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RELATIVISTIC STELLAR STABILITY: AN EMPIRICAL APPROACH

WEI-TOU NI

California Institute of Technology, Pasadena, California

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and by the National Aeronautics and Space Administration [NGR 05-002-256].

Supported in part by the National Science Foundation [GP-27304, GP-28027]

https://ntrs.nasa.gov/search.jsp?R=19730004166 2020-05-20T15:00:22+00:00Z

ABSTRACT

The "PPN formalism" - which encompasses the post-Newtonian

limit of nearly every metric theory of gravity - is used to

analyze stellar stability. This analysis enables one to infer,

for any given gravitation theory, the extent to which post-

Newtonian effects induce instabilities in white dwarfs, in

neutron stars, and in supermassive stars. It also reveals

the extent to which our current empirical knowledge of post-

Newtonian gravity (based on solar-system experiments) actually

guarantees that relativistic instabilities exist. In parti-

cular, it shows that: (i) for "conservative theories of gra-

vity", current solar-system experiments guarantee that the

critical adiabatic index, Pcrit' for the stability of stars

against radial pulsations exceeds the Newtonian value of 4/3:

pcrit = 4/3 + K M/R , K positive and of order unity;

(ii) for "nonconservative theories", current experiments do

not permit any firm conclusion about the sign of rcrit - 4/3;

(iii) in the PPN approximation to every metric theory, the

standard Schwarzschild criterion for convection is valid.

i

I. INTRODUCTION AND SUMMARY

Relativistic corrections to Newtonian gravity should induce dynamical

instabilities in stars with adiabatic indices slightly greater than 4/3.

This fact was first discovered, within the framework of General Relativity

(GR), by Chandrasekhar (1964a,b) and independently by Feynman [unpublished,

but quoted in Fowler (1964)]. More recently Nutku (1969) has shown that

the same type of instability is predicted by the Brans-Dicke theory of

gravity (BDT), but that it is slightly weaker (stars are slightly more

stable) than in GR. If the dynamical relativistic instability actually

exists, as predicted by GR and BDT, then it plays a fundamental role in

white dwarfs, in neutron stars, and in supermassive stars [see e.g. Thorne

(1967) orZel'dovichand Novikov (1971) for a review].

But it is conceivable that neither GR nor BDT is the correct relati-

vistic theory of gravity. If so, might the relativistic instability not

exist? Is it conceivable that relativistic effects would stabilize stars

rather than destabilize them? William A. Fowler has asked this question

of gravitation theorists so often at Caltech, that we have felt compelled

to seek a firm answer. The most firm answer possible is one which relies

heavily on experimental tests of relativistic gravitational effects, while

assuming nothing (or almost nothing) about which relativistic theory of

gravity is correct.

Of course, one cannot work in a complete theoretical vacuum. A minimal

amount of theory is required to link the relativistic instability in stars

to solar-system measurements of perihelion shift, light deflection, radar

time delay, etc. That the amount of theory needed is small, however, one

can see heuristically by noticing that both the perihelion shift and the

1

relativistic instability are caused by a relativistic strengthening of

Newtonian gravitational forces. [Stronger gravity than predicted by Newton

when a star contracts means greater force to pull the star on inward, i.e.

means less stability; stronger gravity than predicted by Newton when a

planet approaches close to the sun (perihelion) means greater force to

"whip" the planet around, and a resultant advance of its perihelion.]

The purpose of this paper is to derive a quantitative measure of the

extent to which solar-system experiments imply the existence of the dynami-

cal relativistic instabilities in stars. The "minimal amount of theory"

to be used in the derivation is the Parametrized Post-Newtonian ("PPN")

Framework of Nordtvedt and Will (Will and Nordtvedt 1972; Will 1971a;

Nordtvedt 1968).

The PPN Framework is a post-Newtonian theory of gravity with adjustable

parameters. In Will's fluid version, it has nine PPN parameters, Y, P a1l

a2, a3, ~11 C2' Aid and 4. The parameter y measured curvature of the space-

geometry; B measures the non-linearity of gravity; a1 , a2 , and a3measure

"preferred-frame" effects; 1' ~2' C3. and % measure the effects resulting

from a breakdown of conservation laws. For theories which have no "preferred-

frame" effects, all a's vanish (Nordtvedt and Will 1972; Will 1971b). For

theories which have conservation laws for energy, momentum, angular momentum,

and center-of-mass motion ("conservative thoeries"), all a's and O's vanish

(Will 1971b). The post-Newtonian limit of every "metric theory of gravity"1

Metric theories of gravity are a wide class of theories including (i)

every theory that satisfies the equivalence principle (laws of physics in

local Lorentz frames the same as in special relativity), and (ii) every

2

theory that the Caltech group has thus far examined and found to be complete,

self-consistent, and in agreement with experiment. See Thorne, Will, and Ni

(1971); Ni (1972) and Will (1972b) for full discussions.

known to us [except Whitehead's theory which is non-viable (Will 1971c] is

a special case of the PPN Framework, corresponding to particular values of

the PPN parameters. Ni (1972) has calculated the values of the parameters

for a variety of theories, including general relativity, the scalar-tensor

theories of Bergmann-Wagoner, Nordtvedt, and Brans-Dicke-Jordan, the

conformally-flat theories of Witrow-Morduch, Littlewood-Bergmann and Nordstr'om,

and the stratified theories of Einstein, Witrow-Morduch, Page-Tupper, Yilmaz,

Papapetrou and Rosen.

