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Gravitational collapse and higher-order gravitational lagrangians
ANNALS
OF PHYSICS:
76, 281-298
(1973)
Gravitational Collapse and Higher-Order Gravitational Lagrangians F. CURTIS MICHEL* Institute for Advanced Study, Princeton, New Jersey 08540 Received January 24, 1972 A general form of higher-order contributions (in &) to the Einstein field equations is displayed. The additional terms may either stabilize or destabilize self-gravitating objects in gravitational collapse depending on the sign of the coefficient introducing the quadratic term. If the quadratic term is stabilizing, intertial mass can be converted to radiation with an efficiency approaching 100 ‘A, and arbitrarily large masses can be stabilized. On the other hand, the resultant field equations are pathological in that they admit gravitons with negative mass-squared (i.e., tachyons). A nonsingular class of vacuum solutions exist in general for the quadratic case (“grey dimples”).
1. INTRODUCTION We consider here the possibility that the Einstein field equations represent only the linear approximation to a more complete but nonlinear (in the curvature tensor, R) theory. Our motivation for examining such nonlinear theories is to see what form the higher order terms must take. In particular, we wish to seeif the gravitational collapse of massive stars might not be modified (circumvented, perhaps) by such corrections. The usual action for the General theory of Relativity is written
where R is the scalar curvature. The most general Lagrangian would replace R with an arbitrary function of all 14 invariants of the curvature tensor. If that function were expanded in a power seriesin the curvature, the next terms beyond that in (I) would naturally be quadratic. Such terms have previously been discussed as well as the justification for their consideration [l]. We will show that such quadratic Lagrangians can lead not only to stability of arbitrarily massive stars, but in addition have apparently undesirable physical “side effects.” *Permanent Copyright All rights
address: Space Science Department,
Q 1973 by Academic Press, Inc. of reproduction in any form reserved.
281
Rice University, Houston, Tex 77001.
282
MICHEL
2. THE LAGRANIAN
AND FIELD EQUATIONS
Consider the action [2]
s= sf(R)(-id”” dQ+Snatter
(1)
variation of which leads to f ‘Rik + v,v’y’
- a,“(+ f
+ v,vlf')
=
KT,.~,
0)
where V denotes the covariant derivative and f’ is df/dR. The usual form of the field equations is given by taking f (R) = R, hence f’ = 1. The left hand side is divergenceless, however, for any function f as can easily be verified explicitly. Formally expanding f in the Taylor series f(R) = c R”f’“‘/n!, where the coefficients are defined by f (n’ = (d*fldR”),=, and therefore (2) becomes f (o)[- &&“I + f’l’[Ri”
- &6i”R]
+ f f2)[RRik + ViVk(iR2 +
V,VR)I
+ ...
+ -!#.+l’ + ... =
R”&” + \J,V”R” - &”
1 R”+l + VLVzRn)] 2(n + 1) (3)
~~~~
with trace 2f to’ + Rf (l’ + 3f @‘V,VIR + ... =
-KT.
(4)
The Einstein field equations are given by truncating the seriesafter f (I), in which which case &f to) = A, the “cosmological” constant, and f (l) = 1 if the constant multiplying the energy-momentum tensor (Tik) is taken to be &-G/c4. With these definitions, we will denote f c2)by 5 and truncate the seriesafter f (2). Thus we have a Lagrangian with both an R and R2 contribution. There are other quadratic invariants that might also be considered, which we will examine after
283
HIGHER-ORDER LAGRANGIANS
first discussing the present simple case. One immediate consequence of any such theory is the introduction of a new nondimensionless parameter (in the present case, [ with the dimensions of R-l). In the Newtonian limit we are then left with a characteristic length (or time or mass) and therefore the possibility that the Newtonian limit does not give the Newtonian gravitational theory for certain distance scales.In fact, it is almost unavoidable that the distance scaleparameterize a graviton mass,as we will see. Equations (3) are already difficult to solve even when truncated after f(“). It might be thought desirable to invert the Taylor series since Tik is often given explicitly from auxiliary physical assumptions.Unfortunately, such inversion leads to complicated expressionsin terms of T,“. SeeAppendix B.
3. STATIC SPHERICALLY SYMMETRIC SOLUTIONS The metric will be taken to have the form (we assumetime dependence and spherical symmtry) ds2= evdt2 - eAdr2 - rz(d02 + sin28 dv2).
(5)
The field Eqs. (3) contain the term V,V’R which may be eliminated with the trace relationship (4). One finds then the relations (1 + c$R)Go0+ (f/4) R2 + $R = K(T,O - &T)
(6)
(1 + c$R)G1 + (f/4) R2 = ~Tll,
(7)
and where these symbols have the familiar [2] definitions: Go0 = e-A((l/r2) - (X/r)) - (l/v2),
(8)
Gll = e-A((v’/r) + (l/r2)) - l/r2,
(9)
and --R
=
+G
=
J+
(e-A
4. VACUUM
-
1)
+
e-~
iv”
I
(v’)’
I
2 ( V’
SOLUTIONS AND OBSERVATIONAL
;
A’ )
-
V’h’ -.
2
1 (10)
TESTS
The basic tests of General Relativity involve experiments to detect higher order corrections to the (first order) predictions of Newtonian gravitation. These higher order corrections (bending of light, periapsis advance, frequency shifts and time
284
MICHEL
delays, spin precession, etc.) are calculated from the Schwarzschild or Kerr metrics, which in turn are solutions to the equations Ri” = 0.
(11)
However, as has already been pointed out in previous work [I], these solutions no longer correspond to the metric outside of physical bodies (e.g., stars). We can see this result from (4) which is, to first order in R, (V,Vz G I?) R f
3.faR
=
(12)
-KT.
Setting T to zero at the surface of the star does not in turn set R to zero, thus R # 0 outside of the source and hence the vacuum metrics only apply asymptotically (as R --f 0). Consider the surface of the earth: There T drops abruptly from a value -pc2 (p = mass density) to essentially zero. If 5 is sufficiently small that we can regard the surface as an infinite plane, then the solution for R is simply R =
-~pc~[r(z)
+
(8
-
q(z))
e-l’l],
where z > 0 is the height above the plane in units of (-3@12, function defined by 2 < 0, 77(z) = 1, zz.z0, z > 0.
