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Hamiltonian formulation of the linearized perturbation equations for the Vlasov-Einstein system of relativistic stellar dynamics
Physica A
160 (1989) 213-224 North-Holland, Amsterdam
HAMILTONIAN FORMULATION OF THE LINEARIZED PERTURBATION EQUATIONS FOR THE VLASOV-EINSTEIN SYSTEM OF RELATIVISTIC STELLAR DYNAMICS
Henry E. KANDP.UP Space Astronomy Laboratory and Institute for Fundamental Theory, University of Florida, Gainesville, FL 32609, USA Received 27 April 1989
Relativistic star clusters are often modeled as static, spherically symmetic solutions to the Vlasov-Einstein system. It is shown here that, for spherical disturbances, the equations for linearized perturbations about such a static equilibrium can be cast into a Hamiltonian form using relativistic analogues o[ Morrison's bracket for the Vlasov-Poisson system and the energy functional of Lynden-Bell and Sanitt for a linearized perturbation.
1. Motivation It is generally acccpted that, at least for sufficiently short times and neglecting any gas which might be present, a system of stars like a galaxy may usually be described by a one-particle distribution function )¢', the evolution of which is governed by the Vlasov-Poisson system. What this mean5 is that )¢" satisfies a Vlasov (i.e., a collisionless Boltzmann) equation, the characteristics of which correspond to trajectories determined by an average gravitational force generated self-consistently via the gravitational Poisson equation. Thus, in particular, one believes that most, if not all, galaxies can be modeled as (nearly) time-independent solutions to this Vlasov-Poisson system. And, furthermore, one knows that many, albeit certainly not all, of these static solutions are linearly stable [1, 2]. One may, however, envision situations in which Newtonian gravity proves inadequate for describing the system of interest, and, in such cases, one presumes that the relevant equations are provided instead by the Vlasov-Einstein system. What this means (cf. ref. [3]) is that the covariant distribution function N satisfies a relativistic Vlasov equation, the characteristics now conesponding to geodesics in an "average spacetime" generated self-consistently via Einstein's equation. 0378-4371 / 89 / $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
When no special symmetries are assumed, the Vlasov-Einstein system is fraught with numerous complications reflecting, e.g., the effects of gravitational radiation. If, however, one restricts attention to spheric~tly symmetric configurations, the situation simplifies dramatically. In this case, it is easy to construct infinitely many static solutions (at least in terms of quadratures), and it is straightforward to formulate the problem of stability with respect to linearized spherical perturbations. Thus, one knows that, in this setting, a static ~lution may be specified as an arbitrary function W = fo(E, j2, m), where m is th ~. rest mass of the, star (one is thus allowing for a nontrivial distribution of massesl), and E and j2 2enote respectivdy the conserved energy and squared angular momentum associated with the time translational and rotational symmetries. If, in particular, f0 is a function only of E and m, this static solution corresponds at each point in space to an isotropic distribution of three-velocities, as measured in a local orthonormal frame. The key physical question is whether any given f0 is linearly stable with respect, first of all, to petlurbations which preserve the spherical symmetry. One is of course also interested in perturbations which destroy the spherical symmetry. However, it is believed (of. refs. [4, 5]) that it is through some spherically symmetric perturbation that a system may become unstable towards a catastrophic collapse to form a massive black hole, a possibility that has been confirmed numerically [6]. In the Newtonian case, isotropic equilibria fo(E, m) are especially convenient to consider in that it is also straightforward to study questions related to linear stability towards nonspherical perturbations [1, 7] and even the question of nonlinear stability [8]. In the corresponding relativistic theory, this restriction again facilitates a consideration of nonspherical perturbations (cf. ref. [9]), although the analysis is now complicated by the possibility of gravitational radiation. The problem of nonlinear stability is even more difficult. No one has as yet formulated the nonlinear perturbation equations in a useful form, and, except for the special case of spherical perturbations, one does not know how to construct a nonlinear generalization of the energy of a linearized perturbation (this entailing the construction of a relativistic analogue of an energy-Casimir [8, 10]). The question of linear stability towards spherically symmetric perturbations can be translated into a well-defined mathematical problem for the special case in which the derivative F e -- 0.1'o/OE is everywhere negative. Let f denote the linearized perturbation, and suppose that it is decomposed into two components, f÷ and f_, respectively even and odd under an inversion of spatial momentum. In this case, both Newtonianly and relativistically, f_ may be shown to satisfy a linear operator equation of the form [4, 11] O~f_ = - i f f _ ,
(1)
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
215
where 0 t denotes a coordinate time derivative, and 3" is a time-independent linear operator, symmetric with respect to an inner product (~, 7/). It then follows that Off_, and hence f_, remains bounded and well-behaved at all later times if and only if the functional W = ½(~, 3.~)
(2)
is strictly positive for all odd test functions ~:~ 0 . And, by combining this with the linear equation Orf+ = - ~ f _
,
(3)
with ~ time independent and antisymmetric, relating f_ and f+, one knows that the positivity of W is both necessary and sufficient for linear stability. One can in fact prove [1] that 3" is a positive operator in the Newtonian theory, so that all f0's with F E < 0 are linearly stable with respect to spherical perturbations. And, by extending that argument somewhat, one can also show [7] that, if fo is a function only of E and m, it will also be stable to linear perturbations that destroy the spherical symmetry. By contrast, one knows through the construction of suitable test functions (cf. ref. [12]) that, in a relativistic setting, W can in fact be negative, so that a spherical solution to the Vlasov-Einstein system, even with F E negative, need not be linearly stable. Unfortunately, though, the consideration of test functions is both tedious and inelegant, so that one might seek an alternative approach for diagnosing stability or instability. One such approach entails a consideration of binding energy curves for sequences of equilibria, which allow the formulation of a sufficient criterion for stability (and thus a necessary criterion for instability). The key points [13-15] are (a) that one can identify a natural energy e for a linearized perturbation f, (b) that the positivity of this energy for all nonvanishing f is a sufficient criterion for stability, and (c) that the sign of the energy of one of the linearized modes must change at any turning point in the binding energy curve. This implies that if the starting point of the sequence of equilibria is known to be stable (as it may be if it is essentially Newtonian), all subsequent configurations up to the first turning point are also guaranteed to be stable. In the Newtonian case, e can be understood as the quadratic form ½ D X H c ' ( f ) 2 constructed from the Hessian of an appropriate energy-Casimir H c. In relativity, this is also known to be true, at least when one restricts attention to perturbations that preserve the spherical symmetry. It is, however, important to recognize that the existence of modes with e < 0 need not a priori imply a linear instability. Thus, e.g., one knows [16, 17] that
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H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
many linearly stable Newtonian configurations admit modes with e < 0. The explanation thereof [8] resides in the fact that these negative energy modes f< in the zero-frequency subspace of perturbations, so that ,9,f< --0. Thus. in particular, one 0bserves that the quadratic form W, acting on a function f_, agrees with e evalUated for the function Otf_, i.e., W[f_] = e[Otf_], so that linear instability requires a time-dependent negative energy mode with f_ f 0 . It has long been believed that, in a relativistic theory, the negative energy modes will acquire a nontrivial time dependence and manifest a linear instability. However, despite a great deal of work, and the absence of any counterexamples, this conjecture remains unproved. These subtleties associated with zero frequency modes suggest the desirability of formulating the problem of stability in yet another way which does not, e.g., cast the perturbation equation for f into a form that is second order in the time derivative #,. Such is the object here. Specifically, the aim is to construct a suitable relativistic "bracket" [A, B] in terms of which the relativistic perturbation equations for f and the "perturbed gravitational force" 0 take the forms # i f = [f, ~l and
0,$ = [0, e].
