Effect of sources on the inner horizon of black holes

(1)

Effect of sources on the inner horizon of black holes

Ozay Gurtug*and Mustafa Halilsoy

Department of Physics, Eastern Mediterranean University, G.Magusa, North Cyprus, Mersin 10 - Turkey

共Received 7 May 2001; published 26 September 2001兲

A single pulse of null dust and colliding null dust both transform a regular horizon into a spacelike singu-larity in the space of colliding waves. The local isometry between such space-times and black holes extrapo-lates these results to the realm of black holes. However, the inclusion of particular scalar fields instead of null dust creates null singularities rather than spacelike ones on the inner horizons of black holes.

DOI: 10.1103/PhysRevD.64.084023 PACS number共s兲: 04.20.Cv, 04.20.Dw

I. INTRODUCTION

In this paper we show, with exact solutions, that the Cauchy horizon 共CH兲 has an indeterminate character with respect to different perturbing potentials and in some cases the CH can be spacelike singular and in others it can be null singular. The first signs of this sort of indeterminacy of char-acter for the CH were seen in the outcome of Chandrasekhar and Xanthopoulos 共CX兲关1兴 and Yurtsever’s 关2兴 analysis of the stability of the horizon共or quasiregular singularity兲 of the Bell-Szekeres 共BS兲关3兴 space-time. These authors used per-turbation methods in their work. In Ref. 关4兴 it has been shown that there is a similar inner horizon instability for black holes共BHs兲 and the horizons change to spacelike sin-gularities. On the other hand, Ori关5兴 found the horizon of a Kerr BH changes to a null singularity. All of these works used perturbation methods.

This lack of consensus for the instability of colliding plane waves 共CPW兲 and BH horizons has attracted much interest and effort. Burko关6,7兴 confirmed Ori’s findings of a regular horizon changing to a null singularity when he ap-plied a scalar field to a Reissner-Nordstro¨m 共RN兲 BH. His work was done using numerical methods.

As an alternative to Burko’s approach we have applied the local isometry between CPW and the region between the two 共event and Cauchy兲 horizons of BHs. This enables us to couple scalar and other fields to CPWs, where there are ana-lytically tractable solutions, and then to transform our results to BH cases. This approach was first introduced by Yurtsever 关8兴. Yurtsever concluded that the instabilities of the CHs in Kerr and RN BHs turns the CH to a spacelike curvature singularity. Yurtsever’s comments were indications of certain possibilities but they remained unsupported.

We exploit the isometry analogy and we consider two sorts of sources in the CX CPW关9兴 space-time. The first is null propagating dust 共with their mutual collision兲. We con-centrate on the CX space-time because it has a nonsingular horizon and it is locally isometric to the Kerr-Newman共KN兲 BH, enabling us to transform through isometry to the BH space-time. We find that propagation 共or collision兲 of null dusts in the CX space-time convert the inner horizon to a spacelike singularity. This conclusion is supported by CX 关10兴 for the Einstein-vacuum problem which is locally

iso-metric to the Kerr BH. CX’s solution supports our conclu-sion applied to the KN BH, because the presence of charge is trivial. We point out that the introduction of coupling in our case is entirely different from the one in CX.

The second is a scalar field in between the horizons. We show that such a field effects the inner horizon differently from the above case and the singularity it creates turns out to be null. In other words our two sorts of coupling show the inner horizon of a BH does not have a unique character in its singularity structure and this character depends on the per-turbing potential. The instabilities of these CHs occurring in CPW times and those of corresponding BH space-times have dual character.

We also make some comments about the Helliwell-Konkowski 共HK兲关11兴 conjecture. HK conjecture was thought to enable us to predict the instability of a horizon and the sort of singularity it changes to. However, our con-clusion that inner horizons have dual character shows that HK conjecture cannot uniquely determine the sort of the out-coming singularity, hence it should be used with caution.

This paper is organized as follows. In Sec. II we review the connection between the CX and the KN space-times. In Sec. III we consider null dust as a test field in the CX space-time. Section IV follows with an exact back-reaction solution to the foregoing section. Section V exposes the role of scalar fields leading to the null singularities in the KN BH space-time. We conclude the paper with a discussion in Sec. VI.

II. CHANDRASEKHAR-XANTHOPOULOS AND KERR-NEWMAN METRICS

1⫺u2, and⌬⫽1⫺␶2,␦⫽1⫺␴2.

