Interior Solutions for Non-Singular Gravity and the Dark Star

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UTPT-94-10, gr-qc/9405065

Interior Solutions for Non-singular Gravity and the Dark Star

alternative to Black Holes.

Neil J. Cornish

Department of Physics, University of Toronto

Toronto, Ontario M5S 1A7, Canada

Abstract

The general equations describing hydrostatic equilibrium are develop ed for

Non-singular Gravity. A new typ e of astrophysical structure, a Sup er Dense

Ob ject SDO or \Dark Star", is shown to exist beyond Neutron star eld

strengths. These structures are intrinsically stable against gravitational col-

lapse and represent the non-singular alternative to General Relativity's Black

Holes.

Typ eset using REVT X

E 1

I. INTRODUCTION

Recently it was discovered that a generalisation of Einstein's theory of General Relativity

GR yields an analog of the Schwarzschild solution which is everywhere non-singular [1].

This remarkable result was found for a hitherto neglected sector of Mo at's Non-symmetric

Gravity Theory NGT [2]. The neglected degrees of freedom were found to a ect drastically

the strong eld b ehaviour of the gravitational eld. The smallest, non-vanishing contribution

from this sector removes event horizons and renders the curvature everywhere nite. This

sector of NGT has b een dubb ed Non-Singular Gravity theory or NSG in order to distinguish

this new physics from the singular truncation of NGT studied in the literature to date.

The results in Ref. [1] have now b een generalised to include the analog of the Reissner-

Nordstrom solution. It was found that the electric eld is everywhere non-singular and b oth

the electromagnetic energy density and spacetime curvature are everywhere nite [3].

These results immediately raise several questions: 1 What do static structures lo ok

like? 2 Can they undergo gravitational collapse? 3 What is the end result of such a

collapse? This pap er is primarily devoted to answering the rst of these questions, although

tentative answers are prop osed for all three questions. It will be shown that two distinct

classes of interior solution exist. The rst class is very similar to the usual GR solutions,

while the second class describ es a new and unusual Sup er Dense Ob ject SDO. The GR

typ e structures are unstable against gravitational collapse, although less so than their GR

counterparts. The SDOs on the other hand are intrinsically stable since their binding energy

tends to zero as they b ecome more compact. Stable SDOs only exist with radii in a small

range near their \Schwarzschild radius" R = 2M . SDOs with radii much less or much

greater than this value are unstable against gravitational expansion. For these reasons,

it is conjectured that SDOs represent the non-singular alternative to GR's Black Holes.

Astrophysical phenomena such as Quasi-Stellar Ob jects QSOs and Active Galactic Nuclei

AGNs may well be SDOs.

The outline of this pap er is as follows: First the eld equations are presented. From these

the equations describing hydrostatic equilibrium are derived. The equilibrium equations are

then studied in a small region near the origin, where the existence of two geometrically

distinct branches of solution is established. Each branch of solution is then studied in turn

using a mixture of analytic approximations and numerical surveys. Physical b ounds on the

\skewness parameter", which controls the departure from GR, are obtained by considering

Neutron stars.

II. FIELD EQUATIONS

Einstein's General Theory is based on the assumption that the metric of spacetime is

a symmetric tensor. By removing this assumption, Mo at was able to construct a non-

symmetric theory of gravity where b oth the metric and connection are non-symmetric.