Experiments to date have placed the following limits on the PPN para-

meters [see Thorne, Will, and Ni (1971) or Will (1972b) for detailed dis-

cussion; see also Nordtvedt and Will (1972)]:

y = 1.04 ± 0.08 [time delay and deflection experiments exceptthat of Sramek (1972)] (1)

y = 0.88 ± 0.12 [Sramek's (1972) deflection experiment] (2)

= 1.14 +0.2 [perihelion shift plus time delay experiments] (3)-0.3

lt4 - C -'2 a| V 0.4 [Kreuzer (1966, 1968) measurementIC4 - 31 ~l - 2 of m /M ]2 o4)

active passive

3 i1 5 0.05 [Kreuzer (1966, 1968) measurement of ie/mpassIve

]

(5)

a1l < 0.2 [Earth rotation rate experiments (Nordtvedt andWill 1972)] (6)

3

IV2 < 0.03 [Earth-tide measurements (Will 1971b)] (7)

ja 31 < 2 X 10- 5

[perihelion shift observations (Nordtvedtand Will 1972)] (8)

2Kreuzer's (1966) analysis of his data was completely correct, despite a

recent claim to the contrary by Gilvarry and Muller (1972). Gilvarry

and Muller err in making a quadratic fit to Kreuzer's data, rather than

restricting themselves to a linear fit as did Kreuzer. Kreuzer measured

the expansion of his liquid relative to teflon over a wide temperature

range and thereby showed experimentally that the quadratic correction to

the linear behavior must be negligibly small over the small temperature

range used for the experiment. Moreover, the magnitude of the quadratic

term obtained by Gilvarry and Muller using their least-squares fits is

ridiculously large for any but pathological materials. We thank R. H. Dicke

for a helpful discussion of these points.

In this paper it is shown that for conservative theories of gravity

current experimental limits on the PPN parameters - based on perihelion

shift, time delay, and deflection experiments - guarantee the existence of

the dynamical relativistic instability in stars; while for non-conservative

theories the present, experimentally undetermined state of the two PPN

parameters t2 and t4 makes it uncertain whether relativistic effects will

actually stabilize or destabilize stars. In quantitative terms, a non-

rotating spherically symmetric star with adiabatic index

in =) Pn ( p s(9)

4

constant throughout its interior is unstable against adiabatic radial per-

turbations if and only if its radius R and geometrized mass (2M - Schwarzschild

radius) satisfy

r - 4/3 K(2M/R) . (10)

Here K is a constant that depends on the star's structure and upon the PPN

parameters. If K is positive, there is a relativistic instability. If K

is negative, relativity stabilizes the star. In the Newtonian limit K = 0.

In GR and BDT K is positive and of order unity. Values of K for polytropic

gas spheres, as evaluated in §IV of this paper, are tabulated in Tables 1

and 2.

Table 1 lists values of K for polytropic stars in the case of conser-

vative theories of gravity. From the positive signs of the minimum values

of K (column 3), we have the following conclusion: for conservative theories

which are compatible with current solar-system experiments, relativistic

corrections to Newtonian theory will always induce dynamical instabilities.

It is interesting to note that y has a positive contribution to K while P

has a negative contribution; the same is true for the perihelion advance.

This, together with the positivity of K, confirms the heuristic argument

given at the beginning of this section.

Table 2 lists the values of K for the general PPN formalism and for

several particular non-conservative theories. The third column gives mini-

mum values of K corresponding to current experimental limits on the PPN

parameters. If ~2 or ~ (which are undetermined by experiments to date)

were sufficiently negative, then K would be negative. For example, for

the currently viable cases { = 0.76, D = 1.3, 2 = - 0.5, 3 = = }

and {Y = = 1, 2 = 2.2, t3 = t4 = 0} the value of K is negative.

5

Therefore we arrive at the following conclusion due to the lack of experi-

mental information on ~2 and ~4, it is inconclusive whether relativistic

effects will actually stabilize or destabilize stars. From the last three

columns, one may notice that the Vector-Metric theory (Will and Nordtvedt

1972) and the Papapetrou (1954a,b,c) theories have the same K-values as

general relativity, while K-values for the Modified Yilmaz theory (Ni 1972)

are all negative.

Other aspects of dynamical stellar pulsations are also investigated

in this paper. The Schwarzschild criterion is found to hold for the onset

of dynamical instability against non-radial oscillations (convection).

Sufficient conditions for self-adjointness of the linearized pulsation

equations are derived. These conditions together with the condition ~1 = 0

coincide with Will's conditions for the existence of ten post-Newtonian

conserved integrals.