(13) and 77 is a step (14)
We note that 5 must be negative to obtain an exponentially decaying solution (we have discarded the growing solution), otherwise an oscillating solution is set up. This behavior seems to be quite general and can be understood from the general properties of higher order dljjkential terms in R either to increase or decrease R from the value -KT [see Eq. (12)]. If the effect is to decrease R, then necessarily oscillating solutions are possible whereas if the effect is to increase R, exponential solutions are possible. In general a harmonic solution of some sort must result. What is the effect of R # 0 outside of the source ? Since to lowest order R = --KTit follows that R # 0 is equivalent in effect roughly to the corresponding mass density (we can and will be more precise when we deal with the specific field equations). Test
If the behavior in (13) were actually present, one could quickly determine f by taking gravimeter readings at different heights above the ground. For 8 = 0 the measured value of g should decline with height at a rate
HIGHER-ORDER
285
LAGRANGIANS
where h is the height above the surface, rE the radius of the earth. On the other hand if 5 is larger than the range over which the experiment is conducted the results should be the same as if the experiment were performed in a well with the surrounding medium having density p/6. Then the lapse rates are given by dg/dh = -(2g/r,)
+ $z-Gp,
or 4W
= (1 - (p/12iNdglW,=o,
(16)
where p is the mean density of the earth. Since ,!i - 2p one would expect to observe a lapse rate for g that was 4 % less than expected, provided that h2 5 -5 S rE2. The consistency between surface gravimeter measurements and satellite observations in determining GA4 appears to rule out such an effect. The upper range for 5 and beyond (.$ > rE2) is almost certainly excluded, since the periodic degeneracy of satellite orbits in a pure l/r2 field would be manifestly broken and readily observed. We will later argue that the observed stability of stars rules out < > rE2 as well. In the likely event that (15) holds, it would follow that -4 must be small, say <1 cm2. Similar considerations apply to +.$; however, other considerations arise here. Similar arguments have been given by Pechlaner and Sex1 [l]. Provided that 14 1 is small, the basic effect of the higher order terms in (3) is to affect the sources of gravitational fields. Thus the sun’s mass would be changed slightly, but this mass is an observationally determined quantity while the initial conditions and history of the sun from which a theoretical mass might be inferred for comparison are at best known only vaguely. Thus the three famous tests of general relativity all follow since the Schwarzschild metric will be a valid representation once one is several characteristic distances, (35)1/2, outside of source. Grey Dimples and Bfuck Holes The s.pherically symmetric vacuum solution to (4), analogous to the Schwarzschild solution is given by R = (l/r)(Ae-B’ = (l/r)(C
+ /?e+B’)
(E < 0) sin /3r + D cos pr) (5 > 01,
(17)
where /3 = (3 1f /)-l/Z.
(18)
The Schwarzschild solution is then the trivial (R = 0) case of these solutions, The nontrivial solutions all mark r = 0 as a special point in space. From (6) we see that Go0 = -GM”
+ $‘W(l
+ M);
(19)
286
MICHEL
thus Go0= -qR,
(20)
where i
(21)
the metric coefficients are given from e-A = 1 + -1 r Goor dr. r Io
Thus for the case R = Ae+/r,
the metric coefficient is
e--+ = 1 + (A(q)/fi)(e+ and has a minimum
(22)
+ (l/fir) e--BT
(1I/+)>
(23)
at /?r = 1.793 *** of (emA&,in= 1 - (3.350 ...) A(q)/p.
(24)
Thus there would exist nonsingular vacuum solutions for sufficiently small effective masses (A). If the Schwarzschild metric can be styled as describing a “black hole” -black in that no light can come from interior points (r < 2m)-and a bottomless hole in that a round trip to interior points cannot be made in a finite time, then the above solutions would correspond analogously to a “grey dimple.” it is to these solutions in general that the interior metric of a spherically symmetric mass distribution must match. We will continue the general discussion of this theory in Section 9, but first we will examine the interior solutions for matter in hydrostatic equilibrium.
5. HYDROSTATIC
EQUATIONS
For the sake of definiteness, we will examine the solutions for the rather oversimplified but widely studied energy-momentum tensor Ti”
=
and for stationary matter we have
(p
+ P) uiuk
+ P&“,
(25)
HIGHER-ORDER
LAGRANGIANS
287
and T = 3p -/A
(27)
Hydrostatic equilibrium (V,T,” = 0) requires P’+g++p)
=o.
(28)
And the adoption of a specific equation of state, together with the field equations, completes the system of equations.
6. APPROXIMATE
SOLUTIONS
If we chose some equation of state relating p with p, then the hydrostatic equations can be integrated, numerically, if necessary,to give the structure of a star. The masscontained in the star is then (subscript D denotes “dispersed,” namely, the massbrought together from infinity to make the star) (29) when R is the radial coordinate distance for which p = 0. It is necessaryto impose one boundary condition on the integration, for example, the pressure at r = 0. Thus, MD is given as a function of central pressure. For realistic equations of state in the linear theory it is observed that MD increaseswith p. up to a maximum and then declines monotonically thereafter. Accordingly, there exist no stalic solutions for stars more massive than some limit. These equations can be solved approximately to obtain the correct qualitative behavior, and to seeif such a limiting mass exists in the nonlinear theory as well. The approximation is based on the fact that the pressure at the center must behave as p = pO(l - oIr2 + ...); (30) thus the characteristic radius of the star is simply 01-l/~.Since the massis concentrated at the center, the massis MD a po~-312. The analysis is simplified by the fact that from (22) one has the limit
(31)
288
MICHEL
Then v’ is given both from the equation of state and the field equations. Consistency between the two requires fll = 4PO + PdPo + M/4P, In the nonrelativistic relationship is
limit
(ordinary
(5 = 0).
(33)
stars), p0 < pO, and the usual adiabatic
P CCpcLy
(P < PIi
(34)
hence the mass is &fD 0~ p(3~-4)/2,
(35)
which recovers the well-known result that a star with y = 4/3 (radiation dominated) is neutrally stable and larger values of y give stability. It is now only necessary to include appropriately the correction terms to see how they effect the stability, the result being a: = K[2&0 + 3Pd - &-G - 3Po)21h -t pd/24[1
+ &Q - 3Pd1
(36)
which reduces to (33) for [ = 0 as required (see appendix A). In the extreme relativistic limit, p ---f 3p, thus y -+ 1 and T + 0 (relative to p,p) as expected. This behavior should not be interpreted as indicating that the higher order corrections are unimportant in the extreme relativistic limit, as we can see as follows. Consider a totally degenerate, extreme relativistic Fermi gas (a plausible assumption for postneutron star matter), then we have [3] a proper energy density I* = p&inh rl - 4 (37) and a proper pressure p = p,(sinh q - 8 sinh(v/2)
+ 3~)/3,
(38)
where sinh(r)/4) is the Fermi momentum divided by mc (i.e., the proper velocity at the Fermi surface in units of c), and the coefficient is pd = m4c5/32.rr2ii3.
(39)
For large value of 7 we find the limit (P - 3~)~ We see that the correction in p; hence the condition
64~
term to the numerator
(‘I -
a>.
in (36) then contributes
26~ + 3po) - S&o - 3pd2 = 4/d
- ~Qhz).
(40) linearly
(41)
HIGHER-ORDER
289
LAGRANGIANS
It is conceivable that 165~~ > 1 for nuclei, since it is only for neutron stars that one encounters densities approaching nucleon degeneracy densities, and there is no observational evidence available at present regarding the masses of neutron stars (indeed, only inferential evidence for their existence). The quantity 01would not be negative if 16.$p, were greater than unity, since then the limit 7 + co would not be valid. While not discounting the above possibility, we will assume that t is so small that degeneracy is not avoided, in which case the proportionality (41) gives us the mass dependence (42) The limiting behaviors are then MD cc py
(p,, small but degenerate)
(43)
MO 0-z py4
(f% -
(44)
a).
The beh.avior in (43) indicates the familiar instability of massive neutron stars; however, the behavior in (44) indicates that a final stable state should exist which is not a “black hole” (i.e., its surface is not asymptotically approaching its Schwarzschild radius). The gravitational mass of the star is simply given by the condition that the vacuum metric be of the Schwarzschild exterior type, with coefficient e-A = 1 - 2GMGIrc2 and the approximate
expression
(at surface)
analogous to (31) is (see Appendix
(45) A)
MG cc --h,a-3/a. In the extreme relativistic
limit we have X1 --f v1 , giving the proportionality
In terms of the characteristic
in contrast
(46)
size (CC-~/~) we have the results MG oc 01-l/~ CC R
(47)
MD cc 1/R
(48)
to
290
MICHEL
and there is no problem of the object having a gravitational physical radius. Note that we have the limit
radius larger than its
McMD + constant as p0 + co.
