Spherically Symmetric Static Solution for Colliding Null Dust

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Spherically symmetric static solution for colliding null dust

Laszlo A. Gergely

KFKI Research Institute for Particle and Nuclear Physics, Budapest 114,

P.O.Box 49, H-1525 Hungary

The Einstein equations are completely integrated in the presence of two (incoming and outgoing)

streams of null dust, under the assumptions of spherical symmetry and staticity. The solution is

also written in double null and in radiation co ordinates and it is reinterpreted as an anisotropic

uid. Interior matching with a static uid and exterior matching with the Vaidya solution along

null hyp ersurfaces is discussed. The connection with two-dimensional dilaton gravity is established.

I. INTRODUCTION

Null dust represents the high frequency (geometrical optics) approximation to the unidirectional radial ow of

unp olarized radiation. This is a reasonable approximation whenever the wavelength of the radiation is negligible

compared to the curvature radius of the background. Various exact solutions of the Einstein eld equations were

found in the presence of pure null dust (for reviews see [1], and more recently [2] and [3]).

In some scenarios even gravitation b ehaves as null dust. Price [4,5] has shown that a collapsing spheroid radiates

away all of its initial characteristics excepting its mass, angular momentum and charge. (This result is known as

the no hair theorem.) The escaping radiation then interacts with the curvature of the background b eing partially

backscattered. Both the escaping and the backscattered radiation can b e mo deled bynull dust [6] as the curvature

radius of the background is larger than the wavelength of the radiation.

Letelier has shown that the matter source comp osed of twonull dust clouds can b e interpreted as an anisotropic

uid [7], giving also the general solution for plane-symmetric anisotropic uid with twonull dust comp onents. Later

Letelier and Wang [8] have discussed the collision of cylindrical null dust clouds. The collision of spherical null dust

streams was discussed byPoisson and Israel [6]. Their analysis yielded the phenomenon of mass in ation.

However, no exact solution in the presence of two colliding null dust streams with spherical symmetry was known

until now. Recently Date [9] tackled this problem under the assumption of staticity,integrating part of the Einstein

equations. It is the purp ose of the present pap er to present the exact solution for the case of two colliding spherically

symmetric null dust streams in equilibrium, in a completely integrated form.

The plan of the pap er is as follows. In Sec. II we derive the eld equations and weintegrate them. The emerging

exact solution is written explicitly in suitable co ordinates adapted to spherical symmetry and staticity. The metric

in radiation co ordinates and in double null co ordinates is also given. We analyze the metric b oth analytically and by

numeric plots. In Sec. III we presentvarious p ossible interpretations of the solution, including the anisotropic uid

picture, and dilatonic gravity.

Finally Sec. IV contains the analysis of the interior and exterior matching conditions on junctions along timelike

and null hyp ersurfaces, resp ectively. We employ the matching pro cedure of Barrabes and Israel [10]. The interior

junction xes the parameters of the solution in terms of physical characteristics of the central star: the mass and

energy density on the junction. The exterior matching with incoming and outgoing Vaidya solutions [11] leads to

the conclusion that no distributional matter is present at the junction. This feature is in contrast with the exterior

matching with the Schwarzschild solution [9].

I I. SOLUTION OF THE EINSTEIN EQUATIONS

2 2

The general form of a spherically symmetric, static metric in a spacetime with top ology R S is

2 2 1 2 2 2

ds = h(r )dt + f (r ) dr + r d : (2.1)

2 2 2

Here t is time, r is the curvature co ordinate (e.g. the radius of the sphere t=const with area 4r ), d = d +

sin ( )d is the square of the solid angle element. The functions f (r ) and h(r ) are p ositivevalued. Weintro duce

the lo cal mass function m(r ), related to the gravitational energy within the sphere of radius r [12]:

2m(r)

f (r )=1 : (2.2)

The energy-momentum tensor in the region of the cross- owing null dust is a sup erp osition of the energy-momentum

tensors of the incoming and outgoing comp onents: 1

(r )

ab a b a b

T = (v v + u u ): (2.3)

All energy conditions are satis ed for (r ) 0. The same linear mass density function was chosen for b oth

comp onents as staticity requires no net ow in either of the null directions. The vector elds

p p

1 1 1 1

a a

p p p p

v = ; f; 0; 0 ; u = ; f; 0; 0 (2.4)

2 2

h h

are the tangents to the (future-oriented) outgoing and incoming null congruences, partially normalized such that

v u = 1. Similarly as for the one-comp onentnull dust, here b oth null congruences are geo desics [9].