In §II the PPN formalism is summarized, the linearized pulsation

equations are derived, and "preferred-frame terms" (which lead to vibrational-

secular and other Machian-type instabilities) are separated out of the pul-

sation equations and reserved for study in a future paper. Section III

derives a variational principle for dynamical stellar stability. Section

IV derives the post-Newtonian conditions for the onset of a dynamical insta-

bility. Section V derives the Schwarzschild criterion for non-radial insta-

bilities. Concluding remarks are make in §VI. An Appendix treats the

problem of self-adjointness.

Throughout this paper, we follow closely the methods of Chandrasekhar

(1965b), and we use geometrized units. The notations and conventions of

this paper are the same as those of Chandrasekhar (1965b), and Will and

6

Nordtvedt (1972) - unless otherwise specified.

II. PPN FORMALISM AND EQUATIONS OF MOTION

FOR SMALL OSCULLATIONS ABOUT EQUILIBRIUM

In the PPN formalism one describes the response of matter to gravity

by the "local law of energy-momentum conservation"

v . T = (11)

(where T is the stress-energy tensor, and 7 is the covariant derivative

with respect to the PPN metric); and one describes the generation of gravity

by matter in terms of the PPN metric (Will and Nordtvedt 1972):

go = 1 - 2U + 2PU 2 - (27 + 2 + o + t1)$1 + 1 6

- 2[ (3y - 2P + 1 + 2)2 + (1 + + ( +

+ (a1 - - a - °)w2 U + 2 wa w Uoo - (2 3 - al)wa va , (12)

= 1I ~ ~ -1g O (1 + 3 + al

- a2 + tl)Vc+1 + a )W

+ 1 -22)wa U + 12 w Uc

gc = - (1 + 2yU)b .

Here w is the velocity of the chosen coordinate frame relative to the "pre-

ferred-frame" of the Universe (if any); and

p(x', t)U(x,t) = X_ ~,T_ T dx' , (13)

I- -

7

(x', t)v 2 (',t )O2(X )~ = I x- tdx' , (14)

p(x', t)U(x', t)0 2 (x't)=J' =- d' (15)

p(x', t) rl(x', t)%3 (xt) = x- d' , (16)

p(x', t)

~4(x,t) = J dx' x(17)

p(x',t) [v(x',t) * (x x')]2

d(x,t) = 'J ' Ix - X ' 3 dx' , (18)O~~~~~~~~~~

va tJ p(x ',t) va(x',t) (19)V t)= jx (19)

p(x',t) v (x',t) (x - x0,) (xa - x,')W(x, t) = 1 dx' (20)

a I - ~' 13

p(x',t) (xB

- x ')(x - xUC'O (XIt) ~ dx' · (21)Uo~(X' t)= f ~ Ix- x' d

Here p is rest-mass density, p is pressure, and H is specific internal

·~~~~~~~~~~energy all measured in the matter's rest frame, and vaC = dx /dt is the

matter's coordinate velocity.

The equations of hydrodynamics governing a perfect fluid follow from

equations (11), (12) and the form of the stress-energy tensor (Will 1972a):

-(av ) + - (avCV) p ±U + jap[ + (37 - 1)U]

+Pdrt[(5 7 ( 1 (47 + a + Cl) VC ~ 21

U wV]

22+ p v -+[(5-a - 1) 1) a a U w -a (22)

8

1- '2CN -(I + l) P (Ww - v )

1+I p[ (y + 4 + ±a 1 ) vo + (al - 2U3 ) we

]

1I

~w

o

- p a 23XC

- 2 1 a - 1 2 w0 U2

- a U [2xWI

1 1- aW' v+ (C2+=3

- a1 ) w2 ] = 0 (22 cont'd.)

and

(23)

where

C = p(l + v2 + 2U + + p/P)~= p(1 +v + 2U +HR+p/p) (24)

1 2 1 1 30 = I-(a3 + 2y + 2 + v 2 + 1 - 2¢) U + -(1 + t3) IT + -(7+4)P/O (25)

1 2 1 30 = -(y + 1) v + (3y7 - 2 + 1) U + 1+ 7 P/p

*12

p* = p(l + - v + 3yU)

x(x,t) p(x',t) x ' - x' dx'

p(x',t) 0(x',t)'D )= fx -' dx'

(26)

(27)

(28)

(29)

Consider an equilibrium spherically symmetric distribution of matter.

