Non-Integrability of Geodesic dynamics of Chazy-Curzon space-time

TL;DR

This paper investigates the geodesic equations of Chazy-Curzon space-time and demonstrates their non-integrability by analyzing the absence of additional analytic first integrals around specific equilibrium points.

Contribution

It provides a novel analysis showing that the geodesic system in Chazy-Curzon space-time lacks extra integrals, indicating non-integrability.

Findings

No additional analytic first integral exists for the system.

Periodic solutions are only found at specific equilibrium points.

The period function's properties imply non-integrability.

Abstract

We study the integrability of the geodesic equations of the Chazy- Curzon space-time. It was established that for the equilibrium point $p_{\rho}=p_z=z=0$ and, $\rho_0 \in (1,\, 2)$, there are only periodic solutions, the Hamiltonian system, describing geodesic motion of Chazy-Curzon space-time has no additional analytic first integral. Our approach is based on the following: if the system has a family of periodic solutions around an equilibrium and if the period function is infinitely branched then the system has no additional analytical first integral.

Equations

$$\frac{dq}{dt}=\frac{\partial H}{\partial p},\frac{dp}{dt}=-\frac{\partial H}{\partial q}.$$

$$\dot{x}=X_H(x),x\in {\mathbf{R}}^{2n},$$

$$\{f,g\}=X_f(g)=\sum \frac{\partial f}{\partial p}\frac{\partial g}{\partial q}-\frac{\partial f}{\partial q}\frac{\partial g}{\partial p}=-\{g,f\}.$$

$$\gamma :I\to M$$

$$E(\gamma )=\frac{1}{2}{\int }_{I}g(\dot{\gamma (t)},\dot{\gamma (t)})dt,$$

$$\frac{d^2{x}^a}{d{t}^2}+{\Gamma }_{bc}^a\frac{d{x}^b}{dt}.\frac{d{x}^c}{dt}=0,$$

$$T^{\ast }M{|}_U\simeq U\times {\mathbf{R}}^{\mathbf{n}}$$

$$H(x,p_x)=\frac{1}{2}{g}^{ab}(x)p_a{p}_b.$$

$$\dot{x^a}=\frac{\partial H}{\partial p_a}=g^{ab}(x)p_b,\dot{p_a}=-\frac{\partial H}{\partial x^a}=-\frac{1}{2}\frac{\partial g^{ab}(x)}{\partial x^a}{p}_b{p}_c.$$

$$s^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }=-e^{2\psi }d{t}^2+e^{-2\psi }(e^{2\gamma }(d{\rho }^2+d{z}^2)+{\rho }^{2}d{\varphi }^2).$$

$$H_2={\Omega }^2(\rho ,z).(\frac{1}{2}(p_{\rho }^2+p_z^2)+V_{eff}(\rho ,z,E^2,L^2)),$$

$$\dot{\rho }=\frac{\partial H_2}{\partial p_{\rho }},\dot{z}=\frac{\partial H_2}{\partial p_z},\dot{p_{\rho }}=-\frac{\partial H_2}{\partial \rho },\dot{p_z}=-\frac{\partial H_2}{\partial z}.$$

$$\psi =-\frac{m}{\sqrt{{\rho }^2+z^2}},\gamma =-\frac{m^2{\rho }^2}{2({\rho }^2+z^2{)}^4}.$$

$$H_2(p_{\rho },p_z,\rho ,z)={\Omega }^2(\rho ,z).(\frac{1}{2}(p_{\rho }^2+p_z^2)+V_{eff}(\rho ,z,E^2,L^2)),$$

$$d\tau ={\Omega }^2(\rho ,z)dt$$

$$F(p_{\rho },p_z,\rho ,z)=\frac{1}{2}(p_{\rho }^2+p_z^2)+\frac{1}{2}-\frac{E^2}{2}{e}^{\frac{2}{\sqrt{{\rho }^2+z^2}}}+\frac{L^2}{2{\rho }^2}{e}^{\frac{-2}{\sqrt{{\rho }^2+z^2}}}$$