The NGT Lagrangian with sources is given by [4]

8 2

L = ; 1 W 8T + W S R + gg

[; ]

3 3 2

where R is the generalized Ricci tensor, W is a Lagrange multiplier, refers to the

torsion-free connection , T is the energy-momentum tensor and S is the conserved

current which gives rise to NGT charge. Square brackets denote anti-symmetrisation and

units are chosen so that G = c = 1 throughout. The eld equations that follow from 1 are

g g g =0; 2

;

p p

[ ]

gg =4 gS ; 3

;

2 1

R = W +8 T g T : 4

[;]

3 2

where = + D , and the tensor D satis es

: 5 S g g g g + g g = + g D g D

[]

The generalised Bianchi identities, which result from the di eomorphism invariance of NGT,

give rise to the matter resp onse equations

p p

1 2

g W S =0 : gT + g gT + T g + g g +

; ; ;

[; ]

; ;

g 3

Taking the matter to b e describ ed by a p erfect uid, the energy momentum tensor is found

to b e [5]

T = + pu u pg ; 7

where u is the uid's four-velo city and and p are the usual internal energy density and

pressure, resp ectively. The Ab elian gauge invariance of the Lagrangian under W ! W +

;

ensures the conservation of NGT current density. This is evidenced in 3 since

p p

[ ]

gS = gg =0: 8 4

;

The NGT charge contained in a sphere of radius r , in spherical co ordinates, is given by

2 0

l r = S g drd d : 9

The static spherically symmetric interior case was studied for the truncated version of

[ ]

NGT bySavaria [6]. In this work the pure divergence free parts of g were dropp ed. Even

with this truncation, the analysis was more complicated than the usual GR case due to the

inclusion of additional source terms arising from the NGT charge. In what follows, matter

will be taken to be NGT charge neutral so that S = 0 and l r = 0. This condition

explicitly enforces the strong equivalence principle.

The most general, static spherically symmetric metric for NGT was found byPapap etrou

[7] to be 3

1 0

r w r 0 0

C B

w r r 0 0

C B

C : 10 g = B

A @

0 0 r f r sin

0 0 f r sin r sin

Taking l r to be zero demands w r = 0. The eld equations are further simpli ed by

making the allowed co ordinate choice r = r . Both of these simpli cations will be made

in the derivation of the hydrostatic equilibrium equations.

III. EQUILIBRIUM EQUATIONS

The eld equations can b e most compactly expressed in terms of the following quantities:

4 2 2

A = log r + f ; B = arctanr =f : 11

The usual GR expressions are recovered when A = 2 log r and B = =2. With w r = 0,

the eld equation 3 is automatically satis ed. This is a generic feature of NSG, as the

[ ]

divergence-free part of g is the NSG sector of NGT. It was these divergence-free degrees

of freedom which were dropp ed in previous work on NGT. The tt, rr, and

comp onents of 4 yield

0 0

0 00

+ A log =8 +3p ; 12 log log log

00 0

00 0 0 0 2 0 2

2A + A log A +B log + =8 p ; 13 log log

! ! !

0 2 0 0 2 0 2 0 0

fB r A fB r A r B fA

2 0

1+ + log =4r p ; 14 + B

2 2 4

! ! !

2 0 0 0 2 0 2 0 0

r B + fA fB r A r B + fA

+ log = 4f p : 15 B q +

2 2 4

The constant q comes from the Lagrange multiplier eld W =3q cos =2, and prime denotes

derivatives with resp ect to r . One of these equations may be replaced by the r comp onent

of the matter resp onse equation 6, which simpli es to read

0 0

p = + plog : 16

This relation is identical to its GR counterpart. The agreement with GR follows from NSG

ob eying the strong equivalence principle when the NGT charge is zero.

The set of equations 12-15 are b est studied as three useful combinations. The rst

combination is f times 14 plus r times 15, which yields

! ! !

4 2 0 4 2 0 4 2

r + f B r + f B r + f

2 0 0 0

f + qr + A B + log =0: 17

2 2 4 4

The second combination is f times 15 minus r times 14, which yields

! ! !

4 2 0 4 2 4 2 0

r + f A r + f r + f A

2 0 2 0 4 2

fq r + A + log =4 r + f p :

2 2 4

4 2

The third useful combination is 13 plus 12 minus 4 =r + f times 18 which gives

2 4 2 0 0 2 0 2 00

r fq 1 r + f A A B 2A

2 + =16 : 19

4 2 4 2

r + f r + f 2

These three equations, along with the conservation equation 16, provide the most useful,

complete set of eld equations describing hydrostatic equilibrium in NSG.