(2)

The metric functions are X⫽ 1 ␣2关共1⫺␣p␶兲 2_⫹_␣2_q2_␴2_兴, Y⫽1⫺p2␶2⫺q2␴2, q2e⫽⫺ q␦ p␣2 1⫹␣2⫺2␣p␶ 1⫺p2_␶2_⫺q2_␴2 共2兲

in which the constants␣, p, and q must satisfy 0⬍␣⭐1,

p2⫹q2⫽1. 共3兲

The metric 共1兲 transforms into the Boyer-Lindquist form of the KN, if the following transformation is used:

t⫽max, y⫽p␾, ␶⫽ m⫺r

冑

m2⫺␣2⫺Q2 , ␴⫽cos␪ 共4兲 with p⫽

冑

m 2_⫺a2_⫺Q2 m␣ , q⫽⫺ a m␣, 共m 2_⬎a2_⫹Q2_{兲共5兲}

so that Q2⫽(1⫺␣2)m2 holds. Note that␣⫽1 removes the charge and reduces the problem from KN to Kerr and in particular the limit a⫽0 yields the Reissner-No¨rdstro¨m case. With these substitutions the line element共1兲 may be written in the form

␣2_m2_ds2_⫽共1⫺_␳⫺2_␻_兲dt2

⫺sin2_␪_关⌬⫹_␻_共1⫹a2_␳⫺2_sin2_␪_兲兴d_␾2

⫺2a␻␳⫺2_sin2_␪_{dt d}_␾_⫺_␳2_共⌬⫺1_dr2_⫹d_␪2_兲

共6兲 with the standard notation

⌬⫽r2_⫺2mr⫹a2_⫹Q2_⬅共r⫺r

⫺兲共r⫺r⫹兲,

␳2_⫽r2_⫹a2_cos2_␪_,

␻⫽2mr⫺Q2_,

in which a and Q stand, respectively, for the constants of rotation and electric charge. Note that ⌬ here is different from the⌬ of the CX metric. The roots of ⌬, r_⫹, and r_⫺are known as the event and Cauchy共inner兲 horizon, respectively. Therefore the colliding wave solution due to CX is locally isometric to the KN metric in between the two horizons.

III. TEST NULL DUST IN THE CX SPACETIME We consider now two null test dusts moving in opposite directions in the interaction region of the CX metric or

equivalently null fields moving in the isometric region of the KN space-time. Such null dusts 共and the following exact solution兲 suffice to expose the nonlinear effect of the back-ground as well as the disturbance of the backback-ground共i.e., the back reaction兲. This is provided by appealing to the null geodesics of the KN and transforming back via Eqs.共4兲 and 共5兲 to the CX metric. For simplicity we choose␴⫽0 in CX 共or ␪⫽␲/2 in KN兲 to obtain the first integrals of the null geodesics as t˙⫽E共r 2_⫹a2_兲 ⌬ , ␾˙⫽⫺ aE ⌬ , r˙⫽E 共7兲 in the KN geometry, and the corresponding first integrals of the CX geometry are

␶˙_⫽ E m␣p, y˙⫽ qE m␣p共1⫺␶2兲, x˙⫽⫺ E m3␣3p2共1⫺␶2兲兵m 2_关共1⫺_␣_p_␶_兲2 ⫹共1⫺␣2_p2_兲兴⫺Q2_其_. _共8兲

In both cases E is the energy constant and dot represents the appropriate parameter for the null geodesics. We insert two null dust congruences with finite densities ␳l and␳n

propa-gating along the null vectors l_␮ and n_␮. In other words the total test energy-momentum tensor is

T_␮␯⫽␳ll␮l␯⫹␳nn␮n␯, 共9兲 where l␮⫽

冉

1,⫺关共1⫺␣p␶兲 2_⫹_␣2_q2_兴 ␣2_p_共1⫺_␶2_兲 , q 1⫺␶2,0

冊

, n␮⫽

冉

1,关共1⫺␣p␶兲 2_⫹_␣2_q2_兴 ␣2_p_共1⫺_␶2_兲 ,⫺ q 1⫺␶2,0

冊

共10兲 in which we have scaled

E

m␣p⫽1. 共11兲

The nontrivial scalar T_␮␯T␮␯ of the criss-crossing null test dust is given by

T_␮␯T␮␯⫽8␳l␳n

共1⫺␣p␶兲4

␣4_共1⫺_␶2_兲2 共12兲

which diverges for ␶→1. This corresponds to a divergence for r→r_⫺ in the KN black hole. As a prediction of the HK conjecture any exact back-reaction solution that is imitated by the foregoing test dust must destroy the horizon. In the next section we present a new exact back-reaction solution which represents a solution of colliding Einstein-Maxwell-null dust that exhibits a spacelike singularity for ␶→1. The

(3)

new solution incorporates a regular conformal factor 共such that it does not diverge as ␶→1) and therefore leaves all Weyl scalars invariant and regular.