$$\frac{\partial F(p_{\rho },p_z,\rho ,z)}{\partial p_{\rho }}=p_{\rho },$$

$$\frac{\partial F(p_{\rho },p_z,\rho ,z)}{\partial p_z}=p_z,$$

$$-\frac{\partial F(p_{\rho },p_z,\rho ,z)}{\partial \rho }=-\frac{E^2{\rho }^2{e}^{\frac{2}{\sqrt{{\rho }^2+z^2}}}}{({\rho }^2+z^2{)}^{3/2}}-\frac{L^2{e}^{\frac{-2}{\sqrt{{\rho }^2+z^2}}}}{({\rho }^2+z^2{)}^{3/2}\rho }+\frac{L^2{e}^{\frac{-2}{\sqrt{{\rho }^2+z^2}}}}{{\rho }^3}$$

$$-\frac{\partial F(p_{\rho },p_z,\rho ,z)}{\partial z}=-z.(\frac{E^2{\rho }^2{e}^{\frac{2}{\sqrt{{\rho }^2+z^2}}}+L^2{e}^{\frac{-2}{\sqrt{{\rho }^2+z^2}}}}{({\rho }^2+z^2{)}^{3/2}{\rho }^2}).$$

$${\rho }^{\prime }$$

$$z^{\prime }$$

$$p_{\rho }^{\prime }$$

$$p_z^{\prime }$$

$$E^2{\rho }^2{e}^{\frac{4}{\rho }}-L^2(\rho -1)=0,$$

$$F(0,0,\rho ,0)=-\frac{E^2}{2}{e}^{\frac{2}{\rho }}+\frac{L^2}{2{\rho }^2}{e}^{\frac{-2}{\rho }}+\frac{1}{2}=0.$$

$$L^2=-\frac{{\rho }^2.e^{\frac{2}{\rho }}}{\rho -2},E^2=-\frac{(\rho -1).e^{\frac{-2}{\rho }}}{\rho -2}.$$

$$v(\rho ,z,E^2,L^2)=\frac{1}{2}-\frac{E^2}{2}{e}^{\frac{2}{\sqrt{{\rho }^2+z^2}}}+\frac{L^2}{2{\rho }^2}{e}^{\frac{-2}{\sqrt{{\rho }^2+z^2}}}$$

$$\frac{{\partial }^{2}v(\rho ,0,-\frac{(\rho -1).e^{\frac{-2}{\rho }}}{\rho -2},-\frac{{\rho }^2.e^{\frac{2}{\rho }}}{\rho -2})}{\partial {\rho }^2}=-\frac{{\rho }^2-6\rho +4}{{\rho }^4(\rho -2)}.$$

$$T=2{\int }_{{\rho }_{-}}^{{\rho }_{+}}\frac{d\rho }{\sqrt{0-v(\rho ,0,L^2,E^2)}},$$

$$T$$

$$v(\rho )$$

Full text

1 Introduction

Let $H$ be a smooth real-valued function in $2n$ real variables $(p,\,q),\,p,\,q\,\in{\bf R^{n}}$. Assume that $dH(a)=0$, where, $a$ is an equilibrium point for the Hamiltonian system $X_{H}$ (with $n$ degrees of freedom), given by

[TABLE]

We write the Hamiltonian systems in the form

[TABLE]

where $X_{H}$ is the hamiltonian vector field. The system is called Liouville - Arnold Integrable near $a$ if there exist $n$ functions in involution $f_{1}=H,\,f_{2},\,\dots f_{n}$, defined around $a$, which are functionally independent. The Poisson bracket of $f$ and $g$ are

[TABLE]

We say that the functions $f$ and $g$ are in involution if the Poisson bracket is equal to 0. This means that $df_{1},\,df_{2},\,\dots df_{n}$ are linearly independent around the equilibrium $a$ and $f_{j}=const$ for all $j$ define a smooth submanifold, this level manifold is invariant under $X_{f_{i}}$. We have $X_{f_{i}}f_{j}=0$ and $[X_{f_{j}},\,X_{f_{k}}]=X_{\{f_{j},f_{k}\}}=0$ - the vector fields commute. The compact and connected component of $M_{c}:=\{f_{j}=c_{j},\,j=1,\,\dots,n\}$ is diffeomorphic to a torus.

