THE MODEL OF OPPENHEIMER AND SNYDER OFA COLLAPSING STAR.

By Rainer BURGHARDT.

Abstract. We work out some interesting features of the model of Oppen-heimer and Snyder concerning a collapsing star. We discuss the geometricalbasis of the theory.

Inroduction. In 1939, Oppenheimer and Snyder [1]1 presented a papernow known as that paper which has given rise to the theory of black holes,although the term ‘black hole’ has been introduced much later. Moreover, theOS approach differs from current methods to implement black holes. The OSmodel is composed of a collapsing interior and an exterior solution, where theexterior part is the static Schwarzschild solution, which remains static due tothe Birkhoff theorem even if the field generating stellar object collapses. Mostapproaches to a black hole do not use an interior solution. In this case theexterior Schwarzschild solution or the exterior Kerr solution is extended beneaththe event horizon. The inner regions of these solutions are to describe blackholes.

The OS model is based on an existing cosmological solution by Tolman[2]. The stellar object is made of pressure-free dust with homogeneous density.Since in this case, the internal resistance against a contraction is missing, theobject cannot be static. It collapses as a consequence of its own gravitationalattraction2.

A completely pressure-free star is not physically realistic, because the par-ticles of a star finally come so close during the collapse that pressure can beexpected at a sufficiently high density. A pressure-free stellar object may ap-proximately describe a dying star. If the thermonuclear processes are exhaustedinside a star, they give way to the star’s own gravitational attraction, and thestar collapses. The just-discussed simplification to p = 0 is primarily on practi-cal grounds. The integration of Einstein’s field equations without this conditionleads to considerable difficulties, and it is hard to find an analytical solution.

In addition to these above-mentioned limitations the OS model permits fur-ther criticism. The star collapses with the velocity of observers coming in freefall from infinity. The stellar object at the time t = 0 would have been infinitelylarge. We will work out this in the following. Mitra [3] has shown that incon-sistencies in the OS model can only be resolved if the mass of a hypotheticalOS black hole is M = 0. However, a massless star is contrary to the widespreadopinion that black holes are supermassive objects. For these and other reasons

1Numbers in brackets refer to the reference at the end of the paper2What could have prevented a pressureless star to contract before its collapse has started

remains an open question. A realistic stellar object is kept in balance by inner nuclear andthermodynamic processes. However, this excludes the absence of pressure.

1

which we will point out the OS model does not provide an appropriate basis fora black hole.

The linking conditions on the boundary surface between the interior andexterior solutions, which are not treated extensively in the literature, are alsoof special interest. One has to deal with the question as to why the OS metricthough time-dependent appears to be flat.

§1. The OS interior solution, basic relations. The first considerationsare closely related to the original paper by OS, but we have to introduce auxil-iary variables which are in close connection with geometrical quantities. Both,the interior and the exterior solutions are treated in two different coordinatesystems: the one is comoving with the collapsing matter and the other one isnot comoving. It is the very transition between the two systems which bringsinsight into how the collapse proceeds, and thus, sheds light onto the inconsis-tencies of the model. Our first effort will be to examine the quantities and therelations of the OS-model, to regroup them, and to get contiguity with familiarmatters, and to prepare a geometric interpretation [4].

For the comoving and for the non-comoving coordinate system we use thenotations

{r′, ϑ, φ, t′} , {r, ϑ, φ, t} .

In the comoving system we write the line element according to OS as

ds2 = eωdr′2+eω(dϑ2 + sin2 ϑdφ2

)−dt′2, ω = ω (r′, t′) , ω = ω (r′, t′) . (1)

For the two metric factors OS put

eω = (G+ Ft′)4/3

, eω =1

4eω(∂ω

∂r′

)2

, (2)

wherein one has for the interior solution

G =√r′3, F = −3

2

√2M

√r′3

r′g3 (3)

with r′g the value of r′ on the surface of the stellar object, i.e. on the boundaryof the interior and exterior solutions. We note the auxiliary variables

R′g =

√r′3g2M

, ρ′g = 2R′g, Λ = 1− 3

ρ′gt′. (4)

As in the non-comoving system the lateral part of the metric has the form

r2(dϑ2 + sin2 ϑdφ2

),

and this form is conserved under a coordinate transformation between these twosystems a comparison with (1) gives

r2 = eω.

2

With (2) and (4) we obtain the relation

r = Λ2/3r′. (5)

From the point of view of the comoving observer the radial coordinate of thesurface does not change

∂r′g∂r′

= 0,∂r′g∂t′

= 0.

For the quantity Λ one gets the relations

∂Λ

∂r′= 0,

∂Λ

∂t′= − 3

ρ′g, (6)

which we need for some calculations. For the radial coordinate of the non-comoving observer system one gains, taking advantage of (5) and the aboveformulae,

∂rg∂r′

=∂

∂r′

(Λ2/3r′g

)= 0,

∂rg∂t′

=∂

∂t′

(Λ2/3r′g

)= − rg

Rg= −

√2M

rg. (7)

In this calculation the relations

Rg = ΛR′g, ρg = Λρ′g (8)

have been used which can be verified with (4) and (5). Thus, we have alsomotivated the introduction of these auxiliary variables.

ρ =

√2r3

M

is the curvature radius of the Schwarzschild parabola. Rg has half the lengthof ρg, the curvature radius on the boundary surface. If one extends the curva-ture vector of the Schwarzschild parabola to the directrix of the Schwarzschildparabola, the resulting distance between the Schwarzschild parabola and thedirectrix has the length

R =

√r3

2M. (9)

(8) refers to the values on the boundary surface. The quantities occurringin the OS model allow a geometric interpretation which facilitates the under-standing of the theory. We leave the elaboration of the geometrical foundationsto a later Section. Since the proper time T ′ coincides with the coordinate timet′ in the comoving system, the second relation (7) can be written as

∂rg∂t′

= vg, vg = −

√2M

rg,

whereby vg is the velocity of an observer who is in free fall in the Schwarzschildfield, coming from the infinite and reaching the surface of the stellar object.

3

However, this means that the surface itself has this speed and must come frominfinity. After a brief calculation we found out an inconsistency of the OS model.This problem will be represented in detail later on.

Using (7) the changes of further variables which relate to the surface can becalculated. We summarize the results with

∂rg∂r′

= 0,∂rg∂t′

= vg,∂Rg

∂r′= 0,

∂Rg

∂t′= −3

2,

∂ρg∂r′

= 0,∂ρg∂t′

= −3. (10)

In the next step we calculate the changes of the relevant variables in the interiorof the stellar object. From (5) and (6) one gets

∂r

∂r′= Λ2/3 =

r

r′,

∂r

∂t′= − r

Rg= vI . (11)

vI is the speed in the interior, i.e. the speed with which the particles draw nearin the interior during the collapse. The velocity decreases linearly inwards, andin the center of the object one has

vI (0) = 0.

