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Gravitational perturbations of spherically symmetric systems. II. Perfect fluid interiors
ANNALS
OF
PHYSICS
Gravitational
88,
343-370 (1974)
Perturbations of Spherically Symmetric Perfect Fluid Interiors* VINCENT
Department
of Mathematics,
University
Systems.
II.
MONCRIEF
of Californiu
at Berkeley,
Berkeley,
California
94720
Received December 14, 1973
Methods developed in a previous paper on perturbations of the Schwarzschild metric are here extended to the treatment of perturbations of perfect fluid stellar models. The perturbations of a perfect fluid sphere are explicitly decomposed into their gauge invariant and gauge dependent parts and a variational principle for the perturbation equations is derived. The Hamiltonian for the perturbations is constructed and a sufficient condition for stability against nonradial, radiative perturbations is derived from it. The stability criterion is applied to two interesting classes of stellar models, polytropic white dwarf models and high-density neutron star cores with pressure proportional to energy density.
1. INTRODUCTION In a recent paper [I] (refered to hereinafter as paper I), the perturbations of the Schwarzschild metric were discussed from a new point of view. The metric perturbations were decomposed into gauge invariant and gauge dependent contributions and a variational integral for the perturbation equations was derived. The ReggeWheeler [2] and Zerilli [3] equations for the odd and even parity perturbations were rederived and shown to have a gauge invariant significance. Finally, the Hamiltonian for the perturbations was constructed and used to discuss the stability properties of the Schwarzschild black hole. The aim of this paper is to extend the methods of paper I to the treatment of the perturbations of a sphere of perfect fluid, an idealized nonrotating stellar model. Our main result is the derivation of a sufficient (but unnecessarily stringent) condition upon such models which ensures their stability against small radiative perturbations. The perturbations of a spherically symmetric, perfect fluid stellar model have already been treated in great detail in a series of papers by Thorne et al. [4-81. Furthermore, Ipser and Detweiler [9] have derived a variational principle for the normal mode solutions of the perturbation equations. Our work differs from theirs * Supported in part by NSF Grant No. GP 31358. Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
343
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VINCENT
MONCRIEF
in several respects. We decompose the metric perturbations into gauge invariant and gauge dependent parts and construct the Hamiltonian for the perturbations. Our main aim is to try to derive a sufficiency condition for stability directly from properties of the Hamiltonian, an idea previously developed by Schuti [IO]. Our starting point is the variational principle for selfgravitating perfect fluids derived by Taub [ll]. We put Taub’s variational integral into Hamiltonian form and take its second variation in order to obtain a variational integral for the perturbations. The perturbations of a static, spherically symmetric system are expanded in the tensor harmonics of Regge and Wheeler [2]. We then transform from the Regge-Wheeler representation of the perturbations to a new representation (introduced in paper I) which is more naturally fitted to the gauge symmetry of the perturbed Einstein equations. The Hamiltonian for the perturbations is expressed in terms of the new variables and used as a basis for the discussion of stability. Stability against odd parity, radiative perturbations is established in general. A condition upon the unperturbed configuration which ensures its stability against the even parity, radiative perturbations is derived and applied to two interesting classes of stellar models. In order to be able to use Taub’s elegant variational formalism we shall restrict our attention to the adiabatic perturbations of isentropic (constant entropy per baryon) stellar models. As Thorne has shown [12], such models are marginally convective: They possess an infinity of possible convective motions which are neutrally stable and which, if excited, do not emit gravitational radiation. Thome derived the properties of these convective motions directly from the perturbation equations and has also given an elementary argument for their occurrence and neutral stability [13]. In our formalism, the convective motions (and also the odd parity differential rotations) obey certain conservation laws (derived by Taub [l 11) related to the occurence of rotation in the fluid. Complementary to (and effectively decoupled from) the convective motions are the pulsations of the fluid which couple to modes of gravitational radiation. In our formalism, these radiative modes are described by the gauge invariant parts of the metric perturbations. They obey equations of motion which are completely independent of any choice of gauge. It is mainly the stability of various stellar models against this class of radiative, gauge invariant perturbations that one would like to establish. Schutz’s idea [lo] is to try to use the perturbation Hamiltonian (which is conserved and gauge invariant) as a Lyapunov function for establishing stability in the linear approximation. To do so one must establish its positivity (if, in fact, it is positive). The positivity of the odd parity Hamiltonian is established in Section 4. The even parity Hamiltonian is constructed in Section 5. Positivity of the kinetic energy is proven in Section 6. Finally the positivity of the potential energy is established for certain classes of stellar models in Section 7. The particular models considered are nearly Newtonian polytropes and high-density neutron star
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cores with pressurep and energy density w related byp = (r - 1)~. For these cases our stability condition depends only upon the value of the polytropic index or upon the value of y, and does not depend upon the value taken for the central density w(0). The analysis given here is not yet complete. There are some technical questions involving the junction conditions on the even parity perturbations that we have not yet studied. While they are not expected to cause any difficulty, one should, nevertheless, regard our conclusions on stability as being somewhat tentative until these details have been worked out. A more interesting (and less routine) problem is the determination of the weakest conditions under which the perturbation Hamiltonian is positive. The particular condition derived in Section 7 is merely the most obvious one ensuring positivity. It seems likely that, with further work, a less restrictive condition could be derived.
2. TAUB'S VARIATIONAL
PRINCIPLE FOR PERFECT FLUIDS
The stress-energy tensor for a perfect fluid is given by TUB = (w +p)u"us
+pg@,
(2.1)
where p is the pressure, w the energy density, P the fluid four-velocity (UV~ = -1) and g,, the metric tensor of space-time. The Einstein field equations for a selfgravitating perfect fluid are 4,
- ikcd
= Es,
(2.2)
where we have chosen units in which 8nG = c = 1 and where we have adopted the sign conventions defined by (2.3) Here and in the following a semicolon (;) shall signify covariant differentiation with respect to the metric g,, , Greek indices shall run from 1 to 4 and Latin indices from 1 to 3, x4 z t being a time coordinate. The Bianchi identities imply T~B;~ = 0,
(2.4)
VW,, + (w +p) U”;, = 0
(2.5)
(w + P) u~;“u*+ P;. + p:,u”u, = 0.
(2.6)
which are equivalent to and
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VINCENT
In addition,
MONCRIEF
we assume the fluid to obey the equation of continuity [nzP];, = 0
(2.7)
P = P(h, 4,
(2.8)
and an equation of state where n is the baryon number density, s the specific entropy, and h the specific enthalpy defined by h = (P + w)/n.
(2.9)
In terms of these variables the second law of thermodynamics
has the form (2.10)
dp = n dh - nTds,
where T is the temperature. From Eqs. (2.9) and (2.10), all thermodynamic variables are determined as functions of h and s provided the equation of state is specified. Equations (2.9) and (2.10) give nTds
=dw-hdn,
which implies that (2.11)
nT@s,, = ZPW,, - hu%,, .
Equation
(2.11) combined with Eqs. (2.5), (2.7), and (2.9) implies that (2.12)
u=s,, = 0.
Thus entropy is conserved along the fluid world lines and there is no heat exchange between the different fluid elements. Taub [Ill analyzes the fluid equations of motion in a comoving system of coordinates for which ui zzz0
(2.13)
2.44= (-g,,)-112.
(2.14)
and In such a system of coordinates Eq. (2.6) takes the form
M-gdl’Yi
- (-g4aY2 Ts,i = -[hgd-gd-1’21,4 ,
(2.15)
where we have used Eqs. (2.10) and (2.12) which implies that s,~ = 0. In case the entropy is constant throughout the fluid interior, Eq. (2.15) reduces to
W(-&1’21,i = -hd--gd-““I
.4
*
(2.16)
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For a static, spherically symmetric configuration one can always introduce a comoving coordinate system in which the field variables are time independent and have g,i = 0. If, in addition, the entropy is constant, Eq. (2.16) gives
v(-idl’%
= 0
(2.17)
or h( -g4#i2
= constant.
(2.18)
Taub shows that condition (2.18) may be imposed, as an additional coordinate condition, upon an arbitrary (not necessarily static) configuration of the fluid. This follows from a consideration of the following class of coordinate transformations:
x** = x*4(x4,xi), x*i
=
x1
(2.19)
?
which give
~~*(-dP2
u*i = 0
(2.20)
= ax4 ax*4 q-g4p*
(2.21)
Thus the new coordinates are also comoving and one can choose the transformation function x*~(+, xi) so that h*(-g&)1/2 is constant throughout the fluid interior. Therefore, we may assume, for an arbitrary fluid configuration, that coordinates may be chosen in which ui zzz0 9
u4 =
(-g&W
(2.22)
and h( -g&p
= 1,
(2.23)
the latter giving a convenient normalization of the time coordinate. We shall call a coordinate system for which (2.22) and (2.23) hold an adapted system of coordinates. Taub now shows that the field equations for a self-gravitating perfect fluid, expressed in an adapted system of coordinates, are derivable from a simple variational principle provided that the entropy s is constant throughout the fluid. The variational integral is taken to be I* =
s
8x (-g)‘/”
[R + 2p],
(2.24)
where p = p(h, s) is expressed purely in terms of g,, through Eq. (2.23) and the restriction s = constant. The (constant) entropy s will be held fixed during the
348
VINCENT MONCRIEF
variation
of I*. Using Eqs. (2.9), (2. lo), (2.22), and (2.23) one can easily show that
6 J d4x pp(-g)lq
= 1 d4x (-tw2
[(w + 14 u4d k,,
+ we sgaBi (2.25)
and thus that Taub’s variational principle provides the correct field equations specialized to an adapted system of coordinates. The restriction to constant’entropy configurations is not an essential one since Schutz [lo] has shown how one may eliminate it. However, Schutz’s principle involves the introduction of a number of additional functions (velocity potential functions) to describe the fluid motion. Rather than increase the number of unknown functions we shall, in this paper, work with Taub’s original principle and study only constant entropy configurations. Thus, as explained in the introduction, we shall deal only with perturbations of marginally convective equilibrium configurations. The use of an adapted system of coordinates for the description of isentropic fluid motions has some interesting consequences. From Eqs. (2.7), (2.16), and (2.23) we find that [nu”( -g)‘/“] *4 = 0 (2.26a) or n( -gyy -g44)-l/2 = f(xi) (2.26b) and (2.27a) [k4i(-g44)-1’21.4 = 0 or, as follows from (2.23), g4d-g44F
(2.27b)
= GW
Thus, in an adapted system of coordinates we have four field functions which are conserved throughout the flow. The three functions C,(xj) are simply related to the,vorticity and thus to the rotation as may be seen by evaluating the vector CfJu= (-g)-l/a
E”Tl”U,,,
(2.28)
in the adapted system of coordinates. 3. HAMILTONIAN
FORMALISM
AND PERTURBATION
METHODS
When the equations of motion for a dynamical system are derivable from a variational principle, one can apply Jacobi’s method of second variation to obtain a new variational principle for the equations of small perturbation of any given exact solution. Taub [I l] has derived such a variational principle for perturbations
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of self-gravitating perfect fluids by taking the second variation of the integral (2.24). Here, we are interested in putting Taub’s variational integral for the perturbations into Hamiltonian form. To do so, we could apply the standard methods for deriving a Hamiltonian from a Lagrangian directly to Taub’s expression for the variational integral. For technical reasons though, we have found it more convenient first to derive the Hamiltonian form for the exact variational integral (2.24), and then to take its second variation. This method was justified in paper I and applied there to derive the vacuum perturbation equations. A. Exact Hamiltonian
Formalism
Since the Lagrangian density Z’, defined by I* = s d4x 2 in Eq. (2.24), differs from the vacuum density sVac G (-g)‘/” R by only a function of the undifferentiated metric, we may use the results of Arnowitt, Deser, and Misner [14] to write immediately the Hamiltonian form of I*. We have I*
=
j-, dhx
[,ii
%
-
NZ4
- Nit@ + 2Nr’l”p
+ j-,d4x [+ - 2[7rijNj
1
- (l/2) n-Ni + Nliy’lz],i],
(3.1)
where we have defined Yij
z
gi5
N
9
3
(-g44y-112,
Ni
g
g4i
-
&2]
n
,
E
yij&i,
t
= x4, (3.2)
and where x4
s
y-1/2[niirrij
*i
=-
-2
-
9
*R,
ij $7 13.