The equation of hydrostatic equilibrium, which follows from (24k), is

di I3 ~ U 1 1 1 2, dU ( ~ dU~[1 + ( - 1) U] p = p[ + 2+ - 1 ) w] - + 2p(dr + d .r (30)

Let the equilibrium configuration be slightly perturbed and describe its

9

+ c2 w xi Po]

P* + 6(pv") = o

perturbation by a Lagrangian displacement of the form

~(x) eiU t (31)

The linearized form of the equations governing the perturbation, as derived

by combining equations (22), (23), and (30), is

+ (37 - 1) U]p-

(3 2 +% 2l +1 2al)0v ] w a +[- l)(,0 au au

ax ax ax ax

-1 + 5 1 v+~~~ ~ %3 ~2(~ w~

=[ ~1 + (57 -1) u]l A p + 2 )p(AU au ~[ + ( C2+ CX Cl

+ -S pa 2 wT w 7 u - 5 AU) p(32)2 PC2ww (ax YE Y 6

and

Ap= p* div (33)

Here and henceforth a, 0, 0, and p are defined by equations (24), (25),

(26), and (27), with all quadratic velocity terms (v terms) omitted; and

Va and Wa are given by definitions (19) and (20) with v,(x') replaced by

a(x'). [New V. and W equal to (l/if) times old Va and W a] The symbol

A denotes the Lagrangian change in the quantity that it qualifies.

Now we must evaluate the Lagrangian changes for various quantities

explicitly in terms of ~. From the definition of p* and equation (33), it

follows that

Ap = - p(div £ + 3AU) , (34)

10

correct to post-Newtonian order. Similarly, the first law of thermodynamics

and the definition of "the adiabatic index" Pr lead to

= (p/p2) Ap , Ap = r 1(p/P) AP (35)

respectively, Therefore,

A¢= 1(37 + 1 + -2 2) AU- - [(3r1 - 1)(7 + O) + 1 + C5)](div ~ + 3yAU) (36)2 C2 - 213) AU 2 p. '143

and

(3y - 2 + 1) AU-2 P (3rly _ 3y + l)(div ~ + 3y7AU) . (37)2 2P 1

Finally, the expressions for AU, AU7 5, AX, and f can be written down from

equations (13), (21), (28), (29) as follows:

AU= vu + p(x') a(x ) lx 1 dx- xO~ Ix~- x'I

p(x') AU(x' )-3y x- l dx' (38)

·6 .+(x xAU Y= r (x- '_-) dx dx'= .. . . c~auo~ 7 + (x)cx') xxV ax Ix- x' I

=+(X x)(x -x x ')-37 ( x ) AU( dx' (39)

V~~~~~~~~

, 7 (IC, 05

Vand

T~~~ x-

x- d 12T b + fv p(X') O(x') c(') e8 x_ l dx'

p( =) ·+(x')~ ~ ~ ' Ix - x'l dx'(41)= v '

11

The last two terms in equations (38), (39), (40), and (41) make up the

Eulerian changes in U, U , X and o, corresponding to their repsective

Lagrangian changes.

Notice that the linearized pulsation equation (32) is not invariant

under rotations. Terms linear in w couple "l-modes" (modes with spherical-

harmonic index f) to (I - 1) and (I + 1) modes; terms quadratic in w couple

1-modes to (I - 2), (e - 1), (e + 1), and (£ + 2) modes; all other terms

are invariant under rotation. The terms linear in w have imaginary coeffi-

cients; therefore they (like viscosity, energy generation, and radiative

transport) contribute to the vibrational-secular stability of the star, but

do not affect its dynamical stability. Terms quadratic in w contribute to

the dynamical stability and couple different angular modes. We will delay

until a later paper all analyses of w-dependent terms ("preferred-frame

terms") - including both the problem of vibrational-secular stability (linear

in w) and preferred-frame influence on dynamical stability (quadratic in

w). Thus, we shall set w = 0 throughout this paper.

III. THE VARIATIONAL PRINCIPLE

Equation (32), when supplemented by the expressions for the various

Lagrangian changes in terms of i, becomes explicitly an equation for ~. As

boundary conditions, we shall demand that Ap = 0 at the surface of the star

(r = R), and that there be no physical singularity at the star's center

(r = 0). Equation (32) together with the boundary conditions then con-

stitutesa characteristic value problem for 2.

If a characteristic value problem is self-adjoint, the orthogonality

relations for its characteristic functions hold and a variational base for

12

determining n can be obtained. The stellar pulsation equations in the post-

Newtonian limits of general relativity and Brans-Dicke theory are self-adjoint

(Chandrasekhar 1956; Nutku 1969) as is the equation for radial oscillations

in the full theory of general relativity (Chandrasekhar 1964b). In the

Appendix, it is shown that the characteristic value problem in the PPN for-

malism is self-adjoint if and only if

al == = 2 = 2 3 =

t4 =

0 (42)

Although, in the general case, the characteristic value problem is not self-

adjoint and the orthogonality relations do not hold, a variational integral

can still be constructed in the following manner:

Take equation (32) with Q replaced bythe characteristic value P(i) for

the i-th normal mode, with ~ replaced by the corresponding characteristic

function (i), and with w-terms deleted. Dot into this equation t(J) the

characteristic function for the j-th mode, and integrate over the interior

of the star. Thereby obtain

[(i)]2 Q(i,j) = S(ij) + R(i,j) (43)

Here Q(\Lj) is expression (A.4) with 1 replaced by i, 2 replaced by j, and

complex conjugations deleted; S (i

) is the symmetric part of the right-hand

side of (A.3), with similar replacements; and R(i,'i) is (A.4) with similar

replacements. Notice that R( i j ) is of post-Newtonian order:

R(i' j) = 0(2) . (44) 3

13

3 By O(n) we mean, in Chandrasekhar's (1965a) language, O(c-n).

From equations (43) and (44), it follows immediately that the standard

orthogonality relation for characteristic functions is valid to Newtonian

order, i.e.