(4)
The key point is that, provided that the bracket satisfies appropriate properties, as will be verified here, it defines a Lie algebra, so that the flows determined by eq. (4) are symplectic. In more conventional language, this means that the values of f and ~, at any two times are connected by a canonical transformation, so that one has at his or her disposal all the sgphisticated apparatus of geometric mechanics. Section 2 of this paper formulates the basic relativistic equations, introduces the bracket operation, shows that it generates the correct equations of motion, and then verifies that the bracket defines a Lie algebra. Section 3 comments on possible modifications and extensions, and then concludes by observing that, in the Newtonian limit, one recovers the linearized Vlasov-Poisson system in a form where the gravitational field is treated as an independent dynamical variable. In this limit, the bracket reduces essentially to the Morrison [18] or Marsden-Weinstein [19] bracket for the Vlasov-MaxweU system, evaluated for a vanishing magnetic field. 2. Construction of the bracket
Consider now the specifics of the Vlasov-Einstein system, using units in which the speed of light c and the gravitational constant G are set equal to unity. Spacetime indices will be denoted by Greek characters /x, v , . . . , whereas spatial indices will be denoted instead by Latin letters i, j, . . . .
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
217
Given some spacetime d / a n d the associated cotangent bundle T(.a), the covariant distribution function N(x ~, p~) is defined by the demand that the quantity N(x ~, p~)d°Fx d°Fp denote the number of stars in the covariant, constant time, seven-dimensional volume element dWx d°Fp. In a specific coordinate system (x i, t), the volume elements d°//'~ and d°Fp reduce to dOFx * (pTm)(_g)~/2 dax - (pt/m) d3Vx and dOFp _ _ ( _ g ) - l / 2 dap * ( m l p , ) ( _ g ) - l / 2 d3p d i n ,
(5)
where p~" - g~,Vp~ and m 2 ~ -g~,vp p , so that
d~f(xa' Pa ) d °~x d o~p * d~f(xi, t, Pi, m) d3x d3p d m .
(6)
It is then assumed that Jf satisfies the Vlasov-Einstein system, so that p~ ON m ox ~'
1 Og~ OJr 2m Ox ~" P~P~ op--~ = 0 ,
(7)
where the metric g~, is determined self-consistently from the Einstein equation
G~[g] = 8,r f d~rp m-lp~pl3jf .
(8)
Suppose now that the system is spherically symmetric. Following, e.g., Ipser and Thorne [4] (note, however, a difference in sign conventions), one can assume a line element of the Schwarzschild form ds 2 = _ e v dt 2 + e a dr 2 + r2(d0 2 + sin20 d $ 2 ) ,
(9)
where V and A are functions only of r and t. The angular coordinates have an obvious geometric significance, as does the radial coordinate, so chosen that a sphere of "radius" r has surface area 4~rr2 One then verifies that a generic static, spherically symmetric solution corresponds to a static N - fo(E, j2, m) and static metric functions V = v(r) and A = A(r) involving only the radial coordinate, t being interpreted now as the cyclic coordinate associated with the time translation symmetry. Here, of course, the conserved energy E = IPtl satisfies - m 2 = g "E2 + g PiPj.
(10)
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H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
To define linearized perturbations away from some static solution, one must introduce a mapping between the cotangent bundles associated with the static and pe~urbed spacetimes. This mapping is not unique. Given, however, the assumption of spefical symmetry, there is a "natural" choice physically (although not necessarily the most elegant mathematically! [20]). Specifically, following Ipser and Thorne [4], define a mapping between the perturbed bundle, with coordinates (x '*, p ' ) and metric g'a, and the static bundle, with 0 coordinates (x", Pa) and metric g,a, by (a) identifying spacetime points, i.e., Xp~ = x fit, and (b) identifying components of momentum as measured in a local F orthonormal tetrad, i.e., P(,~)=P(*), so that, e.g., t,-g"_,,,,~/2)p,, = (_g#)~2p,. Given such an identification, one can then set AC=fo+ f ,
V=v+Sv
and
A=A+SA,
(11)
where f is defined in the static bundle and 8v and 8A in the static spacetime. And, given this decomposition, one can then calculate the linearized equations of motion. Because of the underlying symmetries, there is only one dynamical degree of freedom for the gravitational field, namely 8A, the quantity 8v being determined by a constraint equation. Thus, by implementing the constraint explicitly, and then defining a new function ------ S A / 2 r ,
(12)
one is led to the linearized perturbation equations P_2' / Off + ~ f + p, ~1 + r-~r FEpr=o
(13)
0,~, = 4,rr e ^ f d°l/p m - l p t p r f .