(3.3) (3.4)
In the above formulas, indices are raised and lowered using the spatial metric yi3 and its inverse, a vertical bar signifies covariant differentiation with respect to yu and *R is the curvature scalar constructed from it. The momenta # are related to the second fundamental form Kij induced upon the x4 = t = constant surfaces by the formula .# = +/2[Kii - pykzpZ]. (3.5) Some additional g44 =
-N2
useful relations are + NiNi,
g4i = Ni/(N2),
gij = yij - (NiNj)/(~2),
(3.6)
and (-g)‘/”
= NyVm
(3.7)
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VINCENT
MONCRIEF
We may discard the second integral on the right-hand side of Eq. (3.1) since it may be converted to an integral over the boundary 81;2of the region of integration and thus will not contribute to Hamilton’s equations for the interior of 52. Therefore we define I = I, dJx [&i ??.$ - Ns’E’~ - NiXi
+ 2Ny112p
I
(3.8)
and we shall use I rather than I* as the basis for discussion. The pressure p is here understood to be a given function of g,, determined by the equation of state and the coordinate conditions explained in Section 2. Independent variation of yij , rrii, N, and Ni leads to a system of equations equivalent to the Einstein equations for (the interior of) a self-gravitating perfect fluid of constant entropy expressed in an adapted system of coordinates. In particular, variation of N and Ni leads to the equations X4 = -2N(-g)l/“[(w = -2N( Xi
+ p) u4u4 + pg44]
-g)lj2
T*,
(3.9)
= 2(-g)V2(w + p) u4u4Ni = 2(-g)lP T4i + 2(-g)V
T44Ni,
(3.10)
which are the initial value constraints. In Eqs. (3.9) and (3.10) the functions u4, p, and w are given functions of g,, = -N2 + NiNi determined by the formulas of the previous section. We may regard the constraints as algebraic equations (at each point of an initial surface) for the lapse function N and the shift vector NS . Ordinarily, the lapse and shift functions are undetermined by the initial value equations; they may be specified arbitrarily and any particular choice for them determines the system of coordinates away from the initial surface. Here, however, the choice of coordinates off the initial surface has already been made since Taub’s variational principle produces the Einstein equations (for the fluid interior) specialized to an adapted system of coordinates. B. Static, Spherically Symmetric
Conjigurations
We shall assume the unperturbed system to be static and spherically symmetric. In this case, coordinates may be chosen in which dr2 = -N2dt2
+ exp(2h) dr2 + r2[d02 + sin2 0 d$],
(3.11)
where N and h are functions of r alone. For such systems .#
=
0
(3.12)
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and the independent Einstein equations take the form N,TT - &N,
= NW)
4, + U/2)
e2*(p- Nl,
re-2AN,r = N[l - e-2A(1 - rX,,) + (l/2) r2(p - IV)],
(3.13)
(2/r) e-2Ah,, + (l/r2)(1 - e-““) = w.
Here p and w are known functions of N = (-g&l2 determined by the equation of state, the entropy assumption and the coordinate condition (2.23). The three equations (3.13) for the two unknown functions are consistent by virtue of the Bianchi identities. We shall often use Eqs. (3.13) to simplify expressions arising in the perturbation formalism. C. Variational Principle for the Perturbations
The second variation method for deriving a variational integral for the perturbations of an exact solution was discussed in paper I and will only be briefly reviewed here. One assumes a one parameter family of functions yiiW;
4,
NW; 4,
N&Y
4,
T+(x”; e)
(3.14)
and evaluates Z(e) where Z is given by Eq. (3.8). One then defines J* = (l/2) d2Z(e)/de2 1e=O
(3.15)
and requires g,,(x@; 0) to be an exact solution of the unperturbed field equations (2.2). It follows from the discussion of paper I that J* may be written as J* = J f
I
d4x speu ,
(3.16)
where J is the integral of an expression quadratic in the quantities hii
= %
(x”; e)lGnO,
(3.17)
and
352
VINCENT
MONCRIEF
and where SJ’ is linear in the quantities (Pr,/ae”)(x“; e) jsSO, etc. Independent variation of J* (or J )with respect to hij , pii, N’, and Ni’ leads to a system of equations equivalent to
the perturbed Einstein equations. Taub’s variational s(P; 0) = constant
and
principle
$ s(x”; e)
requires us to take = 0.
(3.19)
f-0
Thus we only consider adiabatic perturbations of a constant entropy exact solution. When the unperturbed solution is static, the second variation of I leads to (3.20) where
=
s
d% Ny-1/2[pi’ppij I
- (1/2)(~“~)~]
+ N’ [-(y’i2
*R)’ + y1i2hw - N-lN’y112(w
+ Ni’[-2p’$
- N-lyllz(w
+ p) ($),l
+ p) yijNi’] (3.21)
+ N7112[(l/2) hi’hup - (l/4) h’p]l + V. The quantities (y112 *R)’ and V are defined by (9P *A)’ = rV”[(l/2)
*fi
- *@jhj,
11 d% N+“[(1/2)
h”“‘&nlilr
+ h,,liJ - h,,V]
(3.22)
and v = -w9
+ hhtJki -
+ h”MW>
+ h”l,hhtJk
(l/2) hhlilt + 3hhji *Rf5 - 2hgkhk, *Rim P + (7/g) w) + h2((5P) P - (13P)
+ j- d8x [2y112Nlr(hhi’lj
+ h”hjklk)]\.
~11 (3.23)
GRAVITATIONAL
In these formulas indices metric yij and its inverse, constructed from yij and respect to yij . In addition,
PERTURBATIONS
353
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are raised and lowered using the unperturbed
spatial
“Rij and *R are the Ricci tensor and curvature scalar
a vertical bar signifies covariant differentiation we have defined h = Yijh..13.
with (3.24)
The unperturbed quantities yij , N, p, and w are related by the exact Einstein equations for a static perfect fluid configuration. In the spherically symmetric case the unperturbed variables may be taken to satisfy Eqs. (3.1 l)-(3.13). In deriving expressions (3.21) and (3.23) from the second variation of (3.8) we have performed a number of integrations by parts and discarded the resulting surface integrals. We have also made use of Eqs. (3.13) to simplify various expressions. Equations (3.20)-(3.23) define a variational principle from which the equations of small perturbation of any static isentropic exact configuration may be derived. Variation of J with respect to N’ and Ni’ leads to -(~l/~
*R)’ + y1j2hw - 2N-lN’y112(w
+ p) $1
s
= 0
(3.25)
and -2pri
- 2N-ly1/2(w
+ p) yiiNj’
= 0,
(3.26)
which are the first variations of the exact constraint equations (3.9) and (3.10) evaluated at a static background solution. Equations (3.25) and (3.26) are simple linear algebraic equations for the perturbed lapse function N’ and shift vector Ni’ within the fluid interior (where (w + p) and (awl+) = (awl+) jSare nonvanishing). Thus, within the fluid interior N’ and Ni’ are determined by the constraints; this is a consequence of the coordinate conditions built into Taub’s variational principle from which J was derived. Outside the fluid, where w = p = 0, N’ and Ni’ remain undetermined. The perturbation functions are subject to certain junction conditions at the boundary surface of the stellar model. These conditions are just the first variations of the well-known exact junction conditions and will be discussed later. D. Conserved Quantities and Gauge Invariance The four conserved functions defined by Eqs. (2.26) and (2.27) have direct analogues in the perturbation problem. The conserved quantities of the linearized theory are simply the first variations of the exact conserved quantities. Thusf’ and Ci’ defined by f’
= a/ae[n( -g)‘/“(
-g44)-1/2]e=0
(3.27)
and
G’ = ~/Mg4j(-g44)-11,=o 595/w2-3
(3.28)
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VINCENT
are conserved. When the unperturbed expression for Ci’ simplifies to
MONCRIEF
metric is static ( gdi = 0, g,,,, = 0) the
Cc’ = (-gJlg&
= N-2Ni’.
Using the constraint Eqs. (3.26) to eliminate Ci’ = -[Ny1/2(w
(3.29)
N,‘, this may be written as
+ p)]-l yircp”jlj .
(3.30)
Since the background is here assumed to be static, it follows that the quantities Vi = -2pi$,
(3.31)
are conserved in the interior of the perturbed fluid (they are constrained to vanish in the exterior). Thus the Vi must commute (at least weakly) with the perturbation Hamiltonian (i.e., the commutator must, at most, be a linear combination of the constraints). It is not difficult to show that conservation of the V$’ is equivalent to (weak) invariance of the Hamiltonian under the transformation with 6hii = ailj + ail, 7
(3.33)
where ai is an arbitrary vector field. But, as explained in paper I, this is just the set of gauge transformations of the hij (i.e., the usual expression for 6hij reduces to the above when the background is static). In the following sections (which extend to the interior problem the computations of paper I) we shall decompose the perturbations hij into gauge invariant and gauge dependent parts. By the above argument, it follows that the gauge dependent contributions must be (at least They become strongly cyclic when the weakly) cyclic in the Hamiltonian. constraints are eliminated (i.e., when they are solved for Ni’ and N’ as functions of hij and pij). The remaining conserved quantityf’ also generates an invariance transformation of the Hamiltonian. This transformation leaves the hii unchanged but induces a change in the pij. In Section 6 we shall show that this is also a gauge transformation and we shall use it to prove the positivity of the kinetic energy. Unfortunately, not all of the conserved quantities commute with one another (although the Vi’ do since they involve only the perturbed momenta). Consequently, it is not possible to choose a set of canonical variables which makes the Hamiltonian cyclic in four different functions simultaneously. By adapting the choice of canonical variables to the invariance transformation (3.33), we shall be able to make the gauge dependent parts of hi? cyclic. This is the most obvious choice and is the one we shall adopt in the following sections. It leads to a considerable simplification
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of the potential energy part of the Hamiltonian but complicates the kinetic energy part. One could use a noncanonical set of field variables, but then the Hamilton equations would have a queer appearance. The complication of the kinetic energy (especially in the even parity case) is not a serious problem since we can prove its positivity by a gauge invariance argument (Section 6). 4. ODD PARITY PERTURBATIONS
We may now apply the methods of the previous sections to obtain a variational principle for perturbations of a static, spherically symmetric perfect fluid configuration. The unperturbed configuration is assumed to obey the equations of Section 3B and the odd parity perturbations are expanded in tensor spherical harmonics. Following Regge and Wheeler [2] we take, for each value of I and m (with I = 2, 3,...; m = -1, -1 + l,..., + I), hii = Mr, Wlh
+ h,(r, o(e,h ,
(4.1)
where e^, and Cz are the tensor harmonics given explicitly by Eqs. (4.2) and (4.3) of paper I. The odd parity shift vector perturbation is again expressed as [N,‘] = h,(r, t) [O, -(l/sin
0) -$- Y,, , sin 0 $ Y,,],
(4.2)
whereas the odd parity N’ vanishes. The notation used here is the same as that of paper I and, as explained there, only the real parts of the perturbations are to be used in the variational integral. The perturbed momenta pij are expanded as
From (4.1) and (4.3) we find that
(4.4)
where $1 = 21(1+ 1) e-“p, ,
j2 = Z(1+ l)(Z -
1)(1 + 2)(29)-l
e”p2)
(4.5)
and in which eAis defined by Eq. (3.11). Thus $I and fiz are the canonical momenta conjugate to h, and h, .
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VINCENT
MONCRIEF
As explained in paper I, a simplification of the perturbation equations results if the perturbed metric is decomposed into its gauge invariant and gauge dependent parts. Therefore, we introduce new perturbation functions k, and k, defined by
and find, as before, that k, is gauge invariant. The momenta conjugate to k, and kz are easily shown to be (4.7) as may be seen by evaluating 0 in terms of the new variables. The complete variational integral for odd parity perturbations may now be obtained by substituting expressions (4. I)-(4.3) into Eqs. (3.20)-(3.23), integrating over angles and reexpressing the result in terms of the functions kI , k, , 7r1 , and 7~~. The result, for any 1 3 2, is J = S dt irn dr [7T1ak,/at + r2 akdat - iq,
(4.8)
where
2Nr2e-A + [ (, - I)(1 + 2)
1[n2- wx771,r
+ I(Z + 1) Irn dr [(1/2r2) Ne-A(l - l)(f + 2)(kJ2] 0
-
I 0 m dr [2h,rr, + Z(I + 1) N-l(w + p) eA(hd2]
(4.9)
and where we have suppressed the labels 1 and m on J, HT , etc., to simplify the notation. Variation of J with respect to ho gives the initial value equation -2Sr, - 21(1 + 1) N-l(w + p) eAho = 0.
(4.10)
If we let r = R designate the stellar boundary, then Eq. (4.10) determines ho in the
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357
interior region (r < R) and requires 7r2 to vanish in the exterior region (I > R). The terms in HT which contain h, are given by s0
Evaluating
O3dr [2h,7r, + l(l + 1) N-l(w + p) e”(h0)2] = C.
(4.11)
C on a solution of Eq. (4.10), we obtain
C = + JoRdr [Ne-A(~2)2/[Z(Z + l)(w + ~$11.