Given a Riemannian manifold $M$, a geodesic may be defined as curve that results from the application on the principle of least action (we follow[1]). A differential equation describing their shape may be derived using variational principles, by finding extremum of the energy of a curve. Given a smooth curve

[TABLE]

this maps an interval $I$ in real line to the $M$, one writes the energy

[TABLE]

where $\dot{\gamma(t)}$ is the tangent vector to the curve $\gamma$ at the point $t\in I$. Here $g(.,.)$ is the metric tensor on the manifold $M$.

In terms of local coordinates on $M$, the geodesic equation is

[TABLE]

where the $x^{a}(t)$ are coordinates of $\gamma(t)$, $\Gamma_{bc}^{a}$ are the Christoffel symbols and indexes up and below means summation convention.

Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term. The geodesic equations are second order differential equations, they can be expressed as a first order equations by additional independent variables. Note that $x_{a}$ is the coordinates on the neighborhood $U$ induces local trivialization of

[TABLE]

by the map which sends a point $\eta\in T^{*}_{x}M|_{U}$ of the form $\eta=p_{a}dx^{a}$ to the point $(x,\,p_{x})\in U\times\bf{R^{n}}$. The Hamiltonian is

[TABLE]

Here $g^{ab}(x)$ is the inverse of the metric tensor $g^{ab}(x)g_{bc}(x)=\delta^{a}_{c}$. The behavior of the metric tensor under transformation of coordinates implies that $H$ is invariant under change of variables. The geodesic equations may be written as

[TABLE]

The question of the integrability of the geodesic equation has been investigated by several in the authors context of Stationary Axially-symmetric Vacuum (SAV) space-time in [2], [3], [8] and [9]. The properties of stationarity and axial-symmetry suggest that every SAV space- time possesses a pair Killing vector fields $\partial_{t}$ and $\partial_{\phi}$, and a pair first integrals $p_{t}=-E$, and $p_{\phi}=L$, corresponding to energy $E$ and azimuthal angular momentum $L$. The problems of geodesic motion reduces to two dimensional Hamiltonian system in the meridian plane $(\rho,\,z)$. The existence of additional independent integral of motion in involution with others integrals $(E,\,L,\,H)$ lead to Liouvillian integrability. The interesting case Kerr space-time - a SAV space-time, describing rotating black hole - admits two-rank irreducible Killing tensor, that is why four constant of motion which is a quadratic in the momenta - the so-called Carter’s constants, [10] and [18]. The important question is when the Carter’s constant exists in other SAV- models.

Geodesic motion on Zipoy-Voorhees space-times known as a $\gamma$-metrics, has been studied via numerical methods, in [15] and [16] and analytically in [13] and [17]. The Zipoy-Voorhees metrics with a parameter $\delta$ have as special cases both Minkowski flat space-time ($\delta=0$) and Schwarzschild ($\delta=1$) space-time. Important for study of Zipoy-Voorhees metrics is that finding obvious regularity in geodesics across large parts of the parameter $\delta$ and the finding of approximate constants of motion, is not enough for proof of the Liouville integrability, i.e. for the existence of fourth constant of motion.

In [4] it was shown numerically that the Chazy-Curzon and Zipoy-Voorhees space-time exhibit obviously-regular dynamics in special parameter $\delta$ values. After further research in [7], Brink suggests that the Zipoy-Voorhees metrics may give rise to integrable systems for $\delta\neq 0;\,1$. In [5] and [6] Brink show that all SAV metrics admits irreducible second-order Killing tensors are necessarily of Petrov type D. This key result implies that for generic Zipoy-Voorhees metrics without Petrov D property, do not permit in general for any new constant of geodesic motion which is quadratic in momentum variables.

In 2012 in [15] using numerical calculations it was shown that the Hamiltonian phase space of the Zipoy-Voorhees system has the features expected of a perturbed non-integrable system such as chaotic layers, Birkhoff chains, and “stickiness” in the rotation number [15] and [16]. Chaotic layers are mos visible near to tha Lyapunov orbit - near to the unstable periodic orbit at the border between bound and plunging orbits (see [9] for details).

