration is positive during the whole course of the motion.

It is easy to notice that this scenario remarkably resembles (even quantitatively!) the one in Ref. [6] gedankenexperiment. This likeness happens due to the remarkable similarity between (45) and the corresponding equation for $\mathit{dr}/\mathit{dt}$ from Ref. [6], which may be written as:

$${\left(\frac{\mathit{dr}}{\mathit{dt}}\right)}^2=\frac{E^2}{4}-{\left[\frac{E}{2}-\frac{1}{E}\left(1-{\omega }^2{r}^2\right)\right]}^2,$$E47

The resemblance is not incidental, of course, but is related to the likeness of the space–time (4) in the rotating pipe:

$${{\mathit{ds}}_p}^2=-\left(1-{\omega }^2{r}^2\right){\mathit{dt}}^2+{\mathit{dr}}^2,$$E48

with the Schwarzschild space–time along a radial geodesic ($\theta =\mathit{\text{const}}$, $\phi =\mathit{\text{const}}$):

$${{\mathit{ds}}_s}^2=-\left(1-2M/r\right){\mathit{dt}}^2+\frac{{\mathit{dr}}^2}{1-2M/r}.$$E49

Comparing, for example, lapse functions for these two metrics (${\alpha }_p\equiv \sqrt{-g_{\mathit{tt}}}=\sqrt{1-{\omega }^2{r}^2}$, and ${\alpha }_s\equiv \sqrt{-g_{\mathit{tt}}}=\sqrt{1-2M/r}$) one can see that ${\alpha }_p$ and ${\alpha }_s$ become infinite at the light cylinder and horizon, respectively. Obviously, the likeness is not complete: the spatial part of the rotating pipe metric is flat, while the spatial part of (49) metric is curved ($g_{\mathit{rr}}\ne 1$). The latter difference is also essential: it ensures finiteness of the time $t^{\ast }$ [6], needed by the bead to reach the light cylinder $r^{\ast }\equiv {\omega }^{-1}$. As it is well known, an analogous time interval for a test particle to reach the Schwarzschild black hole horizon, as measured by the distant observer, is infinitely long!

2.5 A centrifugal force “reversal”?!

It is well known that a generalization of Newton’s second law

$$\overrightarrow{F}=d\overrightarrow{p}/\mathit{dt}=d\left(m\overrightarrow{v}\right)/\mathit{dt}$$E50

is the most useful and convenient definition of a (three-) force $\overrightarrow{F}$ in special relativity. According to Rindler ([24], p. 88) “This definition has no physical content until other properties of force are specified, and the suitability of the definition will depend on these other properties.”

For real applied forces, arising in relativistic dynamics, this definition is physically self-consistent and proved to be the most appropriate. But, how one should define inertial forces in relativity!? Miller and Abramowicz suggested [21] using the same general method for these forces as well. The approach is fully justified but we would like to emphasize one aspect of the problem.

The definition contains a quantity $m\left(t\right)=m_0\gamma \left(t\right)$, which has, clearly, the meaning of the relativistic mass in the LF. However, note that this quantity is used for the definition of another physical quantity—centrifugal force—which exists in another non-inertial frame of reference. A question arises: is it self-consistent to define a physical variable in one frame by means of another variable defined in another frame? Certainly, the time-variable mass may be defined also in RF, but it would not be equal to $m\left(t\right)$. The important point is that $\gamma \left(t\right)$, having in the LF the meaning of Lorentz factor, has not the same meaning in the RF because Lorentz factor of the moving particle is not invariant between frames [24].

This circumstance appears evident in the “$1+1$” formulation of the same problem. In fact, for the two-dimensional curved metric (4) in the RF $V\equiv v/{\alpha }_p$, and $\mathrm{\Gamma }={\left[\left(1-{\omega }^2{r}^2\right)/\left(1-{\omega }^2{r}^2-v^2\right)\right]}^{1/2}$. Accordingly, it is possible to define relativistic mass in the RF as $M\left(t\right)\equiv m_0\mathrm{\Gamma }$ and to write, instead of Eq. (50), the following definition:

$$F^{\ast }=\frac{d}{\mathit{dt}}\left(\mathit{MV}\right)=\frac{M{\omega }^{2}r}{{\alpha }_p}.$$E51

This definition is already made by means of the true Lorentz factor of the bead as measured in the RF. Just like (50) it, also, gives “irreversible” centrifugal force but lacks the attractive simplicity of (50). However, we should always remember that the mass, which the RF observer actually measures, is $M\left(t\right)$ and is not $m\left(t\right)$. It seems, therefore, more consistent to express the physical quantity existing in the particular frame (centrifugal force $f_c$, which exists in the RF) through the other physical quantity $M\left(t\right)$ defined and measured in the same frame.

