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Normal modes of relativistic systems in post Newtonian approximation

We use the post Newtonian (pn) order of Liouville's equation (pnl) to study the normal modes of oscillation of a relativistic system. In addition to classical modes, we are able to isolate a new class of oscillations that arise from perturbations of the space-time metric. In the first pn order; a) their frequency is an order q smaller than the classical frequencies, where q is a pn expansion parameter; b) they are not damped, for there is no gravitational wave radiation in this order; c) they are not coupled with the classical modes in q order; d) in a spherically symmetric system, they are designated by a pair of angular momentum eigennumbers, (j,m), of a pair of phase space angular momentum operators (J^2,J_z). Hydrodynamical behavior of these new modes is also investigated; a) they do not disturb the equilibrium of the classical fluid; b) they generate macroscopic toroidal motions that in classical case would be neutral; c) they give rise to an oscillatory g_{0i} component of the metric tensor that otherwise is zero for the unperturbed system. The conventional classical model are, of course, perturbed to order q. These, however, have not been discussed in this paper.


Introduction
In recent years some authors have debated the existence of a new class of quasi modes in relativistic systems believed to have been originated from the perturbations of the space-time metric. Kokkotas and Schutz (1986) considered a toy model of a finite string (to mimic a fluid) coupled to a semi-infinite one (to substitute the dynamical space-time). Such a system accommodates a family of damped oscillations. Baumgarte and Schmidt (1993) obtained the same results by considering a slightly more realistic toy model. Kojima (1988) verified that strongly damped (w-) modes do indeed exist in realistic stellar models. Subsequently, Kokkotas and Schutz (1992) used a WKB method to study w mode spectra for several polytropic models. Different investigators have proposed different mathematical and numerical schemes to isolate these modes.
Discrepancies exist. Leins, Nollert and Soffel (1993) employed two different approaches for the modes of Schwarzschild black hole. They used Leaver's (1985) continued fraction approach and a Wronskian technique (Nollert and Schmidt 1992). They confirmed the findings of Kokkotas and Schutz. They, however, found other modes with smaller oscillation frequencies and stronger dampings than those that had been found by the WKB scheme. Anderson, Kokkotas and Schutz (1995) used a numerical approach to clarify the reason for the discrepancies. They came to agreement with the findings of Leins et. al. (1993),and showed that the WKB method employed by Kokkotas and Schutz failed to distinguish between these modes. Recently, Lindblom, Mendell and Ipser (1997) have used a formulation of the relativistic pulsation equation in terms of two scalar potentials, one for the fluid and the other for the perturbations in the gravitational field.
They have found a set of even-parity modes for a relativistic star, including the w modes.
In this paper, we approach the problem through relativistic Liouville's equation rather than relativistic hydrodynamics. The reason for doing so, is to avoid thermodynamic concepts incorporated into hydrodynamics. Liouville's equation is a purely dynamical theory and free from such complexities. One, of course, pays the price by having to dwell in the six dimensional phase space an elaborate mathematical task, but not obtrusive. In section 2 we give the pn order of the linearized Liouville equation that governs the evolution of small perturbations from an equilibrium distribution function. In section 3 we extract the evolution equation for the non classical modes and explore some of its properties. In section 4 we explore the O(3) symmetry of the modes and classify them on basis of this symmetries. In section 5 we study hydrodynamics of the pn modes. In section 6 we on a variational approach to the solution of pn modes and give a numerical analysis for the modes of polytropes. Lengthy mathematical manipulations are given in appendices A and B.