(4%
7. OTHER PHYSICAL CONSEQUENCES-RADIATION We have found that the Lagrangian (2) is stabilizing provided that .$ is positive. On the other hand, if [ were positive then we have oscillating vacuum solutions of the form R = (I/r)Re(Ae@‘). (50) Such behavior is somewhat unfamiliar in known physical systems. In Appendix C we develop the weak field limit for the field Eqs. (3) and find that two modes of radiation are available: The usual transverse gravitational wave satisfying, for propagation in the x1 direction, the relations ha, = --ha, # 0, ha, f 0, OkWk = uJowo+ 6Jpl = 0,
(51)
gii = mij + Re{h,, exp(iw,x”)},
(52)
where the metric is
and, in addition,
a second set of solutions satisfying h # 0, WkWk = l/35,
namely, a dispersive scalar wave. The invariant limit to a scalar wave having mass rnhz = -P/3(.
(53) 1135 corresponds in the quantum (54)
For t > 0 we then obtain tuchyon solutions for the scalar graviton. One in fact could have anticipated this result from (50), since there we have a spatially oscillating (w12 > 0) static (w. 2 = 0) field which requires the invariant WkWk to correspond to negative mass squared. We do not wish here to be detoured into the controversy over the physical consistency of tachyon solutions. In any event the reader may understandably judge that massive gravitational tachyons are too high a price to pay for avoiding gravitational singularities.
HIGHER-ORDER
LAGRANGIANS
291
8. MOST GENERAL LAGRANGIAN
Can we still have stabilization but get rid of the tachyon solutions ? Indeed, can we get rid of the massive gravitions at all ? Returning to Eq. (3), we note the following points: (1) If we have terms of orler R2 we must have as well terms of V2R. The divergence free property of additional terms dependson cancelation between terms of order RVR generated by the noncommutivity of the covariant derivatives and those generated by the nonvanishing divergence of the R2 terms. In the weak field limit it is clear that only the derivative terms vanish. In fact the only divergencefree field terms are Gi”
(Einstein tensor),
(S,“o - V”V,) G
(term already considered),
0 Gik
(not yet considered).
and
The last two are divergence free only in the weak field limit; however, by adding appropriate higher order terms they can be made divergence free in the covariant litmit. The last term is generated by addition of a contribution
to the :Lagrangian. In fact, that essentially completes the possible second order contributions. It can be shown that a contribution from RijkLRijkz is redundant [I] to the linear combination 4RijRii - R2. Finally the scalar ~~~~~~~~~~~~~ might be considered, but owing to the totally antisymmetric $i factor both this term and its contributions vanish for simpleproblems treated here (wave propagation and static spherical symmetry). Variation of (55) yields x( ORi” + 2VjVkRij - + SikOR + 2RjkR( - 4 aikRmjRjm),
(56)
or in terms of Gik = Rik - 4 &“R, x[nG<” + 2VjVkG; 3 + (Sikn - ViVk) G + 2GjkGji - 2GGiL + 4 Sik(G2 - GmiGi”)].
(57)
Since V,Gd = 0, the second term vanishes in the weak field limit (since then the covariant derivatives commute), giving us just the expected terms. The trace is just ‘(Trace). +2xUG w 595/76/l-19
292
MICHEL
It follows then that altogether the trace would be given by G + (2~ + 35) OG = KT;
(59)
hence the choice 2% +35 = 0
(60)
would remove the scalar graviton solution and permit G + 0 when T---f 0. Then in the weak field limit, one obtains yik = (1 - ~0) Gik + $&*o
- ViOli) G
(61)
and jf = G.
(62)
Accordingly, the vacuum wave solution has G = 0 (no scalar wave) but a spin 2 wave solution for WkWk= l/x.
(63)
Thus (55) fails to eliminate gravitons with mass.We therefore still have the same modified physical effects: extension of the source and a mass spectrum for the photons. If we assumedthe first corrections to R were third order instead of second order (e.g., a Lagrangian R exp(+R2), say), then in the weak field limit only the usual theory obtains, since all other terms are at least of order R2 (terms such as q 2R are not obtained). On the other hand, we then have no guarantee that R + 0 in the vacuum limit. 9. QUALITATIVE
DISCUSSION AND SUMMARY
The effect of the nonlinear terms in (7) is to give a correction 700to the gravitational massdensity MO = - j (Too + -roe)4,rrr2dr,
(64)
whereasthe massrequired to build such a star is given by MD = -
s
T,OeA 124.rrr 2 dr ,
(65)
with e-A = 1 - (2G/rc2) M,(r). For 7oo= 0, the difference (MD - MG) c2 is just the gravitational binding energy. If we imagine a large dust cloud of massMD collapsing to form a star with massMG , the conservation equations assureus that we will observe an integrated energy flux (MD - MG) c2 away from the cloud/star.
HIGHER-ORDER
LAGRANGIANS
293
If To0 < 0, the mass MG is increased. The coefficient h is increased as well, but the effect is smaller by a factor GM/R?. The net effect should therefore be to decrease the binding energy. Even with TVO = 0, one finds for stars that with fixed MD and increasing central density, the binding energy decreases at first but finally reaches a minimum and thereafter increases. It is recognized that the star is unstable to gravitational collapse at the minimum binding energy. If we now add a correction ~~0 that further decreases the binding energy at least quadratically with central density, it is clear that the minimum binding energy will be shifted towards lower central densities. Thus, we expect 7oo < 0 to have a destabilizing influence on gravitational phenomena. We can put a lower limit then on negative values of 6 simply from the fact that white dwarfs exist that have not collapsed, which they would have, were -j 7oo dV comparable to the binding energy. Thus the inequality -TAO 5 p(GM/r?) follows,
(66)
and since we can estimate 70’ -
-&‘p2,
(67)
we obtain -(
5 (GM/prc2)
*
1015cm2
(white dwarfs).
(68)
Since massive white dwarfs should be quite close to collapse anyway (hence even more sensitive to shifts in binding energy), it seems likely that even more stringent limits could be deduced from specific models. The upper mass limit for neutron stars would be seriously reduced if -5 were as large as loll cm2. Supose instead that Too > 0 hence acts as a negative mass and provides a stabilizing influence. For example, consider the limits Too + Too-+ constant for Too 4 03, then M,; remains finite while M, can become arbitrarily large. Such behavior would avoid entirely gravitational collapse. In the limit M, + co, adding more rest mass produces no increase in gravitational mass. Thus the binding energy approaches the rest energy. Infalling matter would then have to radiate eventually all of its energy. It is possible that a large portion of this radiated energy might be in the forms of gravitational waves, since the gravitational potentials involved would be quite large. The nonlinear terms do, for example, admit the possibility of emitting scalar gravitational waves (Appendix C), particularly in the destabilizing case. A simple alternative way of seeing that 5 > 0 results in stabilization is to note that in the Lagrangian we have the rearrangements R + 5R2 = (1 + .$R) R FZ R/(1 - fR),
294
MICHEL
and therefore the gravitational coupling coefficient (G) is effectively multiplied by (1 - @). In other words, G decreases as R increases, with the result that the gravitational attraction falls below the Newtonian value. Thereby the effective adiabatic index of the relativistic particles is increased above 4/3, and stabilization is obtained. Unfortunately stabilization is obtained at the expense of introducing apparently pathological properties, some of which (the gravimeter experiment, Section 3) are readily restable. APPENDIX
A: SOLVING FOR BEHAVIORNEAR
r =0
The asymptotic behavior near r = 0 for equations of hydrostatic equilibrium can be deduced from the expansions ech = I + Alr2 + ..., v = v. + v1r2 + -.-,
(AlI
P = PO+ Plr2 + .*., p = PO+ pFLlr2 + -*-, R = R, + R,r2 -+ **.,
which gives (1 + 5R,)(3U + f Ro2+ f 4, = - ; @/Jo (1 + @,)(A
+ 2~3
+
$ Ro2 = Ro =
+
3PfA
(A31
~0,
44
042)
+
VA
644)
and 2PI + VdPo + PO) = 0.