Starting with the rst of these combinations, it is straightforward to verify that f = qr

solves equation 17. With this form for f , all the eld equations collapse to be identical

to their GR counterparts. Unfortunately, this simple solution is incompatible with the

external Wyman solution [8,1] the NSG analog of the Schwarzschild metric since having

[ ]

q non-zero leads to a jump discontinuity in the curvature invariant R R at the edge

[ ]

of the structure. This follows from the fact that W = 0 for the Wyman solution, while

W =3q cos =2 in the interior. Consequently, the matching of these solutions at the edge

of the structure demands q =0.

The complexity of the eld equations makes it dicult to combine them into a simple,

fundamental hydrostatic equilibrium equation. However, considerable insightinto the nature

of the interior solutions can b e gained by considering p ower-series solutions ab out the origin.

Taking the following expansions:

p = p + p r + p r + ::: 20

0 1 2

= + r + r + ::: 21

0 1 2

= + r + r + ::: 22

0 1 2

= + r + r + ::: 23

0 1 2

f = f + f r + f r ::: 24

0 1 2

and substituting them into 16-19 leads to two distinct branches of solution. The rst is

characterised by having = 1 and f = 0, while the second branch has = 0 and f 6=0.

0 0 0 0

The rst branchisvery similar to the usual interior solutions found in GR, while the second

branch is unlike anything seen b efore.

Physically, the rst branch can be thought of as a matter elds p erturbing Minkowski

space, while the second branch lo oks like matter elds p erturbing the Wyman metric. The

existence of this second branch will be seen to be intimately related to the non-singular

nature of the theory. The fact that there are two branches of solution do es not lead to any

lack of uniqueness in the description of a structure since the two branches describ e ob jects

of very di erent eld strengths. For example, the Wymanian branch of solutions cannot

describ e the sun, or a white dwarf or even a neutron star. Instead, this branch describ es

sup er dense ob jects SDOs, which typically have radii of R 2M . Each branch will now

be considered in turn. 5

IV. MINKOWSKIAN SOLUTIONS

The Minkowkian-typ e solutions have expansions ab out the origin which are very similar

to their GR counterparts:

2 2 5

3 2 2

f = r 1+ p + r + ::: ; 25

0 0

3 5 8

+ p +3p r + ::: ; 26 p = p

0 0 0 0 0

1+ = +3p r + ::: ; 27

0 0 0

2Mr

; 28 = 1

where

11 5 13 21

2 2 2 2 2

+ p + x + ::: x dx : 29 Mr =4 +

0 0

8 12 4 16

The ab ove quantity is related to the gravitational mass b eneath radius r . The relation is not

exact, however, as the matching at the edge of the structure leads to a complicated relation

between MR and the mass of the structure M . The skew eld, f , increases the mass of

a structure ab ove that of a structure with a similar matter distribution, r , in GR. From

29 it is clear the mass increase results from the energy density of the skew eld. Since

has the dimensions of energy density, it is convenient to parameterise this quantity as

some fraction of the central density:

= : 30

Since the contribution to the e ective energy density from f increases from 21 =4 with

increasing r , it is no surprise that even a small value of causes signi cant changes. Nu-

merical studies reveal that large departures from the usual GR results o ccur once reaches

0:12M=R. The fact that the critical value for scales with the dimensionless \grav-

cr it

itational p otential" 2M=R is in keeping with the general physical picture of NSG. Large

departures from GR o ccur when the gravitational p otential is large. This relationship is

made more precise by considering the matching of the interior solutions to the exterior

Wyman metric.