After eliminating second time derivatives from the nontrivial Einstein equations wehave the system:

r = f +1

= h( f +1) (2.5) rf

rf = ( + f 1) :

The solutions with =const all reduce to = 0, meaning vacuum. They are either the Schwarzschild or the at

solution, in accordance with the Birkho theorem. For 6=const one of the equations is a relation b etween the mass

density (r ) and the metric function h (r ):

h = ; (2.6)

where A > 0 is a constant. This relation can be deduced also from the energy-momentum conservation [9]. The

remaining equations do not contain the constant A:

d ln f

f = f +1 (2.7)

d ln r

d ln

f = + f 1 : (2.8)

d ln r

Inserting (2.2) in (2.7) the mass is found to increase with the radius: dm=dr = =2 > 0.

Nowwe solve the system (2.7),(2.8). Following Date [9] , we eliminate f from (2.7) by its expression taken from

(2.8):

f = : (2.9)

dln

d ln r

The resulting second order ordinary di erential equation in 1= can b e integrated, nding:

1 1 d 1

= D 2ln : (2.10)

d ln r r 1+

Here D is an integration constant. Inserting this expression in (2.9), an algebraic relation b etween the metric comp o-

nents emerges:

(1 + ) Cr

=ln ; (2.11)

where C = exp((1 + D )=2) > 0.

Then we complete the integration of the system (2.7),(2.8). The key remark is that none of the equations in this

system contains explicitly the indep endent variable ln r . Thus one can pass to the new indep endent variable , in

terms of which an ordinary rst order equation can b e written:

f ( + f 1) df

= : (2.12)

d ( f +1) 2

Intro ducing the new p ositivevariables

1=2

P =(2f ) ; L = (2.13)

1=2

(2f )

the equation (2.12) takes the form of a rst order linear (inhomogeneous) ordinary di erential equation:

=2(1PL) : (2.14)

The solution is found byintegrating rst the homogeneous equation, then varying the constant. It is

2 2

L x

P =2e (L) ; (L)=B+ e dx > 0; (2.15)

B B

where B is a third integration constant. The function (L) in (2.15) can b e expressed either in terms of the error

function or in terms of the Dawson function:

(L) B = erf (iL)=e Dawson(L): (2.16)

For prop erties of these transcendental functions see [13].

From Eqs. (2.11),(2.13) and (2.15) b oth the curvature co ordinate r and the metric functions and f are found as

functions of the radial variable L:

+2L (L) (2.17) Cr(L)=e

(L)=1+ 2Le (L) (2.18)

2 (L)

f (L)= : (2.19)

2 2

L L

e e +2L (L)

Then the mass function m = m (L) is obtained from (2.2):

2 2

L L 2

2Cm(L)=e +2L (L)2e (L): (2.20)

It is easy to check that b oth r and m are monotonously increasing functions of L:

d (Cr)

=2 (L) > 0 (2.21)

h i

2 2

d (Cm)

L L

= e (L) e +2L (L) 0: (2.22)

B B

In the last relation the equality holds for r =0.

Nowwehaveeverything together to write the metric in terms of the new radial co ordinate L:

h i h i

L 2 2 2

2 2 2

Ae dt d dL

2 L L L

ds = +2e + e +2L (L) : (2.23) e +2L (L)

B B

2 2

C C

e +2L (L)

There are three parameters in the solution, two of them restricted to be p ositive: A and C . Without loss of

generalitywe can cho ose A =1 by rescaling the time co ordinate. The parameter C provides some distance scale. We

comment on the third parameter in what follows. Both Cr 0 and the energy conditions 0 imply

e dx (2.24) B (L) ; (L)=

valid for all admissible values of L. The equality B = (L ) holds for L = L corresp onding to r = 0. As

0 0

L 2

d =dL = e =2L < 0, the function (L) is monotonously decreasing and the inequality (2.24) will b e satis ed for

any L> L . Thus B gives the lower b oundary L for the range of the radial co ordinate L.

0 0

Next we plot numerically the functions (r ), f (r ) and m (r ) for di erentvalues of the parameter B (Figs.1-3). 3 0.8 β β 5 0.6 4 3 2 0.4 1 0 7 6 5 0.2 4 B 3 2 8 10 4 6 1 0 2 -4 -20 2 4 6 8 10 0 -4 -2 ln r

ln r

FIG. 1. (a) The function = (ln r ) at parameter values B = C =1. The metric is singular at r = 0 and the space-time is

not asymptotically at. (b) Plot of = (ln r;B) for C = 1 and 0 B 7.