(4.12)
Thus C contributes a positive term to the Hamiltonian HT which is, therefore, a positive function of the perturbations. This property of HT is important for the discussion of stability as we shall show below. The function k, is cyclic so 7~~is a conserved quantity (which must vanish in the vacuum region). 7r2 is, in fact, one component of the conserved vector field given by Eq. (3.31); the other two components belong to the even parity class of perturbations. By varying J with respect to rrl and k, we may derive the equation a2k,/3t2 - r2[(l/r2) Ne-A(Ne-“kl),T],T
+ (l/r2) N2(Z - I)(1 + 2) k, = 0.
(4.13)
The function k, is gauge invariant by construction and therefore satisfies Eq. (4.13) in any gauge. In particular, we may evaluate k, is the Regge-Wheeler (RW) gauge which has h,* = 0, the asterisk signifying the RW gauge. In that case k, = h,* and Eq. (4.13) becomes equivalent to the corresponding one derived by Thorne and Campolattaro [4] for odd parity perturbations treated in the Regge-Wheeler gauge. Thus the Thorne-Campolattaro equation has a gauge invariant significance since k, (but not h,) obeys that equation in any gauge. Energy Conservation
and Stability
As pointed out by Thorne and Campolattaro, odd parity perturbations of the fluid consist purely of differential rotations which do not emit gravitational radiation. Furthermore, odd parity gravitational waves do not induce any pulsations of the fluid since all density and pressure perturbations belong to the even parity class of perturbations. Nevertheless, one can still ask whether the unperturbed space-time is stable against odd parity perturbations. This question is analogous to the one raised initially by Regge and Wheeler with regard to the Schwarzschild black hole except that now, owing to the presence of the fluid, the unperturbed space-time is free of singularities. In this section, however, we shall see that, owing to the positivity of the Hamiltonian for the perturbations, no such instabilities can occur.
358
VINCENT
MONCRIEF
Using Hamilton’s equations we compute the time derivative of the Hamiltonian density XT . The result may be written as
~a%at = 41+ l)(l - l)(l + 2)CU/W NC+k,(k,*t+ 2hJl,, -
(4.14)
In the appendix it is shown that the quantity in brackets is continuous across the stellar boundary and vanishes at r = 0. Consequently, we obtain -dHT dt
=
* dr%
Io
= 41 + l)(l - l)(l + 2)[(1/2r2) Ne-Ak,(k2,t + 2ho)],=, ,
(4.15)
which expresses dH,/dt in terms of a flux through a surface at spatial infinity. In the vacuum region (r > R) Eq. (4.13) reduces to the Regge-Wheeler equation which can be written
a2Q --gg+[ at2
(’ -f”“]
[1(I + 1) - 6M/r] Q = 0,
(4.16)
where we have defined Q = (I/r)k,(l
- 2M/r)
(4.17)
and I* = r + 2M ln[r/(2M)
- I].
(4.18)
The asymptotic solutions (as r + m) of Eq. (4.16) which correspond to purely outgoing radiation are Q = A(t - r*),
r* + co,
(4.19)
where A( ) is some function depending upon the particular solution under consideration. A normal mode solution with frequency w and describing purely outgoing waves at spatial infinity has, therefore, the asymptotic form Q = 6 exp[iw(t - r*)],
r* -+ co,
(4.20)
where ci is a constant. Unstable normal mode solutions are those for which the frequency w has a negative imaginary part; they grow exponentially in time. We wish to show that no unstable normal mode solutions exist which have the outgoing wave behavior as r --t co. From Eq. (4.20) we see that an unstable normal mode solution decays
GRAVITATIONAL
PERTURBATIONS
359
II
exponentially in r* (at constant t) as r* -+ co. Furthermore, (for T > R) give
Hamilton’s
equations
Uk2.t + 2hl = k,t + 2hJ.t = 2(1 - 2M/r)[(l
Consequently, both Q (or k,) and k,,t + 2h, decay exponentially thus, from Eq. (4.15), we have that or
dH,/dt = 0
(4.21)
- 2M/r) k,],, .
as r* --f co and
H, = constant.
(4.22)
But, as we have already shown, HT is a sum of positive terms of which several would be growing exponentially in time for an unstable normal mode solution. Thus the only way (4.22) can be satisfied is to have k, =
=1
--
(4.23)
0.
Thus no unstable normal mode solutions with the outgoing wave boundary condition exist and the unperturbed space-time is stable against small odd parity perturbations.
5. EVEN PARITY PERTURBATIONS
The even parity perturbations are expanded, for each value of I and m (with I = 2, 3,...; m = -1, --I + l,..., + I), as hi9 = h(r, t>C& N’ = -W/2)
+ ezAH2@,t)(fA
+ r2K(r, t)U’Jc
+ r2W, t>(j&
Ho@, 0 Km ,
, (5.1)
Wi’l = VW, t> Yz, , Mr, t> aYd@ ho@,t> aYL,/+l, where f; ,..., f” are the tensor harmonics given by Eqs. (5.4) of paper I. The unperturbed metric, which defines N(r) and h(r), is again written as ds2 = -N2dt2
+ e2Adr2 + r2(de2 + sin2 0 dy2)
(5.2)
and is assumed to satisfy the equations of Section 3B. Only the real parts of the perturbations are to be used in the variational integral. However, since the perturbation equations for h, , Hz, etc., are independent of the harmonic index m [2], we may set m = 0 and take the functions h, , Hz, etc., to be real. In that case the perturbations written above are all real valued. As in paper I and in Section 4 of this paper, we may decompose the perturbed
360
VINCENT
MONCRIEF
metric hij into its gauge invariant and gauge dependent parts. For the even parity problem we introduce new functions q1 , q2 , q3, and q4 defined in terms of h1 , H, , K, and G by transformations (5.8), (5.12), and (5.13) of paper I. The functions q1 and q2 are gauge invariant by construction and the transformation equations are invertible provided only that A = Z(Z+ 1) - 2e-2A(1 + r&) # 0.
(5.3)
In part B of Appendix we show that condition (5.3) is satisfied provided only that the energy density w of the unperturbed configuration obeys
w,, G 0,
O
(5.4)
In the exterior /1 reduces to A = (I - 1)(1 + 2) + 644/r,
(5.5)
which is clearly positive for the radiative modes (I 2 2) under consideration. Momentum functions n1 , 7r2, r3 , and rr4 conjugate to q1 ,..., q4 may again be introduced as in paper I by requiring that
(5.6)
In terms of the new momenta, the two independent, even parity components of the vector density pii” are given by pJj
= -(l/2)
e2%rdsin BY,,
(5.7)
and pejlj = r2[1(Z + l)]-l[(1/2)
7rl,+. + r4/r - v3/r2] sin 0 aY,,/ae.
(5.8)
However, in Section 3D we showed that the p$ are conserved quantities. Consequently r3 and 7~~are conserved. It follows that q3 and q4 must be (at least weakly) cyclic in the Hamiltonian for the perturbations. By this we mean that the variation of the Hamiltonian with respect to q3 and q., must be no more than a linear combination of the constraints which, of course, vanish for any actual motion of the fluid. In the following subsection we shall show how the constraints may be eliminated and the variational problem reduced to one for unconstrained canonical variables. After this is done the quantities q3 and q4 will be strictly cyclic. The resultant simplification is the main advantage of using the functions q1 ,..., q4 rather than the original Regge-Wheeler representation of the perturbations.
GRAVITATIONAL
A. Elimination
PERTURBATIONS
361
11
of the Constraints
In the fluid interior (where w + p # 0 and aw/ap # 0) we may solve the initial value equations (3.25) and (3.26) for N’ and NC’ as functions of pij and hij . The Hamiltonian for the interior perturbation equations (- H,) is given by Eq. (3.21) with the integral extending only over the interior of the unperturbed configuration (0 < r < R). By substituting into HI the expressions for N’ and Ni’ obtained by solving the constraints we obtain a function (- HI*) involving only the canonical variables pi’ and hij . However, it is not immediately evident that the Hamilton equations obtained from HI* are equivalent to those one would get by first deriving them from H, and then eliminating N’ and NC’. To justify the use of HI* we must show that [6HI/6pij]*
= 6[H,*]/6pij
[6H,/6h,i]*
= 6[H,*]/Sh,,
(5.9)
and ,
(5.10)
where the asterisk signifies that N’ and Ni’ have been eliminated from the superscripted quantity. To prove this we note that
~[H,*l/%.+(y) = [~H~/&(y)l* + i J d3x [~H~/~Ns’(x)l*s[Ns’(x)*l/&(Y), S=l
(5.11) where N’ z N4’. However, the quantities initial value constraints and thus obey
SH,/aN,‘(x)
= C,(X)
are just
[G(x)l* = 0 identically.
A similar argument for 6[HI*]/6pij
B. The Hamiltonian
the (5.12)
completes the proof.
for the Perturbations
The Hamiltonian HI* for the interior, even parity perturbations may now be derived by substituting the expansions for hii, pij, N’ and NC’ into (3.21), reexpressing the result in terms of the variables q1 ,..., q4 ; rrl ,..., r4 and eliminating the perturbed lapse and shift functions by means of the initial value constraints. The constraints are given explicitly by
HO= FYw + PN-l @plW{q2 + r3w.&dr4 - 2e-VU + 1) 4-l (ch- 40) + e-2AKW)q4- rq3.A11, HI = NeA[2r2(w + p)]-l h, = -Ne-“[Z(l
(5.13)
r4,
+ 1) r2(w + p)]-l
[(l/2) r2m4,2 + rn4 -
v3],
362
VINCENT
MONCRIEF
and are used to eliminate r-I,, HI , and h, from the interior resulting expression for HI* is HI* = Hc* + JoR dr {Ne-, [(l/2)
(/ “‘1;;1
Hamiltonian.
The
‘t 2) [An1 - PW,,~$
+ e-2nr2(p + W>(T - n2.d2 +
4P
+
w> 41
+
1)
771372
+
+ IoRdr 1WW3[(l/2) -
- 1x1 +
u/w
+ u - 1x1+ 2) 4r2(w1+
p)
=
loR dr
1
21crz
+
[n4
KI
+
1) .rr22]
1
rf 2, e-2A(q, - ql,r)2
l)(l + 2)
(~P/W4l2
1)
4
q1q2lr
where Hc* contains all the terms involving and is given explicitly by Hc*
+
(’ ,,‘y
q12[U -
2r2A
+
2)
(P
- r3w,,1
r2wd2]
(5.14)
13
the conserved quantities
r3 and 7r4
2 - (4/r) e-2A7r4l(l + l)[r(?r, - 7r2,r)
+ UP) e2W + 1) - 4 ?i211~ Nr2eAl(l + 1) 2e-” + JoR dr I 2(1 - 1)(1 + 2) [[ r21(l + 1) *3
2
1
+ r41PleS_anl) n3kArl + ~2(r-4.J] 1 + JoR dr 1 r2(~~pj --2ZKU4 + 41 + 1)
[(l/4) 374.r +
e2An42
n4/r - r3/r212]1.
(5.15)
By working out Hamilton’s equations, one can obtain a pair of coupled wave equations for the gauge invariant functions q1 and q2 . A regular solution in the interior region must be matched smoothly across the r = R surface with a regular exterior solution. As shown in paper I, q2 is constrained to vanish in the exterior region while the function Q = q,/A obeys Zerilli’s equation. We shall postpone, for a subsequent paper, the detailed discussion of junction conditions and boundary conditions.
GRAVITATIONAL
6.
POSITIVITY
PERTURBATIONS OF THE
KINETIC
II
363
ENERGY
Since positivity of the total perturbation Hamiltonian provides a sufficient condition for stability of the type discussed in Section IV, it is of interest to determine whether the even parity Hamiltonian is positive. In this section we present a formal argument which establishes the positivity of the kinetic energy part of the Hamiltonian. To do this we shall show that the kinetic energy is gauge invariant and that it is always possible to find a gauge transformation which transforms the kinetic energy to a manifestly positive form. From the general expression (3.21) for the Hamiltonian, we define the kinetic energy T to be T =
s
- (1/2)(pkJ2]
d3x (N+“[pi&.