Finally, the second relation (2) should be resolved. According to (11) onehas

∂eω/2

∂r′=

r

r′

and at last

eω = Λ4/3 =r2

r′2.

Hence all the metric coefficients of the inner OS solution can be written in aform that simplifies the further considerations

1′

e 1′ =r

r′,

2′

e 2′ = r,3′

e 3′ = r sinϑ,4′

e 4′ = 1. (12)

The metric can be written either as

ds2 =r2

r′2dr′2 + r2dϑ2 + r2 sin2 ϑdφ2 − dt′2

or asds2 = Λ4/3

[dr′2 + r′2dϑ2 + r′2 sin2 ϑdφ2

]− dt′2.

The latter form is entirely written in comoving coordinates and the factor Λcontains the time dependence of the otherwise flat 3-dimensional line element.

In order to adapt the notation of authors who deal with collapsing modelswe introduce the scale factor

K (t′) = Λ2/3

with which we can basically describe the relation between comoving and non-comoving coordinates

r = Kr′.

4

With comoving coordinates we obtain instead of (12)

1′

e 1′ = K,2′

e 2′ = Kr′,3′

e 3′ = Kr′ sinϑ,4′

e 4′ = 1. (13)

and instead of (6)

∂K

∂r′= 0,

1

K

∂K

∂t′= − 1

Rg; K|1′ = 0,

1

KK|4′ =

i

Rg. (14)

The last expression we shall meet again in the field equations and it is alsoknown for other time-dependent models. From (11) and (13) we read

∂r

∂r′= K.

With this and with (13) one has for t′ = const.

dx1′ = Kdr′ = dr

from which one recognizes that the radial arc element corresponds to a segmenton the r-axis. Thus, the OS interior geometry appears to be flat. In a laterSection we will critically return to this subject.

§2. The OS interior solution, coordinate and reference systems. Inthe previous Section we have already made use of the variable r which desig-nates the radial coordinate in the non-comoving system. OS have specified therelation of t′ and t. Thus, a matrix can be assembled for the transformationbetween the two coordinate systems. The finding of such a transformation isobviously quite tedious. It is probably for this reason that other authors didnot provide such a coordinate transformation for their models, or the puttingup of a coordinate transformation was not possible because the model has noanalytical solution. The use of different coordinate systems is apparently theonly purpose of providing calculations on the simplest possible basis. The greatadvantage lies in the fact that such a coordinate transformation is accompaniedby a Lorentz transformation which contains the velocity parameters. If one hasfound such a Lorentz transformation, and if one refers to the velocity of thesurface of the stellar object, one has the physical velocity of the collapse athand.

In the last Section we have already prepared the way for a Lorentz transfor-mation. From (11) and (12) we obtain

dr =r

r′dr′ − r

Rgdt′ = dx1′ − ivIdx

4′ , vI = − r

Rg,

wherein the{dx1′ , dx4′

}are the anholonomic tetrad differentials. By means of

an auxiliary variable

y =1

2

[(r′

r′g

)2

− 1

]+

r′g2M

r

r′, dy =

r′

r′g2dr′ − 1

2M

√2M

rgdt′ (15)

5

and from

t =3

2

1√2M

[r′g

3/2 − (2M)3/2]− 4M

√y + 2M ln

√y + 1

√y − 1

one gets, by differentiating,

dt = −2My3/2

y − 1

r′

r′gdr′ +

√2M

rg

y3/2

y − 1dt′.

From this one can read the transformation coefficients of the coordinatetransformation

Λii′ = xi

|i′ .

Since these coefficients are orthogonal, one gains also the reciprocal values

Λ11′ =

rr′ , Λ1

4′ = i rRg

= −ivI , Λ41′ = −2Miy

3/2

y−1r′

r′2g, Λ4

4′ =y3/2

y−1

√2Mrg

,

Λ1′

1 = α2Ir′

r , Λ4′

1 = −iα2IvI , Λ1′

4 = −iα2Iy−1y3/2

r′

rg, Λ4′

4 = α2Iy−1y3/2

√rg2M ,

αI = 1√1− 2M

rr′3r′g

3

= 1√1− r2

R2g

, vI = − rRg

.

(16)

In these expressions all indices are coordinate indices. From

gik = Λi ki′k′gi

′k′

and (12) can be calculated the 4-bein em

i and from this the reciprocal one

1e1 = αI ,

2e2 = r,

3e3 = r sinϑ,

4e4 = αI

√rg2M

y − 1

y3/2. (17)

The indices m number the 4-bein vectors of the reference system which is asso-ciated with the observers at rest. Now it is easy to calculate the correspondingLorentz transformation connecting the comoving observers and the observers atrest

Lmm′ =

me i Λ

ii′ e

m′i′ ,

L11′ = αI , L1

4′ = −iαIvI , L41′ = iαIvI , L4

4′ = αI .

(18)

Thus, one has the physical quantities describing the collapse at hand. Onthe surface one has with

1e1 = αg

I =1√

1− r2gR2

g

=1√

1− 2Mrg

,4e4 = αg

I

√rg2M

rg2M − 1√( rg

2M

)2 =

√1− 2M

rg

6

the Schwarzschild values of the exterior field on the surface of the stellar object.Thus, the two solutions are matched on the boundary surface.

For the velocity on the surface one obtains

vgI = − rgRg

= −

√2M

rg, (19)

the velocity of an observer who is in free fall from infinity. Thus, we have onceagain proved by using physically relevant equations that the surface of the stellarobject collapses in free fall from infinity. From (19) we also recognize that forrg = ∞ the initial velocity is vgI = 0.

In (6) we had developed∂Λ

∂t′= − 3

ρ′g.

Since Λ does not depend on r one has

dt′ = −ρ′g3dΛ, t′ = −

ρ′g3Λ + C = −ρg

3+ C,

also relying on (8). ρg is the radius of curvature of the Schwarzschild parabolaon the boundary surface, which has its minimum value ρg (2M) = 4M at thevertex of the Schwarzschild parabola. From this we determine the constant ofintegration and finally we have

t′ =4M

3− ρg

3.

That is up to sign the formula for the rise time in the Schwarzschild field. Atrg = 2M applies t′ = 0 and at rg = ∞ applies t′ = ∞. Because of the invari-ance under time reversal it can be concluded that the surface of the collapsingobject begins to collapse at the time t′ = 0 at infinity and reaches the eventhorizon after an infinite proper time. Since for a freely falling observer in theSchwarzschild field the coordinate time t′ matches the proper time T ′, the col-lapsing star would take an infinitely long time to contract to the event horizon.