2 ln f ln f 5 4 1.5 3 2 1 1 0 7 0.5 6 5 4 B 3 10 0 246810 2 6 8 ln r 1 2 4

0 0 ln r

FIG. 2. (a) The function ln f =lnf(ln r ) at parameter values B = C =1. (b) Same for 0 B 7.

10 m 8 20 m 6 10 0 -10 4 -20 -30 -40 2 -50 7 5 6 4 5 4 3 -10 1 2 3 4 5 3 2 B 2 1 ln r ln r 1 0

0 -1

FIG. 3. (a) The mass function m = m(ln r ) at parameter values B = C = 1 takes negativevalues close to r =0. (b) The

mass function in the parameter region 0 B 7. The value of r where the mass b ecomes negative dep ends on the parameter

B .

As we see from Fig. 3, the mass function vanishes at some radius r~ and takes negativevalues at r

here that such a p ositive r~ exists irresp ectiveof the choice of the parameters B and C . By combining (2.17) and

2 2

~ ~

(2.20) with the condition m =0 we nd r~ =2 (L)=C exp(L ) > 0. The constant C has a simple scaling e ect on

the value of r~. Figure 3(b) shows that in the domain (0; 7) the negative mass region can b e extended by increasing

B . We emphasize that the energy conditions are still satis ed in the negative mass regions. The interpretation of the

negative mass function is not immediate and dep ends on the measuring pro cedure of the mass in this asymptotically

non- at space-time.

At the end of this section we give the metric in double null and radiation co ordinates. This requires an additional

integration. Weintro duce a \tortoise co ordinate" r in the same manner it can b e intro duced in the Schwarzschild

space-time:

Z Z

r L h i

dr 2

r = = e +2y (y) dy : (2.25)

1=2

0 0

[f (r )h(r )]

The radial null geo desics are t = t + cr , with c = 1 for incoming and c = 1 for outgoing geo desics.

Intro ducing the null co ordinates x = t r the metric takes the simple form:

2 + + 2 + 2

ds = h(x ;x )dx dx + r (x ;x )d : (2.26)

+ + + +

The radial null geo desics are now x =const. The functions h(x ;x )=h(x x ) and r (x ;x )=r(x x ) are

contained in implicit form in Eqs. (2.6),(2.25), (2.17) and ( 2.18).

Finally we cast our static, spherically symmetric solution of cross owing null dust in either the incoming or outgoing

radiation co ordinates (v = t + cr ;r;;). These co ordinates are like the Eddington-Finkelstein co ordinates for the

Schwarzschild solution. If the solution was asymptotically at, they would b e the Bondi-Sachs co ordinates.

" #

1=2

2 2 2

ds = (f ) dv dv 2cdr + r d : (2.27)

Here v = x for c = 1. In these co ordinates the radial null geo desics are given by one of the equations v =const and

1=2

(r )

v = const +2c dr : (2.28)

f (r )

Now it is evident that there is no apparent horizon:

1=2

dr 1 f

= > 0; (2.29)

d (cv ) 2

thus no event horizon either. Thus the singularityin the origin r =0 is naked. This is very similar to the naked

singularity of the negative mass Schwarzschild solution [9].

I I I. INTERPRETATION OF THE SOLUTION

A. 2D dilatonic mo del

In this subsection we present a dilatonic mo del which in 4D has the interpretation of a spherically symmetric

gravitational eld in the presence of two cross owing null dust streams. This dilatonic mo del emerges from the action

Z Z

p p

1 2 1

2 2

g R [g ]+ g r ln r ln + d x gg r 'r ': (3.1) S = d x

2 2

The rst term is the Einstein-Hilb ert action reduced by spherical symmetry (a surface term was dropp ed). The

second term in (3.1) represents a 2D massless scalar eld in minimal coupling. The minus sign assures that all 4D

energy conditions are satis ed. = r is the dilaton and ' is the scalar eld. The conformal atness of the 2 2

metric g = h is manifest in the corresp onding 4D line element (2.26). r and R [g ] are the covariant derivative

and curvature scalar, resp ectively asso ciated with the metric g . Although the equations emerging from this mo del

are quite similar to the corresp onding equations in the Callan-Giddings-Harvey-Strominger (CGHS) mo del [14], they 5

could not b e exactly integrated in general. For our purp oses we take only the equation emerging from variation of

the scalar eld, in double null co ordinates:

'; =0

Here commas denote derivatives. The equation has the D'Alemb ert solution

+ +

; (3.2) x + ' x ' = '

showing that the scalar eld b ehaves like our matter source of cross owing null dust. The 4D interpretation of the

particular solution with either leftmoving or rightmoving matter is the Vaidya solution [11]. Recently Mikovic has

given a solution [15] for the case where b oth comp onents are present, in the form of a p erturbative series in p owers

of the outgoing energy-momentum comp onent.