Q(ij) = 0(2) [j(i) / ( ) ] (45)

We assume, without proof, that the characteristic functions f (i ) from a

complete set; and we normalize them to give

Q(I'I) = 1 (46)

(no summation on capital letters).

Let P(I) be a solution which differs from ~(I) by post-Newtonian order

and has norm 1, i.e.

P~(I) (I)P(I) + 0(2) (47)

Q(Pt(I) P ) = 1 (48)

Expand P( ) in terms of t(J):

P (I) (j)= C ~ j j

and from equations (45), (46), (47), and (48), obtain

CIj = 1 + 0(4), (j = I) (49)

CIj = 0(2), (j i I) (50)

By combining equations (43), (45), (49), and (50), we obtain

S( t Pt( )) + R(P ( ), P()) = [( )]2 CIj Cij, Q(jij)

= [B(I)]2 Q(I,J) + 0(4) ; (51)

14

by.combining equations (48), (49), and (50), we obtain

Q(Pt(I) P(i)) = CIj cij, Q(,Ji) = Q(II) + o(4) ; (52)

and by combining equations (51), (42), we finally conclude that

(1)] Q(2 Pt P t) = S( t(I), Pt(,)) + R( Pt), P )) + o(W) * (53)

Therefore we can use this equation, and any functions P( ) that agree with

() only at Newtonian order, to calculate [ (I)]2 to post-Newtonian order.(~)I

The Newtonian proper solutions N i) are one set of such functions.

By suppressing the prefix "P" and superscript labels, by inserting from

Appendix the values of Q, S, and R, and by performing some reductions, we

bring our variational expression (53) into the form

QQ2 =I

rl p[l + (3 - 1) U](div )2 dx + I (div ) 3 '[l+(3,y - l)U]p dxV ~ ~ x a

+ J (3rIy - 3y + 1) pAU div ~ dx + f (2p - 1)AUt 3 p dxV V ax~

I' (3r 1) dx-u _)U dv(3p'l - 37 + 1) p div u dx -P dxV 1x V a

~V -xa V 3xU- 2 vP0 a dx - 2 J .e ^dx*(4

Here

2 1 ,~~Q = JV |It dx +- 2(a2 - [1 + 1) () j p(x') p Ix x , dx dx'

[ t(x) *(x -x )I][t(x' ) *(x -X ) ]

- 2~a - + ) v V p (X ) 3 dx dx'

~+ (5y - 1) TVS PW p(x,) p ( dx dx' + x~~~~~I -' [2

+ (*-) t(x')p(x) p(x') Ix' x dx dx' ,(55)

15

is a positive definite quantity in the post-Newtonian limit since the domi-

nant, Newtonian part, fv q 2

dx, is positive definite. This equation can

2be used for a variational determination of 2.

We shall now analyze the Lagrangian displacement g into normal modes

belonging to different vector spherical harmonics. (Since the pulsation

equation without w-dependent terms is invariant under rotation, this pro-

cedure is justified.) Following the procedure of Chandrasekhar (1961, 1964c)

and Lebovitz (1965), we define:

r= r Ym(,0) , (56)

1 dX(r) I__ (5 )

Q= i(/ + 1)r dr 80 (57)

and

1 dx(r) (

o ' ¢

¢= £(Q + 1) r sin dr (58)

(tr' ,t, t are physical components, not covariant components). After

manipulations similar to those in §IV of Chandrasekhar (1965b), we obtain

2the following expression for the variational determination of :

~~2 ~R dd 12d= p[l + (3y- l)U]-r (- x)

2

dr

r24ir R ~ dr dJr+ 2 IR [l + (37 - l)U]pj(2-- - ' dr d2

22 + 1 I (1 + 20 + 20) (J d - K dr0

+ (2¢ - 1 - 2)

pR [AU(r)] r dr0

) fR d+ (6rF1y - 67 + 2 + 3rlt4 - 5~% + 5

)~RpAU(r) - (* - X) dr .(59)0p dr

16

where=o r p 1s) sf2 '(s)+ dX(S) ds ,

J2 ' P(S.) s 12 + dsj)I0

KI(r) = fR (s )+1r s

( + 1)

and

ŽU(r) 4= [ +

(s) d(s)ldss ds

- r K (r)* dU

+_ 2 r

IV. THE POST-NEWTONIAN CONDITION FOR THE ONSET

OF DYNAMICAL INSTABILITY

Consider the case of radial pulsations, i.e. pulsations

with

I = 0 and X = 0 .

