P H Y S I C A L R E V I E W D V O L U M E 1 5 , N U M B E R 1 2 1 5 J U N E 1 9 7 7

Neutron-star mass limit in the bimetric theory of gravitation

G. Caporaso and K. Brecher Department of Physics and Center for Space Research, Massachusetts Institute of Techrzology, Cambridge, Massachusetts 02139

(Received 23 February 1977)

The "neutronm-star upper mass limit is examined in Rosen's bimetric theory of gravitation. An exact solution, approximate scaling law, and numerical integration of the hydrostatic equilibrium equation show the dependence of the mass limit on the assumed equation of state. As in general relativity, that limit varies roughly as 1/2/5, where p, is the density above which the equation of state becomes "stiff." Unlike general relativity, the stiffer the equation of state, the higher the mass limit. For p, = 2 X 10''' g/cm' and P = ( p - po)c ', we found M,,,,, = 81 M O . This mass is consistent with causality and experimental tests of gravitation and nuclear physics. Fo r d p / d p > c it appears that the upper mass limit can become arbitrarily large.

I. INTRODUCTION

The m a s s of a neutron s t a r a r i s ing from a part ic- u la r choice of equation of s tate and gravitational theory i s of in te res t f o r two reasons. F i r s t , s ince there a r e now severa l neutron s t a r s with approxi- mately known m a s s e s , one can begin to compare observational data and theoret ical models to s e t constraints on the equation of s ta te P = P ( p ) , the theory of gravi ty, o r both. Second, s ince the bes t c a s e fo r the existence of black holes depends cru- cially on the value of the upper m a s s l imit , i t is important to know what this limiting m a s s M,,, is and on what assumptions i t depends. Many papers have appeared in the l i t e ra ture on this subject (for a recent review s e e Brecher and Caporaso'). In a previous paper2 we have examined the effects of varying P(p) in general relativity. In this paper we wish to examine the effect on M,, of varying P(p) in a n alternative theory of gravity and compare these resu l t s with the general-rela- tivistic case.

The bimetr ic theory proposed by Rosen3 appears to be the only current ly viable al ternat ive theory of gravity to genera l relativity4 [other than the par- ametr ized post-Newtonian ( P P N ) theories cont s t ructed explicitly to satisfy the classic t es t s , but which a r e otherwise without physical foundation]. It conforms to a l l of the experimental t es t s . Fur - thermore , it does not permi t the existence of black holes. It does appear to allow binary sys tems to emi t "negative-energy" gravitational waves causing the s t a r s to move apar t with time.5 A s a mat te r of principle, this may seem m o r e disturb- ing than violations of par i ty o r of t ime- reversa l in- var iance. However, we take the point of view that only logical inconsistencies o r observationally in- c o r r e c t predictions should be used a s arguments against the bimetr ic theory, neither of which exis t a t present . In this paper , we will be concerned only with part icular consequences of the bimetr ic

theory. At the very leas t , they will help to clarify the very s ingular predictions of general relativity. F o r fur ther detai ls and some unique fea tures of the theory itself, we r e f e r the r e a d e r to the original papers of ~ o s e n . ~ ' ~

The m a s s e s of neutron s t a r s in the bimetr ic theory w e r e previously investigated numerically by Rosen and Rosen7 f o r a part icular equation of s tate . They found neutron-star m a s s e s about five t imes l a r g e r than those calculated in general relativity with the same equation of s ta te . In a subsequent numerical calculation8 these resu l t s were extended to "stiffer" equations of s ta te . In the present paper we presen t an exact solution for a specific equation of s ta te , an approximate mass- l imit scaling law f o r any equation of s ta te , and the resu l t s of numer- ical solutions fo r th ree different equations of s ta te in the bimetr ic theory. In addition, we compare these resu l t s to some corresponding general-rela- tivistic resu l t s .