(14)
and
Here 0t denotes a coordinate time derivative O/Ot and FE --Ofo/OE. Given that f and $ live in the static bundle, all indices are raised and lowered with the unperturbed metric, and the volume element d°Fp refers to the unperturbed momentum space. In terms of the operator
~ _ pi m
0 Ox i
1 Ogo~ 0 2m Oxi P~PtJ Opi ,
the unperturbed total time derivative satisfying ~ f o - 0 ,
(15) the operator
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
219
satisfies
m ~ g + 4 xrr e A -~ m F E\(P'P" g~g = -~ ~ f d °t/'p p,pr - -mg
P,P m
'f doll• -°-:,P / Pm
g" (16)
Note that eqs. (13)-(16) have the correct Newtonian limit. In this limit, = O~/Or, where • is the perturbed gravitational potential, and the integral contributions to ~ vanish identically, so that one recovers the equations
O,f + D r - Fep ~ Or = 0
(17)
O,(O~lOr) = -4~r f d3p dm P T ,
t .8)
and
where, in terms of the unperturbed gravitational potential tP0 , i
D -- p 0 m Ox~
m 0~o 0 . Ox~ Op~
(19)
In order to cast eqs. (13) and (14) into a Hamiltonian form, one must first identify an appropriate energy functional, or Hamil.:onian, e and then define a bracket [A, B] which (a) generates a Lie algebra ar.d (b) reproduces correctly the equations of motion in the form
o,f= [f,
and
0,# = [¢, e].
(20)
The "natural" energy functional was constructed by Ipser [15], who showed, translated into a more modern language, that the relativistic analogue of the quadratic form ½OEHc.(f)2 constructed from the natural energy-Casimir for the Vlasov-Poisson system is
,f e= ~
r2 d~r d°/rp _ F e
1 f d3Vx( l + r
d r ) e-A ~2.
(21)
Note that if, for fixed time t, f is viewed as a function of x i, pi, and m, and g, is viewed as a function of x i, one computes the functional derivatives ~e ~-f-
f FE
and
( dr)
~e _ 1 (_g)l/Ee-^ 1 + r ~ ~g, - - 4---~ -~r "
(22)
H.E. Kandrup I Vlasov-Einsteinsystemof relativisticstellardynamics
220
Before defining the functional bracket [A, B], it is useful first to introduce the spatial Poisson bracket
OA OB
{A, B} - ax' ap,
OA OB Opi Ox'"
(23)
One computes immediately that, for any function g, the bracket
{E, g} = -(mlp')@g ,
(24)
so that a Poisson bracket term in the expression [A, B] will yield a desired contribution ~ f in the equation ,af motion for a,f. Nnte also that, for any function x(E, j2, m) and any sphericaUy symmetric g, or, alternatively, for any x(E, m) and an arbitrary g, the bracket
OX { E , g } = {x'g}=aE
m OX p, ~g.
(25)
OE
Given these preliminaries, now define the bracket [A, B] via the formula [A,B]=
d ~ xd~pf0
8f ' 8f
+4 fd "x d~trp
e ^ +P . . Fe(-g) - l / z ( s8f a 8B 80
8A 8B)
80 8f~,
8B + 4rt f d3Vx re~((L ~-ff )(M - - ~ ) - ( M ~ff )(L ~f ))
(26)
where
L g - f dT'pm- 1PiP , Feg and M g - f dT'pm -1PrP"Feg.
(27)
To verify that this bracket implies that eqs. (20) reproduce eqs. (13) and (14) is straightforward if one makes use of eq. (25) and the identities
80(x 'i, t) =8(3)(x,'-x ') 80(Xi, t)
and
(28)
8f(x": p' ) 8f(x i, p~ )
= 8(3)(x ' i - xi)8(3)(p i - pi)6(m' - m). t
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
221
It remains to check that this bracket actually satisfies the appropriate requirements so as to define a Lie algebra. Indeed, three important results follow immediately. 1) It is obvious that, for any constant c, [cA, B] = [A, cB] = c[A, B], and that [A + B, C] = [A, C] + [B, C] and [A, B + C] = [A, B] + [A, C]. These conditions show that the bracket defines an algebra. 2) It is also obvious that the bracket is antisymmetric, i.e., that [A, B] = - [ B , A]. 3) It is less obvious, but also true, that the bracket satisfies the Jacobi identity [A, [B,
ell
+ [B, [C, All + [C, [A, BI] = 0 .
(29)
This is in fact straightforward to verify. By introducing the two-component vector u i with components f and g,, the bracket can be written in the form [A, B] = f tiT"x d~tPp ~A 0
~Buj,
(30)
where O q is an antisymmetric operation. But one verifies immediately (cf. ref. [18]) that any bracket of this form will satisfy eq. (29) if, as is the case here, ~ o i J / ~ u k -~- O.
Any algebra satisfying conditions 2) and 3) is of course a Lie algebra. And thus, one concludes that the linearized perturbation equations can in fact be east into a Hamiltonian form, the bracket of eq. (26) ensuring that the evolution defined by time translation is symplectic and hence generated via a canonical transformation.