Despite this uncertainty, we should like to note that the importance of the mass variation effect, noticed by Miller and Abramowicz, is a very remarkable and important feature of this problem. As it appears, the capability of mass to vary drastically affects the dynamics of the motion. But, is it appropriate and logically justified to describe this secondary dynamical effect as the action of some “negative self-thrust” force!? Saying “secondary” we do not mean its significance but, rather, its status in the causal order of true physical reasons. If one introduces it that would be yet another “apparent” force, like the centrifugal force. Actual dependence of the bead mass on time is governed explicitly by the concrete kind of its motion in the LF, which is entirely determined by the outer real force, tangential pipe reaction force, applied to this moving body.

3. Astrophysical applications and conclusions

In 2003, the problem of motion of particles on relativistically rotating prescribed trajectories considered in Ref. [6] for a straight trajectory case (gedankenexperiment) was generalized. Rogava, Osmanov, and Dalakishvili considered the motion of rotationally driven particles along flat trajectories of arbitrarily curved shape [7]. In this paper, the problem was again studied on the level of the idealized gedankenexperiment, both in the laboratory (LF) and in the rotating (RF) frames of reference. For the simple example of the Archimedes spiral, it was found that the dynamics of these particles may involve both accelerative and decelerative modes of motion. Moreover, it was also found that there are special solutions, which remain force-free during the whole course of the motion.

Additionally, basic equations were outlined and the scheme for the solution of the more general version of the same problem was given, where the angular velocity of the rotating system is not assumed to be constant. Instead, it is required that the full triple system—rotator+pipe+bead—is conservative and the rotator is allowed to exchange perceptible portions of energy with the bead. The latter setup is a significant step closer to real astrophysical situations where, for instance, a rapidly rotating neutron star acts as a rotator and its magnetic field lines play the role of wires (pipes) along which particles are centrifugally accelerated.

One astrophysically important difference from the linear pipe case [6] is that for curved prescribed trajectories and the motion of the accelerated particle is not any more radially bounded: there exist regimes of motion when the bead may reach infinity! This result has a simple physical explanation. For the case of the linear pipe, rotating with the constant angular velocity, the natural limit of the radial motion is given by the light cylinder radius. However, in the case of the curved pipe, even when it rotates at the constant rate, the bead moves both in radial and azimuthal directions, following the curvature of the prescribed trajectory and having a variable angular velocity $\mathrm{\Omega }\left(t\right)$. It means that now the role of the “effective light cylinder is played by the time-dependent quantity $R_L\left(t\right)={\mathrm{\Omega }}^{-1}\left(t\right)$, and all those radial distances for which $r\left(t\right)<R_L\left(t\right)$ become accessible! Therefore, if both $r\left(t\right)$ and $R_L\left(t\right)$ are monotonously increasing functions, but the former stays always smaller than the latter (e.g., that is the case for the Archimedes spiral) then the bead can reach infinity because wherever it is at any given moment of time, the light cylinder is still ahead of it and the centrifugally accelerated bead will never “overtake” the light cylinder!

4. Acceleration of particles by rotating magnetospheres in AGN

It is well-known that some blazars, such as Mrk 421, Mrk 501, PKS 2155–304, 1ES 2344–514, H1426 + 428, and 1ES 1959 + 650, emit ultra-high-energy, several TeV $\left(1\mathrm{TeV}\equiv {10}^{12}\mathrm{eV}\right)$ photons and constitute a special class of so-called “TeV blazars”. The standard model for a blazar assumes the presence of a supermassive black hole, surrounded by an accretion disc and ejecting twin relativistic jets, one of which is seen by a terrestrial observer almost end-on. Usually, broadband emission spectra of these objects contain two components: the low-energy (from radio to optical/UV) part associated with synchrotron radiation and the high-energy (X- and $\gamma$- rays) component formed by the inverse Compton scattering (ICS) of softer photons. The presence of the latter part of the spectrum is normally explained on the basis of the synchrotron self-Compton (SSC) model. However, the origin of accelerated electrons, the physical mechanism responsible for their efficient acceleration is a matter of uncertainty.