+ Ni’[-2piilj
- N-l~l/~(w
+ p) rijlvj’]}, (6.1)
T contains all terms which involve the gravitational momenta pii and the perturbed shift vector Ni’. Using the initial value equations (3.26), one may eliminate Nj’ from T to obtain T* = s
d3x {NY-““[pi’pii
- (1/2)(piJ2]}
‘l+‘E
+ J;, d3x{Nr-““(w + PI-’ pij~jpi”~d,
(6.2)
where V, signifies the fluid interior (r < R) and V, the fluid exterior, and where the asterisk signifies that the constraints have been eliminated (pijlj vanishes on V,>. Hamilton’s equations give ah,,/at = 2Ny-1/2[pij
- (l/2) yijpkk] + N;li + A$,
(6.3)
which can easily be solved for pij in terms of i3hij/at and Ni’ E h,i . Therefore the gauge transformations of the pi9 can be determined from the usual gauge transformations of the metric perturbations, %.v = C,:, + Cv:, ,
(6.4)
where C, is an arbitrary vector field. When the unperturbed metric is static it is easy to show that the spatial components Ci induce no change in pii whereas the time component C, induces the change (6.5)
where (6.6)
364
VINCENT
MONCRIEF
Using the general form of the field equations for a static perfect fluid configuration [derivable from (3.8)]: *GO
_
N-l(Nlij
_
*R = 2w 9
P.i
pNiklk) =
-(W
e-p +P)
= N-‘NIi
0
(6.7a) (6.7b)
9
where *Gij = *Rij
-
(1/2)yij *R,
(6.8)
we may reexpress Eq. (6.5) as ap,U
=
y1/2[-AIij
+
pA,
klk + id(*Rij
- (w +p)
The first order change in T* induced by this transformation 6T*
= I
v,+ v,
d3x [PiW4t
-
~~I&,
y”j)].
(6.9)
is found to be (6.10)
which can be converted to a boundary integral by Gauss’ theorem. The boundary terms will vanish provided the integrand obeys suitable asymptotic conditions and is continuous across the stellar boundary. In that case T* is invariant under transformation (6.5). From Eq. (6.2), it is clear that only the terms quadratic in the trace of pij prevent T* from being manifestly positive. Therefore, if we can, by means of a gauge transformation, transform pii to zero, we shall have established the positivity of T*. From (6.9) we have 6p”, = --y1’2[-2hk,k
+ (3p + w)/l]
(6.11)
and one need only show that, for an arbitrarily chosen 6pk, , Eq. (6.11) is soluble for A = N-lC, . But relative to the standard inner product
(fi 3fz> = j d3x rYf&
>
(6.12)
the operator L 3
-2yiiViVj
+ (3p +
W)
(6.13)
is positive definite and self adjoint on functions which vanish sufficiently rapidly at spatial infinity. Therefore, the elliptic problem defined by equation (6.11) should be soluble for arbitrary 8pkk obeying suitable asymptotic conditions. We hope to treat the technical details of this argument more completely in a subsequent paper by making explicit use of the tensor harmonic expansion. The invariance of the kinetic energy (and thus the total Hamiltonian) under
GRAVITATIONAL
PERTURBATIONS
365
II
transformation (6.5) is closely related to one of the conservation laws discussed in Section 3D. As shown there, the function f’ SE [(-gy
u%z]’
(6.14)
is a conserved quantity. By working out the first variation explicitly and eliminating N’ by means of the initial value equations, one can show that f’ = (N/2) yqw
+p) hii + hijlij - (hi&J”
- hij *lPq.
(6.15)
Consequently, the quantity FA =
s
d”x [(2/N) (lf’]
(6.16)
must commute with the Hamiltonian (and thus with T*) for arbitrarily chosen Il(xi). From the above equations it is not difficult to show that conservation of FA is equivalent to the invariance of T* under transformation (6.5). It is useful to recall that in paper I we found that both the kinetic and potential energies for the vacuum region were positive. Consequently, there is no necessity to transform pi, to zero in the exterior region in order to display the positivity of the total kinetic energy T*. Thus the asymptotic conditions mentioned in the foregoing argument are really unnecessary since one may allow the function A to fall off arbitrarily rapidly in the exterior region. 7. POSITIVITY OF THE POTENTIAL
ENERGY
The potential energy lJ* defined by H” = T* + U*
(7.1)
is quadratic in the functions q1 and q2 (q3 and q4 being cyclic). In terms of new variables Q, and Q2 given by
Q, = 413 (7.2)
the interior (r < R) contribution U,* = I” dr ($)
to U* (-
1 (I - :I::‘=
U,*) is given by
I:’ e-2A [Q2 - (r2ws,/A) Q, - QJ2
0
+ VW
[
(I - l)(Z + 2) r(l Q, - UP) Q2]’
AQ22 a.2 /1 [ + 4r2@ + p)
_ r2(u’ + P> 1 2012
I !’
(7.3)
366
VINCENT
MONCRIEF
where 01is the velocity of sound in the fluid: a2 = (ap/aw), .
(7.4)
The exterior (r > R) contribution to U* was found in paper I to be strictly positive. Therefore, we see from Eq. (7.3) that a sufficient condition for positivity of U* is
r;, SE (1 -
ry’w,a; ‘)
3
3 0,
for all
O
I > 2.
(7.5)
Since, as shown in part B of the Appendix, A obeys (1 = (I - I)(1 + 2) + f(r)
with
f(r)
>, 0
(7.6)
it follows that if (7.5) is satisfied for any Z,it is also satisfied for all larger values of 1. Since condition (7.5) is sufficient for positivity of U* (and, therefore, of H*), one should be able to prove stability of models obeying this condition by an argument analogous to that given in Section 4. In fact, only some technical questions concerning the junction conditions remain to be investigated before we can say conclusively that condition (7.5) is sufficient for stability of a relativistic stellar model. We shall postpone, for a future paper, the detailed discussion of these technical questions. In any case, it is important to remember that H* may be positive under weaker conditions than that given by (7.5). It even seems conceivable that all isentropic stellar models have positive perturbation Hamiltonian. A proof of this conjecture would essentially establish the stability, against radiative perturbations, of all such models. Applications
White dwarf stellar models are often constructed with the assumption polytropic equation of state p = kwv,
of a (7.7)
where y and k are constants. The models are nearly Newtonian so that a nonrelativistic (NR) approximation of the equations of structure is sufficient to determine the details of the unperturbed configurations. These details are determined by the properties of the Lane-Emden functions [ 151. It suffices to evaluate FEin the nonrelativistic approximation where p < w and /1 = (I - 1)(1 + 2). In that case condition (7.5) reduces to Fl --+ FFR = (I - I)(1 + 2) - g-J = (l - 1)U + 2) - ;a;gys
20
for
0 < r < R.
(7.8)
GRAVITATIONAL
PERTURBATIONS
II
367
By introducing the usual dimensionless variables used in deriving the Emden equation [15] one can show that F;NRis in fact independent of both the constant k in the equation of state and of the central density w(O) of the model. Satisfaction of the condition Fv 3 0 (for any given value of I 3 2) depends only upon the value of y. By examining tables of numerical solutions of the Emden equation [16] we have obtained the following results:
FrR > 0
for
I > 3 and
FNR 2 > I 0
for
615 < y 5 413.
615 < y < 2, (7.9)
Spatially
bounded stellar models do not exist for y < (6/5) and the condition
FrR 3 0 always fails (in a neighborhood of the stellar surface) for y > 2. We can, as yet, say nothing about the stability of polytropes with y > 2 and nothing about the quadrupole (I = 2) perturbations of polytropes with y 2 4/3. Neutron star cores are often constructed by assuming an equation of state with
P = (Y - lb
1
(7.10)
Such core solutions must be smoothly fitted to a surrounding envelope (with modified equation of state) to obtain a complete stellar model. Equation (7.10) cannot hold throughout since such models would have infinite size. Here we shall ignore the envelope problem and simply ask whether condition (7.5) can be satisfied throughout the core. By introducing the dimensionless variables x and O(x) defined by I = [w(O)]-112 x, (7.11) w(x) = w(0) O(x), into the relativistic equations of structure and into the expression for Ft , one can show that Ft is independent of the value of the central density w(0). A numerical solution of the structure equations has been given by Gratton and Giannone [17] for the case y = (4/3). By examining their graphical result we find that condition (7.5) is satisfied for all I 3 2 (irrespective of the size of the core). Additional numerical investigation of these models would be of interest. An “asymptotic” solution of the equations of structure for models obeying (7.10) has been found by Misner and Zapolsky [18]. Their solution is
W=m
a (7.12)
with
a = S(y - l)/[(~ + 212- 81.
368
VINCENT MONCRIEF
It is easily shown that condition (7.5) (which has &rG = 1) is satisfied for all values of y in the allowed range: 1 < y G 2. The infinite central density of this solution prevents its being a completely adequate physical model. Nevertheless, our results suggest that condition (7.5) may be satisfied throughout the core for all allowed values of y.
APPENDIX
A.
JUNCTION CONDITIONS FOR ODD PARITY PERTURBATIONS
The junction conditions upon the metric at the stellar boundary are got by requiring continuity of the first and second fundamental forms induced upon the boundary surface. The stellar surface is the boundary of the region of space-time in which the pressure is nonvanishing. The odd parity perturbation of the pressure vanishes so that the boundary surface of the perturbed space-time is still labeled by r = R. The junction conditions upon the unperturbed metric imply continuity of and N,,(r) at Y = R. (4 N(r), 0) Continuity of the first variation of the first fundamental implies continuity of the odd parity functions hdr, t> The second fundamental
and
h,(r, t)
at r = R.
where
(A3)
4 b = (t, 4 d,
N, = (grr)--l/2. of the first variation
of Kab implies continuity
hl,t - ho.r These imply continuity
642)
form of the surfaces r = constant is given by Kab = N + Or:b
Continuity
form across the boundary
and
(A4) of
2h, + h,,, .
645)
of the quantities
k, = hl + (W)[h2,, - (2/r) h21, k, = h, ,
w4
and thus continuity of the energy flux given by Eq. (4.15). To complete the argument of Section 4, we must show that there is no contribution to the energy flux at r = 0 for regular solutions of the odd parity field equations. We consider a normal mode solution of the Thorne-Campolattaro
GRAVITATIONAL
PERTURBATIONS
equation and examine its behavior in the neighborhood regularity conditions for the unperturbed metric: h(O) = 0,
N(0) = 1, we find, near
r
369
II
of
r
= 0. Using the
N,(O) = L(O) = 0
(A7)
- 1)V + 2) = 0.
648)
= 0, that k, obeys: k I.rr - (2/r) k,,
- U/r7 Ml
Assuming a solution of the form k, = k, = and
we obtain:
rn,
(regular solution)
,-(2fO
kl = r(1-Z)
(irregular solution).
The regular solutions give vanishing flux at grounds. B.
W’)
POSITIVITY
WO)
= 0 as one would expect on physical
r
OF (1
The field equations (3.13) for the background spacetime yield the equation r2w = [r(l - e-2A)],T
(JW
which may be used to reexpress A as A = (I - l)(Z + 2) + 3(1 - e-“^) - r2w. Equation
WI
(Bl) has the (regular) solution (1 - e-““) = (l/r) /or r’2w(r’)
dr’
(B3)
so that A = (1 - 1)(1 + 2) + (3/r) Ia’ r’2w(r’) Now, if the energy density obeys w .7 < 0 throughout Jo’
r’2w(r’)
dr’
>
w(r)
I’
r’2 0
dr’ - r2w.
the configuration,
dr’ = w(r)(r3/3).
(B4) we have 035)
Thus II > (I - I)(1 + 2) and is thus positive definite for the radiative modes (I > 2) under consideration. 595/88/2-4
370
VINCENT MONCRIEF ACKNOWLEDGMENT
I am grateful to Professor A. Taub for numerous valuable conversations and suggestions concerning this research.
REFERENCES 1. V. MONCRIEF, Gravitational Perturbations of Spherically Symmetric Systems. I. The Exterior Problem, Ann. Phys. (N.Y.) 88 (1974), 323. 2. T. REGGE AND J. A. WHEELER, Phys. Rev. 108 (1957), 1063. 3. F. ZERILLI, Phys. Rev. Letters 24 (1970), 737. 4. K. THORNE AND A. CAMPOLATTARO, Astrophys. J. 149 (1967), 591. 5. K. THORNE AND R. PRICE, Astrophys. J. 155 (1969), 163. 6. 7. 8. 9.
10. 11. 12. 13. 14.
15.
16. 17. 18.
K. K. K. J.
THORNE, Astrophys. J. 158 (1969), 1. THORNE, Astrophys. J. 158 (1969), 997. THORNE AND A. COMPOLATTARO, Astrophys. J. 159 (1970), 847. IPSER AND S. DETWEILER, A Variational principle and a stability
criterion for the nonradial modes of pulsation of stellar models in general relativity, (Preprint). B. SCHULTZ, Astrophys. J. Suppl. Ser. 24 (1972), 343. A. TAUB, Commun. Math. Phys. 15 (1969), 235. K. THORNE, Astrophys. J. 144 (1966), 201. K. THORNE, Relativistic Stellar Structure and Dynamics, in “High Energy Astrophysics” (C. Dewitt, E. Schatzman and P. Veron, Eds.), Gordon and Breach, New York, 1967. R. ARNOWITT, S. DESER AND C. MISNER, Dynamics of General Relativity, in “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), Chap. 7, Wiley & Sons, New York, 1962. S. WEINBERG, Stellar Equilibrium and Collapse, in “Gravitation and Cosmology,” Chap. 11, Wiley & Sons, New York, 1972. British Association for the Advancement of Science Mathematical Tables, Vol. II, London, 1932. L. GRATTON AND P. GIANNONE, Mem. Sot. Astr. Italy 36 (1965), 445. C. MISNER AND H. ZAPOLSKY, Phys. Rev. Letters 12 (1964), 635.