The substitutions

'= rrq and r = r

reduce equation (59) to the form

QQ2 = R p[l + (3y7 - 1) U ] r (dn) + (3r - 4) d (r )dr

0

+ (2~ - 1 - TR p[AU(r)]2

r 2 dr0

+ (6r17 - 67 + 2 + 3Pl k- 3% + 53) J o

0

_ (3rl1 4 - 3-4 + 53) JoR

+ 52 fR0

dU d( r drP *'-r - r dr

dp 3~¥ U~r dr

where

17

(60)

(61)

(62)

(63)

(64)

(65)

pAU(r) dd ( r 3T) dr

R ds VdUAU(r) = 4it R p(s) '(s) ds + dUr (66)- -- s s 2 + 2 dr

r s r

Recall that p and p are the distributions of pressure and density in the

equilibrium configuration in the post-Newtonian approximation, and they

therefore include terms of 0(2).

The condition for marginal stability (instability) follows from equa-

tion (65) by setting Q2 = 0. In the particular case P1 = const. - which

implies r = const. at the point of onset of instability in the Newtonian

limit, i.e. I = const. + 0(2) - the condition for marginal instability (eq.

[65] with n = 0) involves the structure of the equilibrium configuration in

the Newtonian approximation alone. Under these conditions the criterion

for marginal instability becomes P1 = rcrit' where

rcrP t =4 + - (2 - 1 - 2 ) IR [U(r)] dM(r) + 3[2y + 2 + + ]crit 3 3W1 0 o~

R P AU(r) dM(r)

- 3 + dU r (A r) dM(r) J AuIp r dM(r) (67)4 Op dr dr)+ C2 p \/dl0

Here

W - 12 fR pr2 dr , (68)0

dM = 4irpr2dr , (69)

and

R dUAU(r) =_ 4 iR psds + dr . (70)

r

This result agrees with those in general relativity (Chandrasekhar 1965b)

and in Brans-Dicke-Jordan theory (Nutku 1969) when specialized to the cor-

responding PPN values.

18

Criterion (67) for marginal instability may be reformulated as follows.

A dynamical instability will set in if and only if the following inequality

is satisfied:

2M-~~~ K- . ~~~(71)Pl ' Pcr it 5 3 + K (71)I ~crit 3 R

Here M is the mass and R is the radius of the configuration, and K is a

constant (typically of order unity), depending on the Newtonian structure

of the configuration. If K is positive, there is a relativistic instability;

if K is negative, then relativistic effects stabilize the star.

For polytropes, an explicit expression for K can be obtained, from

equations (67)-(71), in terms of Lane-Emden functions:

1 1K= (+ 7) K- (2 - 1) K

2(t3 + 4)(, K1 3N) +

2(K2 ) (72)

where

12(5 - n) + f n t2 dt (73)

18(n + 1) l 4| '4I Th + 11 l'

K (5 - n) I 1 o + e 2 l1l 2 o (74)K2 =18e l4l I 0

.5 -4n ,l 1n+l 3d (75)18(n + 1) tI [011 0

5 - n t Q ndG( 3(76)KS 4-i- J ItI~ t 7+4 1813 · + 11 l' dt (

and where n is the polytropic index, ~ is the first zero of the Lane-Emden

function n', and Q' is the value of the derivative of at 1 Then ~~~~~~~~n

values of the constant K, evaluated with the aid of the foregoing formula

for various values of n, are listed in Table 1 for conservative theories

19

and in Table 2 for the general PPN formalism and for non-conservative theories.

See §I for discussions of these tables and for the conclusions inferred from

these tables.

V. NON-RADIAL OSCILLATIONS AND THE SCHWARZSCHILD

CRITERION FOR CONVECTION

We shall now obtain the condition for the occurrence of a neutral mode

of non-radial oscillation belonging to I - 1 in the general PPN framework.

As in the last two sections, all "preferred-frame" effects (w-dependent

2terms) will be ignored. By setting i = 0 in equation (32), by following

an analysis parallel to §VI of Chandrasekhar (1965b), and by using the

result of §VIII of Chandrasekhar (1965b), one can show that, to 0(2), a

necessary and sufficient condition for the occurrence of a neutral mode of

non-radial oscillation is that

S(r) I+H r3 p p d-= ; (77)''I1r3lI p dp/drJ

i.e. that

S(r) = 0 (78)

over some finite interval of r. Here

s(r) dP - rl p dp (79)dr l p dr

is the "Schwarzschild discriminant", and r3 and r4 are defined by

P3 = 1 + |(log)J ' (80)

and

_= = 1 p (81)P1

4 - 1 P

20

By following a procedure similar to §VII of Chandrasekhar (1965b), one can

also derive this condition from the variational principle (61).

The proportionality of the Newtonian and the post-Newtonian discrimi-

nants implies that the physical condition for convective instability remains

the same in the PPN formalism as in general relativity and in Newtonian

theory. Although for some PPN values, the characteristic value problem is

not self-adjoint, the Schwarzschild criterion is still valid, and no new

dynamical instabilities occur.