Proposed mechanisms such as the Fermi-type acceleration processes or repetitive acceleration of electron-positron pairs as a feedback mechanism could account for the observed high energy emission up to 20 TeV. However, the Fermi-type acceleration in relativistic jets is efficient only if the seed population of “pre-accelerated” electrons is present, possessing high enough (${\gamma }_{\max }\ge {10}^2$) Lorentz factors. The provenience of the “pre-acceleration” is not well-understood.

Since pioneering work by T. Gold [25, 26] it has been proposed that centrifugally driven outflows (CDOs) consisting of centrifugally accelerated particles may acquire extremely high energies. The idea has often been discussed [8, 27] in the context of pulsar emission theory. Moreover, regarding the AGNs, it has been shown that CDOs from accretion disks may occur if the poloidal magnetic field lines are inclined at an angle $\le {60}^{\circ }$ to the equatorial plane of the disk [28].

Osmanov et al. [9] argued that the presence of the CDO would imply that despite the intense energy losses via UV radiation (when due to ICS, soft photons are upscattered on electrons and gain energy from them), electrons may still reach very high, ${\gamma }_{\max }\sim {10}^5$, Lorentz factors. It was surmised that if robust enough, this mechanism could account not only for the “pre-acceleration” of electrons but could be considered as an independent bona fide alternative mechanism for the generation of high-energy emission (up to $\sim$20 TeV) of TeV blazars!

Gangadhara and Lesch [29] examined the role of the centrifugal force on the dynamics of electrons moving along straight magnetic field lines, situated in the equatorial plane and corotating with the spinning AGN. They showed that an upscattering of low-energy photons against accelerated electrons may readily lead to the generation of the nonthermal x-ray and $\gamma$-ray emission. Later on, the problem was critically re-examined by Rieger and Mannheim [13], checking whether the rotational energy gain of charged particles, moving along straight magnetic field lines, is limited not only by ICS but also by the expected breakdown of “the-bead-on-the-wire” approximation. The latter is a synonymous term for the “bead+pipe” approximation used in our gedankenexperiment. The breakdown happens in the vicinity of the light cylinder, where a Coriolis force acting on the particle and trying to “tear it off” the field line would exceed the Lorentz force “binding” the particle to the field line.

Bearing in mind that in real, three-dimensional astrophysical situations (e.g., jets), the magnetic field lines are not localized in the equatorial plane but are inclined with respect to it. Osmanov et al. [9] examined the same problem considering the wide range of possible inclinations. It was shown that for a wide range of AGN, the mechanism responsible for limiting the maximum Lorentz factors is ICS, but under favorable conditions, it can allow particles to reach quite high Lorentz factors ${\gamma }_{\max }\ge {10}^5$. The breakdown of the-bead-on-the-wire approximation becomes important for the low luminosity (< ${10}^{41}\mathrm{erg}/\mathrm{s}$) AGN, when ${\gamma }_{\max }\sim {10}^8$. For higher luminosities (> ${10}^{41}\mathrm{erg}/\mathrm{s}$), it can be dominant only for relatively small inclination angles of the magnetic field lines with respect to the rotation axis ($\le {10}^{\circ }$).

Therefore, it was argued [9] that CDOs could indeed be amply efficient to account for the TeV blazar emission. For the wide range of cases, ICS remains the dominant factor in limiting the maximum Lorentz factor of accelerated particles, letting them produce very high-energy photons [1–20] TeV. Obviously, an important restriction of the [9] approach is that only straight magnetic field lines were considered, whereas, in realistic astrophysical situations, the magnetic field lines are curved. It might be especially important if one considers particle dynamics on a longer time scale and over larger length scales when the curvature of the field lines cannot be neglected. As a matter of fact, the mathematical formalism for such a study is described in [7].

4.1 Centrifugal acceleration and gamma flares in Crab nebula

The Crab nebula is a source of almost steady high-energy emissions. Observations made from orbital probes (Fermi, SWIFT, and RXTE) showed evidence of its variability in the X\x-ray range. Several years ago the Fermi and AGILE satellites detected dazzling, brief, and strong bursts of gamma radiation above 100 MeV, with its source located in the Crab Nebula. Since then, several gamma-ray flares have been reported.