OF
PHYSICS
Gravitational
88,
343-370 (1974)
Perturbations of Spherically Symmetric Perfect Fluid Interiors* VINCENT
Department
of Mathematics,
University
Systems.
II.
MONCRIEF
of Californiu
at Berkeley,
Berkeley,
California
94720
Received December 14, 1973
Methods developed in a previous paper on perturbations of the Schwarzschild metric are here extended to the treatment of perturbations of perfect fluid stellar models. The perturbations of a perfect fluid sphere are explicitly decomposed into their gauge invariant and gauge dependent parts and a variational principle for the perturbation equations is derived. The Hamiltonian for the perturbations is constructed and a sufficient condition for stability against nonradial, radiative perturbations is derived from it. The stability criterion is applied to two interesting classes of stellar models, polytropic white dwarf models and high-density neutron star cores with pressure proportional to energy density.
1. INTRODUCTION In a recent paper [I] (refered to hereinafter as paper I), the perturbations of the Schwarzschild metric were discussed from a new point of view. The metric perturbations were decomposed into gauge invariant and gauge dependent contributions and a variational integral for the perturbation equations was derived. The ReggeWheeler [2] and Zerilli [3] equations for the odd and even parity perturbations were rederived and shown to have a gauge invariant significance. Finally, the Hamiltonian for the perturbations was constructed and used to discuss the stability properties of the Schwarzschild black hole. The aim of this paper is to extend the methods of paper I to the treatment of the perturbations of a sphere of perfect fluid, an idealized nonrotating stellar model. Our main result is the derivation of a sufficient (but unnecessarily stringent) condition upon such models which ensures their stability against small radiative perturbations. The perturbations of a spherically symmetric, perfect fluid stellar model have already been treated in great detail in a series of papers by Thorne et al. [4-81. Furthermore, Ipser and Detweiler [9] have derived a variational principle for the normal mode solutions of the perturbation equations. Our work differs from theirs * Supported in part by NSF Grant No. GP 31358. Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
343
344
VINCENT
MONCRIEF
in several respects. We decompose the metric perturbations into gauge invariant and gauge dependent parts and construct the Hamiltonian for the perturbations. Our main aim is to try to derive a sufficiency condition for stability directly from properties of the Hamiltonian, an idea previously developed by Schuti [IO]. Our starting point is the variational principle for selfgravitating perfect fluids derived by Taub [ll]. We put Taub’s variational integral into Hamiltonian form and take its second variation in order to obtain a variational integral for the perturbations. The perturbations of a static, spherically symmetric system are expanded in the tensor harmonics of Regge and Wheeler [2]. We then transform from the Regge-Wheeler representation of the perturbations to a new representation (introduced in paper I) which is more naturally fitted to the gauge symmetry of the perturbed Einstein equations. The Hamiltonian for the perturbations is expressed in terms of the new variables and used as a basis for the discussion of stability. Stability against odd parity, radiative perturbations is established in general. A condition upon the unperturbed configuration which ensures its stability against the even parity, radiative perturbations is derived and applied to two interesting classes of stellar models. In order to be able to use Taub’s elegant variational formalism we shall restrict our attention to the adiabatic perturbations of isentropic (constant entropy per baryon) stellar models. As Thorne has shown [12], such models are marginally convective: They possess an infinity of possible convective motions which are neutrally stable and which, if excited, do not emit gravitational radiation. Thome derived the properties of these convective motions directly from the perturbation equations and has also given an elementary argument for their occurrence and neutral stability [13]. In our formalism, the convective motions (and also the odd parity differential rotations) obey certain conservation laws (derived by Taub [l 11) related to the occurence of rotation in the fluid. Complementary to (and effectively decoupled from) the convective motions are the pulsations of the fluid which couple to modes of gravitational radiation. In our formalism, these radiative modes are described by the gauge invariant parts of the metric perturbations. They obey equations of motion which are completely independent of any choice of gauge. It is mainly the stability of various stellar models against this class of radiative, gauge invariant perturbations that one would like to establish. Schutz’s idea [lo] is to try to use the perturbation Hamiltonian (which is conserved and gauge invariant) as a Lyapunov function for establishing stability in the linear approximation. To do so one must establish its positivity (if, in fact, it is positive). The positivity of the odd parity Hamiltonian is established in Section 4. The even parity Hamiltonian is constructed in Section 5. Positivity of the kinetic energy is proven in Section 6. Finally the positivity of the potential energy is established for certain classes of stellar models in Section 7. The particular models considered are nearly Newtonian polytropes and high-density neutron star
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345
cores with pressurep and energy density w related byp = (r - 1)~. For these cases our stability condition depends only upon the value of the polytropic index or upon the value of y, and does not depend upon the value taken for the central density w(0). The analysis given here is not yet complete. There are some technical questions involving the junction conditions on the even parity perturbations that we have not yet studied. While they are not expected to cause any difficulty, one should, nevertheless, regard our conclusions on stability as being somewhat tentative until these details have been worked out. A more interesting (and less routine) problem is the determination of the weakest conditions under which the perturbation Hamiltonian is positive. The particular condition derived in Section 7 is merely the most obvious one ensuring positivity. It seems likely that, with further work, a less restrictive condition could be derived.
2. TAUB'S VARIATIONAL
PRINCIPLE FOR PERFECT FLUIDS
The stress-energy tensor for a perfect fluid is given by TUB = (w +p)u"us
+pg@,
(2.1)
where p is the pressure, w the energy density, P the fluid four-velocity (UV~ = -1) and g,, the metric tensor of space-time. The Einstein field equations for a selfgravitating perfect fluid are 4,
- ikcd
= Es,
(2.2)
where we have chosen units in which 8nG = c = 1 and where we have adopted the sign conventions defined by (2.3) Here and in the following a semicolon (;) shall signify covariant differentiation with respect to the metric g,, , Greek indices shall run from 1 to 4 and Latin indices from 1 to 3, x4 z t being a time coordinate. The Bianchi identities imply T~B;~ = 0,
(2.4)
VW,, + (w +p) U”;, = 0
(2.5)
(w + P) u~;“u*+ P;. + p:,u”u, = 0.
(2.6)
which are equivalent to and
346
VINCENT
In addition,
MONCRIEF
we assume the fluid to obey the equation of continuity [nzP];, = 0
(2.7)
P = P(h, 4,
(2.8)
and an equation of state where n is the baryon number density, s the specific entropy, and h the specific enthalpy defined by h = (P + w)/n.
(2.9)
In terms of these variables the second law of thermodynamics
has the form (2.10)
dp = n dh - nTds,
where T is the temperature. From Eqs. (2.9) and (2.10), all thermodynamic variables are determined as functions of h and s provided the equation of state is specified. Equations (2.9) and (2.10) give nTds
=dw-hdn,
which implies that (2.11)
nT@s,, = ZPW,, - hu%,, .
Equation
(2.11) combined with Eqs. (2.5), (2.7), and (2.9) implies that (2.12)
u=s,, = 0.
Thus entropy is conserved along the fluid world lines and there is no heat exchange between the different fluid elements. Taub [Ill analyzes the fluid equations of motion in a comoving system of coordinates for which ui zzz0
(2.13)
2.44= (-g,,)-112.
(2.14)
and In such a system of coordinates Eq. (2.6) takes the form
M-gdl’Yi
- (-g4aY2 Ts,i = -[hgd-gd-1’21,4 ,
(2.15)
where we have used Eqs. (2.10) and (2.12) which implies that s,~ = 0. In case the entropy is constant throughout the fluid interior, Eq. (2.15) reduces to
W(-&1’21,i = -hd--gd-““I
.4
*
(2.16)
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For a static, spherically symmetric configuration one can always introduce a comoving coordinate system in which the field variables are time independent and have g,i = 0. If, in addition, the entropy is constant, Eq. (2.16) gives
v(-idl’%
= 0
(2.17)
or h( -g4#i2
= constant.
(2.18)
Taub shows that condition (2.18) may be imposed, as an additional coordinate condition, upon an arbitrary (not necessarily static) configuration of the fluid. This follows from a consideration of the following class of coordinate transformations:
x** = x*4(x4,xi), x*i
=
x1
(2.19)
?
which give
~~*(-dP2
u*i = 0
(2.20)
= ax4 ax*4 q-g4p*
(2.21)
Thus the new coordinates are also comoving and one can choose the transformation function x*~(+, xi) so that h*(-g&)1/2 is constant throughout the fluid interior. Therefore, we may assume, for an arbitrary fluid configuration, that coordinates may be chosen in which ui zzz0 9
u4 =
(-g&W
(2.22)
and h( -g&p
= 1,
(2.23)
the latter giving a convenient normalization of the time coordinate. We shall call a coordinate system for which (2.22) and (2.23) hold an adapted system of coordinates. Taub now shows that the field equations for a self-gravitating perfect fluid, expressed in an adapted system of coordinates, are derivable from a simple variational principle provided that the entropy s is constant throughout the fluid. The variational integral is taken to be I* =
s
8x (-g)‘/”
[R + 2p],
(2.24)
where p = p(h, s) is expressed purely in terms of g,, through Eq. (2.23) and the restriction s = constant. The (constant) entropy s will be held fixed during the
348
VINCENT MONCRIEF
variation
of I*. Using Eqs. (2.9), (2. lo), (2.22), and (2.23) one can easily show that
6 J d4x pp(-g)lq
= 1 d4x (-tw2
[(w + 14 u4d k,,
+ we sgaBi (2.25)
and thus that Taub’s variational principle provides the correct field equations specialized to an adapted system of coordinates. The restriction to constant’entropy configurations is not an essential one since Schutz [lo] has shown how one may eliminate it. However, Schutz’s principle involves the introduction of a number of additional functions (velocity potential functions) to describe the fluid motion. Rather than increase the number of unknown functions we shall, in this paper, work with Taub’s original principle and study only constant entropy configurations. Thus, as explained in the introduction, we shall deal only with perturbations of marginally convective equilibrium configurations. The use of an adapted system of coordinates for the description of isentropic fluid motions has some interesting consequences. From Eqs. (2.7), (2.16), and (2.23) we find that [nu”( -g)‘/“] *4 = 0 (2.26a) or n( -gyy -g44)-l/2 = f(xi) (2.26b) and (2.27a) [k4i(-g44)-1’21.4 = 0 or, as follows from (2.23), g4d-g44F
(2.27b)
= GW
Thus, in an adapted system of coordinates we have four field functions which are conserved throughout the flow. The three functions C,(xj) are simply related to the,vorticity and thus to the rotation as may be seen by evaluating the vector CfJu= (-g)-l/a
E”Tl”U,,,
(2.28)
in the adapted system of coordinates. 3. HAMILTONIAN
FORMALISM
AND PERTURBATION
METHODS
When the equations of motion for a dynamical system are derivable from a variational principle, one can apply Jacobi’s method of second variation to obtain a new variational principle for the equations of small perturbation of any given exact solution. Taub [I l] has derived such a variational principle for perturbations
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of self-gravitating perfect fluids by taking the second variation of the integral (2.24). Here, we are interested in putting Taub’s variational integral for the perturbations into Hamiltonian form. To do so, we could apply the standard methods for deriving a Hamiltonian from a Lagrangian directly to Taub’s expression for the variational integral. For technical reasons though, we have found it more convenient first to derive the Hamiltonian form for the exact variational integral (2.24), and then to take its second variation. This method was justified in paper I and applied there to derive the vacuum perturbation equations. A. Exact Hamiltonian
Formalism
Since the Lagrangian density Z’, defined by I* = s d4x 2 in Eq. (2.24), differs from the vacuum density sVac G (-g)‘/” R by only a function of the undifferentiated metric, we may use the results of Arnowitt, Deser, and Misner [14] to write immediately the Hamiltonian form of I*. We have I*
=
j-, dhx
[,ii
%
-
NZ4
- Nit@ + 2Nr’l”p
+ j-,d4x [+ - 2[7rijNj
1
- (l/2) n-Ni + Nliy’lz],i],
(3.1)
where we have defined Yij
z
gi5
N
9
3
(-g44y-112,
Ni
g
g4i
-
&2]
n
,
E
yij&i,
t
= x4, (3.2)
and where x4
s
y-1/2[niirrij
*i
=-
-2
-
9
*R,
ij $7 13.