GW
Equations (A2) and (A3) can be rewritten, using (A4), as h, - 2l’, - 9[(h12
-
V12)
=
--gK(2j.Lo
3f(h12 -
VI”)
=
Kpo
+
3p,)
W’)
and h, +
23
+
.
(A37
Solving for X, and eliminating could give a quartic equation in v1 to be solved; however, in fact, the resultant equation is only linear in v1 , being (after some straightforward algebra) --12vSl + 5&o - 3Po)l + M/-40 + 3Po)- &PO - 3Po121 = 0.
646)
HIGHER-ORDER
295
LAGRANGIANS
Substitution in (A5) then yields p1 which is --cyp,, in the notation of the text. Note that u1 returns to the value given by linear theory if either e = 0 or p0 = 3p,, since in either case the higher order terms do not contribute. The remaining constants are given from A, = -Q - (~0 - 3~86
(A7)
and (A4). The constant y0 is irrelevant to the static structure, and is determined by the boundary condition that Y + h = 0 at the stellar surface (in order to fit onto the vacuum exterior solution).
APPENDIX
B: INVERSION OF THE TAYLOR SERIES
Often Tik is given by assumption. In these cases, it would be much easier to solve for Rik given by a nonlinear function of Ti”, rather than vice versa [Eq.(3)]. Thus we would like to write the nonlinear equations in the more familiar form Rik - $$iikR =
rcTik
+ higher order terms in T.
(Bl)
To second order, the trace of (3) gives R+~.$~R=-KT-KK~,
032)
where q = V,V”, h = if(O), and 4 = f (2) with f(i) = 1. Thus expressing R in terms of T leads to the potentially nonconvergent series R = -4h
-
K(T
-
35nT
+ 9e2n2T
-
27f3n3T
+ ...).
(B3)
So long as .$ is small compared to the scale of the system and the system itself is free of singularities, the series might reasonably be expected to converge. The term X, as we have noted before, is observed to be quite small, and henceforth we will ignore it. Then to order .$ we have Rik -
+ 6ikR =
~~~~
+
KTik
+
o(f2)
+
o(h)
(B4)
where rik =
-K<(TT~~
+ ; &“T2 - V,V”T f V,VlT).
(B5)
Note that f-l has units of an energy density, thus Tik contributes importantly when the physical energy density becomes comparable to 5-l. However, (B4) is only approximately divergence free, owing to the above simplifying approximations, since vkTik = -(Tik
- 4 8,“T) v,T,
036)
296
MICHEL
which is of order T3. Thus the inversion accomplished in Eqs. (B4) and (B5) is a useful approximation so long as one is restricted to low densities and only retains terms to order T2 (the mass of a pulsating star might fluctuate in order T3, for example, but that would indicate a failure of the approximation rather than the existence of a monopole source; see Appendix C).
APPENDIX
C: RADIATION AND THE WEAK FIELD LIMIT
From Eq. (3) we see that in the weak field limit, the only higher-order terms that contribute are &ViVkR
- &“OR) + O(R2),
(Cl)
which is itself divergenceless to the same order (the V-operators commute zeroth order in R). If we write gii = mij + hij(x) = mij + hijeiWkxk with mij being the Minkowski
to (C2)
(Flat Space) metric, we have for (3) to first order in h
yik = Gik + f(c.opk - 6,“~“) G,
(C3)
Gi” = $(w”wih + W2hik- WiWshsk- WkW,his+ 6ikG)y
(C4)
where
with G = -co2h + w swthts.
We will restrict ourselves to the case of a wave propagating then only o,, and w1 are nonzero, and we find
in the X-direction,
Go0= - &01d(h22 + h33), Glo = $t+a”(h22 + h33), Gll = - +wooo(h22+ h33), G22= $(co2hz2- w2h + wswthts), G20= +(co1dho2- coodh12), G,2 = &,2,‘j392
etc.
((35)
HIGHER-ORDER
Transforming
the coordinates
297
LAGRANGIANS
using
to change gi, (but not the physics) according to gives
gij+ gij+ vi!fj+ vjfi
(C7)
In the latter equation the t’s are just numbers; hence we have only 4 conditions that we can impose. The amplitudes h,O, h,O, and Ir11 appear only in the combination G = -c~P(h,~ + h33) + 2q,dhl”
- w,w0h,’
- w~w~~~O.
(C9)
In the usual case (Gzk = 0), the first term is zero, since Go0 + G,l = 0; hence two coordinate transforms (i.e., choice of to and E,) are sufficient to set h,O = h,l = ho0 = 0, provided that w. or w1 # 0 (otherwise we could transform away the gravitational potential of weak static field as well). The choices of rz and t3 can be used to set ho2 = h12 = ho3 = Al3 = 0, leaving only the two orthogonal transverse polarization amplitudes /I,~ = -h33 and h32, both satisfying the dispersion relation w2 = wow0 + wlwl Returning
= 0.
to (C3), we see from the trace relationship (1 - 35~~) G = 0.
If we were to chose wz = 0, the same argument If instead we chose u2 = +1/(3t-L
(ClO)
that K-11)
given above would go through. (CW
then G need no longer vanish. It follows then that we will be unable to eliminate all three of /zoo,h,O, A,‘. Since (C12) corresponds to a particle with mass, it follows that a purely transverse wave would not be Lorentz invariant; thus we need, in fact, contributions from the longitudinal amplitudes. The simplest invariant choice is just h = ho” = /z,l = h,2 == h 3,3 W3) and this choice satisfies the full set of field Eqs. (C4). The appropriate coordinate transforms are to set ho1 = ho2 = ho3 = 0 and ho0= h,l. Substitution into yoo + c,@l1 = 0 gives ho0+ h11 = hz2 + hs3 and finally $22 - $33 = 0 gives i122= h33.
298
MICHEL ACKNOWLEDGMENT
The author benefitted from the many valuable comments offered by Professors J. N. Bahcall and F. Dyson. This research was supported in part by the National Science Foundation, under Grant GP-25854, and in part under Grant GP-16147 A # 1. The author would &o like to thank the Director of the Institute for Advanced Study, Carl Kaysen, and the members of the School of Natural Sciences for their kind hospitality.
REFERENCES AND FOOTNOTES 1. E. PECHLANER AND R. SEXL, Comm. Math. Phys. 2 (1966), 165; also contains references to earlier work. 2. Cf. L. D. LANDAU AND E. M. LIFSHITZ, “Classical Theory of Fields,” Addison-Wesley Reading, MA, 1962. Usual notation; however, we set c = 1. 3. L. D. LANDAU AND E. M. LIFSHITZ, “Statistical Physics,” Section 58, Addison-Wesley, Reading, MA, 1958.