3. Discussion
One obvious question about the equations of motion (20) and the bracket (26) concerns the origin of the inelegant integral contributions reflected in the operator ~, which vanish identically in the Newtonian limit. No analogue of these terms arise when the Vlasov-Maxwell system is formulated in fiat space, even though that is also a relativistic field theory, and they would not be present even if one were to formulate the Vlasov-Maxweil system in a fixed curved spacetime. These contributions arise explicitly as a consequence of the fact that, since the Vlasov equation is determining the underlying spacetime itself, the perturbed and unperturbed configurations are defined in different
222
H.E. Kandrup / Vlasov-Einstein system t f relativistic stellar dynamics
cotangent bundles. Only by constructing a nontrivial, and nonunique, mapping between these bundles can one isolate upon a perturbation f that satisfies an equation like (13) defined in the unperturbed bundle. The specific form of this equation, and hence the operator ~ , reflects the choice of the mapping between the bundles, and, indeed, other prescriptions are known to lead to different equations [20,21]. Thus, e.g., the IsraelKandrup prescription [20] entails identifying spacetime points, i.e., setting x ' ~ = x ~, and then identifying momenta by a rescaling p ' " = o--lp ", with or chosen so as to preserve the mass shell constraints and cross sections of the tangent bundle. This leads to a perturbation equation of the form pt
0
-- 0if + ~f= - ~ m 0p~ (F~ f0),
(31)
where F,, denotes a linearized "gravitational force" generated by the perturbed Einstein. Eq. (31) seems more elegant than eq. (13), but this is perhaps a swindle. Because of the choice of mapping between the bundles, the perturbed field equations become more complicated, so that, e.g., the perturbed component of stress energy 8T/[f]
= f d°t/'p m-'p,p'f
(32)
appropriate for eq. (13) is modified by additional contributions reflecting the fact that, with this alternative identification, p'tp 't ~ ptp t. Another obvious, and as yet unanswered, question is whether this analysis could be extended to facilitate a consideration of perturbations that destroy the spherical symmetry or, alternatively, are not assumed to satisfy the linearized evolution equations. Both these objectives would be difficult to realize. When considering perturbations that destroy the spherical symmetry, additional field degrees of freedom are triggered, so that the perturbed equations become much more complicated, including in particular the possibility of gravitational radiation [9]. Alternatively, at least for the special case of spherically symmetric perturbations of an isotropic equilibrium f0(E, m), one knows how [15] to generate a relativistic analogue of an energy-Casimir Hc, for which e = !2DEltic "(f)2, but one has not yet computed the perturbed evolution equations in anything beyond a linear approximation. Given these comments, one may now conclude by considering the Newtonian limit of the bracket [A, B] and the Hamiltonian e. In this limit, q/= 8~/Or, so that
eN[f, Vt~*,]= ~I f dWx d°/fp -f2 Fe
lfd3 8~r
Vx IV~l 2 "
(33)
H.E. Kandrup / Viasov-Einstein system of relativistic stellar dynamics
And, similarly, noting that [A,
=
f
d%
223
p rFe = m(Of/Op,), one concludes that
fo
Sf '
+ 4,rrf d~,, d~p (_g)_t/z Ofo . ( SA SB 0r
~f 8 v ¢
8A SB ) 8v~
~f
(34) '
where v ~ is the ordinary three-velocity. Here one has introduced a vector notation, proceeding as if the perturbations f and V~ need not be spherically symmetric. And indeed, one verifies [8, 18] that this eN and this bracket reproduce the correct Newtonian equations even for nonspherical perturbations, provided that the equilibrium f0 corresponds to an isotropic distribution of velocities. It should be observed that es is nothing other than the Lynden-BeU and Sanitt [7] energy of a linearized perturbation, with the potential energy reexpressed as a local functional of the field V~ rather than as a nonlocal functional of f. And similarly, the bracket (34) is nothing other than the Mordson [18] or Marsden-Weinstein [19] bracket for the Vlasov-Maxwell system with electric charge e = ira, evaluated for a vanishing magnetic field and linearized about the static f0, as is appropriate when considering a linearized perturbation. If one chooses to reexpress the potential • as a nonlocal functional of f by means of a Green function representation, one is then led to the true Lynden-Bell and Sanitt energy eN[f] =
1 f dT'x dT'p
f2 t
, mr( xi, Pi, m ) m ' f ( x'i, Pi, m ' )
•
(35)
Ix -- X'[
And, given this eN, the linearized Newtonian equations follow simply from the Vlasov-Poisson bracket [A,B]=
f
dV xdT'pfo
~f , ~f
•
(36)
References
[1] H.E. Kandrup and J.-F. Sygnet, Astrophys. J. 298 (1985) 27. [2] J. Barnes, J. Goodman and P. Hut, Astrophys. J. 300 (1986) 112. [3] Ya. B. Zel'dovich and I.D. Novikov, Relativistic Astrophysics, vol. 1 (University of Chicago Press, Chicago, 1971).
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H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
[4] J.R. lpser and K.S. Thorne, Astrophys. J. 154 (1968) 251. [5] Ya. B. Zerdovich and M.A. Podurets, Astron. Zh. 42 (1965) 963 [trans. Soy. Astron. AJ 9 (1966) 742]. [6] S,L. Shapiro and S.A. Teukolsky, Astrophys. J. 298 (1985) 58. [7] D, LyndewBell and N~ Sanitt, Mon. Not. S. Astr. Soc. 143 (1969) 167. [8] H.E. Kandrup, University of Florida preprint (1989). [9] J.R. Ipser and R. Semenzato, Astrophys. J. 229 (1979) 1098. [10] D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep. 123 (1985) 1. [111 V.A. Antonov, Astron. Zh. 37 (1960) 918 [trans. Sov. Astron. AJ 4 (1960) 859]. it2] J.R. Ipser, in: IAU Symposium, no. 69, A. Hayli, ed. (Reidel, Dordrecht, 1975). [131 H. Poincare, Ann. Math. 7 (1885) 259. [14] J.R. Ipser and G. Horwitz, Astrophys. J. 232 (1979) 863. [15] J.R. Ipser, Astrophys. J. 238 (1980) 1101. [16] V.A. Antonov, Vest. Leningrad. Univ. 7 (1962) 135. [171 J. Katz. Mon. Not. R. Astr. Soc. 190 (1980) 497. [18] P.J. Morrison, in: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, M. Tabor and Y.M. Treve, eds. (American Institute of Physics, New York, 1982). [19] J.E. Marsden and A. Weinstein, Physica D 4 (1982) 394. [201 W. Israel and H.E. Kandrup, Ann. Phys. (New York) 152 (1984) 30. [21l P. Droz-Vincent and R. Hakim, Ann. Inst. Henri Poincare A 9 (1968) 17.