(3.3) (3.4)
In the above formulas, indices are raised and lowered using the spatial metric yi3 and its inverse, a vertical bar signifies covariant differentiation with respect to yu and *R is the curvature scalar constructed from it. The momenta # are related to the second fundamental form Kij induced upon the x4 = t = constant surfaces by the formula .# = +/2[Kii - pykzpZ]. (3.5) Some additional g44 =
-N2
useful relations are + NiNi,
g4i = Ni/(N2),
gij = yij - (NiNj)/(~2),
(3.6)
and (-g)‘/”
= NyVm
(3.7)
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VINCENT
MONCRIEF
We may discard the second integral on the right-hand side of Eq. (3.1) since it may be converted to an integral over the boundary 81;2of the region of integration and thus will not contribute to Hamilton’s equations for the interior of 52. Therefore we define I = I, dJx [&i ??.$ - Ns’E’~ - NiXi
+ 2Ny112p
I
(3.8)
and we shall use I rather than I* as the basis for discussion. The pressure p is here understood to be a given function of g,, determined by the equation of state and the coordinate conditions explained in Section 2. Independent variation of yij , rrii, N, and Ni leads to a system of equations equivalent to the Einstein equations for (the interior of) a self-gravitating perfect fluid of constant entropy expressed in an adapted system of coordinates. In particular, variation of N and Ni leads to the equations X4 = -2N(-g)l/“[(w = -2N( Xi
+ p) u4u4 + pg44]
-g)lj2
T*,
(3.9)
= 2(-g)V2(w + p) u4u4Ni = 2(-g)lP T4i + 2(-g)V
T44Ni,
(3.10)
which are the initial value constraints. In Eqs. (3.9) and (3.10) the functions u4, p, and w are given functions of g,, = -N2 + NiNi determined by the formulas of the previous section. We may regard the constraints as algebraic equations (at each point of an initial surface) for the lapse function N and the shift vector NS . Ordinarily, the lapse and shift functions are undetermined by the initial value equations; they may be specified arbitrarily and any particular choice for them determines the system of coordinates away from the initial surface. Here, however, the choice of coordinates off the initial surface has already been made since Taub’s variational principle produces the Einstein equations (for the fluid interior) specialized to an adapted system of coordinates. B. Static, Spherically Symmetric
Conjigurations
We shall assume the unperturbed system to be static and spherically symmetric. In this case, coordinates may be chosen in which dr2 = -N2dt2
+ exp(2h) dr2 + r2[d02 + sin2 0 d$],
(3.11)
where N and h are functions of r alone. For such systems .#
=
0
(3.12)
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and the independent Einstein equations take the form N,TT - &N,
= NW)
4, + U/2)
e2*(p- Nl,
re-2AN,r = N[l - e-2A(1 - rX,,) + (l/2) r2(p - IV)],
(3.13)
(2/r) e-2Ah,, + (l/r2)(1 - e-““) = w.
Here p and w are known functions of N = (-g&l2 determined by the equation of state, the entropy assumption and the coordinate condition (2.23). The three equations (3.13) for the two unknown functions are consistent by virtue of the Bianchi identities. We shall often use Eqs. (3.13) to simplify expressions arising in the perturbation formalism. C. Variational Principle for the Perturbations
The second variation method for deriving a variational integral for the perturbations of an exact solution was discussed in paper I and will only be briefly reviewed here. One assumes a one parameter family of functions yiiW;
4,
NW; 4,
N&Y
4,
T+(x”; e)
(3.14)
and evaluates Z(e) where Z is given by Eq. (3.8). One then defines J* = (l/2) d2Z(e)/de2 1e=O
(3.15)
and requires g,,(x@; 0) to be an exact solution of the unperturbed field equations (2.2). It follows from the discussion of paper I that J* may be written as J* = J f
I
d4x speu ,
(3.16)
where J is the integral of an expression quadratic in the quantities hii
= %
(x”; e)lGnO,
(3.17)
and
352
VINCENT
MONCRIEF
and where SJ’ is linear in the quantities (Pr,/ae”)(x“; e) jsSO, etc. Independent variation of J* (or J )with respect to hij , pii, N’, and Ni’ leads to a system of equations equivalent to
the perturbed Einstein equations. Taub’s variational s(P; 0) = constant
and
principle
$ s(x”; e)
requires us to take = 0.
(3.19)
f-0
Thus we only consider adiabatic perturbations of a constant entropy exact solution. When the unperturbed solution is static, the second variation of I leads to (3.20) where
=
s
d% Ny-1/2[pi’ppij I
- (1/2)(~“~)~]
+ N’ [-(y’i2
*R)’ + y1i2hw - N-lN’y112(w
+ Ni’[-2p’$
- N-lyllz(w
+ p) ($),l
+ p) yijNi’] (3.21)
+ N7112[(l/2) hi’hup - (l/4) h’p]l + V. The quantities (y112 *R)’ and V are defined by (9P *A)’ = rV”[(l/2)
*fi
- *@jhj,
11 d% N+“[(1/2)
h”“‘&nlilr
+ h,,liJ - h,,V]
(3.22)
and v = -w9
+ hhtJki -
+ h”MW>
+ h”l,hhtJk
(l/2) hhlilt + 3hhji *Rf5 - 2hgkhk, *Rim P + (7/g) w) + h2((5P) P - (13P)
+ j- d8x [2y112Nlr(hhi’lj
+ h”hjklk)]\.
~11 (3.23)
GRAVITATIONAL
In these formulas indices metric yij and its inverse, constructed from yij and respect to yij . In addition,
PERTURBATIONS
353
II
are raised and lowered using the unperturbed
spatial
“Rij and *R are the Ricci tensor and curvature scalar
a vertical bar signifies covariant differentiation we have defined h = Yijh..13.
with (3.24)
The unperturbed quantities yij , N, p, and w are related by the exact Einstein equations for a static perfect fluid configuration. In the spherically symmetric case the unperturbed variables may be taken to satisfy Eqs. (3.1 l)-(3.13). In deriving expressions (3.21) and (3.23) from the second variation of (3.8) we have performed a number of integrations by parts and discarded the resulting surface integrals. We have also made use of Eqs. (3.13) to simplify various expressions. Equations (3.20)-(3.23) define a variational principle from which the equations of small perturbation of any static isentropic exact configuration may be derived. Variation of J with respect to N’ and Ni’ leads to -(~l/~
*R)’ + y1j2hw - 2N-lN’y112(w
+ p) $1
s
= 0
(3.25)
and -2pri
- 2N-ly1/2(w
+ p) yiiNj’
= 0,
(3.26)
which are the first variations of the exact constraint equations (3.9) and (3.10) evaluated at a static background solution. Equations (3.25) and (3.26) are simple linear algebraic equations for the perturbed lapse function N’ and shift vector Ni’ within the fluid interior (where (w + p) and (awl+) = (awl+) jSare nonvanishing). Thus, within the fluid interior N’ and Ni’ are determined by the constraints; this is a consequence of the coordinate conditions built into Taub’s variational principle from which J was derived. Outside the fluid, where w = p = 0, N’ and Ni’ remain undetermined. The perturbation functions are subject to certain junction conditions at the boundary surface of the stellar model. These conditions are just the first variations of the well-known exact junction conditions and will be discussed later. D. Conserved Quantities and Gauge Invariance The four conserved functions defined by Eqs. (2.26) and (2.27) have direct analogues in the perturbation problem. The conserved quantities of the linearized theory are simply the first variations of the exact conserved quantities. Thusf’ and Ci’ defined by f’
= a/ae[n( -g)‘/“(
-g44)-1/2]e=0
(3.27)
and
G’ = ~/Mg4j(-g44)-11,=o 595/w2-3
(3.28)
354
VINCENT
are conserved. When the unperturbed expression for Ci’ simplifies to
MONCRIEF
metric is static ( gdi = 0, g,,,, = 0) the
Cc’ = (-gJlg&
= N-2Ni’.
Using the constraint Eqs. (3.26) to eliminate Ci’ = -[Ny1/2(w
(3.29)
N,‘, this may be written as
+ p)]-l yircp”jlj .
(3.30)
Since the background is here assumed to be static, it follows that the quantities Vi = -2pi$,
(3.31)
are conserved in the interior of the perturbed fluid (they are constrained to vanish in the exterior). Thus the Vi must commute (at least weakly) with the perturbation Hamiltonian (i.e., the commutator must, at most, be a linear combination of the constraints). It is not difficult to show that conservation of the V$’ is equivalent to (weak) invariance of the Hamiltonian under the transformation with 6hii = ailj + ail, 7
(3.33)
where ai is an arbitrary vector field. But, as explained in paper I, this is just the set of gauge transformations of the hij (i.e., the usual expression for 6hij reduces to the above when the background is static). In the following sections (which extend to the interior problem the computations of paper I) we shall decompose the perturbations hij into gauge invariant and gauge dependent parts. By the above argument, it follows that the gauge dependent contributions must be (at least They become strongly cyclic when the weakly) cyclic in the Hamiltonian. constraints are eliminated (i.e., when they are solved for Ni’ and N’ as functions of hij and pij). The remaining conserved quantityf’ also generates an invariance transformation of the Hamiltonian. This transformation leaves the hii unchanged but induces a change in the pij. In Section 6 we shall show that this is also a gauge transformation and we shall use it to prove the positivity of the kinetic energy. Unfortunately, not all of the conserved quantities commute with one another (although the Vi’ do since they involve only the perturbed momenta). Consequently, it is not possible to choose a set of canonical variables which makes the Hamiltonian cyclic in four different functions simultaneously. By adapting the choice of canonical variables to the invariance transformation (3.33), we shall be able to make the gauge dependent parts of hi? cyclic. This is the most obvious choice and is the one we shall adopt in the following sections. It leads to a considerable simplification
GRAVITATIONAL
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355
of the potential energy part of the Hamiltonian but complicates the kinetic energy part. One could use a noncanonical set of field variables, but then the Hamilton equations would have a queer appearance. The complication of the kinetic energy (especially in the even parity case) is not a serious problem since we can prove its positivity by a gauge invariance argument (Section 6). 4. ODD PARITY PERTURBATIONS
We may now apply the methods of the previous sections to obtain a variational principle for perturbations of a static, spherically symmetric perfect fluid configuration. The unperturbed configuration is assumed to obey the equations of Section 3B and the odd parity perturbations are expanded in tensor spherical harmonics. Following Regge and Wheeler [2] we take, for each value of I and m (with I = 2, 3,...; m = -1, -1 + l,..., + I), hii = Mr, Wlh
+ h,(r, o(e,h ,
(4.1)
where e^, and Cz are the tensor harmonics given explicitly by Eqs. (4.2) and (4.3) of paper I. The odd parity shift vector perturbation is again expressed as [N,‘] = h,(r, t) [O, -(l/sin
0) -$- Y,, , sin 0 $ Y,,],
(4.2)
whereas the odd parity N’ vanishes. The notation used here is the same as that of paper I and, as explained there, only the real parts of the perturbations are to be used in the variational integral. The perturbed momenta pij are expanded as
From (4.1) and (4.3) we find that
(4.4)
where $1 = 21(1+ 1) e-“p, ,
j2 = Z(1+ l)(Z -
1)(1 + 2)(29)-l
e”p2)
(4.5)
and in which eAis defined by Eq. (3.11). Thus $I and fiz are the canonical momenta conjugate to h, and h, .
356
VINCENT
MONCRIEF
As explained in paper I, a simplification of the perturbation equations results if the perturbed metric is decomposed into its gauge invariant and gauge dependent parts. Therefore, we introduce new perturbation functions k, and k, defined by
and find, as before, that k, is gauge invariant. The momenta conjugate to k, and kz are easily shown to be (4.7) as may be seen by evaluating 0 in terms of the new variables. The complete variational integral for odd parity perturbations may now be obtained by substituting expressions (4. I)-(4.3) into Eqs. (3.20)-(3.23), integrating over angles and reexpressing the result in terms of the functions kI , k, , 7r1 , and 7~~. The result, for any 1 3 2, is J = S dt irn dr [7T1ak,/at + r2 akdat - iq,
(4.8)
where
2Nr2e-A + [ (, - I)(1 + 2)
1[n2- wx771,r
+ I(Z + 1) Irn dr [(1/2r2) Ne-A(l - l)(f + 2)(kJ2] 0
-
I 0 m dr [2h,rr, + Z(I + 1) N-l(w + p) eA(hd2]
(4.9)
and where we have suppressed the labels 1 and m on J, HT , etc., to simplify the notation. Variation of J with respect to ho gives the initial value equation -2Sr, - 21(1 + 1) N-l(w + p) eAho = 0.
(4.10)
If we let r = R designate the stellar boundary, then Eq. (4.10) determines ho in the
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357
interior region (r < R) and requires 7r2 to vanish in the exterior region (I > R). The terms in HT which contain h, are given by s0
Evaluating
O3dr [2h,7r, + l(l + 1) N-l(w + p) e”(h0)2] = C.