IV. A NEW EXACT BACK-REACTION SOLUTION Our aim now is to present an exact solution which in-volves collision of Einstein-Maxwell fields coupled with null shells. The relations between the mathematical theory of BHs and of colliding waves allows us to find exact solutions which describe the region where two plane waves interact. Then, by a suitable extension it becomes possible to give a complete picture of the incoming space-time before the col-lision. In this section, we have used this fact to obtain a new solution to the EM fields coupled with null shells. The shells are added as a conformal factor and our method can be sum-marized as follows.

Let ds₀2 represent the CX metric共1兲 which is isometric to the KN. Then, the new metric 关12兴

ds2⫽ 1

␾2ds0 2

. 共13兲

Verification of the metric共13兲, follows from the substitution of the following into the Ricci scalars 共see the Appendix of Ref. 关13兴兲

M⫽M0⫹2 ln␾,

U⫽U₀⫹2 ln␾, V⫽V0,

W⫽W₀, 共14兲

where ( M0,U0,V0,W0) correspond to the metric functions

of the CX solution and ␾⫽1⫹␣0u␪(u)⫹␤0v␪(v), with

(␣0,␤0) positive constants and␪ standing for the step

func-tion. Equation 共13兲 represents colliding Einstein-Maxwell fields coupled with null shells. This metric has some advan-tages over the back-reaction solution of CX. First, while Ricci components and scalar curvature 共if any兲 are affected by inclusion of the conformal factor the Weyl scalars remain invariant 共because M⫺U⫽M0⫺U0 is the combination that

arises in those scalars兲, i.e., they are finite on the horizon. It turns out as shown in Appendix A explicitly that the scalar curvature and some Ricci components diverge on the hori-zon. Thus it is misleading to judge the behavior of a horizon by looking only at the Weyl scalars. Our approach gives the clue: it is more reliable to investigate the behavior of the scalar curvature and the Ricci components. In this sense the solution adopted in Eq. 共13兲 as the exact version of the test null dust is stronger共and simpler兲 than the implication of the source added CX solution关9兴.

The acceptability of the inclusion of this conformal factor is shown by checking the null and dominant energy condi-tions of the new solution. The details are given in Appendix B.

The collision of these shells in the background of CX space-time modifies the background in the sense that the curvature scalar, which was zero in the case of CX is now nonzero and becomes divergent as we approach the horizon. The method employed here should not be confused with the characteristic initial value problem. Extending the solution to the incoming regions to see the waves that participate in the collision is possible.

To find the wave profiles in the incoming regions for ex-ample in region II (u⬎0,v⭐0), we substitute v⫽0 in the obtained metric functions M, U, V, and W, and using Ricci and curvature scalars given in Appendix A, we obtain

2⌽22⫽ ␣2_共1⫺_␣2_兲共1⫺␣u兲4 ␪共u兲⫹␣ 2_␣ 0␦共u兲, ⌿4⫽共⌿4兲CX, ⌽00⫽⌽02⫽⌽11⫽⌳⫽0. 共15兲

Similarly in region III (v⬎0,u⭐0), the nonvanishing Ricci

and Weyl scalars are

2⌽00⫽ ␣2_共1⫺_␣2_兲共1⫺␣v兲4 ␪共v兲⫹␣ 2_␤ 0␦共v兲, ⌿0⫽共⌿0兲CX, ⌽22⫽⌽02⫽⌽11⫽⌳⫽0. 共16兲

The continuity of the metric functions across the null bound-aries u⫽0,v⫽0 makes these scalars continuous across the null boundary too.

As we can see from the incoming waves, in addition to gravitational and electromagnetic fields a matter共shell兲 field represented by an impulsive component is coupled to the system. Choosing ␣0⫽␤0⫽0 removes the matter field and

the resulting solution represented by the metric共13兲 reduces to CX solution. Since the metric 共13兲 satisfies all these boundary and continuity conditions it must be the correct physical solutions to the present collision problem.

As a second advantage we point out that the metric ds2⫽共1⫹␣0u⫹␤0v兲⫺2共2 du dv⫺dx2⫺dy2兲共17兲

represents the de Sitter space with scalar curvature and cos-mological constant as the only nonzero physical quantities 关13兴. The transformation

1⫹␣0u⫹␤0v⫽e␭t,

␣0u⫺␤0v⫽␭z 共18兲

takes this metric into

ds2⫽dt2⫺e⫺2␭t共dx2⫹dy2⫹dz2兲共19兲 which is identified as the de Sitter metric. Similarly by the choice of the conformal factor and transformation

(4)

1⫹␣0u⫺␤0v⫽␭z,

␣0u⫹␤0v⫽␭t, 共20兲

our metric becomes

ds2⫽ 1 ␭2_z2共dt

2_⫺dx2_⫺dy2_⫺dz2_兲, _共21兲

which is the anti–de Sitter metric. In both cases the constant ␭ is defined by ␭⫽