VI. CONCLUSIONS

In this paper, stability criteria for stellar pulsations were found

using the general PPN formalism. These criteria are valid for almost all

metric theories of gravity in the post-Newtonian approximation, when one

ignores preferred-frame effects. As in general relativity, so also for

conservative theories (conservative theories do not have preferred-frame

effects), the relativistic corrections do actually induce dynamical insta-

bilities in stars. But in the general case the present experimental uncer-

tainty in the PPN parameters t2' k4 makes it inconclusive

whether relativistic effects will actually stabilize or destabilize stars.

As experimental tests are performed to higher precision, the answer may

become definite. The differences in the dynamical stability criterion for

various theories may affect the evolution of white dwarfs and supermassive

stars; such effects are worth exploration. The relationship of non-self-

adjointness to the non-existence of conservation laws is intriguing and

should be examined further.

A subsequent paper will deal with the problem of Machian instabilities

21

due to preferred-frame effects (w-dependent terms). Those instabilities,

when combined with astronomical observations on white-dwarf pulsations, may

lead to tight experimental limits on the "preferred-frame parameters" a1,

a2, and c.

ACKNOWLEDGMENTS

I would like to thank S. Chandrasekhar for helpful discussions. I

thank B. A. Zimmerman for assistance in the computer programming. Finally

I wish to thank Kip S. Thorne, whose moral support and valuable advice made

this paper possible and whose critical comments about presentation made it

readable.

22

APPENDIX

SELF-ADJOINTNESS OF THE CHARACTERISTIC VALUE PROBLEM

We shall here derive the constraints which the PPN parameters must

satisfy for the characteristic value problem [eq. (32)] to be self-adjoint.

For this prupose we do not delete the w-dependent terms ab initio (cf. end

of §II).

By bringing the right-hand side over to the left, write equation (32)

in the form ~ = 0, where e is a linear operator. This equation is self-

adjoint [or can be made so by multiplication with some weighting function

>(x)] if and only if

() (2) dx = · (2) t(l) dx , (A.1)

V V ~

where the complex conjugation "*" is not permitted to act on the eigenvalue

Q (which is contained in ;). In this equation E(1) and (2) are arbitrary

functions satisfying the boundary conditions at r = 0 and r = R (but not

necessarily satisfying z (1) = 0 or ; (2) = 0); V is the interior of the

star; "*" denotes complex conjugation; and dx denotes dx dy dz = dx1 dx2 dx3.

From this definition one readily verifies that (i) if the weighting function

is chosen real, then the if-terms prevent self-adjointness; (ii) if 9 has

any imaginary part, then the 2 -terms prevent self-adjointness. It is pos-

sible to get rid of the in-terms by demanding a1 = a2 = a3= 0; but it is

not possible to get rid of the Q2 -terms. Therefore, to have any hope of

self-adjointness one must choose q(x) real and

a1 a2 3 = ° *(A.2)

Insist, then, that a1 = a2 = 3 = 0; and try, for the moment to prove

23

self-adjointness with the trivial weighting function >(x) = 1. Then the

terms on the left-hand side of equation (32) give, when integrated,

Q l ~ (2 dx + 2 p(x) p(x') - x' dx dx'

[ I] (1 ~(1)xa x(2) dx) + 21 x,~~_

- 2(a2 - 1 + 1) ' V ¾ p(x) p(X') f(l)a*( )(x - X' a) (2)5(')(x -x)

dx dx'

-~' 13Ix - x'1

+ (57 - 1) fv 'P( ) P(') g(l).a*() (2)a() dx dx'

1 ~ ~ 1 ¾ + 4 +' 4lC*) TV)ax dx dx' I2Q(l1,2) (A 3)- (a + 4y + 4) Pv~ ( )P(X') O()*~ Q()(' I( ~-;'3I

[For use in the text of the paper we retain all a terms except those that

depend on w; but we keep in mind that henceforth in this appendix the a's

vanish.] Notice that Q(1,2) is manifestly self-adjoint in the superscripts

and , ie. (1,2) Q(2,1)*1 and 2, i.e. Q(,2) Q(2) Reduction of the right-hand side of the

integrated equation is less straightforward; it requires integrations by

parts, followed by substitutions for various Lagrangian changes, simplifica-

tions, and rearrangements. After some work, the right-hand side is brought

into the form

2Q(1'2) = r [ 1 + (37 - 1)U] p div (1 ) div (2) dxfV

. d (1)*][''

~

(2)V [x 1 x. I

+ 1 dp d I[1 + (37- 1)U] p 2 - dx (A.4)V p dr dr 2~~~

x ' 9(1 )*V f .[, + (3y _ )U] dp-r div t(2)

v dr~~~~~~~~~~

x * (2)+ div

rM(1)* dx

I -

- ~Je {41 + 2[11(x) +V V

[ A(O)*(x)(1 _(- )*(x) . vp(x)][AI ( )(x')

+JV P d-- (53yU + 20)V d~r'* t(2)

+. rr

r E(1)*AU a ( 2 ) Au(1)I dx

(37 -1) r pdU x ·.( )* (2)-(3s - 1).f P -T v d~r r

- (3yr1 - 3y +

2VjdU d p (

0

-d2r dr (

1) y p[Au l ) * <(2)v

- dr +drP

x. (2) r ~H~dx

+ AU( 2 ) Ai(1)*]dx

(1)* [ x E (2)]