2 Formulation of the problem

The Chazy-Curzon space-time model are part of the class of Static Axially-symmetric Vacuum (StAV) metrics - a subclass of SAV metrics. In cylindrical coordinates $(t,\,\rho,\,z,\,\phi)$ the StAV line element can be written in Weyl form as

[TABLE]

The geodesic motion is governed by a hamiltonian $H_{4}(x^{\mu},\,p_{\nu})=1/2g^{\mu\nu}p_{\mu}p_{\nu}=-\displaystyle{\frac{1}{2}}$ with $g^{\mu\nu}$ being the inverse of metric. $H_{4}$ is a constant of motion which we set to $-\displaystyle{\frac{1}{2}}$ to test the bodies of unit mass. The momentum variables are given by $p_{\mu}=g^{\mu\nu}\dot{x}^{\nu}$, where the dot stands for differentiation with respect to the proper time.

From Hamiltonian equations we have $\dot{p_{t}}=\dot{p_{\phi}}=0$ and we obtain two first integrals $E=-p_{t}$ and $L=p_{\phi}$. Motion on the meridian plane is governed by the reduced Hamiltonian $H_{2}$, with the restriction, $H_{2}=0$ given by

[TABLE]

where $\Omega=e^{(\psi-\gamma})$, $V_{eff}=-\displaystyle{\frac{1}{2}}.\Phi$ with $\Phi=e^{-2\gamma}.(E^{2}-e^{2\psi}-\rho^{-2}e^{4\psi}L^{2})$. The contour $V_{eff}=0$ defines a curve of zero velocity. The geodesics are found by solving Hamiltonian equations

[TABLE]

with a choice of $E$ and $L$ and initial conditions for $p_{\rho},\,p_{z},\,\rho,\,z$ such that $H_{2}=0$.

Einstein’s vacuum field equations for the line part reduce to Laplace equation

$\partial_{\rho\rho}\psi+\rho^{-1}\partial_{\rho}\psi+\partial_{zz}\psi=0$ with $\gamma_{,\rho}=\rho(\psi^{2}_{,\rho}+\psi^{2}_{,z})$ and $\gamma_{,z}=2\rho\psi_{,\rho}\psi_{,z}$.

The solution of the Laplace equation is

[TABLE]

This is a single-particle Chazy-Curzon solution. In this paper, we establish, that the Hamiltonian system, describing this motion is not Liouville- integrable .

Next we will find the points of equilibrium of the system, describing geodesic motion of Chazy-Curzon space-time. We take $m=1$, by scaling, and we have

[TABLE]

where $\Omega(\rho,\,z)=\displaystyle{e^{(\frac{-1}{\sqrt{\rho^{2}+z^{2}}}+\frac{\rho^{2}}{2(\rho^{2}+z^{2})^{2}})}}$ and change the time

[TABLE]

for the Hamiltonian and the equations (2.2).

We found the new “Hamiltonian” - change of the time variable is not a canonical transformation, but preserves the integrability properties (see [19] for details)

[TABLE]

and we obtain

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

The substitution (2.3), transforms the Hamiltonian system to the system ($`=\frac{d}{d\tau}$)

[TABLE]

Then the conditions for the equilibrium are $p_{\rho}=p_{z}=z=0$ and

[TABLE]

from $F=0$ we get

[TABLE]

Solving this linear system for $L^{2}$ and $E^{2}$ we obtain for the equilibrium

[TABLE]

Finally, if the point $(p_{\rho},\,p_{z},\,z,\,\rho)$ is an equilibrium for Chazy- Curzon space-time Hamiltonian system, then $p_{\rho}=p_{z}=z=0$ and $\rho$ satisfies $L^{2}=-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2}$ and $E^{2}=-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2}$. For real $E^{2}$ and $L^{2}$ we have $1<\rho<2$.

3 Main result

Proposition 1**.**

On the manifold $P:=\{p_{\rho}=p_{z}=z=0,\,\rho\}$, invariant under $X_{F}$ the system (2) has only periodical solutions around equilibrium $p_{\rho}=p_{z}=z=0$, $\rho_{0}\in(1,\,2)$ with restrictions $L^{2}=-\frac{\rho_{0}^{2}.e^{\frac{2}{\rho_{0}}}}{\rho_{0}-2}$, $E^{2}=-\frac{(\rho_{0}-1).e^{\frac{-2}{\rho_{0}}}}{\rho_{0}-2}$ .