(4.11)
C on a solution of Eq. (4.10), we obtain
C = + JoRdr [Ne-A(~2)2/[Z(Z + l)(w + ~$11.
(4.12)
Thus C contributes a positive term to the Hamiltonian HT which is, therefore, a positive function of the perturbations. This property of HT is important for the discussion of stability as we shall show below. The function k, is cyclic so 7~~is a conserved quantity (which must vanish in the vacuum region). 7r2 is, in fact, one component of the conserved vector field given by Eq. (3.31); the other two components belong to the even parity class of perturbations. By varying J with respect to rrl and k, we may derive the equation a2k,/3t2 - r2[(l/r2) Ne-A(Ne-“kl),T],T
+ (l/r2) N2(Z - I)(1 + 2) k, = 0.
(4.13)
The function k, is gauge invariant by construction and therefore satisfies Eq. (4.13) in any gauge. In particular, we may evaluate k, is the Regge-Wheeler (RW) gauge which has h,* = 0, the asterisk signifying the RW gauge. In that case k, = h,* and Eq. (4.13) becomes equivalent to the corresponding one derived by Thorne and Campolattaro [4] for odd parity perturbations treated in the Regge-Wheeler gauge. Thus the Thorne-Campolattaro equation has a gauge invariant significance since k, (but not h,) obeys that equation in any gauge. Energy Conservation
and Stability
As pointed out by Thorne and Campolattaro, odd parity perturbations of the fluid consist purely of differential rotations which do not emit gravitational radiation. Furthermore, odd parity gravitational waves do not induce any pulsations of the fluid since all density and pressure perturbations belong to the even parity class of perturbations. Nevertheless, one can still ask whether the unperturbed space-time is stable against odd parity perturbations. This question is analogous to the one raised initially by Regge and Wheeler with regard to the Schwarzschild black hole except that now, owing to the presence of the fluid, the unperturbed space-time is free of singularities. In this section, however, we shall see that, owing to the positivity of the Hamiltonian for the perturbations, no such instabilities can occur.
358
VINCENT
MONCRIEF
Using Hamilton’s equations we compute the time derivative of the Hamiltonian density XT . The result may be written as
~a%at = 41+ l)(l - l)(l + 2)CU/W NC+k,(k,*t+ 2hJl,, -
(4.14)
In the appendix it is shown that the quantity in brackets is continuous across the stellar boundary and vanishes at r = 0. Consequently, we obtain -dHT dt
=
* dr%
Io
= 41 + l)(l - l)(l + 2)[(1/2r2) Ne-Ak,(k2,t + 2ho)],=, ,
(4.15)
which expresses dH,/dt in terms of a flux through a surface at spatial infinity. In the vacuum region (r > R) Eq. (4.13) reduces to the Regge-Wheeler equation which can be written
a2Q --gg+[ at2
(’ -f”“]
[1(I + 1) - 6M/r] Q = 0,
(4.16)
where we have defined Q = (I/r)k,(l
- 2M/r)
(4.17)
and I* = r + 2M ln[r/(2M)
- I].
(4.18)
The asymptotic solutions (as r + m) of Eq. (4.16) which correspond to purely outgoing radiation are Q = A(t - r*),
r* + co,
(4.19)
where A( ) is some function depending upon the particular solution under consideration. A normal mode solution with frequency w and describing purely outgoing waves at spatial infinity has, therefore, the asymptotic form Q = 6 exp[iw(t - r*)],
r* -+ co,
(4.20)
where ci is a constant. Unstable normal mode solutions are those for which the frequency w has a negative imaginary part; they grow exponentially in time. We wish to show that no unstable normal mode solutions exist which have the outgoing wave behavior as r --t co. From Eq. (4.20) we see that an unstable normal mode solution decays
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359
II
exponentially in r* (at constant t) as r* -+ co. Furthermore, (for T > R) give
Hamilton’s
equations
Uk2.t + 2hl = k,t + 2hJ.t = 2(1 - 2M/r)[(l
Consequently, both Q (or k,) and k,,t + 2h, decay exponentially thus, from Eq. (4.15), we have that or
dH,/dt = 0
(4.21)
- 2M/r) k,],, .
as r* --f co and
H, = constant.
(4.22)
But, as we have already shown, HT is a sum of positive terms of which several would be growing exponentially in time for an unstable normal mode solution. Thus the only way (4.22) can be satisfied is to have k, =
=1
--
(4.23)
0.
Thus no unstable normal mode solutions with the outgoing wave boundary condition exist and the unperturbed space-time is stable against small odd parity perturbations.
5. EVEN PARITY PERTURBATIONS
The even parity perturbations are expanded, for each value of I and m (with I = 2, 3,...; m = -1, --I + l,..., + I), as hi9 = h(r, t>C& N’ = -W/2)
+ ezAH2@,t)(fA
+ r2K(r, t)U’Jc
+ r2W, t>(j&
Ho@, 0 Km ,
, (5.1)
Wi’l = VW, t> Yz, , Mr, t> aYd@ ho@,t> aYL,/+l, where f; ,..., f” are the tensor harmonics given by Eqs. (5.4) of paper I. The unperturbed metric, which defines N(r) and h(r), is again written as ds2 = -N2dt2
+ e2Adr2 + r2(de2 + sin2 0 dy2)
(5.2)
and is assumed to satisfy the equations of Section 3B. Only the real parts of the perturbations are to be used in the variational integral. However, since the perturbation equations for h, , Hz, etc., are independent of the harmonic index m [2], we may set m = 0 and take the functions h, , Hz, etc., to be real. In that case the perturbations written above are all real valued. As in paper I and in Section 4 of this paper, we may decompose the perturbed
360
VINCENT
MONCRIEF
metric hij into its gauge invariant and gauge dependent parts. For the even parity problem we introduce new functions q1 , q2 , q3, and q4 defined in terms of h1 , H, , K, and G by transformations (5.8), (5.12), and (5.13) of paper I. The functions q1 and q2 are gauge invariant by construction and the transformation equations are invertible provided only that A = Z(Z+ 1) - 2e-2A(1 + r&) # 0.
(5.3)
In part B of Appendix we show that condition (5.3) is satisfied provided only that the energy density w of the unperturbed configuration obeys
w,, G 0,
O
(5.4)
In the exterior /1 reduces to A = (I - 1)(1 + 2) + 644/r,
(5.5)
which is clearly positive for the radiative modes (I 2 2) under consideration. Momentum functions n1 , 7r2, r3 , and rr4 conjugate to q1 ,..., q4 may again be introduced as in paper I by requiring that
(5.6)
In terms of the new momenta, the two independent, even parity components of the vector density pii” are given by pJj
= -(l/2)
e2%rdsin BY,,
(5.7)
and pejlj = r2[1(Z + l)]-l[(1/2)
7rl,+. + r4/r - v3/r2] sin 0 aY,,/ae.
(5.8)
However, in Section 3D we showed that the p$ are conserved quantities. Consequently r3 and 7~~are conserved. It follows that q3 and q4 must be (at least weakly) cyclic in the Hamiltonian for the perturbations. By this we mean that the variation of the Hamiltonian with respect to q3 and q., must be no more than a linear combination of the constraints which, of course, vanish for any actual motion of the fluid. In the following subsection we shall show how the constraints may be eliminated and the variational problem reduced to one for unconstrained canonical variables. After this is done the quantities q3 and q4 will be strictly cyclic. The resultant simplification is the main advantage of using the functions q1 ,..., q4 rather than the original Regge-Wheeler representation of the perturbations.
GRAVITATIONAL
A. Elimination
PERTURBATIONS
361
11
of the Constraints
In the fluid interior (where w + p # 0 and aw/ap # 0) we may solve the initial value equations (3.25) and (3.26) for N’ and NC’ as functions of pij and hij . The Hamiltonian for the interior perturbation equations (- H,) is given by Eq. (3.21) with the integral extending only over the interior of the unperturbed configuration (0 < r < R). By substituting into HI the expressions for N’ and Ni’ obtained by solving the constraints we obtain a function (- HI*) involving only the canonical variables pi’ and hij . However, it is not immediately evident that the Hamilton equations obtained from HI* are equivalent to those one would get by first deriving them from H, and then eliminating N’ and NC’. To justify the use of HI* we must show that [6HI/6pij]*
= 6[H,*]/6pij
[6H,/6h,i]*
= 6[H,*]/Sh,,
(5.9)
and ,
(5.10)
where the asterisk signifies that N’ and Ni’ have been eliminated from the superscripted quantity. To prove this we note that
~[H,*l/%.+(y) = [~H~/&(y)l* + i J d3x [~H~/~Ns’(x)l*s[Ns’(x)*l/&(Y), S=l
(5.11) where N’ z N4’. However, the quantities initial value constraints and thus obey
SH,/aN,‘(x)
= C,(X)
are just
[G(x)l* = 0 identically.
A similar argument for 6[HI*]/6pij
B. The Hamiltonian
the (5.12)
completes the proof.
for the Perturbations
The Hamiltonian HI* for the interior, even parity perturbations may now be derived by substituting the expansions for hii, pij, N’ and NC’ into (3.21), reexpressing the result in terms of the variables q1 ,..., q4 ; rrl ,..., r4 and eliminating the perturbed lapse and shift functions by means of the initial value constraints. The constraints are given explicitly by
HO= FYw + PN-l @plW{q2 + r3w.&dr4 - 2e-VU + 1) 4-l (ch- 40) + e-2AKW)q4- rq3.A11, HI = NeA[2r2(w + p)]-l h, = -Ne-“[Z(l
(5.13)
r4,
+ 1) r2(w + p)]-l
[(l/2) r2m4,2 + rn4 -
v3],
362
VINCENT
MONCRIEF
and are used to eliminate r-I,, HI , and h, from the interior resulting expression for HI* is HI* = Hc* + JoR dr {Ne-, [(l/2)
(/ “‘1;;1
Hamiltonian.
The
‘t 2) [An1 - PW,,~$
+ e-2nr2(p + W>(T - n2.d2 +
4P
+
w> 41
+
1)
771372
+
+ IoRdr 1WW3[(l/2) -
- 1x1 +
u/w
+ u - 1x1+ 2) 4r2(w1+
p)
=
loR dr
1
21crz
+
[n4
KI
+
1) .rr22]
1
rf 2, e-2A(q, - ql,r)2
l)(l + 2)
(~P/W4l2
1)
4
q1q2lr
where Hc* contains all the terms involving and is given explicitly by Hc*
+
(’ ,,‘y
q12[U -
2r2A
+
2)
(P
- r3w,,1
r2wd2]
(5.14)
13
the conserved quantities
r3 and 7r4
2 - (4/r) e-2A7r4l(l + l)[r(?r, - 7r2,r)
+ UP) e2W + 1) - 4 ?i211~ Nr2eAl(l + 1) 2e-” + JoR dr I 2(1 - 1)(1 + 2) [[ r21(l + 1) *3
2
1
+ r41PleS_anl) n3kArl + ~2(r-4.J] 1 + JoR dr 1 r2(~~pj --2ZKU4 + 41 + 1)
[(l/4) 374.r +
e2An42
n4/r - r3/r212]1.
(5.15)
By working out Hamilton’s equations, one can obtain a pair of coupled wave equations for the gauge invariant functions q1 and q2 . A regular solution in the interior region must be matched smoothly across the r = R surface with a regular exterior solution. As shown in paper I, q2 is constrained to vanish in the exterior region while the function Q = q,/A obeys Zerilli’s equation. We shall postpone, for a subsequent paper, the detailed discussion of junction conditions and boundary conditions.
GRAVITATIONAL
6.
POSITIVITY
PERTURBATIONS OF THE
KINETIC
II
363
ENERGY
Since positivity of the total perturbation Hamiltonian provides a sufficient condition for stability of the type discussed in Section IV, it is of interest to determine whether the even parity Hamiltonian is positive. In this section we present a formal argument which establishes the positivity of the kinetic energy part of the Hamiltonian. To do this we shall show that the kinetic energy is gauge invariant and that it is always possible to find a gauge transformation which transforms the kinetic energy to a manifestly positive form. From the general expression (3.21) for the Hamiltonian, we define the kinetic energy T to be T =
s
- (1/2)(pkJ2]
d3x (N+“[pi&.