A. Matching Conditions

The b oundary between the interior and exterior solutions o ccurs at a radius r = R

de ned by pR =0. The exterior Wyman metric is describ ed by

= e ; 31

2 0 2 2

M e 1 + s

= ; 32

cosha cosb 6

2M e [sinh a sin b +s1 cosh a cos b ]

f = ; 33

cosha cosb

where

s s

p p

2 2

1+s +1 1+s 1

; b = ; 34 a =

2 2

and is given implicitly by the relation:

= cosha cos b 1+s sinha sin b : 35 e cosha cosb

The dimensionless quantity s arises as a constant when solving the vacuum eld equations [8].

Unlike the constantofintegration M , which is related to a Gaussian integral of a conserved

charge, the constant s is not connected with any conserved quantity in the theory. Unlike

M , there is nothing in the theory which could explain why s might vary from one body

to another. For this reason, it is reasonable to exp ect that s is a fundamental constant of

nature. This conjecture is supp orted in the weak eld regime where s is seen to b e a coupling

constant setting the relative strength of the skew and symmetric elds. In what follows, s

will be taken to b e a universal constant of nature.

Formally matching the solutions at r = R is not very enlightening, since is only known

as an implicit function of r . More can be learnt by matching onto a small M=r expansion

of the Wyman metric, where the metric functions take the near-Schwarzschild form:

2 5 2 6

2M s M 4s M

=1 + + + ::: ; 36

5 6

r 15r 15r

2 5 2 6 2 4

7s M 87s M 2M 2s M

+ + + ::: = 1 ; 37 +

4 5 6

r 9r 9r 45r

2 3 4

sM 2sM 6sM

+ ::: : 38 f = + +

3 3r 5r

These expansions are only valid for 2M=r 1, which is not a serious restriction seeing as

most structures have 2M=R << 1.

Matching and f at r = R reveals:

2 4

s M 15 7

M = M + 11M + R +3p R + ::: ; 39

0 0 0 0

18R 4 15

2 4

4M 4 s M

2 2

1+ + 3p 5 R + ::: ; 40 =

0 0

9R R 15

where M is the usual GR expression for the mass

GR 2

M = M =4 r dr : 41

0 7

Since the corrections to the GR expression for the mass of the structure are always p ositive,

the gravitational mass of a body in NSG will exceed that of a body with a similar density

distribution in GR. This increase in mass is primarily due to a decrease in the binding energy,

which o ccurs as a result of the repulsive contributions to the gravitational force coming from

the skew sector. If s is to o large, the repulsive contributions from the skew sector can cause

a structure to b ecome unb ound. More mo derate values of s serve to stabilise a structure

relative to its GR counterpart. The leading term in 40 can be used to express in terms

of s:

M 4

= s ; 42

27 R

where is the \average density" of the structure,

= : 43

The ab ove expression relating and s indicates that the critical value of s for which large

departures from GR b egin to be seen will scale as s R=M . This indicates that the

cr it

most stringent b ounds on s will come from the study of Neutron stars, which is no surprise

considering Neutron stars are the only observationally veri ed astrophysical ob jects which

require a relativistic description. The numerical results will show that if s exceeds s 20,

Neutron stars b ecome unb ound.

One unusual feature of the matching at r = R is that the matching of the metric functions

do es not ensure the matching of the gradients of the metric functions. In GR, any pressure

and density pro le which smo othly approaches zero at the edge of the structure leads, via

0 0

the eld equations, to a smo oth matching of and once and have b een matched.

This in turn ensures that all curvature invariants smo othly match at r = R , which is the

physically imp ortant condition. In NSG the curvature invariants can smo othly match at

the edge of the structure even though there are jump discontinuities in the gradients of the

metric functions, so long as these jumps ob ey certain relations which can be derived from

the eld equations. For example, the jump in is given by the relation

! " "

4 2 3 0 0 0

h i

R f 8R f

0 0 3

= f 2f ; 44 [f ] 4R

4 2 4 2

R + f R + f

where square brackets denote the jump in the enclosed quantity at r = R . In the GR limit,

0 0

f = 0 and the ab ove equation yields [ ]=0. In NSG there mayormay not b e a jump in ,

0 0

dep ending on whether there is a jump in f . For the Minkowskian branch of solutions f is

always p ositive in the interior while f is always negative in the exterior if r>M, leading

to a jump in f . Equation 44 serves as a useful check on the accuracy of the numerical

solutions, and was found to b e satis ed within error tolerances.