OF PHYSICS:
76, 281-298
(1973)
Gravitational Collapse and Higher-Order Gravitational Lagrangians F. CURTIS MICHEL* Institute for Advanced Study, Princeton, New Jersey 08540 Received January 24, 1972 A general form of higher-order contributions (in &) to the Einstein field equations is displayed. The additional terms may either stabilize or destabilize self-gravitating objects in gravitational collapse depending on the sign of the coefficient introducing the quadratic term. If the quadratic term is stabilizing, intertial mass can be converted to radiation with an efficiency approaching 100 ‘A, and arbitrarily large masses can be stabilized. On the other hand, the resultant field equations are pathological in that they admit gravitons with negative mass-squared (i.e., tachyons). A nonsingular class of vacuum solutions exist in general for the quadratic case (“grey dimples”).
1. INTRODUCTION We consider here the possibility that the Einstein field equations represent only the linear approximation to a more complete but nonlinear (in the curvature tensor, R) theory. Our motivation for examining such nonlinear theories is to see what form the higher order terms must take. In particular, we wish to seeif the gravitational collapse of massive stars might not be modified (circumvented, perhaps) by such corrections. The usual action for the General theory of Relativity is written
where R is the scalar curvature. The most general Lagrangian would replace R with an arbitrary function of all 14 invariants of the curvature tensor. If that function were expanded in a power seriesin the curvature, the next terms beyond that in (I) would naturally be quadratic. Such terms have previously been discussed as well as the justification for their consideration [l]. We will show that such quadratic Lagrangians can lead not only to stability of arbitrarily massive stars, but in addition have apparently undesirable physical “side effects.” *Permanent Copyright All rights
address: Space Science Department,
Q 1973 by Academic Press, Inc. of reproduction in any form reserved.
281
Rice University, Houston, Tex 77001.
282
MICHEL
2. THE LAGRANIAN
AND FIELD EQUATIONS
Consider the action [2]
s= sf(R)(-id”” dQ+Snatter
(1)
variation of which leads to f ‘Rik + v,v’y’
- a,“(+ f
+ v,vlf')
=
KT,.~,
0)
where V denotes the covariant derivative and f’ is df/dR. The usual form of the field equations is given by taking f (R) = R, hence f’ = 1. The left hand side is divergenceless, however, for any function f as can easily be verified explicitly. Formally expanding f in the Taylor series f(R) = c R”f’“‘/n!, where the coefficients are defined by f (n’ = (d*fldR”),=, and therefore (2) becomes f (o)[- &&“I + f’l’[Ri”
- &6i”R]
+ f f2)[RRik + ViVk(iR2 +
V,VR)I
+ ...
+ -!#.+l’ + ... =
R”&” + \J,V”R” - &”
1 R”+l + VLVzRn)] 2(n + 1) (3)
~~~~
with trace 2f to’ + Rf (l’ + 3f @‘V,VIR + ... =
-KT.
(4)
The Einstein field equations are given by truncating the seriesafter f (I), in which which case &f to) = A, the “cosmological” constant, and f (l) = 1 if the constant multiplying the energy-momentum tensor (Tik) is taken to be &-G/c4. With these definitions, we will denote f c2)by 5 and truncate the seriesafter f (2). Thus we have a Lagrangian with both an R and R2 contribution. There are other quadratic invariants that might also be considered, which we will examine after
283
HIGHER-ORDER LAGRANGIANS
first discussing the present simple case. One immediate consequence of any such theory is the introduction of a new nondimensionless parameter (in the present case, [ with the dimensions of R-l). In the Newtonian limit we are then left with a characteristic length (or time or mass) and therefore the possibility that the Newtonian limit does not give the Newtonian gravitational theory for certain distance scales.In fact, it is almost unavoidable that the distance scaleparameterize a graviton mass,as we will see. Equations (3) are already difficult to solve even when truncated after f(“). It might be thought desirable to invert the Taylor series since Tik is often given explicitly from auxiliary physical assumptions.Unfortunately, such inversion leads to complicated expressionsin terms of T,“. SeeAppendix B.
3. STATIC SPHERICALLY SYMMETRIC SOLUTIONS The metric will be taken to have the form (we assumetime dependence and spherical symmtry) ds2= evdt2 - eAdr2 - rz(d02 + sin28 dv2).
(5)
The field Eqs. (3) contain the term V,V’R which may be eliminated with the trace relationship (4). One finds then the relations (1 + c$R)Go0+ (f/4) R2 + $R = K(T,O - &T)
(6)
(1 + c$R)G1 + (f/4) R2 = ~Tll,
(7)
and where these symbols have the familiar [2] definitions: Go0 = e-A((l/r2) - (X/r)) - (l/v2),
(8)
Gll = e-A((v’/r) + (l/r2)) - l/r2,
(9)
and --R
=
+G
=
J+
(e-A
4. VACUUM
-
1)
+
e-~
iv”
I
(v’)’
I
2 ( V’
SOLUTIONS AND OBSERVATIONAL
;
A’ )
-
V’h’ -.
2
1 (10)
TESTS
The basic tests of General Relativity involve experiments to detect higher order corrections to the (first order) predictions of Newtonian gravitation. These higher order corrections (bending of light, periapsis advance, frequency shifts and time
284
MICHEL
delays, spin precession, etc.) are calculated from the Schwarzschild or Kerr metrics, which in turn are solutions to the equations Ri” = 0.
(11)
However, as has already been pointed out in previous work [I], these solutions no longer correspond to the metric outside of physical bodies (e.g., stars). We can see this result from (4) which is, to first order in R, (V,Vz G I?) R f
3.faR
=
(12)
-KT.
Setting T to zero at the surface of the star does not in turn set R to zero, thus R # 0 outside of the source and hence the vacuum metrics only apply asymptotically (as R --f 0). Consider the surface of the earth: There T drops abruptly from a value -pc2 (p = mass density) to essentially zero. If 5 is sufficiently small that we can regard the surface as an infinite plane, then the solution for R is simply R =
-~pc~[r(z)
+
(8
-
q(z))
e-l’l],
where z > 0 is the height above the plane in units of (-3@12, function defined by 2 < 0, 77(z) = 1, zz.z0, z > 0.