11. THE GENERAL PROBLEM

If we choose yp,=q,, = d i a g ( l , -1, - 1 , - l ) , then the line element appropriate to the study of a non- rotating, spherically symmetr ic neutron s t a r may be taken a s 3

ds2 =eZ0dt2 - e2#(dx + dy2 +dz2) , (1)

where + = + ( Y ) , Q = $ ( Y ) , Taking T," a s the s t ress -energy tensor of a perfect fluid, namely

Too = p , T,, = T,' = TS3 = -P , a l l o thers 0 , with density p = p(v) and p r e s s u r e P = P(Y) , Rosen's field equations become7 (in units where G = c 2 = 1)

2 v ~ @ = @ # + - $' = 4 ~ e ' + ~ ~ ( ~ + 3 ~ ) , Y

(2)

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The energy-momentum relations6 TUV,,=O yield the hydrostatic equilibrium equation

It should be noted a t this point that numerical solu- tion of the field equations for a neutron s t a r would involve correct ly choosing the two initial param- e t e r s @, and #, until a point i s found such that both boundary conditions (7) a r e satisfied. However, by making appropriate u se of information contained in (14), this two-dimensional space of t r i a l initial conditions can be reduced to a line.

To accomplish this we define a new variable O(r), following ~ o s e n , ~

Outside the s tar6*? (for r > R , R i s the neutron-star radius)

where

With this definition the field equations (2) and (3) become

and

1 dM 1 d @dB =--=4ne"'3$+@(R)(p+3p), -- r2 d r ( d r ) r2 d r

(Note that M' is not the derivative of M but i s an- other m a s s distinct f r om M which contributes to the deflection of light near the s ta r . ) The boundary conditions for the field equations a r e

@ ' = $ ' = 0 a t r = O , (6) (18) If we now make the transformation

Now, for r <R , where a is a dimensionless constant, we can re - duce Eqs. (17) and (18) to

With these f o r m s for M and M ' we note that (2) and (8) and (3) and (9) yield, respectively,

where B(T) = O(r) ; $(T) = # ( r ) , P(T) = P(r) ; p ( ~ ) = p ( r ) , aM(T) = M ( r ) , and a R f ( T ) = Mb(r), and these func- tions a r e given by

and

Solving for @ ' and q? ' in these expressions and in- tegrating gives

if we choose

= @ - r n ( R ) / Z Now, defining @, = @(0) and $, = $(0) and integrat- ing the hydrostatic equilibrium equation (4) f rom r = O to r = R , we have With (16) and (19) the hydrostatic equilibrium

equation (4) can be written as

where p, is the central density (a t r = 0) and p, is the surface density (at r = R ) and where P is related to p by an equation of s ta te

while (14) and (16) imply that

3538 G . C A P O R A S O A N D K . B R E C H E R 15 -

The boundary condition ( 7 ) on $ now becomes

- d m ) + $ ( X ) = 0 R - dT ( 2 9 )

so that the boundary conditions a r e now 8(W) = 0 and Eqs. ( 2 7 ) and ( 2 8 ) .

To solve a neutron-star s tructure problem an equation of state along with a central and surface density i s chosen. This gives 8, through ( 2 7 ) . Next a +, i s chosen and the equations for p , P , R , A', - 0 , and $ a r e iterated radially ouhvard from P= 0 until a(P) = 0. At this point the boundary condition on $, Eq. (381, is tested. If the boundary condition i s not satisfied to within the desired precision rji, i s changed and the procedure repeated a s often a s necessary until the boundary condition i s met. At this point R , R ( R ) , and R 1 ( R ) a r e known. Now ( 2 6 ) becomes

, , e - ~ ~ ) / z = e E ( X ) / z E . ( 3 0 )

Thus we have

~ ( ~ ) = m ( ~ ) ~ B ( i i ) l 2 B j

M ' ( R ) = i q / ( ~ ) , B ( B ) / z E , R =XeP(i i ) l2B

completing the solution of the problem.