The OS stellar object in its initial state would have an infinitely large ex-tension, collapses in free fall and leaves empty space behind it in which aSchwarzschild field spreads. However, the collapse velocity would reach thespeed of light at rg = 2M which has to be ruled out by the principle of rel-ativity. At this location the gravity and tidal forces would be infinitely large.Under these conditions a star cannot exist. The OS model is afflicted with allthose problems which are known from the Schwarzschild theory. Consequently,the OS model cannot be used as a base model for a black hole.

§3. The OS exterior solution. For the exterior solution OS start from

7

the same metric (1) with the ansatz (2) where now we have3

G =√r′3, F = −3

2

√2M, Λ = 1− 3

ρ′t′, ρ′ =

√2r′3

M(20)

and the following auxiliary formulae

∂Λ

∂r′=

9

4

√2M

r′5t′,

∂Λ

∂ t′= − 3

ρ′,

∂ω

∂r′=

2

Λr′

apply. After similar calculations such as we have performed for the interiorsolution we obtain the tetrads in the comoving system

1′

e 1′ =

√r′

r= Λ1/3,

2′

e 2′ = r,3′

e 3′ = r sinϑ,4′

e 4′ = 1

and fromr = Λ2/3r′ (21)

with (8) and (9) the auxiliary formulae

∂r

∂r′=

√r′

r,

∂r

∂t′= − r

R= −

√2M

r= vE , (22)

in which we recognize the Schwarzschild velocity of free fall from infinity. OSput for the time of the system at rest

t =3

2

1√2M

(r′3/2 − r3/2

)− 2

√2Mr − 2M ln

√r −

√2M

√r +

√2M

.

With (21) and (20), last equation, one has

r =

(1− 3

√M

2r′3t′

)2/3

r′.

Isoliating t′

t′ =2

3

1√2M

(r′3/2 − r3/2

)we finally obtain

t′ = t+ 2√2Mr + 2M ln

1−√

2Mr

1 +√

2Mr

, (23)

the formula for the transition from the static Schwarzschild time coordinate toa time coordinate which refers to an observer who is in free fall from infinity.

3Note that the quantity ρ′ = ρ′ (r′, t′) is used while the analogous quantity for the interiorsolution is the constant quantity ρ′g , the value of ρ′ on the boundary surface between theinterior and exterior solutions.

8

The relation originally was found by Lemaıtre. By a gauge transformation ofthe radial coordinate

r′′ =2

3

1√2M

r′3/2, t′′ = t′

one obtains with

1′

e 1′dr′ =

√r′

r

√2M

r′dr′′ =

√2M

rdr′′

the relation

dx1′′ = −vEdr′′, vE = −

√2M

r,

well-known from the Lemaıtre transformation as well. Thus, it is ensured thatin spite of different coordinate systems the radial arc elements are the same:dx1′ = dx1′′ .

After recasting one can specify r as a function of the Lemaıtre coordinates

r =3√2M

[3

2(r′′ − t′′)

]2/3and can write the curvature radius of the Schwarzschild parabola and the quan-tity Λ as

ρ = 3 (r′′ − t′′) , Λ = 1− t′′

r′′.

Thus, we have explained the OS coordinate transformation as a Lemaıtre trans-formation. With the help of (22) and by differentiating (23) we evaluate thecoefficients of this transformation

Λ11′′ = −vE , Λ1

4′′ = −ivE , Λ41′′ = −iα2

Ev2E , Λ4

4′′ = α2E ,

Λ1′′

1 = −α2E

vE, Λ1′′

4 = −i, Λ4′′

1 = −iα2EvE , Λ4′′

4 = 1.

(24)

By transvecting with the 4-bein of the comoving system

1′′

e 1′′ = −vE ,4′′

e 4′′ = 1

and the 4-bein of the non-comoving Schwarzschild system

1e1 = αE ,

4e4 =

1

αE, αE =

1√1− 2M

r

(25)

we obtain the coefficients of a Lorentz transformation

L11′′ = αE , L1

4′′ = −iαEvE , L41′′ = iαEvE , L4

4′′ = αE . (26)

In these coefficients are included the physical components of the relative velocityvE of the two systems which we refer again to the surface of the stellar object.One has

vgE = −

√2M

rg.

9

In accordance with previous results the surface has always the speed of an objectcoming in free fall from infinity. At rg = 2M arise the well-known Schwarzschildproblems.

We want to modify slightly the calculus. If we introduce the scale factor

K = Λ2/3

we arrive atr = Kr′ (27)

instead of (21) and at

1′

e 1′ =1√K

,2′

e 2′ = Kr′,3′

e 3′ = Kr′ sinϑ,4′

e 4′ = 1. (28)

By differentiating (27) we obtain, in addition, the relation

dr =1√K

dr′ = dx1′ .

It should also be noted that according to (20) the scale factor K is a functionK = K (r′, t′) in contrast to the analogous quantity of the interior OS solution.Thus, one has

1

rdr =

1

r′dr′ +

1

KdK,

wherein1

KK|1′ = (1− Λ)

1

r,

1

KK|4′ =

i

R.

The result of the second expression is known from earlier Sections. Just compareit with the corresponding formulae of the interior OS solution (14). A transfor-mation matrix for the OS coordinate system akin to the Lemaıtre-matrix (24)can be set up

Λ11′ = − 1√

K, Λ1

4′ = −ivE , Λ41′ = −i

√Kα2

EvE , Λ44′ = α2

E ,

Λ1′

1 =√Kα2

E , Λ1′

4 = −i√KvE , Λ4′

1 = −iα2EvE , Λ4′

4 = 1.

It leads with (28) and (25) to the same Lorentz transformation (26). To calculatethese expressions the relations (21), (27), and (23) can be used. A furtherextension of the procedure can be omitted because the free fall has already beendiscussed in detail [4].

§4. The field equations of the interior OS solution The OS solutiondescribes the collapse of an infinitely large stellar object and is therefore notphysically expedient. However, we examine the field equations because themodel is mathematically quite interesting and can bring some insights whichare supportive for other models.

10

First, we calculate the stress-energy tensor contained in

Gmn = Rmn − 1

2gmn = −κTmn.

The pressure-free model comprises only the time-dependent energy density

T44 = µ0 (t′) ,

whereby OS have specified

κµ0 =4

3

1(t′ + G

F

)(t′ +

∂G/∂r′

∂F/∂r′

) .

With (3) we computeG

F=

∂G/∂r′

∂F/∂r′

and

κµ0 =4

3

1(t′ + G

F

)2 , G

F= −2

3

√r′g

3

2M= −

ρ′g3.