Our static solution (2.26) represents the rst explicit exact solution for this mo del, when none of the null dust

comp onents are neglected.

B. Anisotropic uid

Following Letelier [7] we reinterpret the energy-momentum tensor (2.3) as describing an anisotropic uid with a

pressure comp onent equaling its energy density:

T = (U U + ) (3.3)

ab a b a b

a a a

U U = =1; U =0 : (3.4)

a a a

A straightforward comparison with (2.3) gives

= ; (3.5)

1 1 @

a a a

p p

( v + u )= U = (3.6)

1 @

a a a

( v u )= = f : (3.7)

Thus the solution represents an anisotropic uid at rest with the energy density and radial pressure p = : No

tangential pressures in the spheres r =const are present. The uid is isotropic only ab out a single p oint, the origin.

C. Radiation atmosphere

The solution can b e interpreted as the outer region of a radiating star, receiving radiation from the surrounding

region either. If equilibrium is achieved b etween the two comp onents, wehave the static solution (2.23). This was

the initial interpretation prop osed by Date [9] .

We write the energy-momentum tensor of the static solution in the double null co ordinate system (x ;x ;;).

Inserting the null covectors

! !

r r

h h

0; v = ; 0; 0 ; u = ; 0; 0; 0 (3.8)

a a

2 2

in the covariant form of (2.3) and taking account of (2.6) the energy-momentum tensor b ecomes

a b + +

T dx dx = dx dx + dx dx ; (3.9)

16r

a sup erp osition of two cross- owing null dust streams with equal and constant mass density functions. However one

a a

can freely rescale the null vectors v and u to have arbitrary mass density functions either. After such a rescaling

the mass density function of the incoming null dust dep ends only on the outgoing co ordinate x and viceversa, a

prop erty p ertinent to the mass functions of the Vaidya solution [11], characterized by the energy-momentum tensor 6

c dM (V )

a b

T dx dx = dV dV : (3.10)

4r dV

M (V ) is the mass function and the co ordinate V is outgoing for incoming radiation and incoming for outgoing

radiation.

In the next section we will study the interior junction with a static star, and the exterior junction with incoming

and outgoing Vaidya solutions.

IV. JUNCTION CONDITIONS

A. Matching with interior spherically symmetric static solutions

We discuss the junction with an interior solution with accent on the anisotropic uid interpretation given previously.

A similar treatment was given in [9]. Both our analysis and the one in [9] reveals that the junction with a static

interior matter can b e done without a regularizing thin shell. Our treatment is more general, however. We formulate

the junction conditions for two generic static spherically symmetric space-times, following the standard Darmois-Israel

junction pro cedure [16,17] and we establish a constraint on the matter pressures implied by the matching conditions.

An other improvement over [9] is due to the fact that we disp ose of the exact solution (2.23), thus the explicit

computation of the the matching conditions with an arbitrary particular interior b ecomes p ossible.

An orthonormal basis is given by the vectors U and de ned by Eqs. (3.6) and (3.7) together with the spacelike

vectors

1 1 @ @

E = ; E = : (4.1)

3 4

r @ r sin @

Any spherically symmetric static energy-momentum tensor has the form

ab a b a b a b a b

T = U U + p + p E E + E E : (4.2)

1 2

3 3 4 4

By inserting p = =8r and in the form (3.5) in the Einstein equations for the metric (2.1) we nd

i i

= f +1 r

= h( f +1) (4.3) rf

2rf = + ( + f 1) + (f 1)+4 f:

1 2

The induced metric of the surface r =const is

2 2 2 2

ds = hdt + r d : (4.4)

Without loss of generalitywe can cho ose the time co ordinates in b oth static space-times such that they are continuous

on the junction. Then the continuity of the rst fundamental form requires the metric function h to b e continuous.