B. Binding Energy

The internal energy of a body is de ned in the usual way to be E = M m N , where

m is the rest mass of the N constituent and N is the conserved particle number for this

constituent: 8

Z Z

4 2

N = gJ r r + f r nr dr ; 45 dr d d = 4

and nr is the prop er number density for this constituent. The total internal energy can

then b e broken up into its thermal and gravitational comp onents as E = T + V . The thermal

energy is given in terms of the prop er internal material energy density, er = r m nr ,

4 2

T = 4 r r + f r er dr ; 46

while the gravitational p otential energy V is given by

4 2

V = M 4 r r + f r r dr : 47

The gravitational binding energy, = V , is always smaller in NSG than it would be for

the same mass distribution in GR. To leading order, the binding energy is given by

2 3 GR

R ::: =

= M :::

M 15Ms

::: ; 48 =

72 R

which explains why a large value of or s causes structures to b ecome unb ound. For

p olytrop e equations of state:

1+1=n

p = ; 49

the Newtonian expression for the binding energy b ecomes

3 M

Newt:

= ; 50

5 n R

which can be combined with 48 to give an expression for the value of s for which the

structure b ecomes unb ound:

R 72

s = : 51

55 n M

The largest mass White Dwarfs are well approximated by a n = 3 p olytrop e, and have

4 3

M=R 4 10 , which b ounds s to be b elow s = 6:7 10 . The ma jor source of error

in 51 comes from the approximate expression for the binding energy, which can be o by

a factor of two for White Dwarfs. The second term in 48 is essentially exact for White

Dwarfs and agrees with the numerical results within 0.1. A plot of the dep endence of =M

on s is displayed in Fig. 1. for a White Dwarf with M=R =2 10 . 9

Ω/M FIGURES 3

2.5 (b)

1.5 (a)

0.5

0 0 50000 100000 s

FIG. 1. =M 10 as a function of the skewness constant s for a White Dwarf with central

density =1:45 10 km . The lower line, a, is the exact numerical result while the upp er

line, b, is the analytic result employing the Newtonian approximation for the binding energy.

The largest mass Neutron stars can be approximated by a n =3=2 p olytrop e, and have

M=R 0:1, which gives the far more stringent b ound on s of s = 20. This value of s will

1 1

be roughly 30 to o large due to the approximate value used for the binding energy. Since

Neutron stars are known to exist, and since they are well describ ed by GR, a tighter b ound

on s can be obtained by requiring that the repulsive force causes at most a 10 reduction

in the binding energy. This then gives a maximum allowed value of s =6. Limiting NSG

crit

to b e a1 correction over GR for Neutron stars gives s = 2, and so on. crit

f 0.025 (a)

0.02

0.015 (b)

0.01

0.005

r 0

00.2 0.4 0.6 0.8

1=2

FIG. 2. f r in the interior of a Neutron star of radius R = 0:8322 = 12:5 km when

s =8:9. The upp er line, a, is an analytic approximation, while the lower line, b, is the exact

numerical result.