(13) and 77 is a step (14)
We note that 5 must be negative to obtain an exponentially decaying solution (we have discarded the growing solution), otherwise an oscillating solution is set up. This behavior seems to be quite general and can be understood from the general properties of higher order dljjkential terms in R either to increase or decrease R from the value -KT [see Eq. (12)]. If the effect is to decrease R, then necessarily oscillating solutions are possible whereas if the effect is to increase R, exponential solutions are possible. In general a harmonic solution of some sort must result. What is the effect of R # 0 outside of the source ? Since to lowest order R = --KTit follows that R # 0 is equivalent in effect roughly to the corresponding mass density (we can and will be more precise when we deal with the specific field equations). Test
If the behavior in (13) were actually present, one could quickly determine f by taking gravimeter readings at different heights above the ground. For 8 = 0 the measured value of g should decline with height at a rate
HIGHER-ORDER
285
LAGRANGIANS
where h is the height above the surface, rE the radius of the earth. On the other hand if 5 is larger than the range over which the experiment is conducted the results should be the same as if the experiment were performed in a well with the surrounding medium having density p/6. Then the lapse rates are given by dg/dh = -(2g/r,)
+ $z-Gp,
or 4W
= (1 - (p/12iNdglW,=o,
(16)
where p is the mean density of the earth. Since ,!i - 2p one would expect to observe a lapse rate for g that was 4 % less than expected, provided that h2 5 -5 S rE2. The consistency between surface gravimeter measurements and satellite observations in determining GA4 appears to rule out such an effect. The upper range for 5 and beyond (.$ > rE2) is almost certainly excluded, since the periodic degeneracy of satellite orbits in a pure l/r2 field would be manifestly broken and readily observed. We will later argue that the observed stability of stars rules out < > rE2 as well. In the likely event that (15) holds, it would follow that -4 must be small, say <1 cm2. Similar considerations apply to +.$; however, other considerations arise here. Similar arguments have been given by Pechlaner and Sex1 [l]. Provided that 14 1 is small, the basic effect of the higher order terms in (3) is to affect the sources of gravitational fields. Thus the sun’s mass would be changed slightly, but this mass is an observationally determined quantity while the initial conditions and history of the sun from which a theoretical mass might be inferred for comparison are at best known only vaguely. Thus the three famous tests of general relativity all follow since the Schwarzschild metric will be a valid representation once one is several characteristic distances, (35)1/2, outside of source. Grey Dimples and Bfuck Holes The s.pherically symmetric vacuum solution to (4), analogous to the Schwarzschild solution is given by R = (l/r)(Ae-B’ = (l/r)(C
+ /?e+B’)
(E < 0) sin /3r + D cos pr) (5 > 01,
(17)
where /3 = (3 1f /)-l/Z.
(18)
The Schwarzschild solution is then the trivial (R = 0) case of these solutions, The nontrivial solutions all mark r = 0 as a special point in space. From (6) we see that Go0 = -GM”
+ $‘W(l
+ M);
(19)
286
MICHEL
thus Go0= -qR,
(20)
where i
(21)
the metric coefficients are given from e-A = 1 + -1 r Goor dr. r Io
Thus for the case R = Ae+/r,
the metric coefficient is
e--+ = 1 + (A(q)/fi)(e+ and has a minimum
(22)
+ (l/fir) e--BT
(1I/+)>
(23)
at /?r = 1.793 *** of (emA&,in= 1 - (3.350 ...) A(q)/p.
(24)
Thus there would exist nonsingular vacuum solutions for sufficiently small effective masses (A). If the Schwarzschild metric can be styled as describing a “black hole” -black in that no light can come from interior points (r < 2m)-and a bottomless hole in that a round trip to interior points cannot be made in a finite time, then the above solutions would correspond analogously to a “grey dimple.” it is to these solutions in general that the interior metric of a spherically symmetric mass distribution must match. We will continue the general discussion of this theory in Section 9, but first we will examine the interior solutions for matter in hydrostatic equilibrium.
5. HYDROSTATIC
EQUATIONS
For the sake of definiteness, we will examine the solutions for the rather oversimplified but widely studied energy-momentum tensor Ti”
=
and for stationary matter we have
(p
+ P) uiuk
+ P&“,
(25)
HIGHER-ORDER
LAGRANGIANS
287
and T = 3p -/A
(27)
Hydrostatic equilibrium (V,T,” = 0) requires P’+g++p)
=o.
(28)
And the adoption of a specific equation of state, together with the field equations, completes the system of equations.
6. APPROXIMATE
SOLUTIONS
If we chose some equation of state relating p with p, then the hydrostatic equations can be integrated, numerically, if necessary,to give the structure of a star. The masscontained in the star is then (subscript D denotes “dispersed,” namely, the massbrought together from infinity to make the star) (29) when R is the radial coordinate distance for which p = 0. It is necessaryto impose one boundary condition on the integration, for example, the pressure at r = 0. Thus, MD is given as a function of central pressure. For realistic equations of state in the linear theory it is observed that MD increaseswith p. up to a maximum and then declines monotonically thereafter. Accordingly, there exist no stalic solutions for stars more massive than some limit. These equations can be solved approximately to obtain the correct qualitative behavior, and to seeif such a limiting mass exists in the nonlinear theory as well. The approximation is based on the fact that the pressure at the center must behave as p = pO(l - oIr2 + ...); (30) thus the characteristic radius of the star is simply 01-l/~.Since the massis concentrated at the center, the massis MD a po~-312. The analysis is simplified by the fact that from (22) one has the limit
(31)
288
MICHEL
Then v’ is given both from the equation of state and the field equations. Consistency between the two requires fll = 4PO + PdPo + M/4P, In the nonrelativistic relationship is
limit
(ordinary
(5 = 0).
(33)
stars), p0 < pO, and the usual adiabatic
P CCpcLy
(P < PIi
(34)
hence the mass is &fD 0~ p(3~-4)/2,
(35)
which recovers the well-known result that a star with y = 4/3 (radiation dominated) is neutrally stable and larger values of y give stability. It is now only necessary to include appropriately the correction terms to see how they effect the stability, the result being a: = K[2&0 + 3Pd - &-G - 3Po)21h -t pd/24[1
+ &Q - 3Pd1
(36)
which reduces to (33) for [ = 0 as required (see appendix A). In the extreme relativistic limit, p ---f 3p, thus y -+ 1 and T + 0 (relative to p,p) as expected. This behavior should not be interpreted as indicating that the higher order corrections are unimportant in the extreme relativistic limit, as we can see as follows. Consider a totally degenerate, extreme relativistic Fermi gas (a plausible assumption for postneutron star matter), then we have [3] a proper energy density I* = p&inh rl - 4 (37) and a proper pressure p = p,(sinh q - 8 sinh(v/2)
+ 3~)/3,
(38)
where sinh(r)/4) is the Fermi momentum divided by mc (i.e., the proper velocity at the Fermi surface in units of c), and the coefficient is pd = m4c5/32.rr2ii3.
(39)
For large value of 7 we find the limit (P - 3~)~ We see that the correction in p; hence the condition
64~
term to the numerator
(‘I -
a>.
in (36) then contributes
26~ + 3po) - S&o - 3pd2 = 4/d
- ~Qhz).
(40) linearly
(41)
HIGHER-ORDER
289
LAGRANGIANS
It is conceivable that 165~~ > 1 for nuclei, since it is only for neutron stars that one encounters densities approaching nucleon degeneracy densities, and there is no observational evidence available at present regarding the masses of neutron stars (indeed, only inferential evidence for their existence). The quantity 01would not be negative if 16.$p, were greater than unity, since then the limit 7 + co would not be valid. While not discounting the above possibility, we will assume that t is so small that degeneracy is not avoided, in which case the proportionality (41) gives us the mass dependence (42) The limiting behaviors are then MD cc py
(p,, small but degenerate)
(43)
MO 0-z py4
(f% -
(44)
a).
The beh.avior in (43) indicates the familiar instability of massive neutron stars; however, the behavior in (44) indicates that a final stable state should exist which is not a “black hole” (i.e., its surface is not asymptotically approaching its Schwarzschild radius). The gravitational mass of the star is simply given by the condition that the vacuum metric be of the Schwarzschild exterior type, with coefficient e-A = 1 - 2GMGIrc2 and the approximate
expression
(at surface)
analogous to (31) is (see Appendix
(45) A)
MG cc --h,a-3/a. In the extreme relativistic
limit we have X1 --f v1 , giving the proportionality
In terms of the characteristic
in contrast
(46)
size (CC-~/~) we have the results MG oc 01-l/~ CC R
(47)
MD cc 1/R
(48)
to
290
MICHEL
and there is no problem of the object having a gravitational physical radius. Note that we have the limit
radius larger than its
McMD + constant as p0 + co.