After some further reshaping one finally obtains with (4) and (8)

κµ0 =3

R2g

. (29)

This is the expression for the energy density and it is identic with the one ofthe interior Schwarzschild solution. At the beginning of the collapse (t′ = 0)

Λ (0) = 1, rg = r′g

is valid, whereby the surface of the stellar object is situated at infinity (rg = ∞).For (29) one can also write

κµ0 = 32M

r3g.

Thus, the infinitely extended object has vanishing mass density

µ0 (0) = 0.

Otherwise holds the relation

m =4π

3r3gµ0

by means of

κ =8πk

c4, M = m

k

c4.

Therein m is the mass of the object which is enclosed by the sphere with theradius rg.

11

For further calculations we need several auxiliary formulae. We use themetric of the comoving system in the form with the scale factor

(A) ds2 = K2[dr′2 + r′2dϑ2 + r′2 sin2 ϑdφ2

]+ dit′2

and we read off the 4-bein system

(A)1′

e 1′ = K,2′

e 2′ = Kr′,3′

e 3′ = Kr′ sinϑ,4′

e 4′ = 1. (30)

The scale factor is a function of the time t′, and illustrates the relation betweenthe comoving and non-comoving coordinates

r = Kr′. (31)

With (11)4 we have already prepared the quantities needed for further calcula-tions. With the help of tetrads (30) we write

r|m′ = {1, 0, 0,−iv} = {α, 0, 0,−iαv} a, a = 1/α. (32)

With the Lorentz transformation (18) one calculates the change of the non-comoving radial coordinate

r|m = Lm′

m r|m′ = {a, 0, 0, 0} .

Likewise, one determines the values on the surface, whereby use is made of thecompilation (10)

rg|m = {−αvvg, 0, 0,−iαvg} . (33)

The 4-velocity of a mass point in the interior of the object relative to thecomoving and non-comoving systems is

′um′ = {0, 0, 0, 1} , ′um = {−iαv, 0, 0, α} .

With this and (10) one has

Rg|m′ =3

2i ′um′ , ρg|m′ = 3i ′um′ , Rg|m =

3

2i ′um, ρg|m = 3i ′um.

The values of α and v are taken from (16) last line. Thus, we can calculate

v|m′ =

{− 1

Rg, 0, 0, iv

1

ρg

}=

{vgrg

, 0, 0, iv1

ρg

}.

The change of v consists of a circular part and a parabolic one. The spatialchange of v occurs if one moves on a radial line in the interior of the object ata given moment. The time change takes place, however, because the boundaryof the interior geometry is sliding down on the Schwarzschild parabola withthe radius of curvature ρg. If we write the two components separately and if we

4The marker I of v and α is omitted because in this Section we only deal with the interiorOS solution.

12

calculate the components of the system at rest with the Lorentz transformation,we have

v|m′ = {α, 0, 0,−iαv}(− a

Rg

)+ {0, 0, 0, 1}

(−3iv 1

ρg

),

v|m = {1, 0, 0, 0}(− a

Rg

)+ {−iαv, 0, 0, α}

(−3iv 1

ρg

).

(34)

It can be seen that the circular quantity is defined in the system at rest. Itappears in the comoving system Lorentz transformed while the parabolic com-ponent is derived in the comoving system and must be determined for the non-comoving system with a Lorentz transformation. With

1

αα|m = α2vv|m

one has the values for the change of the Lorentz factor

α|m′ ={−α3v 1

Rg, 0, 0,−iα3v2 1

ρg

},

α|m ={−α4v 1

Rg− α4v3 1

ρg, 0, 0,−3iα4v2 1

ρg

}.

(35)

For the conservation law we note

µ0|m′ =

{0, 0, 0,−3i

µo

Rg

}, µ0|m = −3i

µo

Rg

′um. (36)

Thus, we are prepared to set up the field equations in both systems. In thecomoving system results a similar simple structure of the field equations. Withthe tetrads (12) we calculate the Ricci-rotation coefficients5

′U4′ =′A1′4′

1′= − 1′

e 1′ e1′|4′1′ ,

Bm′ = ′A2′m′2′ = − 2′

e 2′ e2′|4′2′ , Cm′ = ′A3′m′

3′= − 3′

e 3′ e3′|4′3′ .

(37)

With (32) this results in

′Um′ ={0, 0, 0, i

Rg

},

Bm′ ={

1r , 0, 0,−

ivr

}, Cm′ =

{1r ,

1r cotϑ, 0,−

ivr

}.

(38)

Thus, the quantity ′U4′ describes the change in time of the scale factor. With(31), (32), and v = −r/Rg one has

′U4′ =i

Rg,

1

KK = − 1

Rg, K =

∂K

∂t′

5We use the following notation: A prime on a kernel of a quantity indicates a quantity ofthe comoving system. A prime on the index of a quantity indicates that it is measured in thecomoving system. Thus, for example Φm′ is a quantity of the non-comoving system which ismeasured in the comoving, ′Φm a quantity of the co-moving system which is measured in thenon-comoving system.

13

and thus also′U4′

∗=B4′ =

∗=C4′ =

i

Rg. (39)

The common expression for the ‘expansion’ results with ′un′ = {0, 0, 0, 1} in

′un′

||n′ = ′An′′un′

= ′U4′ +B4′ + C4′ = 3i

Rg(40)

and is here to be interpreted as a spatially uniform contraction which has thesame value in all points in space. Nothing changes if one arbitrarily rotates orshifts the local coordinate system. The model has spherical symmetry.

According to (30) the metrical factor of the time-like part of the line element(A) is equal to 1, indicating the free fall from infinity. Thus, no other forces canbe derived from the metric (A), in particular, no effect of gravity. Therefore onehas

′U1′ =′A4′1′

4′= − 4′

e 4′ e4′|1′4′ = 0, ′U4′ =

′A1′4′1′= − 1′

e 1′ e1′|4′1′ =

1

KK|4′ .

Thus, the 4-vector

′Um′ =

{0, 0, 0,

i

Rg

}(41)

has only a single component, the time-like one.With the unit vectors

′mm′ = {1, 0, 0, 0} , ′bm′ = {0, 1, 0, 0} , ′cm′ = {0, 0, 1, 0}

and the Ricci-rotation coefficients

′Am′n′s′= ′Um′n′

s′+Bm′n′

s′ + Cm′n′s′ ,

′Um′n′s′= ′mm′ ′Un′ ′ms′ − ′mm′ ′mn′ ′Us′ ,

Bm′n′s′ = ′bm′Bn′ ′bs

′ − ′bm′ ′bn′Bs′ ,

Cm′n′s′ = ′cm′ Cn′ ′cs

′ − ′cm′ ′cn′ Cs′

we form the graded derivatives

′Um′ ||1

n′ = ′Um′|n′ , Bm′ ||2

n′ = Bm′|n′ − ′Un′m′s′Bs′ ,

Cm′ ||3

n′ = Cm′|n′ − ′Un′m′s′Cs′ −Bn′m′

s′Cs′ .(42)

With the submatrices of the metric

hm′n′ =

1

00

1

, 3gm′n′ =

1

11

0

14

one finally obtains the Ricci

Rm′n′ = −[

′Us′||1

s′ +′Us′ ′Us′

]hm′n′

−[Bn′ ||

2

m′ +Bn′Bm′

]− ′bn′ ′bm′

[Bs′

||2

s′ +Bs′Bs′

]−[Cn′||

3

m′ + Cn′Cm′

]− ′cn′ ′cm′

[Cs′

||3

s′ + Cs′Cs′

]= 3

21R2

g

[3gm′n′ − ′um′ ′un′

](43)

and

R =3

R2g

.