The most obvious source of radiation power in the nebula is the rotational energy damping $\mathit{dE}/\mathit{dt}=I\mathrm{\Omega }d\mathrm{\Omega }/\mathit{dt}=5\cdot {10}^{38}\mathit{erg}/\mathit{sec}$ of the pulsar PSR 0532. Machabeli et al. [30] demonstrated that via the centrifugal acceleration mechanism, it is possible to pump energy from the neutron star’s vast rotational kinetic energy storage, $\sim 5\times {10}^{38}$ erg s − 1, to proper electrostatic plasma (Langmuir) oscillations. Furthermore, it was shown that the growth rate of the perturbations is maximum in the “superluminal” area, where the phase velocity of perturbations exceeds the speed of light. That is why in this region, the condensate of plasmons is formed, which is transferred to the Crab Nebula. The transfer of the energy of the plasmon condensate from the pulsar magnetosphere to the nebula over a huge distance, $3\times {10}^{17}\mathit{cm}$, takes place practically without any tangible losses.

But in the nebula, unlike the pulsar magnetosphere, apart from electrons and positrons, there are also protons. That is why a modulation instability is developed, which leads to the collapse of Langmuir waves; viz. a cavity is formed, which collapses, and on the final stage of the collapse, involved particles attain very high Lorentz factors, resulting in the powerful synchrotron emission of the nebula. The collapse stops at the scale of a few Debye radii.

It was also shown that in the course of the active phase of the collapse in the cavity due to the influence of nonlinear processes on the polarization properties of the medium, self-trapping [31] of the synchrotron radiation (generated within the cavity) takes place. If the conditions for the appearance of the self-trapping are fulfilled for certain values of emitted wave frequencies, for others, both higher and lower values of the frequency, they are not satisfied. It means that giant bursts of energy release have to happen in narrow energy (frequency) ranges. Therefore, it would naturally explain abrupt, burst-like increases of the radiation intensity within narrow bands, impressively explained by self-trapping in the framework of nonlinear optics (so-called “Askaryan effect” [32]). Waves propagating in the non-parallel direction to the line of sight are bent and directed to the observer due to self-trapping. Ultimately it leads to the required increase of the radiation intensity and, at the same time, unlike other mechanisms, it does not require any additional sources of energy.

4.2 Self-trapping as a beaming mechanism for Fast Radio Bursts!?

Bearing in mind what we have just said about self-trapping, it is most reasonable to assume [33] that the principal reason for the beaming of electromagnetic radiation leading to Fast Radio Bursts (FRBs) could be the nonlinear self-trapping phenomenon. This mechanism implies that the part of the radiation beam directed toward the observer is augmented by its outer part, which in the absence of the self-trapping would not be focused toward the same point. As a result, the observer, while the beam is being self-trapped, sees an enhanced intensity of radiation in a very narrow frequency range. This scenario is robust and fully autonomous because unlike many other mechanisms, it does not require additional external sources of energy. Besides, self-trapping depends on quite a large number of parameters, in particular, on a specific proper value of the ratio of the wave amplitude to the amplitude of an incident electrostatic wave, on the temperature of the medium, and on the direction of these waves relative to the line of sight. Any, even the slightest, deviation of any of these parameters from ‘favorable’ values may lead to the disappearance of an observable radiation burst. This is why this is an extremely finely tuned, random, and rarely occurring event. This circumstance guarantees that an FRB event, if indeed being caused by the arrival of the self-trapped self-focused enhanced beam to the observer, is a totally random and extremely rare natural phenomenon.

If our model is correct and relevant to actual FRBs, the self-trapping condition may hold in a given direction only for a very short interval of time. Hence, it is logical to surmise that the probability of the coincidence between the line of sight and the direction of self-trapping has to be quite small. Paradoxically enough, what could be a serious drawback for a commonly occurring phenomenon that in this case ‘works’ just in the opposite way, it strengthens our confidence that self-trapping could be an important physical factor contributing to the appearance of this extremely rare and energetic phenomenon—fast radio bursts or FRBs.