160 (1989) 213-224 North-Holland, Amsterdam
HAMILTONIAN FORMULATION OF THE LINEARIZED PERTURBATION EQUATIONS FOR THE VLASOV-EINSTEIN SYSTEM OF RELATIVISTIC STELLAR DYNAMICS
Henry E. KANDP.UP Space Astronomy Laboratory and Institute for Fundamental Theory, University of Florida, Gainesville, FL 32609, USA Received 27 April 1989
Relativistic star clusters are often modeled as static, spherically symmetic solutions to the Vlasov-Einstein system. It is shown here that, for spherical disturbances, the equations for linearized perturbations about such a static equilibrium can be cast into a Hamiltonian form using relativistic analogues o[ Morrison's bracket for the Vlasov-Poisson system and the energy functional of Lynden-Bell and Sanitt for a linearized perturbation.
1. Motivation It is generally acccpted that, at least for sufficiently short times and neglecting any gas which might be present, a system of stars like a galaxy may usually be described by a one-particle distribution function )¢', the evolution of which is governed by the Vlasov-Poisson system. What this mean5 is that )¢" satisfies a Vlasov (i.e., a collisionless Boltzmann) equation, the characteristics of which correspond to trajectories determined by an average gravitational force generated self-consistently via the gravitational Poisson equation. Thus, in particular, one believes that most, if not all, galaxies can be modeled as (nearly) time-independent solutions to this Vlasov-Poisson system. And, furthermore, one knows that many, albeit certainly not all, of these static solutions are linearly stable [1, 2]. One may, however, envision situations in which Newtonian gravity proves inadequate for describing the system of interest, and, in such cases, one presumes that the relevant equations are provided instead by the Vlasov-Einstein system. What this means (cf. ref. [3]) is that the covariant distribution function N satisfies a relativistic Vlasov equation, the characteristics now conesponding to geodesics in an "average spacetime" generated self-consistently via Einstein's equation. 0378-4371 / 89 / $03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
When no special symmetries are assumed, the Vlasov-Einstein system is fraught with numerous complications reflecting, e.g., the effects of gravitational radiation. If, however, one restricts attention to spheric~tly symmetric configurations, the situation simplifies dramatically. In this case, it is easy to construct infinitely many static solutions (at least in terms of quadratures), and it is straightforward to formulate the problem of stability with respect to linearized spherical perturbations. Thus, one knows that, in this setting, a static ~lution may be specified as an arbitrary function W = fo(E, j2, m), where m is th ~. rest mass of the, star (one is thus allowing for a nontrivial distribution of massesl), and E and j2 2enote respectivdy the conserved energy and squared angular momentum associated with the time translational and rotational symmetries. If, in particular, f0 is a function only of E and m, this static solution corresponds at each point in space to an isotropic distribution of three-velocities, as measured in a local orthonormal frame. The key physical question is whether any given f0 is linearly stable with respect, first of all, to petlurbations which preserve the spherical symmetry. One is of course also interested in perturbations which destroy the spherical symmetry. However, it is believed (of. refs. [4, 5]) that it is through some spherically symmetric perturbation that a system may become unstable towards a catastrophic collapse to form a massive black hole, a possibility that has been confirmed numerically [6]. In the Newtonian case, isotropic equilibria fo(E, m) are especially convenient to consider in that it is also straightforward to study questions related to linear stability towards nonspherical perturbations [1, 7] and even the question of nonlinear stability [8]. In the corresponding relativistic theory, this restriction again facilitates a consideration of nonspherical perturbations (cf. ref. [9]), although the analysis is now complicated by the possibility of gravitational radiation. The problem of nonlinear stability is even more difficult. No one has as yet formulated the nonlinear perturbation equations in a useful form, and, except for the special case of spherical perturbations, one does not know how to construct a nonlinear generalization of the energy of a linearized perturbation (this entailing the construction of a relativistic analogue of an energy-Casimir [8, 10]). The question of linear stability towards spherically symmetric perturbations can be translated into a well-defined mathematical problem for the special case in which the derivative F e -- 0.1'o/OE is everywhere negative. Let f denote the linearized perturbation, and suppose that it is decomposed into two components, f÷ and f_, respectively even and odd under an inversion of spatial momentum. In this case, both Newtonianly and relativistically, f_ may be shown to satisfy a linear operator equation of the form [4, 11] O~f_ = - i f f _ ,
(1)
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
215
where 0 t denotes a coordinate time derivative, and 3" is a time-independent linear operator, symmetric with respect to an inner product (~, 7/). It then follows that Off_, and hence f_, remains bounded and well-behaved at all later times if and only if the functional W = ½(~, 3.~)
(2)
is strictly positive for all odd test functions ~:~ 0 . And, by combining this with the linear equation Orf+ = - ~ f _
,
(3)
with ~ time independent and antisymmetric, relating f_ and f+, one knows that the positivity of W is both necessary and sufficient for linear stability. One can in fact prove [1] that 3" is a positive operator in the Newtonian theory, so that all f0's with F E < 0 are linearly stable with respect to spherical perturbations. And, by extending that argument somewhat, one can also show [7] that, if fo is a function only of E and m, it will also be stable to linear perturbations that destroy the spherical symmetry. By contrast, one knows through the construction of suitable test functions (cf. ref. [12]) that, in a relativistic setting, W can in fact be negative, so that a spherical solution to the Vlasov-Einstein system, even with F E negative, need not be linearly stable. Unfortunately, though, the consideration of test functions is both tedious and inelegant, so that one might seek an alternative approach for diagnosing stability or instability. One such approach entails a consideration of binding energy curves for sequences of equilibria, which allow the formulation of a sufficient criterion for stability (and thus a necessary criterion for instability). The key points [13-15] are (a) that one can identify a natural energy e for a linearized perturbation f, (b) that the positivity of this energy for all nonvanishing f is a sufficient criterion for stability, and (c) that the sign of the energy of one of the linearized modes must change at any turning point in the binding energy curve. This implies that if the starting point of the sequence of equilibria is known to be stable (as it may be if it is essentially Newtonian), all subsequent configurations up to the first turning point are also guaranteed to be stable. In the Newtonian case, e can be understood as the quadratic form ½ D X H c ' ( f ) 2 constructed from the Hessian of an appropriate energy-Casimir H c. In relativity, this is also known to be true, at least when one restricts attention to perturbations that preserve the spherical symmetry. It is, however, important to recognize that the existence of modes with e < 0 need not a priori imply a linear instability. Thus, e.g., one knows [16, 17] that
216
H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
many linearly stable Newtonian configurations admit modes with e < 0. The explanation thereof [8] resides in the fact that these negative energy modes f< in the zero-frequency subspace of perturbations, so that ,9,f< --0. Thus. in particular, one 0bserves that the quadratic form W, acting on a function f_, agrees with e evalUated for the function Otf_, i.e., W[f_] = e[Otf_], so that linear instability requires a time-dependent negative energy mode with f_ f 0 . It has long been believed that, in a relativistic theory, the negative energy modes will acquire a nontrivial time dependence and manifest a linear instability. However, despite a great deal of work, and the absence of any counterexamples, this conjecture remains unproved. These subtleties associated with zero frequency modes suggest the desirability of formulating the problem of stability in yet another way which does not, e.g., cast the perturbation equation for f into a form that is second order in the time derivative #,. Such is the object here. Specifically, the aim is to construct a suitable relativistic "bracket" [A, B] in terms of which the relativistic perturbation equations for f and the "perturbed gravitational force" 0 take the forms # i f = [f, ~l and
0,$ = [0, e].