The extrinsic curvature of the junction surface r = const is de ned as

c c d d

K =( ) r : (4.5)

ab a b c d

a b

The nonvanishing comp onents are:

f h

= K = ( + f 1) ; K : (4.6) K =

1 tt

2r f

In the ab ove expressions the derivatives were eliminated by use of Eq. (4.3).

Continuity of the extrinsic curvature across the junction hyp ersurface r = r is achieved, provided that the metric

and the function (thus also p ) are continuous. Wehave proved the following result:

1 1

Any two spherical ly symmetric static solutions can be matched along hypersurfaces r = const provided the radial

pressures are continuous. This is similar to the theorem given by Fayos, Jaen, Llanta and Senovilla [18] for the 7

matching of the Vaidya solution with a generic spherically symmetric solution along timelikehyp ersurfaces. A generic

discussion on matching spherically symmetric space-times along thin spherical timelike shells can b e found in [19].

For the double null dust solution = . In consequence the interior uid should have a radial pressure equal to

the energy density (3.5) of the double null dust solution on the junction. However, no conditions on the pressures

tangent to the spheres emerge.

We see from Eqs. (2.18), (2.20) and (3.5) that the integration constants B and C app ear in the radial pressure and

mass function of the static double null dust solution. Continuity of these functions on the junction xes the value of

the constants, once the interior solution is chosen. For a realistic star, the mass M should b e p ositive. This implies

alower b oundary for the p ossible values of r , as follows from Fig. 3(a).

2 2

Let us illustrate the junction with the interior Schwarzschild solution [20] ds = a b 1 r =R dt +

2 2 2 2 2 2

dr + r d , with the energy density =3=R =const and pressure p given by 1 r =R

S S

3b 1 a

8p = : (4.7)

S q

R a b 1

Several relations among the Schwarzschild parameters a; b; R , radius of the star r and the parameters B and C

emerge from the junction conditions:

3 2 2

p p

r 3 R r 1 R

p p

2m = ; 2a = + ; 2b 1 = + : (4.8)

1 1 1

2 2

R r R r

1 1

Wehave denoted by and m the values of the functions and m at the junction r = r . The rst two relations (4.8)

1 1 1

determine the constant B and the value of the radial co ordinate L at the junction in terms of r =R and a, when Eqs.

1 1

(2.17), (2.18) and (2.20) are inserted. Eliminating from the last two relations of (4.8), a constraint b = b(a; r =R )

1 1

on the p ossible values of characteristics of the interior emerges. Finally (2.17) implies C = C (a; r =R ; r ).

1 1

In the light of the ab ove relations we see that after cho osing some value for C (a scale), the constant B is determined

exclusively by the radius and density (or mass) of the star.

B. Matching with Vaidya solutions

In this subsection we study the junctions with the incoming and outgoing Vaidya solutions, which are at the

exterior of the static double null dust solution. There are only three p oints (in fact spheres) in common with exterior

Schwarzschild regions (Fig. 4), and the matching can b e done without intro ducing regularizing thin shells.

The high-frequency approximation to the unidirectional radial ow of spherically symmetric unp olarized radiation,

characterized by the energy-momentum tensor (3.10) is represented by the Vaidya solution [11]:

2M (V )

2 2 2

ds = dV 1 dV 2cdr + r d : (4.9)

The radial null geo desics are the lines V =const and the curves given by

dr c 2M (V )

= 1 : (4.10)

dV 2 r

We would like to match our static solution with incoming and outgoing Vaidya solutions along the outgoing,

resp ectively incoming radial null geo desics, e.g. along lines of constant incoming, resp ectively outgoing co ordinate

(Fig 4.). These are given by Eq. (4.10) in the Vaidya solution and by Eq. (2.28) in the static solution.

A convenient formalism for matching solutions along null surfaces, which do es not require co ordinates that match

continuously on the shell, was develop ed by Barrabes and Israel [10]. Their discussion on the particular case of

spherical symmetry requires the metric in b oth space-times written in the form:

2 2 2

ds = e dz fe dz 2dr + r d : (4.11)

Here f and dep end on b oth r and z . The null co ordinate z is always outgoing but it can b e either increasing or

decreasing with time. Then the junction is done along the null hyp ersurfaces z =const. 8

To apply this formalism it would b e desirable to express the Vaidya solution in the other set of radiation co ordinates

[U (V; r);r;;] in which Eq. (4.10) takes the form U =const. However, this is p ossible only when double null

co ordinates are found. It was demonstrated in [21] that double null co ordinates exist when the mass is a linear or

exponential function of the advanced or retarded time V . We argue in what follows that the mass function of that

particular Vaidya solution, which is matching continuously to the static solution has a more complicated dep endence.