Even if s is large enough to cause a 20 reduction in the binding energy, numerical 10

evaluation shows that the density pro le is almost identical to that in GR, with the di erence

in r never exceeding 0:1 anywhere in the structure. These studies were done using the

equation of state

p = ; 52

where =1=72 km is a critical density. Ab ove this density the equation of state go es

over to a n = 1 p olytrop e. Cho osing a central densityof =0:2 and taking s = 0 yields

0 c

a Neutron star with mass M =0:0782, radius R =0:8325 and binding energy =0:0074

1=2

in units of . From 51, these values lead to the prediction that s = 9:7 will cause

a 20 reduction in the binding energy. The numerical results showed a 20 change was

caused byhaving s =8:9, and gave rise to a Neutron star with mass M =0:0798 and radius

R = 0:8322. Since the graphs of ; ; and p are essentially the same for b oth GR and

NSG, the only function worthy of display is the skew eld f . A plot of f is displayed in

Fig. 2. for the ab ove mentioned Neutron star. The exact numerical result is shown, along

with the power series approximation given in 25. Even for relativistic situations, such as

Neutron stars, the approximate solution for f gives remarkably go o d results.

It is interesting to note that the strongest b ounds on the coupling of matter to NGT

charge came from Neutron star calculations, where the NGT charge caused an increase in

the binding energy [9]. The tightest b ound corresp onded to a 25 increase in the binding

energy. If b oth sectors of NGT are taken into account, this b ound is removed since a

mo derate value of s can cause a comp ensatory 25 decrease in the binding energy. This

p ossibility is worthy of further investigation.

If s is a fundamental, dimensionless constant of nature, like the ne structure constant,

it is not unreasonable to exp ect s to lie in the range 0:001 < s < 1. For this range of

values, NSG has essential ly no e ect on the description of any known astrophysical object,

save Black Holes. Since b oth the Chandrasekar and Opp enheimer-Volko limits for the

mass of White Dwarfs and Neutron stars rely on quasi-equilibrium reasoning, and since the

equilibrium description of these structures is unchanged, the usual sequence of gravitational

collapse will o ccur. The departure of NSG from GR comes as the collapse continues, not to

a Black Hole, but to the SDOs to b e describ ed in the next section.

V. SUPER DENSE OBJECTS

The second branch of interior solution is characterised by having a rich geometrical

structure near the origin. The metric near r =0 is of the form

2 4

= + r + O r ; 53

0 2

2 4

= r + O r ; 54

2 4

f = f + f r + O r ; 55

0 2

so that the prop er volume scales as V r , while two-surfaces of constant r have surface area

S = 4r . The curvature invariants are all constant near the origin, and are prop ortional

to S=V . For the vacuum Wyman metric the leading co ecients are given by

p 11

2 + O s ; 56 = exp

s 8

= 1+O s ; 57

2 2

M s

2 3 2

+ s + O s ; 58 f = M 4

where a small s expansion has b een employed. It should be noted that the limit f ! 0

is achieved by taking M ! 0, not s ! 0. The Wymanian interior solutions p ossess the

same geometry near the origin, although the co ecients in the expansion are a ected by

2 2

the presence of matter. Interestingly, the ratio 4 = = s M remains almost completely

0 2

unchanged in the presence of matter. This serves to highlight a ma jor technical dicultyin

obtaining numerical solutions to the eld equations for this branch of solution. The standard

metho d for obtaining numerical solutions relies on power series solutions near r = 0 to

initiate the integration out from the centre of the structure. This is usually no problem

since the region near r = 0 is essentially Minkowski space, and no a priori knowledge of

global parameters, such as the structure's mass, is required. For the Wymanian branch of

solutions this is not the case since the mass of the structure must b e known b efore it can b e determined!

8*10^6

6*10^6

4*10^6

2*10^6

r 0

00.2 0.4 0.6 0.8 1 1.2

FIG. 3. The density pro le r for a typical \neutron SDO".