(4)
The key point is that, provided that the bracket satisfies appropriate properties, as will be verified here, it defines a Lie algebra, so that the flows determined by eq. (4) are symplectic. In more conventional language, this means that the values of f and ~, at any two times are connected by a canonical transformation, so that one has at his or her disposal all the sgphisticated apparatus of geometric mechanics. Section 2 of this paper formulates the basic relativistic equations, introduces the bracket operation, shows that it generates the correct equations of motion, and then verifies that the bracket defines a Lie algebra. Section 3 comments on possible modifications and extensions, and then concludes by observing that, in the Newtonian limit, one recovers the linearized Vlasov-Poisson system in a form where the gravitational field is treated as an independent dynamical variable. In this limit, the bracket reduces essentially to the Morrison [18] or Marsden-Weinstein [19] bracket for the Vlasov-MaxweU system, evaluated for a vanishing magnetic field. 2. Construction of the bracket
Consider now the specifics of the Vlasov-Einstein system, using units in which the speed of light c and the gravitational constant G are set equal to unity. Spacetime indices will be denoted by Greek characters /x, v , . . . , whereas spatial indices will be denoted instead by Latin letters i, j, . . . .
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
217
Given some spacetime d / a n d the associated cotangent bundle T(.a), the covariant distribution function N(x ~, p~) is defined by the demand that the quantity N(x ~, p~)d°Fx d°Fp denote the number of stars in the covariant, constant time, seven-dimensional volume element dWx d°Fp. In a specific coordinate system (x i, t), the volume elements d°//'~ and d°Fp reduce to dOFx * (pTm)(_g)~/2 dax - (pt/m) d3Vx and dOFp _ _ ( _ g ) - l / 2 dap * ( m l p , ) ( _ g ) - l / 2 d3p d i n ,
(5)
where p~" - g~,Vp~ and m 2 ~ -g~,vp p , so that
d~f(xa' Pa ) d °~x d o~p * d~f(xi, t, Pi, m) d3x d3p d m .
(6)
It is then assumed that Jf satisfies the Vlasov-Einstein system, so that p~ ON m ox ~'
1 Og~ OJr 2m Ox ~" P~P~ op--~ = 0 ,
(7)
where the metric g~, is determined self-consistently from the Einstein equation
G~[g] = 8,r f d~rp m-lp~pl3jf .
(8)
Suppose now that the system is spherically symmetric. Following, e.g., Ipser and Thorne [4] (note, however, a difference in sign conventions), one can assume a line element of the Schwarzschild form ds 2 = _ e v dt 2 + e a dr 2 + r2(d0 2 + sin20 d $ 2 ) ,
(9)
where V and A are functions only of r and t. The angular coordinates have an obvious geometric significance, as does the radial coordinate, so chosen that a sphere of "radius" r has surface area 4~rr2 One then verifies that a generic static, spherically symmetric solution corresponds to a static N - fo(E, j2, m) and static metric functions V = v(r) and A = A(r) involving only the radial coordinate, t being interpreted now as the cyclic coordinate associated with the time translation symmetry. Here, of course, the conserved energy E = IPtl satisfies - m 2 = g "E2 + g PiPj.
(10)
218
H.E. Kandrup I Vlasov-Einstein system of relativistic stellar dynamics
To define linearized perturbations away from some static solution, one must introduce a mapping between the cotangent bundles associated with the static and pe~urbed spacetimes. This mapping is not unique. Given, however, the assumption of spefical symmetry, there is a "natural" choice physically (although not necessarily the most elegant mathematically! [20]). Specifically, following Ipser and Thorne [4], define a mapping between the perturbed bundle, with coordinates (x '*, p ' ) and metric g'a, and the static bundle, with 0 coordinates (x", Pa) and metric g,a, by (a) identifying spacetime points, i.e., Xp~ = x fit, and (b) identifying components of momentum as measured in a local F orthonormal tetrad, i.e., P(,~)=P(*), so that, e.g., t,-g"_,,,,~/2)p,, = (_g#)~2p,. Given such an identification, one can then set AC=fo+ f ,
V=v+Sv
and
A=A+SA,
(11)
where f is defined in the static bundle and 8v and 8A in the static spacetime. And, given this decomposition, one can then calculate the linearized equations of motion. Because of the underlying symmetries, there is only one dynamical degree of freedom for the gravitational field, namely 8A, the quantity 8v being determined by a constraint equation. Thus, by implementing the constraint explicitly, and then defining a new function ------ S A / 2 r ,
(12)
one is led to the linearized perturbation equations P_2' / Off + ~ f + p, ~1 + r-~r FEpr=o
(13)
0,~, = 4,rr e ^ f d°l/p m - l p t p r f .