+ Ni’[-2piilj
- N-l~l/~(w
+ p) rijlvj’]}, (6.1)
T contains all terms which involve the gravitational momenta pii and the perturbed shift vector Ni’. Using the initial value equations (3.26), one may eliminate Nj’ from T to obtain T* = s
d3x {NY-““[pi’pii
- (1/2)(piJ2]}
‘l+‘E
+ J;, d3x{Nr-““(w + PI-’ pij~jpi”~d,
(6.2)
where V, signifies the fluid interior (r < R) and V, the fluid exterior, and where the asterisk signifies that the constraints have been eliminated (pijlj vanishes on V,>. Hamilton’s equations give ah,,/at = 2Ny-1/2[pij
- (l/2) yijpkk] + N;li + A$,
(6.3)
which can easily be solved for pij in terms of i3hij/at and Ni’ E h,i . Therefore the gauge transformations of the pi9 can be determined from the usual gauge transformations of the metric perturbations, %.v = C,:, + Cv:, ,
(6.4)
where C, is an arbitrary vector field. When the unperturbed metric is static it is easy to show that the spatial components Ci induce no change in pii whereas the time component C, induces the change (6.5)
where (6.6)
364
VINCENT
MONCRIEF
Using the general form of the field equations for a static perfect fluid configuration [derivable from (3.8)]: *GO
_
N-l(Nlij
_
*R = 2w 9
P.i
pNiklk) =
-(W
e-p +P)
= N-‘NIi
0
(6.7a) (6.7b)
9
where *Gij = *Rij
-
(1/2)yij *R,
(6.8)
we may reexpress Eq. (6.5) as ap,U
=
y1/2[-AIij
+
pA,
klk + id(*Rij
- (w +p)
The first order change in T* induced by this transformation 6T*
= I
v,+ v,
d3x [PiW4t
-
~~I&,
y”j)].
(6.9)
is found to be (6.10)
which can be converted to a boundary integral by Gauss’ theorem. The boundary terms will vanish provided the integrand obeys suitable asymptotic conditions and is continuous across the stellar boundary. In that case T* is invariant under transformation (6.5). From Eq. (6.2), it is clear that only the terms quadratic in the trace of pij prevent T* from being manifestly positive. Therefore, if we can, by means of a gauge transformation, transform pii to zero, we shall have established the positivity of T*. From (6.9) we have 6p”, = --y1’2[-2hk,k
+ (3p + w)/l]
(6.11)
and one need only show that, for an arbitrarily chosen 6pk, , Eq. (6.11) is soluble for A = N-lC, . But relative to the standard inner product
(fi 3fz> = j d3x rYf&
>
(6.12)
the operator L 3
-2yiiViVj
+ (3p +
W)
(6.13)
is positive definite and self adjoint on functions which vanish sufficiently rapidly at spatial infinity. Therefore, the elliptic problem defined by equation (6.11) should be soluble for arbitrary 8pkk obeying suitable asymptotic conditions. We hope to treat the technical details of this argument more completely in a subsequent paper by making explicit use of the tensor harmonic expansion. The invariance of the kinetic energy (and thus the total Hamiltonian) under
GRAVITATIONAL
PERTURBATIONS
365
II
transformation (6.5) is closely related to one of the conservation laws discussed in Section 3D. As shown there, the function f’ SE [(-gy
u%z]’
(6.14)
is a conserved quantity. By working out the first variation explicitly and eliminating N’ by means of the initial value equations, one can show that f’ = (N/2) yqw
+p) hii + hijlij - (hi&J”
- hij *lPq.
(6.15)
Consequently, the quantity FA =
s
d”x [(2/N) (lf’]
(6.16)
must commute with the Hamiltonian (and thus with T*) for arbitrarily chosen Il(xi). From the above equations it is not difficult to show that conservation of FA is equivalent to the invariance of T* under transformation (6.5). It is useful to recall that in paper I we found that both the kinetic and potential energies for the vacuum region were positive. Consequently, there is no necessity to transform pi, to zero in the exterior region in order to display the positivity of the total kinetic energy T*. Thus the asymptotic conditions mentioned in the foregoing argument are really unnecessary since one may allow the function A to fall off arbitrarily rapidly in the exterior region. 7. POSITIVITY OF THE POTENTIAL
ENERGY
The potential energy lJ* defined by H” = T* + U*
(7.1)
is quadratic in the functions q1 and q2 (q3 and q4 being cyclic). In terms of new variables Q, and Q2 given by
Q, = 413 (7.2)
the interior (r < R) contribution U,* = I” dr ($)
to U* (-
1 (I - :I::‘=
U,*) is given by
I:’ e-2A [Q2 - (r2ws,/A) Q, - QJ2
0
+ VW
[
(I - l)(Z + 2) r(l Q, - UP) Q2]’
AQ22 a.2 /1 [ + 4r2@ + p)
_ r2(u’ + P> 1 2012
I !’
(7.3)
366
VINCENT
MONCRIEF
where 01is the velocity of sound in the fluid: a2 = (ap/aw), .
(7.4)
The exterior (r > R) contribution to U* was found in paper I to be strictly positive. Therefore, we see from Eq. (7.3) that a sufficient condition for positivity of U* is
r;, SE (1 -
ry’w,a; ‘)
3
3 0,
for all
O
I > 2.
(7.5)
Since, as shown in part B of the Appendix, A obeys (1 = (I - I)(1 + 2) + f(r)
with
f(r)
>, 0
(7.6)
it follows that if (7.5) is satisfied for any Z,it is also satisfied for all larger values of 1. Since condition (7.5) is sufficient for positivity of U* (and, therefore, of H*), one should be able to prove stability of models obeying this condition by an argument analogous to that given in Section 4. In fact, only some technical questions concerning the junction conditions remain to be investigated before we can say conclusively that condition (7.5) is sufficient for stability of a relativistic stellar model. We shall postpone, for a future paper, the detailed discussion of these technical questions. In any case, it is important to remember that H* may be positive under weaker conditions than that given by (7.5). It even seems conceivable that all isentropic stellar models have positive perturbation Hamiltonian. A proof of this conjecture would essentially establish the stability, against radiative perturbations, of all such models. Applications
White dwarf stellar models are often constructed with the assumption polytropic equation of state p = kwv,
of a (7.7)
where y and k are constants. The models are nearly Newtonian so that a nonrelativistic (NR) approximation of the equations of structure is sufficient to determine the details of the unperturbed configurations. These details are determined by the properties of the Lane-Emden functions [ 151. It suffices to evaluate FEin the nonrelativistic approximation where p < w and /1 = (I - 1)(1 + 2). In that case condition (7.5) reduces to Fl --+ FFR = (I - I)(1 + 2) - g-J = (l - 1)U + 2) - ;a;gys
20
for
0 < r < R.
(7.8)
GRAVITATIONAL
PERTURBATIONS
II
367
By introducing the usual dimensionless variables used in deriving the Emden equation [15] one can show that F;NRis in fact independent of both the constant k in the equation of state and of the central density w(O) of the model. Satisfaction of the condition Fv 3 0 (for any given value of I 3 2) depends only upon the value of y. By examining tables of numerical solutions of the Emden equation [16] we have obtained the following results:
FrR > 0
for
I > 3 and
FNR 2 > I 0
for
615 < y 5 413.
615 < y < 2, (7.9)
Spatially
bounded stellar models do not exist for y < (6/5) and the condition
FrR 3 0 always fails (in a neighborhood of the stellar surface) for y > 2. We can, as yet, say nothing about the stability of polytropes with y > 2 and nothing about the quadrupole (I = 2) perturbations of polytropes with y 2 4/3. Neutron star cores are often constructed by assuming an equation of state with
P = (Y - lb
1
(7.10)
Such core solutions must be smoothly fitted to a surrounding envelope (with modified equation of state) to obtain a complete stellar model. Equation (7.10) cannot hold throughout since such models would have infinite size. Here we shall ignore the envelope problem and simply ask whether condition (7.5) can be satisfied throughout the core. By introducing the dimensionless variables x and O(x) defined by I = [w(O)]-112 x, (7.11) w(x) = w(0) O(x), into the relativistic equations of structure and into the expression for Ft , one can show that Ft is independent of the value of the central density w(0). A numerical solution of the structure equations has been given by Gratton and Giannone [17] for the case y = (4/3). By examining their graphical result we find that condition (7.5) is satisfied for all I 3 2 (irrespective of the size of the core). Additional numerical investigation of these models would be of interest. An “asymptotic” solution of the equations of structure for models obeying (7.10) has been found by Misner and Zapolsky [18]. Their solution is
W=m
a (7.12)
with
a = S(y - l)/[(~ + 212- 81.
368
VINCENT MONCRIEF
It is easily shown that condition (7.5) (which has &rG = 1) is satisfied for all values of y in the allowed range: 1 < y G 2. The infinite central density of this solution prevents its being a completely adequate physical model. Nevertheless, our results suggest that condition (7.5) may be satisfied throughout the core for all allowed values of y.
APPENDIX
A.
JUNCTION CONDITIONS FOR ODD PARITY PERTURBATIONS
The junction conditions upon the metric at the stellar boundary are got by requiring continuity of the first and second fundamental forms induced upon the boundary surface. The stellar surface is the boundary of the region of space-time in which the pressure is nonvanishing. The odd parity perturbation of the pressure vanishes so that the boundary surface of the perturbed space-time is still labeled by r = R. The junction conditions upon the unperturbed metric imply continuity of and N,,(r) at Y = R. (4 N(r), 0) Continuity of the first variation of the first fundamental implies continuity of the odd parity functions hdr, t> The second fundamental
and
h,(r, t)
at r = R.
where
(A3)
4 b = (t, 4 d,
N, = (grr)--l/2. of the first variation
of Kab implies continuity
hl,t - ho.r These imply continuity
642)
form of the surfaces r = constant is given by Kab = N + Or:b
Continuity
form across the boundary
and
(A4) of
2h, + h,,, .
645)
of the quantities
k, = hl + (W)[h2,, - (2/r) h21, k, = h, ,
w4
and thus continuity of the energy flux given by Eq. (4.15). To complete the argument of Section 4, we must show that there is no contribution to the energy flux at r = 0 for regular solutions of the odd parity field equations. We consider a normal mode solution of the Thorne-Campolattaro
GRAVITATIONAL
PERTURBATIONS
equation and examine its behavior in the neighborhood regularity conditions for the unperturbed metric: h(O) = 0,
N(0) = 1, we find, near
r
369
II
of
r
= 0. Using the
N,(O) = L(O) = 0
(A7)
- 1)V + 2) = 0.
648)
= 0, that k, obeys: k I.rr - (2/r) k,,
- U/r7 Ml
Assuming a solution of the form k, = k, = and
we obtain:
rn,
(regular solution)
,-(2fO
kl = r(1-Z)
(irregular solution).
The regular solutions give vanishing flux at grounds. B.
W’)
POSITIVITY
WO)
= 0 as one would expect on physical
r
OF (1
The field equations (3.13) for the background spacetime yield the equation r2w = [r(l - e-2A)],T
(JW
which may be used to reexpress A as A = (I - l)(Z + 2) + 3(1 - e-“^) - r2w. Equation
WI
(Bl) has the (regular) solution (1 - e-““) = (l/r) /or r’2w(r’)
dr’
(B3)
so that A = (1 - 1)(1 + 2) + (3/r) Ia’ r’2w(r’) Now, if the energy density obeys w .7 < 0 throughout Jo’
r’2w(r’)
dr’
>
w(r)
I’
r’2 0
dr’ - r2w.
the configuration,
dr’ = w(r)(r3/3).
(B4) we have 035)
Thus II > (I - I)(1 + 2) and is thus positive definite for the radiative modes (I > 2) under consideration. 595/88/2-4
370
VINCENT MONCRIEF ACKNOWLEDGMENT
I am grateful to Professor A. Taub for numerous valuable conversations and suggestions concerning this research.
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10. 11. 12. 13. 14.
15.
16. 17. 18.
K. K. K. J.
THORNE, Astrophys. J. 158 (1969), 1. THORNE, Astrophys. J. 158 (1969), 997. THORNE AND A. COMPOLATTARO, Astrophys. J. 159 (1970), 847. IPSER AND S. DETWEILER, A Variational principle and a stability
criterion for the nonradial modes of pulsation of stellar models in general relativity, (Preprint). B. SCHULTZ, Astrophys. J. Suppl. Ser. 24 (1972), 343. A. TAUB, Commun. Math. Phys. 15 (1969), 235. K. THORNE, Astrophys. J. 144 (1966), 201. K. THORNE, Relativistic Stellar Structure and Dynamics, in “High Energy Astrophysics” (C. Dewitt, E. Schatzman and P. Veron, Eds.), Gordon and Breach, New York, 1967. R. ARNOWITT, S. DESER AND C. MISNER, Dynamics of General Relativity, in “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), Chap. 7, Wiley & Sons, New York, 1962. S. WEINBERG, Stellar Equilibrium and Collapse, in “Gravitation and Cosmology,” Chap. 11, Wiley & Sons, New York, 1972. British Association for the Advancement of Science Mathematical Tables, Vol. II, London, 1932. L. GRATTON AND P. GIANNONE, Mem. Sot. Astr. Italy 36 (1965), 445. C. MISNER AND H. ZAPOLSKY, Phys. Rev. Letters 12 (1964), 635.