Formulation of the problem
Liouville's equation in the post-Newtonian approximation pnl for the one particle distribution of a gas of collisionless particles maybe written as follows where (x, u) are phase space coordinates, q is a small post-Newtonian expansion parameter, the ratio of Schwarzchild radius to a typical spatial dimension of the system, say. The classical and post-Newtonian operators, L cl and L pn , respectively, are The imaginary number i is included for later convenience. The potentials θ(x, t), Θ(x, t) and η η(x, t), solutions of Einstein's equations in pn approximation, are where dΓ = d 3 xd 3 u. See Rezania and Sobouti (1998, hereafter paper I) for details. In an equilibrium state, F (x, u) is time-independent. If, further, it is isotropic in u, macroscopic velocities along with the vector potential η η vanish. It is also shown in paper I that the following generalizations of the classical energy and classical angular momentum are integrals of pnl: l i = ε ijk x j u k exp(qθ) ≈ l cl i (1 + qθ), f or spherically symmetric θ(r) and Θ(r). (4b) Equilibrium distribution functions in pn approximation can be constructed as appropriate functions of these integrals. In paper I the pn models of polytrope were studied in this spirit.
Here we are interested in the time evolution of small deviations from a static solution. Let F → F (e) + δF (x, u, t), | δF |≪ F ∀(x, u, t). Accordingly, the potentials split into large and small components, θ(r) + δθ(x, t), Θ(r) + δΘ(x, t) and δη η(x, t) where r = |x|. Both, the large and small components, can be read out from Eqs. (3). Substituting this splitting in Eq. (1) and keeping terms linear in δF gives where L cl and L pn are now calculated from Eqs.
(2) with θ(r), Θ(r) and η η = 0, and δL cl and δL pn with δθ(x, t), δΘ(x, t) and δη η(x, t). For the latters, Eqs. (2) give where F e = dF (e)/de. The classical limit of Eq. (5) is Equation (7) is the linearized Liouville-Poisson equation. It was studied briefly by Antonov (1962). He separated δF and Eq. (7) into even and odd components in u, extracted the equation governing the odd component and showed that δF odd satisfies an eigenvalue equation. Sobouti (1984Sobouti ( , 1985Sobouti ( , 1986Sobouti ( , 1989a elaborated on this eigenvalue problem, studied some of its symmetries and approaches to its solution. Sobouti and Samimi (1989), and Samimi and Sobouti (1995) showed that Eq. (7) has an O(3) symmetry and its oscillation modes can be classified by the eigennumber j and m of certain phase space angular momentum operator.
The effect of the pn corrections in Eq. (5) on the known solutions of the classical Eq. (7) can be analyzed by the usual perturbation techniques. Whatever the procedure, the first order corrections on the known oscillation modes will be small and will not change their nature. We

The post-Newtonian modes of oscillations
Theorem: Any function G(e cl , l cl ) of classical energy and angular momentum, e cl and l cl , respectively, for which δρ(x) = Gd 3 u = 0, is a static solution of the linearized Liouville-Poisson, Eq.
One class of such functions are those with odd parity in u or, equivalently, in l, i. e., G(e cl , l cl ) = −G(e cl , −l cl ). We will see in Eqs. (17) and (18) below that there exists a class of odd parity functions that not only fulfill the requirements of the theorem but also give δΘ = 0.
With these preliminaries, we come back to solutions of Eq. (5).
Furthermore the term containing δη η is an order q smaller than δF and can be dropped. Thus see Eq. (A. 1) for details. We owe the reader a demonstration of δΘ = 0. In appendix A we show that ω is real. There follows that δF is complex; for, L pn is purely imaginary. Thus, δF = δF R + iδF I . Equation (9) can be decomposed accordingly: Upon eliminating the imaginary component one obtains The same equation is satisfied by δF I . The latter will, however, be obtained from Eq. (10a) once δF R is calculated from Eq. (11). In appendix A we show that L pn2 is Hermitian, in spite of the fact that L pn is not. Furthermore, L pn2 is even in both x and u. Therefore, the eigensolutions δF R can be chosen either odd or even in x and u. Even parities, however, are not acceptable.
For they give rise to non vanishing changes in the macroscopic density, δρ, and the potential δθ and δΘ, which violates the validity of Eq. (8). The x and u-parities of δF I , however, will be opposite to those of δF R ; for they are connected by Eqs. (10a, b), where L pn is odd in x and u.

O(3) symmetry of L pn
In appendix A we show that L pn and L pn2 commute with the phase space angular momentum . Thus, they can have simultaneous eigensolutions with J 2 and J z . Eigensolutions of the latter pair of operators, Λ jm ; j and m integers, are worked out in appendix B. They are complex polynomials of the components of the classical angular momentum vector. Their x and u parity is that of j.
An expression of the form f (e cl , l cl 2 ) Λ jm is already an eigensolution of J 2 and J m belonging to eigennumbers j and m. See appendix B, Eqs. (B. 2-5). Solutions of Eq. (11) for δF R will, therefore, reduce to substituting an expression of this sort with an odd j value in Eq. (11) and solving it for f (e cl , l cl 2 ). In section 6 we will take a varitional approach to such solutions, and as variational trial functions will consider functions of e cl only. For the moment let be a solution of Eq. (11) belonging to the eigenvalue ω. Combining this with its corresponding imaginary component from Eq. (10a) we obtain With no lost of generality we will adopt the real part. Thus, This, however, is the sum of two eigenfunctions belonging to ω and −ω and is not a simple eigensolutions of Eq. (9).