Obviously, the self-trapping mechanism does not exclude other physically plausible, repetitive or non-repetitive and catastrophic or non-catastrophic mechanisms proposed for FRBs. We tend to believe that self-trapping may be one of a number of very efficient ‘beaming’ mechanisms that might be needed for interpreting FRBs as narrowly beamed radio bursts [34]. We suppose that the prime reason for any single burst appearance could be related to its generic source, located in a distant galaxy, and causing a giant outburst in the radio range. The mechanism of this primary outburst may be related to one of the previously suggested plasma mechanisms.

Presumably, these outbursts might be observed without any additional amplification if they occur on intergalactic distance scales. That was Machabeli et al. guessed in Ref. [33]. Shortly afterward first-ever FRB from within our galaxy was detected [35], confirming our expectation since it was associated with a rapidly rotating neutron star, magnetar SGR 1935 + 2154 about 30,000 light-years away in the Vulpecula constellation. The STARE2 team independently observed the burst [36], detected its fluence, and confirmed the connection between this burst (FRB 200428) and FRBs at extragalactic distances. Needless to say, a magnetar is a plausible candidate for the centrifugally driven relativistic acceleration of electrons with subsequent Langmuir collapse and self-trapping as the beaming mechanism. It only strengthened our expectation that FRB could be akin to, for instance, giant radio pulses occasionally observed from pulsars within our own Galaxy [33]. Actually, for intergalactic giant pulses, the beaming mechanism may not be even necessary for the burst to be detectable. But in order to make FRBs visible on extragalactic distance scales, some powerful additional beaming/amplification is required and this is where we believe self-trapping could play its important role.

With improved capabilities of several wide-field, broad-band surveys that recently became operational the observational situation may change dramatically. It is expected that the FRB field will change from the presently available small-number statistics up to hundreds or even thousands [37] of new FRBs per year. Therefore, in forthcoming years, one might expect that the number of detected FRBs will increase by orders of magnitude, and we could be able to see primary outbursts leading to FRBs without a need for their additional ‘self-trapping-related’ or any other kind of amplification. But within this large number of events, there will be a relatively small fraction of peculiar bursts undergoing secondary self-trapping amplification and having distinctly peaked structures.

5. Conclusions and future prospects

Above given three examples of supposed astrophysical appearance were meant to emphasize that centrifugal acceleration may certainly be one of most efficient converters of rotational energy into the energy of directed motion and may lead to the generation of powerful observational signatures. The universe is full of rotating plasma flows containing enormous amounts of kinetic energy. There are a few physical mechanisms able to transform a considerable part of the mechanical energy into radiation energy. For instance, so-called “nonmodal” processes [38, 39], taking place in flows with nonuniform velocity fields (shear flows), are capable to infuse a part of the regular rotation-related energy in waves and instabilities appearing in these flows [40, 41]. However, none of these nonmodal mechanisms has efficiency even remotely comparable with the efficiency of the centrifugal acceleration mechanism.

Obviously, centrifugal acceleration has to be of particular importance in objects with ultra-strong magnetic fields: pulsars, magnetars, AGN, etc. In these astronomical situations, magnetic field lines provide proper “prescribed trajectories” for relativistic particles and can accelerate them up to very high Lorentz factors. That is why the issue of centrifugal acceleration occupies an important role in the domain of relativistic astrophysics.

Theoretically speaking the most principal, crucial challenge is to move from highly idealized, “gedankenexperiment” setups, and various “toy” problems up to more realistic models, replacing mechanical “bead-on-wire” approximation with real, strong magnetic fields serving as genuine accelerators of relativistic particles in related astronomical objects. We could easily envisage that with the development of observational capabilities in relativistic astrophysics possible areas of applications for the centrifugal acceleration mechanism in high-energy astrophysics will widen further. Obviously, a number of intriguing problems remain to be understood. For instance, how centrifugal acceleration operates in conjunction with space–time “rotation” nearby black holes!? In Refs. [10, 20], a theoretical formalism was developed allowing to study of centrifugally induced phenomena in general static and isotropic metrics, associated with rotating relativistic bodies. One can imagine, for example, that in a Kerr black hole ergosphere, the interlace of centrifugal acceleration with Penrose process [42] may lead to interesting physical peculiarities, some of which may turn out to be important in the astronomical-observational context.

Acknowledgments

Andria Rogava’s research was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) grant FR-18-14747.