This results in

Gα′β′ = 0, Gα′4′ = 0, G4′4′ = −κµ0, α′ = 1′, 2′, 3′.

The conservation law leads to

T 4′n′

||n′ = µ0|4′ +′A4′µ0 = 0, ′A4′ =

′U4′ +B4′ + C4′ = 3i

Rg, ∂4′ =

∂

i∂t′.

Thus one obtains

µ0 = µ03

Rg.

(29) is also verified with (10). It is the expression for the mass density whichwe have taken from the corresponding relation of OS.

The calculation of the field equation in the non-comoving system is consid-erably more difficult. We start with the metric

(B) ds2 = α2dr2 + r2dϑ2 + r2 sin2 ϑdφ2 + a2T dit2,

α = 1√1− r2

R2g

, aT = α√

rg2M

y−1y3/2 ,

y = 12

[(r′

r′g

)2− 1

]+

r′g2M

rr′ , αT = 1/aT

(44)

and with the 4-bein system (17)

(B)1e1 = α,

2e2 = r,

3e3 = r sinϑ,

4e4 = aT , (45)

and we calculate the field strengths similar to (37). Taking from (15), secondequation, the values for

y|1′ = e1′1′ ∂y

∂r′, y|4′ = e

4′4′ ∂y

i∂t′

15

and performing a Lorentz transformation using (18) one obtains

y|1 = 0 (46)

which makes further calculations much easier.The lateral field quantities

Bm =

{1

αr, 0, 0, 0

}, Cm =

{1

αr,1

rcotϑ, 0, 0

}(47)

are easy to calculate with the 4-bein system (45). Alternatively, one obtains thequantities from (38) with the Lorentz transformation (18)

Bm = Lm′

m Bm′ , Cm = Lm′

m Cm′ .

The radial metric factor of the metric (B) contains the curvature quantity αwhich is time-dependent in accordance with (13) and (12). With (35) one has

U4 = A141 = − 1

e1e1|41 =

1

αα|4 = −3iα3v2

1

ρg.

Taking into account (46) one gets from (44)

U1 = A414 = − 4

e 4 e4|14 =

1

aTaT |1 =

1

αα|1 +

1√rg2M

(√rg2M

)|1.

We calculate the first term with (35). For the second term the relationrg|1 = −αvvg derived in (33) is used. Thus, one has in sum

Um = UCm + UP

m, U ′m = UC

m′ + UPm′ ,

UCm = {1, 0, 0, 0}αv 1

Rg, UP

m = {α, 0, 0, iαv}(−3α2v 1

ρg

),

UCm′ = {α, 0, 0,−iαv}αv 1

Rg, UP

m′ = {1, 0, 0, 0}(−3α2v 1

ρg

) (48)

and we have already performed a separation into the circular and into theparabolic part. With the Lorentz transformation (18) we have calculated thesecond line. The circular part has its origin in the non-comoving system. Itis a negative quantity which lies in the radial direction and corresponds to thegravitational force. The parabolic part is positive, has its origin in the comovingsystem and is a result of the collapse. Both parts are regular for r > 2M . Be-neath the Schwarzschild event horizon UC is imaginary and the transformation(18) unphysical.

The field equations take a form such as (43) in the non-comoving system.They are form invariant

Rmn = −[Us

||1

s + UsUs

]hmn

−[Bn||

2

m +BnBm

]− bnbm

[Bs

||2

s +BsBs

]−[Cn||

3

m + CnCm

]− cncm

[Cs

||3

s + CsCs

],

(49)

16

wherein the graded derivatives are constructed similarly to (42)

Um||1

n = Um|n, Bm||2

n = Bm|n−UnmsBs, Cm||

3

n = Cm|n−UnmsCs−Bnm

sCs.

With

−R

2=

[Us

||1

s + UsUs

]+

[Bs

||2

s +BsBs

]+

[Cs

||3

s + CsCs

]one can calculate the Einstein tensor and with the Lorentz transformation

Tmn = Lm′n′

m n Tm′n′

we obtain the components of the stress-energy-momentum tensor for the systemat rest

T11 = −α2v2µ0, T14 = −iα2vµ0, T44 = α2µ0.

To verify the field equations one needs a set of formulae. From the last twoequations (10) one gains after the transition to the non-comoving system

ρg|m = {3αv, 0, 0, 3iα} .

For further calculations we need the relations (34) and (35). If we analyze thefield equations it is of particular interest how the components of the stress-energy-momentum tensor can be extracted from the Einstein tensor. For thispurpose, the second term (35) is decomposed into a circular and a parabolicpart

1αα|m = EC

m + EPm,

ECm =

{−αv 1

Rg, 0, 0, 0

}, EP

m ={−3α3v3 1

ρg, 0, 0− 3iα3v2 1

ρg

}.

From (48) one needs U P1 . With (29) we finally have

(B1 + C1)EP1 = 2a

r

(−3α3v3 1

ρg

)= α2v2κµ0 = −κT11,

(B1 + C1)EP4 = 2a

r

(−3α3v2 1

ρg

)= iα2vκµ0 = −κT14,

− (B1 + C1)UP1 = 2a

r

(3α3v 1

ρg

)= −α2κµ0 = −κT44.

The left-hand side terms of the above expression are included in the Einsteintensor. It can be seen that the parabolic parts of the field quantities have a sig-nificant role in establishing the stress-energy-momentum tensor. The remainingterms in the Einstein tensor cancel each other.

With a little algebra one also finds

Bs||2

s +BsBs = − 2

ρ2g, Cs

||3

s + CsCs = − 6

ρ2g, Us

| s + UsUs =2

ρ2g

17

and thereby verifies Einstein’s field equations

Rmn − 1

2Rgmn = −κTmn.

In the next Section we will use all the relations which we have found here.

§5. Discussion of the OS-field equations. In the previous Sections, theOS model was put into a covariant form, although the question of the geometricinterpretation of the model has not been presented in sufficient detail. This willbe made up now.