.2r

dx

- (6y + 1 + ;2 - 2¢) v PAU( 1

)*

AU(2) dx + R(1,2) (A.4 cont'd.)

where

R(1,2) = [I- s3(ri - 1) pc(l)* AU(2) dx

+())* dU ) dx

[(2)]1drU dx* 2 dr r

1 (2)* 2 ~~~(l)* ,(2) + U(i)* dp

+(2) dU Ix- (t(1)]d r r x

25

- (2) (x' ) *VP(x')]0dxdx'

(A.5)

-4{x- x'I

- C3] ,JV

Equation (A.4) reduces to equation (A.6) of Chandrasekhar (1965b), if we

substitute in the PPN parameter values for general relativity. Aside from

R( 1'2) the terms on the right-hand side of equation (A.4), like Q(1 ,2) of

the left-hand side, are manifestly self-adjoint in the superscripts 1 and 2.

Therefore, the condition for equation (32) to be self-adjoint with weighting

function >(x) = 1 is

R(1l2) = R(2,1 )* (A.6)

or, equivalently (since one insists on this equality for all choices of

(1) and (2)

:2 =t2 = C4 = ° *(A.7)

Might some other choice of weighting function aside from constant per-

mit one to relax these constraints and still retain self-adjointness? No;

because any other choice of >(x) will destroy the self-adjointness of the

left-hand side (2 Q(1,2)), and the arbitrariness of Q2 will prevent the

non-self-adjoint terms thus created from always cancelling non-self-adjoint

terms on the right-hand side.

In summary, equations (A.2) and (A.7) - i.e. a 1 = 52 = 2 = t2 = 3 =

C4 = 0 - are necessary and sufficient conditions for the self-adjointness

of the linearized pulsation problem in the PPN formalism. These conditions,

together with the condition that 1 = 0, are just Will's (1971b) conditions

for theories of gravity to have post-Newtonian conserved integrals for

energy, momentum, angular momentum, and center-of-mass motion.

C 26

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FOOTNOTES FOR TABLE 2

aThese are minimum values compatible with current experimental limits on I,

~, and 3: < 1.34, ' > 0.76, t3 > - 0.05.

bWill and Nordtvedt (1972).

CPapapetrou (1954a, b, c).

dNi (1972). The values of K depend on which "matter density" one chooses as source

ijfor the gravitational field: p = Tij u u = component of stress-energy tensor

ij~~~~~~~ialong four-velocity of matter [upper values]; or p = trace (Tij) [lower values].

(cf. Ni 1972.)

29

REFERENCES

Chandrasekhar, S. 1961, Hydrodynamic and Hydromagnetic Stability (Oxford:

Clarendon Press).

1964a, Phys. Rev. Letters, 12, 114 and 437.

1964b, Ap. J., 140, 417.

1964c, ibid., 139, 664.

1965a, ibid., 142, 1488.

1965b, ibid., 142, 1519.

Fowler, W. A. 1964, Rev. Mod. Phys., 36, 545.

Gilvarry, J. J., and Muller, P. M. 1972, Phys. Rev. Letters, 28, 1665.

Kreuzer, L. B. 1966, Ph.D. thesis, Department of Physics, Princeton University

(unpublished).

1968, Phys. Rev., 169, 1007.

Lebovitz, N. R. 1965, Ap. J., 142, 1257.

Ni. W.-T. 1972, Ap. J. 000, 000.

Nordstrom, G. 1913, Ann. Physik, 42, 533.

Nordtvedt, K., Jr. 1968, Phys. Rev., 169, 1017.

Nordtvedt, K., Jr., and Will, C. M. 1972, Ap. J., in press.

Nutku, Y. 1969, Ap. J., 155, 999.

Papapetron, A. 1954a, Math. Nachr., 12, 129.

1954b, Math. Nachr., 12, 143.

1954 c, Z. Phys., 139, 518.

Sramek, R. A. 1972, reported in the Fourth Cambridge Conference on Experimental

Relativity .

Thorne, K. S. 1967, in High Energy Astrophysics, vol. 3, eds. C. DeWitt,

P. Veron, E. Schatzman (Gordon and Breach: New York).

30

Thorne, K. S., Will, C. M., and Ni, W.-T. 1971, in Proceedings of the

Conference on Experimental Tests of Gravitational Theories, ed.

R. W. Davies (NASA-JPL Technical Memorandum 33-499).

Will, C. M. 1971a, Ap. J., 163§, 611.

1971b, Ap. J., 169, 125.

1971c, Ap. J., 169 141.

1972a, private communication.

1972b, Lectures, International School of Physics "Enrico Fermi"

Varenna, Italy, July 17-July 29, 1972.

Will, C. M., and Nordtvedt, K., Jr. 1972, Ap. J., in press.

Zel'dovich, Ya. B., and Novikov, I. D. 1971, Relativistic Astrophysics

Volume 1 (Chicago: University of Chicago Press).

31