111. AN EXACT SOLUTION

Consider the field equations fo r the equation of state

P = $ ~ . ( 3 4 )

Adding Eq. ( 2 ) to three times ( 3 ) yields

v 2 @ + 3 v 2 + = v 2 ( @ + 3 + ) = 0 . ( 3 5 )

The problem is spherically symmetric so only v dependence i s possible. Thus

@ + ~ $ = c , + c , / Y . ( 3 6 )

Now from ( 1 0 ) we have that

If we integrate and assume that

then

Y " ' ( v ) = M (v) . ( 3 9 )

Similarly

Y 2$'(v) = - I u ' ( ~ ) . ( 4 0 )

Now adding ( 3 9 ) to three t imes ( 4 0 ) and recalling ( 8 ) and ( 9 ) we have

From ( 3 6 ) it i s seen that C , must vanish, that i s ,

@ (v ) + 3 +(v) = C , . ( 4 2 )

The hydrostatic equilibrium equation ( 4 ) and the equation of state ( 3 4 ) yields

Noting the similarity of this equation to the gener- al-relativistic and Newtonian cases for P = p p and reca l l ing that p l / ' v2 i s a solution in both in- stances, we seek a solution of the form

Substituting this into ( 4 3 ) we find that

and

M ( Y ) = $ Y .

Now from ( 1 0 )

Similarly,

so that

@(v) = A + $ l n r ,

$(v) = B - $ lny . Now at some v = R , @ and $ will both satisfy their respective boundary conditions ( 7 ) if a p, greater than zero i s chosen. Then ( 7 ) becomes

Adding ( 5 1 ) and three t imes ( 5 2 ) we obtain A + 3B = 0 . But, by ( 4 2 ) , A + 3B = C,, thus

and

Note from ( 4 9 ) that

This violates the boundary condition (6), but con- dition ( 3 8 ) which was assumed in the derivation i s satisfied. It i s not too surprising that condition ( 6 ) i s violated in this special case a s p, is infinite.

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Now, selecting a density pN below which nuclear physics i s assumed known [i.e., where (34) no long- e r applies] we may find the "core" mass . Insert- ing the appropriate physical constants, (45) and (46) become

Thus (31) becomes

M(R) = i%'(E)e-eo12 . (64)

If we now define an average density 0 by

- - 47rPR3 M(R) z-

3 ' (65)

then

Now if rN is the core radius and

and the core mass M(pN) i s thus

Using (64) and inserting appropriate physical con- stants the mass becomes

where po= 2 x 1014 g/cm3. For P = a p in general relativity, the core mass (for p, = .a) is2

and the radius i s

(60) Fo r P = ip and P= p, respectively, this i s As an illustration consider the equation of s tate

P = apc2 . (70)

Then 8, becomes

Equation (60) has a maximum when a = 1. Thus we immediately see that the core mass for P = ip in the bimetric theory is twice a s large a s the largest core m a s s for P = a p in general relativity. This reinforces the notion7 that neutron s t a r s have la rger masses in the bimetric theory than in genr era1 relativity. As we shall show, we have found that this i s the case for many numerical solutions.

If we put P=p, then M and R can be written a s

where

IV. APPROXIMATE SCALING LAW

Qualitatively, we mav obtain some idea of how the maximum mass of a neutron s t a r depends on the equation of s tate in the bimetric theory. By Eqs. ( l o ) , (12), and (16) we see that

and

f ( X ) = X 3 / 2 e - x / 2 ( l + a ) 7 (75)

g(x) = x l / 2 e - ~ / 2 ( l + a ) . (76)

Maximizing these functions yields

We expect 6, A?, and J? to be related approximately by

-

(63) R .

3540 G . C A P O R A S O A N D K . B R E C H E R 15 -

So the maxima for M and R a r e

1 / 2

Rmax=24.3(z) a l l 2 k m ,

where po = 2 X loi4 g/cm3. With p, = 2 X 1014 g/cm3 we find for a = $ that M G6iZ.1, and R ~ 1 4 km. For a = l , one has Ms31M,, R g 2 4 km.

Note that the disagreement between M here for a = $ and the exact solution found ear l ie r i s due to the fact that the exact solution corresponds to the case of infinite central density, while here the maximum mass occurs when p, = p, exp[3(1+ a ) ] .