An obvious solution to this quandary is to break with convention and integrate from

the edge of a structure in toward the centre. This can be successfully accomplished in GR

by sp ecifying a mass and radius and integrating to nd the corresp onding central density

although unphysical choices for the mass and radius will give rise to negative central den-

sities. The success of this op eration in GR is predicated on the fact that smo oth matching

of and automatically results in the smo oth matching of all gradients. In NSG the jump

in the gradients must also be sp eci ed. For the Minkowskian branch of solutions non-zero

jumps could not be avoided, making this metho d dicult to implement. Thankfully, the

Minkowskian branch can be treated using the conventional metho d. For the Wymanian

branch of solutions f has the same sign in the interior and exterior solutions, so it is not 12

0 0

imp erative that there be a jump in f . In the absence of a pro of that the magnitudes of f

should also match, it is dicult to pro ceed. While there is no guarantee that the derivatives

match, there is no doubt that cho osing all gradients to match is one acceptable p ossibility.

This assumption will now be made in order to pro ceed.

A wide numerical survey was made in order to cover a range of equations of state and

values of s. While the details di er, all SDO solutions studied did t a standard picture. The

generic features include a decrease in binding energy for radii ab ove or b elow a central value,

R , which typically lay in the range 1:1 M < R < 2:1 M . The central density exhibited

similar b ehaviour, but always p eaked at a slightly smaller value of radius, R < R , and

dropp ed more rapidly for larger radii. SDOs with radii slightly larger than R had density

pro les which grew and then dropp ed, while SDOs with radii b elow R had monotonically

increasing density pro les. A comparison of binding energies at xed for SDOs with

radii on either side of R , revealed that the SDOs with radii less than R were more tightly

b ound and thus energetically favoured. This is in agreement with the usual exp ectation

that stable structures must have monotonically increasing density pro les. Of course, usual

exp ectations should not b e relied up on for these unusual ob jects. Varying the value of s did

not change this overall picture, although it did increase R and R somewhat, and tended

to reduce p eak binding energies which could reach huge values for small values of s.

f 3.85

3.8

3.75

3.7

3.65

3.6

3.55 r

0 0.2 0.4 0.6 0.8 1 1.2

FIG. 4. The skew function f r inside a typical \neutron SDO".

The most imp ortant of the ab ove prop erties is the intrinsic stability of all SDOs against

gravitational collapse. If one tries to crush a SDO into a smaller radius its binding energy

decreases, and like a cosmic rubb er ball, the SDO will spring back to its equilibrium radius.

This intrinsic stability allows stable SDOs to exist for any mass. SDOs represent a non-

singular endp oint to gravitational collapse, and provide nature with a sane alternative to

GR's Black Holes. While a SDO resides deep inside a gravitational well, and can have a

surface redshift so large that it is essentially a \black star", the gravitational gradients are

molli ed to the extent that matter pressure is able to supp ort the structure.

In order to illustrate these general features, the density pro le and skew eld are displayed

for a \neutron SDO". The equation of state is designed to roughly mo del a relativistic core 13

coated with a crust of non-relativistic neutrons:

; 1

p = 59

; <1 :

Here units have b een chosen where = 1. The density pro le shown in Fig. 3. is for an

SDO with s =0:1, M =1, R =1:298. This SDO had a central density of =9:63 10 ,

and a binding energy of =26:02. The skew eld f is displayed in Fig. 4., and illustrates

the primary di erence between the Wymanian and Minkowskian branch of solutions. The

Wymanian branch is characterised by having a large central value of f , with f 4M .

The eld undergo es a small initial increase b efore dropping sharply toward the edge of

the structure. The Minkowskian branch, on the other hand, has f starting from zero and

sharply increasing. Interestingly, the energetically disfavoured SDOs with radii ab ove R

have skew functions which start near zero and increase to a p eak value b efore decreasing.

This suggest a series of quasi-equilibrium con gurations may exist b etween the Minkowskian

and Wymanian branch of solutions, although time dep endent collapse calculations will have

to be p erformed to verify this conjecture. Work is currently in progress [10] to study the

evolution of SDOs through gravitational collapse in NSG.

The aforementioned neutron SDO has a surface redshift of z =2:09 10 , which suggests

the name Dark Star is appropriate for these ob jects. As s tends to zero, the surface redshifts

asso ciated with SDOs tend to in nity, making Dark Stars as black as Black Holes.