(14)
and
Here 0t denotes a coordinate time derivative O/Ot and FE --Ofo/OE. Given that f and $ live in the static bundle, all indices are raised and lowered with the unperturbed metric, and the volume element d°Fp refers to the unperturbed momentum space. In terms of the operator
~ _ pi m
0 Ox i
1 Ogo~ 0 2m Oxi P~PtJ Opi ,
the unperturbed total time derivative satisfying ~ f o - 0 ,
(15) the operator
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
219
satisfies
m ~ g + 4 xrr e A -~ m F E\(P'P" g~g = -~ ~ f d °t/'p p,pr - -mg
P,P m
'f doll• -°-:,P / Pm
g" (16)
Note that eqs. (13)-(16) have the correct Newtonian limit. In this limit, = O~/Or, where • is the perturbed gravitational potential, and the integral contributions to ~ vanish identically, so that one recovers the equations
O,f + D r - Fep ~ Or = 0
(17)
O,(O~lOr) = -4~r f d3p dm P T ,
t .8)
and
where, in terms of the unperturbed gravitational potential tP0 , i
D -- p 0 m Ox~
m 0~o 0 . Ox~ Op~
(19)
In order to cast eqs. (13) and (14) into a Hamiltonian form, one must first identify an appropriate energy functional, or Hamil.:onian, e and then define a bracket [A, B] which (a) generates a Lie algebra ar.d (b) reproduces correctly the equations of motion in the form
o,f= [f,
and
0,# = [¢, e].
(20)
The "natural" energy functional was constructed by Ipser [15], who showed, translated into a more modern language, that the relativistic analogue of the quadratic form ½OEHc.(f)2 constructed from the natural energy-Casimir for the Vlasov-Poisson system is
,f e= ~
r2 d~r d°/rp _ F e
1 f d3Vx( l + r
d r ) e-A ~2.
(21)
Note that if, for fixed time t, f is viewed as a function of x i, pi, and m, and g, is viewed as a function of x i, one computes the functional derivatives ~e ~-f-
f FE
and
( dr)
~e _ 1 (_g)l/Ee-^ 1 + r ~ ~g, - - 4---~ -~r "
(22)
H.E. Kandrup I Vlasov-Einsteinsystemof relativisticstellardynamics
220
Before defining the functional bracket [A, B], it is useful first to introduce the spatial Poisson bracket
OA OB
{A, B} - ax' ap,
OA OB Opi Ox'"
(23)
One computes immediately that, for any function g, the bracket
{E, g} = -(mlp')@g ,
(24)
so that a Poisson bracket term in the expression [A, B] will yield a desired contribution ~ f in the equation ,af motion for a,f. Nnte also that, for any function x(E, j2, m) and any sphericaUy symmetric g, or, alternatively, for any x(E, m) and an arbitrary g, the bracket
OX { E , g } = {x'g}=aE
m OX p, ~g.
(25)
OE
Given these preliminaries, now define the bracket [A, B] via the formula [A,B]=
d ~ xd~pf0
8f ' 8f
+4 fd "x d~trp
e ^ +P . . Fe(-g) - l / z ( s8f a 8B 80
8A 8B)
80 8f~,
8B + 4rt f d3Vx re~((L ~-ff )(M - - ~ ) - ( M ~ff )(L ~f ))
(26)
where
L g - f dT'pm- 1PiP , Feg and M g - f dT'pm -1PrP"Feg.
(27)
To verify that this bracket implies that eqs. (20) reproduce eqs. (13) and (14) is straightforward if one makes use of eq. (25) and the identities
80(x 'i, t) =8(3)(x,'-x ') 80(Xi, t)
and
(28)
8f(x": p' ) 8f(x i, p~ )
= 8(3)(x ' i - xi)8(3)(p i - pi)6(m' - m). t
H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
221
It remains to check that this bracket actually satisfies the appropriate requirements so as to define a Lie algebra. Indeed, three important results follow immediately. 1) It is obvious that, for any constant c, [cA, B] = [A, cB] = c[A, B], and that [A + B, C] = [A, C] + [B, C] and [A, B + C] = [A, B] + [A, C]. These conditions show that the bracket defines an algebra. 2) It is also obvious that the bracket is antisymmetric, i.e., that [A, B] = - [ B , A]. 3) It is less obvious, but also true, that the bracket satisfies the Jacobi identity [A, [B,
ell
+ [B, [C, All + [C, [A, BI] = 0 .
(29)
This is in fact straightforward to verify. By introducing the two-component vector u i with components f and g,, the bracket can be written in the form [A, B] = f tiT"x d~tPp ~A 0
~Buj,
(30)
where O q is an antisymmetric operation. But one verifies immediately (cf. ref. [18]) that any bracket of this form will satisfy eq. (29) if, as is the case here, ~ o i J / ~ u k -~- O.
Any algebra satisfying conditions 2) and 3) is of course a Lie algebra. And thus, one concludes that the linearized perturbation equations can in fact be east into a Hamiltonian form, the bracket of eq. (26) ensuring that the evolution defined by time translation is symplectic and hence generated via a canonical transformation.