(4%
7. OTHER PHYSICAL CONSEQUENCES-RADIATION We have found that the Lagrangian (2) is stabilizing provided that .$ is positive. On the other hand, if [ were positive then we have oscillating vacuum solutions of the form R = (I/r)Re(Ae@‘). (50) Such behavior is somewhat unfamiliar in known physical systems. In Appendix C we develop the weak field limit for the field Eqs. (3) and find that two modes of radiation are available: The usual transverse gravitational wave satisfying, for propagation in the x1 direction, the relations ha, = --ha, # 0, ha, f 0, OkWk = uJowo+ 6Jpl = 0,
(51)
gii = mij + Re{h,, exp(iw,x”)},
(52)
where the metric is
and, in addition,
a second set of solutions satisfying h # 0, WkWk = l/35,
namely, a dispersive scalar wave. The invariant limit to a scalar wave having mass rnhz = -P/3(.
(53) 1135 corresponds in the quantum (54)
For t > 0 we then obtain tuchyon solutions for the scalar graviton. One in fact could have anticipated this result from (50), since there we have a spatially oscillating (w12 > 0) static (w. 2 = 0) field which requires the invariant WkWk to correspond to negative mass squared. We do not wish here to be detoured into the controversy over the physical consistency of tachyon solutions. In any event the reader may understandably judge that massive gravitational tachyons are too high a price to pay for avoiding gravitational singularities.
HIGHER-ORDER
LAGRANGIANS
291
8. MOST GENERAL LAGRANGIAN
Can we still have stabilization but get rid of the tachyon solutions ? Indeed, can we get rid of the massive gravitions at all ? Returning to Eq. (3), we note the following points: (1) If we have terms of orler R2 we must have as well terms of V2R. The divergence free property of additional terms dependson cancelation between terms of order RVR generated by the noncommutivity of the covariant derivatives and those generated by the nonvanishing divergence of the R2 terms. In the weak field limit it is clear that only the derivative terms vanish. In fact the only divergencefree field terms are Gi”
(Einstein tensor),
(S,“o - V”V,) G
(term already considered),
0 Gik
(not yet considered).
and
The last two are divergence free only in the weak field limit; however, by adding appropriate higher order terms they can be made divergence free in the covariant litmit. The last term is generated by addition of a contribution
to the :Lagrangian. In fact, that essentially completes the possible second order contributions. It can be shown that a contribution from RijkLRijkz is redundant [I] to the linear combination 4RijRii - R2. Finally the scalar ~~~~~~~~~~~~~ might be considered, but owing to the totally antisymmetric $i factor both this term and its contributions vanish for simpleproblems treated here (wave propagation and static spherical symmetry). Variation of (55) yields x( ORi” + 2VjVkRij - + SikOR + 2RjkR( - 4 aikRmjRjm),
(56)
or in terms of Gik = Rik - 4 &“R, x[nG<” + 2VjVkG; 3 + (Sikn - ViVk) G + 2GjkGji - 2GGiL + 4 Sik(G2 - GmiGi”)].
(57)
Since V,Gd = 0, the second term vanishes in the weak field limit (since then the covariant derivatives commute), giving us just the expected terms. The trace is just ‘(Trace). +2xUG w 595/76/l-19
292
MICHEL
It follows then that altogether the trace would be given by G + (2~ + 35) OG = KT;
(59)
hence the choice 2% +35 = 0
(60)
would remove the scalar graviton solution and permit G + 0 when T---f 0. Then in the weak field limit, one obtains yik = (1 - ~0) Gik + $&*o
- ViOli) G
(61)
and jf = G.
(62)
Accordingly, the vacuum wave solution has G = 0 (no scalar wave) but a spin 2 wave solution for WkWk= l/x.
(63)
Thus (55) fails to eliminate gravitons with mass.We therefore still have the same modified physical effects: extension of the source and a mass spectrum for the photons. If we assumedthe first corrections to R were third order instead of second order (e.g., a Lagrangian R exp(+R2), say), then in the weak field limit only the usual theory obtains, since all other terms are at least of order R2 (terms such as q 2R are not obtained). On the other hand, we then have no guarantee that R + 0 in the vacuum limit. 9. QUALITATIVE
DISCUSSION AND SUMMARY
The effect of the nonlinear terms in (7) is to give a correction 700to the gravitational massdensity MO = - j (Too + -roe)4,rrr2dr,
(64)
whereasthe massrequired to build such a star is given by MD = -
s
T,OeA 124.rrr 2 dr ,
(65)
with e-A = 1 - (2G/rc2) M,(r). For 7oo= 0, the difference (MD - MG) c2 is just the gravitational binding energy. If we imagine a large dust cloud of massMD collapsing to form a star with massMG , the conservation equations assureus that we will observe an integrated energy flux (MD - MG) c2 away from the cloud/star.
HIGHER-ORDER
LAGRANGIANS
293
If To0 < 0, the mass MG is increased. The coefficient h is increased as well, but the effect is smaller by a factor GM/R?. The net effect should therefore be to decrease the binding energy. Even with TVO = 0, one finds for stars that with fixed MD and increasing central density, the binding energy decreases at first but finally reaches a minimum and thereafter increases. It is recognized that the star is unstable to gravitational collapse at the minimum binding energy. If we now add a correction ~~0 that further decreases the binding energy at least quadratically with central density, it is clear that the minimum binding energy will be shifted towards lower central densities. Thus, we expect 7oo < 0 to have a destabilizing influence on gravitational phenomena. We can put a lower limit then on negative values of 6 simply from the fact that white dwarfs exist that have not collapsed, which they would have, were -j 7oo dV comparable to the binding energy. Thus the inequality -TAO 5 p(GM/r?) follows,
(66)
and since we can estimate 70’ -
-&‘p2,
(67)
we obtain -(
5 (GM/prc2)
*
1015cm2
(white dwarfs).
(68)
Since massive white dwarfs should be quite close to collapse anyway (hence even more sensitive to shifts in binding energy), it seems likely that even more stringent limits could be deduced from specific models. The upper mass limit for neutron stars would be seriously reduced if -5 were as large as loll cm2. Supose instead that Too > 0 hence acts as a negative mass and provides a stabilizing influence. For example, consider the limits Too + Too-+ constant for Too 4 03, then M,; remains finite while M, can become arbitrarily large. Such behavior would avoid entirely gravitational collapse. In the limit M, + co, adding more rest mass produces no increase in gravitational mass. Thus the binding energy approaches the rest energy. Infalling matter would then have to radiate eventually all of its energy. It is possible that a large portion of this radiated energy might be in the forms of gravitational waves, since the gravitational potentials involved would be quite large. The nonlinear terms do, for example, admit the possibility of emitting scalar gravitational waves (Appendix C), particularly in the destabilizing case. A simple alternative way of seeing that 5 > 0 results in stabilization is to note that in the Lagrangian we have the rearrangements R + 5R2 = (1 + .$R) R FZ R/(1 - fR),
294
MICHEL
and therefore the gravitational coupling coefficient (G) is effectively multiplied by (1 - @). In other words, G decreases as R increases, with the result that the gravitational attraction falls below the Newtonian value. Thereby the effective adiabatic index of the relativistic particles is increased above 4/3, and stabilization is obtained. Unfortunately stabilization is obtained at the expense of introducing apparently pathological properties, some of which (the gravimeter experiment, Section 3) are readily restable. APPENDIX
A: SOLVING FOR BEHAVIORNEAR
r =0
The asymptotic behavior near r = 0 for equations of hydrostatic equilibrium can be deduced from the expansions ech = I + Alr2 + ..., v = v. + v1r2 + -.-,
(AlI
P = PO+ Plr2 + .*., p = PO+ pFLlr2 + -*-, R = R, + R,r2 -+ **.,
which gives (1 + 5R,)(3U + f Ro2+ f 4, = - ; @/Jo (1 + @,)(A
+ 2~3
+
$ Ro2 = Ro =
+
3PfA
(A31
~0,
44
042)
+
VA
644)
and 2PI + VdPo + PO) = 0.