A further inconvenience is that there is no obvious choice for the intrinsic co ordinates in terms of the space-time

co ordinates in the null junction hyp ersurface. For these reasons we pro ceed as follows. First we nd co ordinates

that match continuously on the junction and in terms of which the metric is continuous. Then we employ the

Barrabes-Israel junction formalism to nd the distributional stress-energy tensor on the junction.

FIG. 4. The intersection of two cross owing null dust streams with mass functions M (V ) is the static solution characterized

by the mass function m(r ). Here V = X stands for c = 1 and X are the advanced (retarded) time in the Vaidya solution.

The 2-dust solution touches in three p oints exterior Schwarzschild regions with masses M and M . At the interior junction

1 2

there is a static uid representing a star with mass M . The mass functions M (V ) of the Vaidya regions change monotonously

from M to M .

1 2

As a radial variable of b oth metrics we cho ose L by extending the expression (2.17) of r = r (L) to the Vaidya

regions to o. An appropriate null co ordinate z is de ned by the values of L = L on the junction. In these co ordinates

the junction hyp ersurfaces are characterized by the null geo desic equations z = L in b oth space-times. We pro ceed

in deriving the expressions of the co ordinates v and V and of the mass function M in terms of the co ordinate z .

Identifying the corresp onding part of Eq. (3.9) with Eq. (3.10 ), we nd the relation b etween the null co ordinates

V and v ,valid on the junction :

1=2

dM (V )

dv =2 c dV : (4.12)

Then we extend this relation over b oth the static and the Vaidya regions.

The null geo desic equation (2.28) of the static solution, evaluated on the junction (where z = L) gives

c dM dz

= (4.13)

dV 2r dz

Inserting Eq. (4.13) in the null geo desic equation (4.10) of the Vaidya solution we nd its mass function in terms of

the new null co ordinate z :

2 2

z z 2

(z): (4.14) 2CM (z)= e +2z (z)2e

Thus the mass function is continuous at the junction. The geo desic equation (4.13) together with (4.14) gives the

relation b etween the null co ordinates V and z : 9

h i

2 2

y y

e +2y (y) dy e

V (z )=const + : (4.15)

C (y )

The relations (4.14) and (4.15) contain in implicit form the dep endence of the mass function M of the Vaidya solution

on the radiation co ordinate V . Despite the complicated dep endence M (V ) it is straightforward to check that M

satis es the required monotonicity condition:

dM (z )

2 = > 0 : (4.16)

d(cV )

Finally Eqs. (4.12) and (4.15) give v = v ( z ):

z h i

2 2c

v (z )=const + e +2y (y) dy : (4.17)

Nowwe express b oth metrics in terms of the co ordinates (z; r(L);;). They take the form (4.11) with

h i

2m(L) 2 e

f =1 ; e = e +2z (z) (4.18)

r(L) C (L)

h i

2M(z) 2 e

f =1 ; e = e +2z (z) : (4.19)

V B

r(L) C (z)

Here the index V refers to the Vaidya solution and the expressions r (L) ;m(L) and M (z ) are given by Eqs. (2.17),

(2.20) and (4.14), resp ectively. Wehave completed the task of writing b oth metrics in co ordinates which are continuous

on the junction and in terms of which the metric is continuous.

2 2 2

It is immediate to check the continuity of the induced metric given by ds = r d . The other junction condition

is a somewhat subtle issue as the conventional extrinsic curvature tensor for null hyp ersurfaces carries no transversal

information.

We de ne a pseudo-orthonormal basis [22] (n; N ; E ;E ), where E and E are given in Eq. (4.1) and

3 4 3 4

@ f @ @

n = e + ; N = : (4.20)

@z 2 @r @r

The vector n is orthogonal (and also tangent) to the hyp ersurfaces fe dz 2dr =0, along which the two space-

times are glued together. The vector N is the other radial null vector, transversal to these surfaces. They are

a a

related to the previously intro duced vectors v and u as follows. For the junction with the incoming Vaidya region

1=2 1=2 1=2

a a a a a a

v =(2=f ) n and u =(f=2) N ; while for the junction with the outgoing Vaidya region u = (2=f ) n

1=2

a a

and v = (f=2) N .