3. Discussion
One obvious question about the equations of motion (20) and the bracket (26) concerns the origin of the inelegant integral contributions reflected in the operator ~, which vanish identically in the Newtonian limit. No analogue of these terms arise when the Vlasov-Maxwell system is formulated in fiat space, even though that is also a relativistic field theory, and they would not be present even if one were to formulate the Vlasov-Maxweil system in a fixed curved spacetime. These contributions arise explicitly as a consequence of the fact that, since the Vlasov equation is determining the underlying spacetime itself, the perturbed and unperturbed configurations are defined in different
222
H.E. Kandrup / Vlasov-Einstein system t f relativistic stellar dynamics
cotangent bundles. Only by constructing a nontrivial, and nonunique, mapping between these bundles can one isolate upon a perturbation f that satisfies an equation like (13) defined in the unperturbed bundle. The specific form of this equation, and hence the operator ~ , reflects the choice of the mapping between the bundles, and, indeed, other prescriptions are known to lead to different equations [20,21]. Thus, e.g., the IsraelKandrup prescription [20] entails identifying spacetime points, i.e., setting x ' ~ = x ~, and then identifying momenta by a rescaling p ' " = o--lp ", with or chosen so as to preserve the mass shell constraints and cross sections of the tangent bundle. This leads to a perturbation equation of the form pt
0
-- 0if + ~f= - ~ m 0p~ (F~ f0),
(31)
where F,, denotes a linearized "gravitational force" generated by the perturbed Einstein. Eq. (31) seems more elegant than eq. (13), but this is perhaps a swindle. Because of the choice of mapping between the bundles, the perturbed field equations become more complicated, so that, e.g., the perturbed component of stress energy 8T/[f]
= f d°t/'p m-'p,p'f
(32)
appropriate for eq. (13) is modified by additional contributions reflecting the fact that, with this alternative identification, p'tp 't ~ ptp t. Another obvious, and as yet unanswered, question is whether this analysis could be extended to facilitate a consideration of perturbations that destroy the spherical symmetry or, alternatively, are not assumed to satisfy the linearized evolution equations. Both these objectives would be difficult to realize. When considering perturbations that destroy the spherical symmetry, additional field degrees of freedom are triggered, so that the perturbed equations become much more complicated, including in particular the possibility of gravitational radiation [9]. Alternatively, at least for the special case of spherically symmetric perturbations of an isotropic equilibrium f0(E, m), one knows how [15] to generate a relativistic analogue of an energy-Casimir Hc, for which e = !2DEltic "(f)2, but one has not yet computed the perturbed evolution equations in anything beyond a linear approximation. Given these comments, one may now conclude by considering the Newtonian limit of the bracket [A, B] and the Hamiltonian e. In this limit, q/= 8~/Or, so that
eN[f, Vt~*,]= ~I f dWx d°/fp -f2 Fe
lfd3 8~r
Vx IV~l 2 "
(33)
H.E. Kandrup / Viasov-Einstein system of relativistic stellar dynamics
And, similarly, noting that [A,
=
f
d%
223
p rFe = m(Of/Op,), one concludes that
fo
Sf '
+ 4,rrf d~,, d~p (_g)_t/z Ofo . ( SA SB 0r
~f 8 v ¢
8A SB ) 8v~
~f
(34) '
where v ~ is the ordinary three-velocity. Here one has introduced a vector notation, proceeding as if the perturbations f and V~ need not be spherically symmetric. And indeed, one verifies [8, 18] that this eN and this bracket reproduce the correct Newtonian equations even for nonspherical perturbations, provided that the equilibrium f0 corresponds to an isotropic distribution of velocities. It should be observed that es is nothing other than the Lynden-BeU and Sanitt [7] energy of a linearized perturbation, with the potential energy reexpressed as a local functional of the field V~ rather than as a nonlocal functional of f. And similarly, the bracket (34) is nothing other than the Mordson [18] or Marsden-Weinstein [19] bracket for the Vlasov-Maxwell system with electric charge e = ira, evaluated for a vanishing magnetic field and linearized about the static f0, as is appropriate when considering a linearized perturbation. If one chooses to reexpress the potential • as a nonlocal functional of f by means of a Green function representation, one is then led to the true Lynden-Bell and Sanitt energy eN[f] =
1 f dT'x dT'p
f2 t
, mr( xi, Pi, m ) m ' f ( x'i, Pi, m ' )
•
(35)
Ix -- X'[
And, given this eN, the linearized Newtonian equations follow simply from the Vlasov-Poisson bracket [A,B]=
f
dV xdT'pfo
~f , ~f
•
(36)
References
[1] H.E. Kandrup and J.-F. Sygnet, Astrophys. J. 298 (1985) 27. [2] J. Barnes, J. Goodman and P. Hut, Astrophys. J. 300 (1986) 112. [3] Ya. B. Zel'dovich and I.D. Novikov, Relativistic Astrophysics, vol. 1 (University of Chicago Press, Chicago, 1971).
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H.E. Kandrup / Vlasov-Einstein system of relativistic stellar dynamics
[4] J.R. lpser and K.S. Thorne, Astrophys. J. 154 (1968) 251. [5] Ya. B. Zerdovich and M.A. Podurets, Astron. Zh. 42 (1965) 963 [trans. Soy. Astron. AJ 9 (1966) 742]. [6] S,L. Shapiro and S.A. Teukolsky, Astrophys. J. 298 (1985) 58. [7] D, LyndewBell and N~ Sanitt, Mon. Not. S. Astr. Soc. 143 (1969) 167. [8] H.E. Kandrup, University of Florida preprint (1989). [9] J.R. Ipser and R. Semenzato, Astrophys. J. 229 (1979) 1098. [10] D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep. 123 (1985) 1. [111 V.A. Antonov, Astron. Zh. 37 (1960) 918 [trans. Sov. Astron. AJ 4 (1960) 859]. it2] J.R. Ipser, in: IAU Symposium, no. 69, A. Hayli, ed. (Reidel, Dordrecht, 1975). [131 H. Poincare, Ann. Math. 7 (1885) 259. [14] J.R. Ipser and G. Horwitz, Astrophys. J. 232 (1979) 863. [15] J.R. Ipser, Astrophys. J. 238 (1980) 1101. [16] V.A. Antonov, Vest. Leningrad. Univ. 7 (1962) 135. [171 J. Katz. Mon. Not. R. Astr. Soc. 190 (1980) 497. [18] P.J. Morrison, in: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, M. Tabor and Y.M. Treve, eds. (American Institute of Physics, New York, 1982). [19] J.E. Marsden and A. Weinstein, Physica D 4 (1982) 394. [201 W. Israel and H.E. Kandrup, Ann. Phys. (New York) 152 (1984) 30. [21l P. Droz-Vincent and R. Hakim, Ann. Inst. Henri Poincare A 9 (1968) 17.