The neutron SDOs with s = 0:1 b ecome unb ound for radii less than R 1:264. The

dep endence of binding energy and central density on radii was studied in detail for the sti

equation of state:

: 60 p =

This involved nding the density pro les for over fty di erent radii in order to pro duce

plots of R and R . 0

Ω

0 R

1.6 1.7 1.8 1.9 2

FIG. 5. The binding energy as a function of radius R for a SDO with a sti equation of state. 14

For most choices of s and equation of state fewer radii were studied, since it was sucient

to check that the general picture was the same. In Fig. 5. the binding energy is plotted

against radius for this sti equation of state with s = 0:1. A similar plot of the central

density versus R for these SDOs is displayed in Fig. 6. For radii beyond R = 2:04 a new

complication enters the picture since the central density is zero and there is a region of

empty space near the origin. This would require an inner and outer matching to the vacuum

Wyman solution, although this p ossibilitywas not pursued since such annular distributions

b ecome unb ound.

ρ 0 1600 1400 1200 1000 800 600 400 200 0

1.6 1.7 1.8 1.9 2 R

FIG. 6. The central density as a function of radius R for a SDO with a sti equation of state.

The ab ove description of SDOs is at b est a rough sketch, based on a wide, but by no

means exhaustive, numerical survey. Since little is known ab out the b ehaviour of matter for

the kinds of densities b eing considered for SDOs, it is dicult to cho ose a sensible equation

of state. On the p ositive side, the generic features describ ed for SDOs do not seem to be

greatly a ected by the choice of equation of state. The p ossibility that the gradients of ;

and f exhibit jumps at the edge of some SDOs remains to be investigated. It is p ossible

that a pro of can be found to show the Wymanian branch of solutions never exhibit jumps

in metric gradients. No work has yet b een done to study the feasibility of describing QSOs

or AGNs in terms of SDOs, although the prosp ects are very good. In short, much work

remains to b e done to understand these fascinating ob jects.

VI. DISCUSSION

The non-singular gravitational theory describ ed in this pap er repro duces every prediction

of GR for weak to mo derate gravitational elds to any desired accuracy taking s small

enough. Any exp erimental prediction based on systems with 2M=R < 1 will fail to tell

NSG and GR apart, unless s is larger than 1. If s is larger than this value there is some

hop e that detailed studies of Neutron stars may tell the two theories apart.

The only unique signal for anyvalue of s would come for the sup er dense ob jects b elieved 15

to exist at the centre of most galaxies. If these galactic nuclei have masses ab ove 10 M ,

GR predicts that a star passing by will be torn apart by tidal forces once it is within the

horizon of the massive Black Hole, so the star would vanish without a trace. If the galactic

nuclei was a SDO on the other hand, the star's death would b e visible [11]. Unfortunately,if

s is to o small, the radiation given o as the star is torn apart would b e so heavily redshifted

that no signal could be seen ab ove background.

In the absence of exp erimental tests to tell NSG and GR apart, there is only the fol-

lowing aesthetic distinction: Any non-zero value of s leads to a non-singular endp oint to

gravitational collapse - a Sup er Dense Ob ject or Dark Star. In contrast, GR is always fated

to have a singular endp oint to collapse - a Black Hole. Why cling to the concept of a Black

Hole when the smallest imaginable change to GR banishes these ob jects from our picture of

the universe?

In nities b elong in mathematics, not nature.

ACKNOWLEDGMENTS

I am grateful for the supp ort provided by a Canadian Commonwealth Scholarship. I

thank Norm Frankel, Janna Levin, John Mo at and Pierre Savaria for their interest in this

work and for carefully reading the manuscript. I would also like to thank Dick Bond and

Glenn Starkmann for several useful discussions concerning NSG. 16

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[5] Vincent D., Class. Quant. Grav. 2, 409, 1985.

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[11] I thank Scott Tremaine for this suggestion. 17