Proof: For the potential $v(\rho,\,z,\,E^{2},\,L^{2})$ we have

[TABLE]

around the points of equilibrium we have $v(\rho,\,0,\,-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2},\,-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2})=0$. We compute the second derivative near the equilibrium,

[TABLE]

We have only periodic solutions around the equilibrium if

$v(\rho)=v(\rho,\,0,\,-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2},\,-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2})=0$, $v^{\prime}(\rho)=\displaystyle{\frac{\partial v(\rho,\,0,\,-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2},\,-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2})}{\partial\rho}}=0$ and

$-v^{\prime\prime}(\rho)=-\displaystyle{\frac{\partial^{2}v(\rho,\,0,\,-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2},\,-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2})}{\partial\rho^{2}}}=\displaystyle{\frac{\rho^{2}-6\rho+4}{\rho^{4}(\rho-2)}}>0$, which is true for $\rho\in(1,\,2)$.

Important for the integrability of the system is the behavior of the period function,

[TABLE]

where $\rho_{-}$ and $\rho_{+}$ are the roots of $0-v(\rho,\ 0,L^{2},\ \,E^{2})=0$, $v(\rho,\ 0,L^{2},\ \,E^{2})$ is the potential function, and $L^{2}=-\frac{\rho^{2}.e^{\frac{2}{\rho}}}{\rho-2}$, $E^{2}=-\frac{(\rho-1).e^{\frac{-2}{\rho}}}{\rho-2}$ are conditions for equilibrium.

Proposition 2**.**

$T=2g(\rho_{0})\log{\eta}+\Phi(\epsilon,\,\delta)$, where $\Phi(\epsilon,\,\delta)$ is analytical function, $\eta=\frac{\epsilon}{\delta}$, and $g(\rho_{0})=\frac{\sqrt{2}}{\sqrt{-v^{\prime\prime}(\rho_{0})}}$.

Proof: Around the point of equilibrium $(p_{\rho},\,p_{z}\,\rho,\,z)=(0,\,0,\,\rho_{0},\,0)$, the period is

[TABLE]

where $\Phi(\epsilon,\,\delta)$ is analytical function, $\eta=\frac{\epsilon}{\delta}$, $g(\rho_{0})=\frac{\sqrt{2}}{\sqrt{-v^{\prime\prime}(\rho_{0})}}$

and $-v^{\prime\prime}(\rho_{0})=-\displaystyle{\frac{\partial^{2}v(\rho_{0},\,0,\,-\frac{(\rho_{0}-1).e^{\frac{-2}{\rho_{0}}}}{\rho_{0}-2},\,-\frac{\rho_{0}^{2}.e^{\frac{2}{\rho_{0}}}}{\rho_{0}-2})}{\partial\rho^{2}}}>0$ for $\rho_{0}\in(1,\,2)$.

The last condition follows from Taylor’s formula applied to $v(\rho)$ around $\rho_{0}$. We have

[TABLE]

because it $v(\rho_{0})=v^{\prime}(\rho_{0})=0$ for the points of equilibrium. Then,

[TABLE]

around equilibrium point.

Our goal is the following:

Theorem 1**.**

The system with Hamiltonian (2.1) is not integrable by means of analytical first integral.

We need an equivalent formulation to prove Theorem 1.

Proposition 3**.**

The system (2) has no additional analytical first integral.

Proof: For the proof we know that there are only a periodical of solutions on $P$, of the Hamiltonian system (2) on the hypersurface $F=0$. If $T$ is the period function then complex continuation of the manifolds $T=const$ turns out to be infinitely branched - this excludes the existence of a nontrivial analytic integral on any open subset of the complex domain where this infinite branching is true. We need to show that if $G$ is a smooth function on open set $U$ such that $V=U\cap(F=0)$ is $X_{F}$ invariant, $\{F,\,G\}=0$ and derivative $d\{F,\,G\}=0$ on $V$, then $G$ is a function of $F$ and $T$ on $V$.

The proof is similar like in the remarkable paper of J. J. Duistermaat [14].

Acknowledgements

The author has been supported by contract No. RD-22-719/29.03.2019 and grant 80-10-132 / 15.04.2019 from Sofia University “St. Kliment Ohridski”.