GW
Equations (A2) and (A3) can be rewritten, using (A4), as h, - 2l’, - 9[(h12
-
V12)
=
--gK(2j.Lo
3f(h12 -
VI”)
=
Kpo
+
3p,)
W’)
and h, +
23
+
.
(A37
Solving for X, and eliminating could give a quartic equation in v1 to be solved; however, in fact, the resultant equation is only linear in v1 , being (after some straightforward algebra) --12vSl + 5&o - 3Po)l + M/-40 + 3Po)- &PO - 3Po121 = 0.
646)
HIGHER-ORDER
295
LAGRANGIANS
Substitution in (A5) then yields p1 which is --cyp,, in the notation of the text. Note that u1 returns to the value given by linear theory if either e = 0 or p0 = 3p,, since in either case the higher order terms do not contribute. The remaining constants are given from A, = -Q - (~0 - 3~86
(A7)
and (A4). The constant y0 is irrelevant to the static structure, and is determined by the boundary condition that Y + h = 0 at the stellar surface (in order to fit onto the vacuum exterior solution).
APPENDIX
B: INVERSION OF THE TAYLOR SERIES
Often Tik is given by assumption. In these cases, it would be much easier to solve for Rik given by a nonlinear function of Ti”, rather than vice versa [Eq.(3)]. Thus we would like to write the nonlinear equations in the more familiar form Rik - $$iikR =
rcTik
+ higher order terms in T.
(Bl)
To second order, the trace of (3) gives R+~.$~R=-KT-KK~,
032)
where q = V,V”, h = if(O), and 4 = f (2) with f(i) = 1. Thus expressing R in terms of T leads to the potentially nonconvergent series R = -4h
-
K(T
-
35nT
+ 9e2n2T
-
27f3n3T
+ ...).
(B3)
So long as .$ is small compared to the scale of the system and the system itself is free of singularities, the series might reasonably be expected to converge. The term X, as we have noted before, is observed to be quite small, and henceforth we will ignore it. Then to order .$ we have Rik -
+ 6ikR =
~~~~
+
KTik
+
o(f2)
+
o(h)
(B4)
where rik =
-K<(TT~~
+ ; &“T2 - V,V”T f V,VlT).
(B5)
Note that f-l has units of an energy density, thus Tik contributes importantly when the physical energy density becomes comparable to 5-l. However, (B4) is only approximately divergence free, owing to the above simplifying approximations, since vkTik = -(Tik
- 4 8,“T) v,T,
036)
296
MICHEL
which is of order T3. Thus the inversion accomplished in Eqs. (B4) and (B5) is a useful approximation so long as one is restricted to low densities and only retains terms to order T2 (the mass of a pulsating star might fluctuate in order T3, for example, but that would indicate a failure of the approximation rather than the existence of a monopole source; see Appendix C).
APPENDIX
C: RADIATION AND THE WEAK FIELD LIMIT
From Eq. (3) we see that in the weak field limit, the only higher-order terms that contribute are &ViVkR
- &“OR) + O(R2),
(Cl)
which is itself divergenceless to the same order (the V-operators commute zeroth order in R). If we write gii = mij + hij(x) = mij + hijeiWkxk with mij being the Minkowski
to (C2)
(Flat Space) metric, we have for (3) to first order in h
yik = Gik + f(c.opk - 6,“~“) G,
(C3)
Gi” = $(w”wih + W2hik- WiWshsk- WkW,his+ 6ikG)y
(C4)
where
with G = -co2h + w swthts.
We will restrict ourselves to the case of a wave propagating then only o,, and w1 are nonzero, and we find
in the X-direction,
Go0= - &01d(h22 + h33), Glo = $t+a”(h22 + h33), Gll = - +wooo(h22+ h33), G22= $(co2hz2- w2h + wswthts), G20= +(co1dho2- coodh12), G,2 = &,2,‘j392
etc.
((35)
HIGHER-ORDER
Transforming
the coordinates
297
LAGRANGIANS
using
to change gi, (but not the physics) according to gives
gij+ gij+ vi!fj+ vjfi
(C7)
In the latter equation the t’s are just numbers; hence we have only 4 conditions that we can impose. The amplitudes h,O, h,O, and Ir11 appear only in the combination G = -c~P(h,~ + h33) + 2q,dhl”
- w,w0h,’
- w~w~~~O.
(C9)
In the usual case (Gzk = 0), the first term is zero, since Go0 + G,l = 0; hence two coordinate transforms (i.e., choice of to and E,) are sufficient to set h,O = h,l = ho0 = 0, provided that w. or w1 # 0 (otherwise we could transform away the gravitational potential of weak static field as well). The choices of rz and t3 can be used to set ho2 = h12 = ho3 = Al3 = 0, leaving only the two orthogonal transverse polarization amplitudes /I,~ = -h33 and h32, both satisfying the dispersion relation w2 = wow0 + wlwl Returning
= 0.
to (C3), we see from the trace relationship (1 - 35~~) G = 0.
If we were to chose wz = 0, the same argument If instead we chose u2 = +1/(3t-L
(ClO)
that K-11)
given above would go through. (CW
then G need no longer vanish. It follows then that we will be unable to eliminate all three of /zoo,h,O, A,‘. Since (C12) corresponds to a particle with mass, it follows that a purely transverse wave would not be Lorentz invariant; thus we need, in fact, contributions from the longitudinal amplitudes. The simplest invariant choice is just h = ho” = /z,l = h,2 == h 3,3 W3) and this choice satisfies the full set of field Eqs. (C4). The appropriate coordinate transforms are to set ho1 = ho2 = ho3 = 0 and ho0= h,l. Substitution into yoo + c,@l1 = 0 gives ho0+ h11 = hz2 + hs3 and finally $22 - $33 = 0 gives i122= h33.
298
MICHEL ACKNOWLEDGMENT
The author benefitted from the many valuable comments offered by Professors J. N. Bahcall and F. Dyson. This research was supported in part by the National Science Foundation, under Grant GP-25854, and in part under Grant GP-16147 A # 1. The author would &o like to thank the Director of the Institute for Advanced Study, Carl Kaysen, and the members of the School of Natural Sciences for their kind hospitality.
REFERENCES AND FOOTNOTES 1. E. PECHLANER AND R. SEXL, Comm. Math. Phys. 2 (1966), 165; also contains references to earlier work. 2. Cf. L. D. LANDAU AND E. M. LIFSHITZ, “Classical Theory of Fields,” Addison-Wesley Reading, MA, 1962. Usual notation; however, we set c = 1. 3. L. D. LANDAU AND E. M. LIFSHITZ, “Statistical Physics,” Section 58, Addison-Wesley, Reading, MA, 1958.