In the last Section we have discovered that the OS-expression for the massdensity can be converted into the familiar form κµ0 = 3

/R2

g. We have en-countered such a relation in the cosmological models and in the Schwarzschildinterior solution. That implies that the space portion of the interior OS solutioncan be represented as a spherical cap. We want to pursue this.

We consider a snapshot of the spatial 3-dimensional part of the model duringthe collapse at an arbitrary time in the non-comoving system. The metric

′ds2 = α2dr2 + r2dϑ2 + r2 sin2 ϑdφ2

applies. By this we separate the time-dependent variables which describe thecollapse from those describing the basic geometry. Thus, for the Ricci remainsonly

∗Rαβ = −[Bβ||

2

α +BβBα

]− bαbβ

[Bγ

||2

γ +BγBγ

]−[Cβ||

3

α + CβCα

]− cαcβ

[Cγ

||3

γ + CγCγ

]with α = 1, 2, 3 and

Bβ||2

α = Bβ|α, Cβ||3

α = Cβ|α −BαβγCγ .

With the lateral field quantities defined in (46) we obtain after a simple calcu-lation

∗Rαβ =2

R2g

gαβ .

This corresponds to a geometry with constant positive curvature.According to these considerations we are able to interpret the geometry of

the OS model. The metric (B), whose structure is known from other models,suggests to interpret the interior spatial geometry as a cap of a sphere which isplaced at a suitable position from the bottom of Flamm’s paraboloid up to theSchwarzschild geometry. Rg is the radius the cap, and we use the relation

r = Rg sin η, (50)

where η is the polar angle of the cap. According to the metric (B) is α = 1/ cos ηand with (50) is dr = Rg cos ηdη. We have for the metric (B)

ds2 = R2gdη

2 +R2g sin

2 ηdϑ2 +R2g sin

2 η sin2 ϑdφ2 + a2T dit2.

18

Thus, we have found a simple geometric interpretation of the OS-model.During the collapse the cap of a sphere slides down on Flamm’s paraboloid ofthe Schwarzschild environment. While the cap of a sphere shrinks during thisprocess Flamm’s paraboloid remains unchanged. This is also a very insightfulpresentation of the Birkhoff theorem. A collapse has no effect on an observerlocated in the exterior field. It can also be seen that the cap of a sphere cannotshrink beneath the event horizon. It would uncouple from Flamm’s paraboloid.Thus, the OS model does not allow the formation of black holes. Mitra [3] hasbrought forward another argument for this subject.

All the years since the publication of the paper by Oppenheimer and Snyderhave obviously been little investigated the linking conditions of the geometries.On the boundary surface two conditions must be satisfied to ensure the matchingof two geometries.

i The metric or the tetrads, respectively, must match, so that the geometriescontact.

ii The first derivatives of the metric or the field strengths, respectively, mustmatch. This means that the surfaces which are described by the geometrieshave common tangents, the surfaces merge smoothly into each other and donot buckle.

Nariai [5, 6] and Tomita are of opinion that the interior OS solution does notmatch the exterior Schwarzschild solution. They have replaced the Schwarzschildsolution by a complicated one maintaining the interior OS solution. Nariai [7]has discussed linking conditions of models which consist of interior and exteriorsolutions in general terms. Israel [7], O’Brien [8], Synge and Robson [9] havewritten extensively about the linking condition of two regions. Nakao [10] hasextended the OS model with a cosmological constant.

We want to examine the linking condition for the OS model more closely. InSection 4, Eqs. (18) ff we have calculated the values of the metric coefficientsof the metric (B) on the boundary surface of the two geometries

αg =1√

1− 2Mrg

, agT =

√1− 2M

rg.

They coincide with the values of the exterior Schwarzschild geometry, if rg isnot less than 2M. Condition (i) is satisfied, the spaces are connected to eachother as long one does not under-run the event horizon.

The first derivatives of the metric coefficients correspond to the field strengthspresented by us. From (46) one immediately recognizes that the quantities Band C have Schwarzschild values on the boundary. This is certainly not thecase with the quantities Um of (48). This can be understood quite well. Thequantity Um contains the portion UC

m which corresponds to the body geometryand takes the Schwarzschild value on the boundary surface. The second part UP

m

19

is allocated to the collapse. UPm does not arise immediately from the geometry,

but from its change. In accordance with (10) this change comprises the relation

Rg = −3

2.

If one has separated the two quantities, it becomes clear that only the sur-face of the body geometry can have the same tangents with the surface of theSchwarzschild geometry. Thus, the condition (ii) is satisfied.

The particle velocity in the interior is closely related to the angle of ascentof the radial curves on the cap of the sphere

v = − sin η = − r

Rg.

In (19) we have calculated the value of v for the boundary surface. It is

sin ηg =

√2M

rg.

But this is exactly the value of the angle of ascent of Flamm’s paraboloid ofthe Schwarzschild geometry on the boundary surface to the interior geometry.Thus the condition (ii) is satisfied.

A look at the field strengths of the comoving system immediately shows thatthe field strengths of the interior and exterior solutions match on the boundarysurface. The exterior field is described by a freely falling observer. In severalpapers ([12] - [14]) we have written down the relation between the OS coordi-nates and the comoving Lemaıtre coordinates and have discussed the free fallin detail in [4].

It is now to be clarified why the metric (A) appears to be flat. A reference-system transformation, ie, a transition from an observer being in rest to a co-moving observer is not expected to change the geometric structure of the space.The flat appearance of the metric arises because the motion of the freely fallingsystem is tightly coupled to the geometric structure of the model. This willnow be examined in more detail. We consider the flat 5-dimensional embeddingspace, and at first we suppress all dimensions except the first two ones. The

global Cartesian coordinates in this space are{x0′′ , x1′′

}with x0′′ as the extra

dimension and the local coordinates are{x0, x1

}. The relations between the

reference systems provide local rotations

dx0 = dx0′′ cos η − dx1′′ sin η,

dx1 = dx0′′ sin η + dx1′′ cos η.