Note also, however, that unlike the general- relativistic case,2 where Ill(a) peaks for a = 1 , in the bimetric theory M -a 3 / 2 SO that there does not appear to be an upper mass limit (independent of the equation of s tate) to a neutron s t a r in the bi- met r ic theory. This a 3 I 2 dependence also a r i s e s for Newtonian s t a r s with Y = a p and infinite central densities. Rosen and Rosen7 found that the masses calculated for an equation of s tate with one param- e ter in the bimetric theory were about 5.6 t imes the corresponding maxima in general relativity. Their equation of state was "soft" a t low density but dp/dp approached 1 a s p increased to infinity. That i s , the effective a for their equation of state was l e s s than 1 . Thus they obtained lower masses than the present results.

In Rosen's theory there a r e two distinct masses iZ.1 and M'. As has been noted,' M corresponds to the Newtonian mass and influences orbits of bodies a t large distances from the s t a r , while M' contrib- utes to the deflection of light rays passing close to the s t a r . In fact , to f i r s t order in G M/r0c2 and GM'/Y,C the angular deflection of a ray incident from infinite distance with "impact" parameter ro is, following ~ e i n b e r g , ~

Now recalling Eq. (9) ,

for the case P = a p we see that

As a - 1 (P approaches p) , M ' shrinks relative to M. When a = 1 (the so-called "causality limit"), M ' vanishes and for "superluminal" equations of state,1° where P > p (a > I ) , M ' can become nega- tive. In any case, M > M f s o that (81) will always be smal ler than i t s general-relativisitic counterpart

and, in the extreme case P >> p, M - -3121 ' and

o r $ the general-relativistic value. In this case, however, ~ n / l / r , c ~ i s likely to be of order 1, and thus more t e rms in t he expansion for Acp must be examined.

V. NUMERICAL TECHNIQUES AND RESULTS

We now turn to the numerical solution of the neu- tron-star s tructure problem for any equation of s tate. We f i r s t find 8, from Eq. (28):

Next we choose a radial increment AT and i terate the equations for 8, $, iV, M', and p. A AT of 10 meters was used for the calculations. Equations (22) and (23) in iterated form a r e

Similarly, Eqs. (24) and (25) for IT? and M' become

The hydrostatic equilibrium equation (27) becomes

where we have used the fact that d ~ / d r = (dp/dp) x (dp/dr). The last relation needed i s the equation of state

P = F ( p ) . (91)

F i r s t the equation of state was selected, then p, and p , were chosen, and 8, was found from (85). A +, was chosen and the equations were iterated from v= 0 outward until B(7) vanished. At that point, boundary condition (29) was tested

The prime denotes differentiation with respect to v. By (11) and (23) this i s

If this condition were not satisfied to the desired precision Q0 would be indexed to a new value and the equations would be iterated again until

15 . - N E U T R O N - S T A R M A S S L I M I T I N T H E B I M E T R I C T H E O R Y ... 3541

----- Bimetric Theory

General Relativity

0.1 I I 1 1 1 1 1 1 1 1 I I I l l l l l l I 1 I l l l l l l I 1 1 1 1 1 1 l 1 I I 1 1 1 1 1 1 1

1014 1015 loi6 loi7 10 l8 1019 Central Density ( g r a m s / c m 3 )

FIG. 1. Results of numerical solutions to neutron-star structure for the three equations of state mentioned in the text in Rosen's bimetric theory (dashed) and in general relativity (solid). Note that the masses in the bimetric theory are much larger than in general relativity and that M vs p, peaks at much higher central densities than in general relativity.

(92) was satisfied. The masses M, M' and the ra- sulting masses were o rde r s of magnitude grea ter dius R would then be found from Eqs. (31)) (32)) than that obtained for P = pc2 and the quantitiy and (33). e x p ( ~ ~ ~ / E c ~ ) became on the order of 10'-lo3. The

Curves of M v s p, were found for 3 equations of s tep size was probably too large a t that point to state which have been studied previously in general give accurate results. relativity.' The equations1' used were

P = C ~ ( P - 2 x 1014) , VI. DISCUSSION

P=c2(p -4 .6 x1014) , P = + C ' ( ~ - 2.28 x 1014).

The resulting curves a r e shown in Fig. 1. For comparison the corresponding general-relativistic curves a r e also shown.