The pro jector to these null hyp ersurfaces with tangent space spanned by E ;E and n is

3 4

a a 3 a 4 a a a

P = E E + E E n N = + N n : (4.21)

b b

b 3 b 4 b b

Following [10] we de ne the transverse or oblique extrinsic curvature tensor with the aid of the transverse vector N :

c d

K = P P r N : (4.22)

ab c d

a b

are The only nonvanishing comp onents of the transverse extrinsic curvature tensor

C 1 dL 1

e @ fe = e @ fe ; K = K = : (4.23) K =

r L zz

2 2 d (Cr) r

Thus when matching any two metrics of the form (4.11) on the nul l hypersurfaces fe dz 2dr =0, the jump in the

extrinsic curvature is given by the jump in @ fe , provided the metric is continuous across the junction.

The comp onents of the extrinsic curvature tensor de ned in [10] are found from K by contracting with the three basis

vectors E ;E and n tangent to the hyp ersurface.

3 4 10

A straightforward computation employing (4.18) and (4.19) gives on the junction hyp ersurfaces of the static double

null dust region with the Vaidya regions

i h

2 2 2

L L L

j j j

(L ) +2L (L )e e 4e

j j B j

h i

= @ f e = @ fe : (4.24)

L V L

C e +2L (L )

j B j

In conclusion the extrinsic curvature is also continuous on the junction. There is no need for a thin regularizing

shell separating the two domains of the space-time, in contrast to the exterior junction prop osed in [9].

V. CONCLUDING REMARKS

We have integrated the Einstein equations in the presence of cross owing null dust under the assumptions of

spherical symmetry and staticity and analyzed various asp ects related to the prop erties of the emerging exact solution

(2.23). The solution has dilatonic gravity connections and it can b e reinterpreted as an anisotropic uid with radial

pressure equal to its energy density and no pressures along the spheres r =const. This can b e a radiation atmosphere

for a star with its radial pressure equal to the energy density of the athmosphere on its surface. No constraint on the

pressures along the spherical junction surface was found. On the exterior, the study of the matching conditions with

the Vaidya solution revealed no thin shells on the junction.

As a bypro duct, wehave derived general conditions for the junction of two spherically symmetric solutions. Matching

of two static space-times (2.1) along r =const hyp ersurfaces is assured by the continuity of the metric functions and

of the radial pressure. Matching of generic spherically symmetric space-times (4.11) along the null hyp ersurfaces

are continuous. fe dz 2dr = 0 is p ossible whenever the metric and @ fe

The negativity of the mass function in some neighb ourho o d of the r = 0 singularity raises the p ossibility of matching

this solution to a negative mass core. This may b e dicult due to the nontrivial top ology of the known negative mass

black hole solutions [23]. Despite the lack of exp erimental evidence for negative mass ob jects, presumably of quantum

origin [9], their microlensing e ect [24] on radiation from Active Galactic Nuclei was shown to pro duce features similar

to some observed Gamma Ray Bursts [25].

Equally interesting would b e to intro duce dynamics in the picture byaninterior matching with a collapsing star.

We defer this topic to a forthcoming study.

An intriguing op en question remains whether exact solutions describing the collision of spherically symmetric null

dust streams, whichhave not reached equilibrium, can b e found.

Note added. After the submission of this pap er a relevantwork in the sub ject was published by Kramer [26]. The

solution presented there is the particular case of the metric (2.23) with the parameter values B = 0 and C = e. Our

L 2 2

metric functions h and f , radial variable L and transcendental function e +2L corresp ond to e ; e ; 1+2

and e(1+2)J, resp ectively of this pap er, when the parameters values B = 0 and C = e are chosen. Keeping

the parameters arbitrary enabled us in Sec. IV A to match the solution (2.23) with an interior Schwarzschild solution

with arbitrary mass and radius. In contrast with our analysis relying on the Barrabes-Israel matching pro cedure,

in [26] the junction with the Vaidya space-time was discussed by imp osing the continuity of the four-metric on the

junction.

VI. ACKNOWLEDGMENTS

The author is grateful to JirBicak, Gyula Fo dor, Karel Kuchar and Zoltan Perjes for discussions on the sub ject

and helpful references. This work has b een supp orted by OTKA no. W015087 and D23744 grants. The algebraic

packages REDUCE and MAPLEV were used for checking computations and numerical plots. 11

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