Since the radial arc element of the line element (A) lies entirely in the local1-direction

dx0 = 0, dx0′′ = tan η dx1′′ , dx1′′ = dr,

dx12 = dx0′′2 + dx1′′2 = tan2η dr2 + dr2 = 1cos2 ηdr

2

20

applies. According to (50) on has

sin η =r

Rg, cos η =

√1− r2

Rg= a =

1

α, (51)

whereby the radial part of the metric (A) is geometrically deepened. The par-allels of the cap of a sphere have the radius r, and its curvature is definedby

Ba′′ =

{0,

1

r, 0, 0, 0

}, a′′ = 0′′, 1′′, ..., 4′′. (52)

In the local system it has the form

Ba =

{1

rsin η,

1

rcos η, 0, 0, 0

}, a = 0, 1, ..., 4. (53)

Using the Lorentz angle the Lorentz transformation (18) can be written as

L11′ = cos iχ, L1

4′ = sin iχ, L41′ = − sin iχ, L4

4′ = cos iχ

and is a pseudo-rotation through the imaginary angle iχ in the local [1,4]-plane.If one operates herewith on (53), one obtains with

Ba′ =

{1

rsin η,

1

rcos η cos iχ, 0, 0,

1

rcos η sin iχ

}an expression whose 1-component will appear flat because the radial metriccoefficient in (B) corresponds to the Lorentz factor. One has

cos η cos iχ = 1. (54)

From (26) and (51) one gathers

v = − sin η, a = cos η

and one obtains

Ba′ =

{1

Rg,1

r, 0, 0,

i

Rg

}. (55)

The quantity Ba′ is generated from (52) by two transformations. A rotationin the [0,1]-plane and a pseudo-rotation in the [1′, 4′]-plane. It should also benoted that this deceptive appearance also occurs in the exterior Schwarzschildfield concerning the free fall. The Schwarzschild metric can be brought with aLemaıtre transformation into a form which appears flat. This is only valid forthe free fall from infinity. If the body starts at a finite position, the metric takeson a much more complicated form.

The metric (A) of the comoving OS model is obtained by means of a Lemaıtre-like transformation which we have discussed in detail in §2

gi′k′ = Λi ki′k′gik, Λi

i′ =m′

e i′Lmm′ e

m

i.

21

A coordinate transformation can never change the geometrical or physicalbackground of the model, so that one cannot make any direct conclusions aboutthe geometric structure of the model by means of the apparently flat represen-tation (A). The field quantities, however, show that the basic structure of thegeometrical model is invariant under a transformation of the coordinates andunder a Lorentz transformation as well.

Let us show now how the field equations of the comoving system and thenon-comoving system can be converted into one another. Therefore one mustrecall that the Ricci-rotation coefficients transform inhomogeneously

′Am′n′s′ = Lm n s′

m′n′s Amns + ′Lm′n′

s′ ,

Amns = Lm′n′s

m n s′′Am′n′

s′ + Lmns.

We call the respective second terms Lorentz terms. They are defined by

′Lm′n′s′ = Ls′

s Lsn′|m′ , Lmn

s = Lss′L

s′

n|m, ′Lm′n′s′ = −Lm n s′

m′ n′s Lmns. (56)

UsingL11′ = α, L1

4′ = −iαv, L41′ = iαv, L4

4′ = α

one can express the components of the Lorentz terms in the following way

′L4′1′4′ = ′L1′ = iα2v|4′ ,

′L1′4′1′ = ′L4′ = −iα2v|1′ ,

L414 = L1 = −iα2v|4, L14

1 = L4 = iα2v|1

(57)

and one can write the Lorentz terms in the form

′Lm′n′s′ = hs′

m′′Ln′ − hm′n′

′Ls′ , Lmns = hs

mLn − hmnLs. (58)

With this arrangement results from

Amns = Umn

s +Bmns + Cmn

s,

Umns = hs

mUn − hmnUs,

Bmns = bmBnb

s − bmbnBs, Cmn

s = cmCncs − cmcnC

s

(59)

the transformation law for the U-quantities

′Um′ = Um′+′Lm′ , Um′ = Lmm′Um, Um = ′Um+Lm, ′Um = Lm′

m′Um′ (60)

if one uses (57) with (34). According to (56)

Lm = −Lm′

m′Lm′ (61)

is to be considered. The relation (60) simplifies the conversion of the fieldequations of the system (A) into the ones of the system (B) and vice versa. In

22

addition we want to decompose the Lorentz terms into circular and parabolicparts. First we define in each case two circular and parabolic quantities

LCm = {1, 0, 0, 0}

(αv 1

Rg

), LP

m = {α, 0, 0, iαv}(−3α2v 1

ρg

),

′LCm′ = {α, 0, 0,−iαv}

(−αv 1

Rg

), ′LP

m′ = {1, 0, 0, 0}(3α2v 1

ρg

) (62)

and we again note that the quantity ′U has no parabolic part. With this we candeepen the transformation (60). It turns out that the circular and parabolicparts of the U -quantities transform separately. Taking into account (62) onecan rearrange (60)

′Um′ =[UCm′ + ′LC

m′

]+[UPm′ + ′LP

m′

]+ ′Um′ ,

′Um =[UCm − LC

m

]+[UPm − LP

m

]+ ′Um.

The lateral field quantities B and C transform as vectors, as we have alreadynoted above. On the basis of

Ln′m′

n m Bn′|m′ = Bn|m − LmnsBs, Umn

s = Lm′n′sm n s′

′Um′n′s′+ Lmn

s (63)

results

Ln′m′

n m

[Bn′|m′ − ′Um′n′

s′Bs′ +Bm′Bn′

]= Bn|m − Umn

sBs +BmBn (64)

and a corresponding relation for the quantity for C. With (60) and (61) weobtain

′Us′

||1

s′ +′Us′ ′Us′ =

1

2

1

R2g

. (65)

After these preliminaries we are able to investigate the transformation propertiesof the field equations. We start with the Ricci

Rmn = Amns|s −An|m −Arm

sAsnr +Amn

sAs

and perform the decomposition according to (59). From this one gets the rela-tion (49) for the non-comoving system and for the comoving system the Ricci(43). As seen from (63) - (65) all the subequations of the Ricci are invariant.Thus, the Ricci is invariant as well

Lm′n′

m n Rm′n′ = Rmn,

which was to be expected.The stress-energy-momentum tensor of the unprimed system can be calcu-

lated from the unprimed field equations or with

Tmn = Lm′n′

m n Tm′n′ = µ0′um

′un,′um = {−iαv, 0, 0, α} .

Thus, the conservation law results in

Tmn||n = ′um

[µ0|n

′un + µ0′un

||n]+ µ0

′um||n′un.

23

With the results of previous Sections it can be shown that the motion of theparticles in the collapsing object is geodesic

′um||n′un= 0. (66)

Therefore remainsµ0|n

′un + µ0′un

||n = 0.

One has µ0||n′′un′

= µ0|4′ and ′un||n = ′un′

||n′ . If one reads from (36) one hasverified the conservation law. With (40) one has

µ0 =3

Rgµ0

in accordance with (10). Let’s take a look at the equation of motion (66). Afterreshaping a little we get

(−iαv)|4′ = −α (αU1 + iαv U4)

and with m = m0αdmv

dt′= −mU1′ .

On the other hand one has with (57)

(−iαv)|4′ = −iα3v|4′ = −α ′L1′

and with (60)′L1′ + U1′ =

′U1′ ≡ 0.