Several points were also computed using 1-meter iterations in 7 to test the sensitivity of the results to step size. The points generally agreed to within 1%. The equation of s tate used by ~ o s e n ~ with h = 40 was also used. As a check of the technique the point corresponding to p,= 1.8 x 105p, (the M v s p, maximum) was computed. The resulting M , M ', and R al l agreed with Rosen's results to within a few percent.

The maximum mass obtained was 81 M,. When computations for P = 2pc2 were attempted, the re-

The importance of these results l ies in the fact that they clearly demonstrate that the upper mass limit of a neutron s t a r is critically dependent on the theory of gravity used to calculate it .

It i s conventionally assumed that if a collapsed object with a mass greater than a few M, is detect- ed i t must be ablack hole. Since this is the only way currently to find a black hole, i t i s now c lear that one must simultaneously know which theory of gravity i s correct .

Current gravitational theories have only been tested in the weak-field domain. However, al l vi- able theories of gravity a r e constructed so that they will agree in this domain. But the upper mass limit depends critically on the behavior of gravity in the strong-field limit where there have been no tes t s of gravitational theories. The mass deter-

3542 G . C A P O R A S O A N D K . B R E C H E R 15 -

mined by doing orbital mechanics on a neutron s t a r in a binary system is I\.:!. Therefore in the case of CYG X-1, whose mass is estimatedL2 at between 3 and 10 M,, we see that one cannot draw the im- mediate conclusion that i t must be a black hole.

If one could study the deflection of light, say by a neutron s t a r in a binary system, one should ob- serve a smaller effect than in generalrelativity since M J < n / l ; indeed M ' may even become negative for sufficiently massive neutron s t a r s . Will4 has noted that the binary pulsar PSR1913 + 16 could emit gravitational dipole radiation at a sufficient ra te to produce a detectable change in the orbital period. In the absence of ei ther of these observa- tional tes t s of the theory the m a s s l imits derived here must be taken a s allowable within the present experimentally tested laws of p h y ~ i c s . On the

other hand, if a collapsed object is ever discovered whose m a s s i s grea te r than the mass limit s e t by general relativity, and the object i s found to be a neutron s t a r (because i t is a pulsar , for example), one may reverse the above arguments to support the bimetric theory a s more viable than general relativity.

ACKNOWLEDGMENTS

One of u s (Kenneth Brecher) would like to ex- p r e s s h is thanks to Professor Nathan Rosen for having encouraged this research through his hos- pitality while K.B. was a visi tor at the Technion I s r ae l Institute of Technology. This research i s supported in par t by the National Science Founda- tion through Grant No. AST 74-19213 A02.

'K. Brecher and G. Caporaso, in Proceedings of the EighthTexas Symposium on Relativistic Astrophysics, 1977 (unpublished).

'K. Brecher and G . Caporaso, Nature (London) 259, 377 (1976).

3 ~ . Rosen, Gen. Relativ. Gravit. A, 935 (1973). 4 ~ . Will, 1977 (unpublished). 'c. Will and D. Eardley, Astrophys. J . Lett. 212, L1

(1977). 6 ~ . Rosen, Ann. Phys. (N.Y.) 2, 455 (1974). '5. Rosen and N. Rosen, Astrophgs. J. 202, 782 (1975). 8J. Rosen and N. Rosen, Astrophys. J. 212, 605 (1977). $s. Weinberg, Gravitation and Cosmology (Wiley, New

York, 1972), p. 188. We use Weinberg's equations

(8.5.5), (8.3.8), and (8.3.9) andnotethat (8.4.1) i s equalto (1) to first order in Gnil/rc%nd G M ' / ~ C ~ . Then we identify B (r) and A ( r ) with e-2 and eZM"+ , re- spectively. We then replace y M by M' in (8.5.7) and put r =r , .

"s. A. Bludman and M . A. Ruderman, Phys. Rev. D I, 3243 (1970). Although i t i s clear that no signal velocity can be greater than c it has still not been satisfactorily determined whether o r not P can exceed pc2.

H ~ o r matching to a realist ic equation of state at nuclear densities, neglecting the matching pressure P, makes a negligible change in the resulting mass as we have previously found in general relativity.

12y. Avni, 1977 (unpublished).