Hydrodynamics of pn modes
In this section we calculate the density fluctuations, macroscopic velocities, and the perturbations in the space-time metric generated by a pn mode.
In appendix B we show that ρv is a toroidal spherical harmonic vector field. In spherical polar coordinates it has the following form where and v es = √ 2θ is the escape velocity from the potential θ(r). The macroscopic density, generated by even component of Eq. (14), is The second integral is obtained by an integration by parts. The vanishing of it comes about because of the fact that the radial vector x is orthogonal to the toroidal vector ρv. One also notes that ∇ · (ρv) = 0. It can further be verified taht, the continuity equation is satisfied at both classical and pn level. To complete the proof of validity of Eq. (9) one should also show that δΘ = 0. From the definition Eq. (3c) for δΘ and Eq. (14) for δF one has The first two terms are zero because δρ and δθ are zero. The third term vanishes because the integral over u ′ has the same form as in δρ except for the additional scalar factor u ′ 2 in the integrand. Like δρ it can be reduce to the inner product of the radial vector x and a toroidal vector that are orthogonal to each other. QED.
The toroidal motion described here slides one spherical shell of the fluid over the other without perturbing the density, the Newtonian gravitational field and, therefore, the hydrostatic equilibrium of the classical fluid. In doing so, it does not affect and is not affected by the conventional classical modes of the fluid at this first pn order.

Variational solutions of pn modes
In Eq. (12) we suppress the subscript jm, for brevity. It will be remembered that f will be chosen odd in both x and u and will be a function of the phase coordinates through the classical energy and angular momentum integrals. We multiply Eq. (11) by f * and integrate over the volume of the phase space available to the system The adjoint operator L pn † is given in Eq. (A. 5). Substituting the latter in Eq. (21a) and simple manipulations gives It will be remembered that ω is real and ω 2 is positive. Therefore, in spite of one negative term, the right hand side of Eq. (21b) has to be positive. That is, L pn2 is positive in addition to being We substitute Eq. (22) in (21b), carry out the integrations, and minimize ω 2 with respect to variations of c n 's. We arrive at the following matrix equation to solve for ω 2 and c n 's: where C is the column vector of the variational parameters, {c n }, and elements of the S and W matrices are Eigen ω's are the roots of the characteristic equation a pq = pq(p + q + 2.5) (p + q)(p + q − 1) , b pq = 4p + 4q + 11 p + q + 3.5 , p, q = 2, 3, · · · . (26c) Polytropic potentials θ and Θ were obtained from integrations of Lane Emden equation and Eqs.
(28) of paper I, respectively. Eventually, the matrix elements of Eqs. (26), the characteristic Eq. Appendix A: Some properties of L pn and L pn2 The Hilbert space: let H be the space of complex square integrable functions of phase coordinates (x, u) that vanish at the phase space boundary of the system, Integrations in H are over the volume of phase space available to the system. In particular, the boundedness of the system in velocity space sets the upper limit of u at the escape velocity, √ 2θ, where θ(x) is the potential at x. Thus, g(x, 2θ(x)) = 0.
The operator L pn is defined in H. For a spherically symmetric, isotropic and static system, We have set η η and all time derivatives equal to zero, used ∂/∂x i = (x i /r)d/dr and denoted d/dr by "′ ".
Theorem 1. Eigenvalues of L pn are real: This is in spite of the fact that L pn is not Hermitian.

Proof:
Let Noting that L pn is purely imaginary, the complex conjugate of this equation is The x and u parities of f * f are even and those of x · u are odd. Therefore, the integral vanishes and ω = ω * = real. QED.
Hermitian adjoint of L pn : For f, g ∈ H, by an integration by parts one finds: With θ and Θ of Eq. (A. 1) being spherically symmetric, it is easy to show that L pn commutes The operators J i , however, obey the angular momentum algebra. Therefor, the three operators L pn , J 2 and J z commute pairwise. The same is true of L pn2 , J 2 and J z . In appendix B we elaborate on the eigenfunctions of J 2 and J z to prepare the path for simultaneous eigensolutions of the latter trio.
where f (e, l 2 ) is an arbitrary function of its arguments, and a and b are normalization constants. Examples: Theorem 2: The vector field V jm = Λ jm udΩ is a toroidal vector field belonging to the spherical harmonic numbers (j, m), where integration is over the directions of u.