It can be seen that the force of gravity acting on the particles in the interiorof the stellar object is canceled by the counter force ′L in the comoving system.A comoving observer is not exposed to gravity.

Due to the absence of the parabolic field strengths the possibility to inspectthe geometric mechanism of the collapse is limited for the comoving observer.This is reserved for the non-comoving observer. If we isolate the purely spatialparts of the field quantities by means of

∗Bα′ =

{1

r, 0, 0

}, ∗Cα′ =

{1

r,1

rcotϑ, 0

}, α′ = 1′, 2′, 3′, (67)

we obtain for the Ricci and the Einstein tensor of these quantities

∗Rα′β′ = 0, ∗Gα′β′ = 0.

For the comoving observer the spatial part of the geometry appears to be flat.But that this is not the case as we have shown above. We process the remainingcomponents of the 4-dimensional variables to

′Am′n′s′ = ∗Am′n′

s′ +Qm′s′ ′un′ −Qm′n′ ′us′ , ′As′n′

s′ = ′An′ = ∗An′ +Qs′s′ ′un′ ,

Q[m′n′] = 0, Qm′n′ ′un′= 0.

24

Included in ∗A are the quantities (67) and the new symmetric quantity is definedby

Q1′1′ =′U4′ , Q2′2′ = B4′ , Q3′3′ = C4′ . (68)

The unit vector ′um′= {0, 0, 0, 1} is orthogonal to the surfaces t′ = const. of

the interior geometry, it is the rigging vector of this shrinking surface. On thebasis of

′um′||n′ = Qm′n′

the Qs are the second fundamental forms of the surface theory. For the Riccithen only remains

Rm′n′ = −[Qm′n′∧s′

′us′ +Qm′n′Qs′s′]

−′un′

[Qs′

s′

∧m′ −Qm′s′

∧s′

]−′um′

[∗An′|s′

′us′ +Qr′s′∗As′n′

r′]

−′um′ ′un′

[Qs′

s′

∧r′′ur′ +Qr′s′Q

r′s′].

In it is ′um′ = {0, 0, 0, 1} and

Φm′∧n′ = Φm′|n′ − ∗An′m′s′Φs′ , m′ = 1′, 2′, 3′

the 3-dimensional covariant derivative of a 3-dimensional quantity.The third brackets in the above block vanish identically and so do the con-

tracted Codazzi equation

Qs′s′

∧m′ −Qm′s′

∧s′ = 0

in the second row of the block. Both relations express that in the comovingsystem must be Rm′4′ = 0.

From this the Einstein tensor

Gα′β′ = 0, κµ0 = −Qα′β′Qα′β′, α′ = 1′, 2′, 3′

can be calculated. The mass density is made up of the field energy density.From (68) and (39) one gets for it the more familiar term

κµ0 =3

R2g

=6M

r3g

with rg the value associated with the surface of the collapsing star. Since thestar collapses in free fall from infinity, the mass density is zero at the beginningof the collapse due to rg → ∞. On the other hand one has

Q1′1′ = −i1

KK, Q2′2′ = Q3′3′ = −i

1

rr.

If one pulls the master switch and if one stops the collapse, the energy densityvanishes and so does the mass. The star disappears and the space is empty.

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From the perspective of a fictional observer the space also appears to be flat.More seriously, one can say that the star of the OS model is gaining its massonly from the collapse.

The solution of Oppenheimer and Snyder is an analytical solution of Ein-stein’s field equations which describes in the same way the interior and theexterior of a stellar object. It has the deficiency to be limited only to pressure-free matter. The model is physically unrealistic because the stellar object isinfinitely large at the time t′ = 0 and its matter is infinitely thinned. The col-lapse takes place in free fall. OS claim that the stellar object can contract belowthe event horizon during the collapse. However, our results of previous Sectionssuggest that the surface of the object can only asymptotically approach theevent horizon. The formation of a black hole is not possible. We again refer tothe papers of Mitra [15] and his theory of ECOs and MECOs. Mitra has shownthat OS have miscalculated by a less important factor 1/4 at the critical part ofthe respective calculation. In addition, the crucial quantity has the wrong signfor penetrating the event horizon.

Thus, the physical usability of the model has been questioned. A pressure-free star which fills the vastness of the universe and has vanishing density at thebeginning of the collapse and leaves an empty space with a Schwarzschild fieldduring the collapse behind it, does not exist.

Rainer BURGHARDTA-2061 Obritz 246

AustriaE-mail arg@aon.at

Homepage: http//arg.or.at

REFERNCES

[1] J. R. Oppenheimer and H. Snyder : On continued gravitational contraction.Phys. Rev., 56(1939), 455-459.

[2] R. C. Tolman : Effect of inhomogeneity on cosmological models. Proc. Nat.Ac. Sci., 20(1934), 169-176.

[3] A. Mitra : The mass of the Oppenheimer Snyder black hole. gr-qc/9904163

[4] R. Burghardt :Spacetime curvature. http://members.wavenet.at/arg/Emono.htm,Raumkrummung. http://members.wavenet.at/arg/Mono.htm.

[5] H. Nariai and K. Tomita : On the applicability of a dust-like model to acollapsing or anti-collapsing star at high temperature. Prog. Theor. Phys.,35(1966), 777-758.

[6] H. Nariai and K. Tomita : On the problem of gravitational collapse. Prog.Theor. Phys., 34(1965), 155-172.

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[7] H. Nariai : On the boundary conditions in general relativity. Prog. Theor.Phys., 34(1965), 173-186.

[8] W. Israel : Discontinuities in spherically symmetric gravitational fields andshells of radiation. Proc. Roy. Soc. Lond. A, 248(1958), 404-414.

[9] S. O’Brien and J. L. Synge : Jump conditions at discontinuities in generalrelativity. Comm. Dublin Inst. Adv. Stud. A, No 9(1952), 1-20.

[10] E. H. Robson : Junction condition in general relativity theory. Ann. Inst.H. Poinc., 16(1972), 41-50.

[11] K.-i. Nakao : The Oppenheimer-Snyder space-time with a cosmologicalconstant. GRG, 24(1992), 1069-1081.

[12] R. Burghardt : Remarks on the model of Oppenheimer and Snyder I.http://arg.or.at/Report ARG-2012-03.

[13] R. Burghardt : Remarks on the model of Oppenheimer and Snyder II.http://arg.or.at/Report ARG-2012-04.

[14] R. Burghardt : Remarks on the model of Oppenheimer and Snyder III.http://arg.or.at/Report ARG-2013-03.

[15] A. Mitra : The fallacy of Oppenheimer Snyder collapse: No general rela-tivistic collapse at all, no black hole, no physical singularity. Astrophys. Sp.Sci., 332(2010), 43-50.

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