冑

2␣0␤0 in which ␣0⬎0,␤0⬎0. Now

instead of the flat metric by substituting the CX metric共1兲 it can be interpreted as colliding Einstein-Maxwell fields in a de Sitter background. This is the alternative interpretation that our metric admits when we remove the step functions in the conformal factor. By this interpretation 共and of course through the isometry兲 it says that a KN BH endowed with a de Sitter background in between the horizons gives rise to a divergent scalar curvature besides some of the Ricci’s. In turn according the HK conjecture such a horizon converts into a singularity. This singularity is necessarily spacelike since a normal vector to the horizon turns out to be timelike. As a final advantage we recall that in the CX metric the null fields are added to the vacuum problem whereas in our case we make the addition to the electrovacuum problem. Such an extension was a missing link in the study of CX.

V. NULL SINGULARITIES IN THE PRESENCE OF SCALAR FIELDS

Recently, we have shown the existence of null singulari-ties in the CPW space-time for a class of linearly polarized metrics 关14兴. Our main objective in this section is to inves-tigate the singularity structure of the CHs that exist in charged spinning BHs, namely, KN BH in the presence of scalar fields. This is achieved through the isometry existing between CPW space-times and BH space-times.

The space-time line element we adopted to describe the collision of plane waves with nonparallel polarization is given by

ds2⫽2e⫺Mdu dv⫺e⫺U关共eV_dx2_⫹e⫺V_{d y}2_{兲cosh W}

⫺2 sinh W dx dy兴. 共22兲

The complete set of partial differential equations for the met-ric functions is given elsewhere 关15兴. The u,v dependent massless scalar field equation

⳵␮共g␮␯

冑

g␾␯兲⫽0 共23兲

reads as

2␾_u_v⫽U_u␾_v⫹U_v␾_u, 共24兲 where␾ is the scalar field. Given any EM solution we can generate an Einstein-Maxwell-scalar 共EMS兲 solution in ac-cordance with the shift.

M→M˜⫹⌫, 共25兲

where M˜ is any EM solution and ⌫uUu⫽2␾u

2_,

⌫vUv⫽2␾v

2_. _共26兲

The integrability conditions for the latter equations imply that

共␾uUv⫺␾vUu兲共2␾uv⫺Uu␾v⫺Uv␾u兲⫽0. 共27兲

Now any solution of the scalar equation 共24兲 helps us to construct the extra metric function ⌫ by the line integral

⌫⫽2

冕

␾u 2 U_udu⫹2

冕

␾v 2 U_vdv. 共28兲 As an EM solution, we use the metric 共1兲 obtained by CX which was shown to be locally isometric to the KN BH that uses␶,␴ coordinates instead of u,v.

In terms of ␶,␴ the scalar field equation 共24兲 and the condition共26兲 are equivalent to

关共1⫺␶2_兲_␾ ␶兴␶⫺关共1⫺␴2兲␾␴兴␴⫽0 共29兲 and 共␶2_⫺_␴2_兲⌫ ␶⫽2⌬␦

冉

␶␾␶2⫹␶␦_⌬ ␾␴2⫺2␴␾␶␾␴

冊

共␴2_⫺_␶2_兲⌫ ␴⫽2⌬␦

冉

␴␾␴2⫹ ␴⌬ ␦ ␾␶2⫺2␶␾␶␾␴

冊

, 共30兲 respectively. For the present problem we choose the scalar field␾ as ␾⫽k1 2 ln 1⫹␶ 1⫺␶⫹ k2 2 ln 1⫹␴ 1⫺␴ 共31兲

in which k₁and k₂are constant parameters. Using Eqs.共30兲, the metric function ⌫, due to the scalar field is found as

e⌫⫽⌬⫺k1 2

␦⫺k2 2

共␶⫹␴兲(k₁⫹k₂)2_共_␶_⫺_␴_兲(k₁⫺k₂)2_. _共32兲

The new metric that describes the collision of EMS fields is expressed by ds2⫽Xe⫺⌫

冉

d␶ 2 ⌬ ⫺ d␴2 ␦

冊

⫺⌬␦ X Yd y 2_⫺Y X共dx⫺q2edy兲 2_, 共33兲 where X, Y, ⌬, ␦, and q2e are given in Eq. 共2兲. It is well

known that the CX solution is a regular solution. However coupling a scalar field ␾ transforms the CH into a scalar curvature singularity共SCS兲. This can be seen from the scalar

4␲T_␮␮⫽ e

⌫

⌬␦X共⌬k2 2_⫺_␦

(5)

and as ␶→1 it becomes divergent; hence it is a SCS. The interesting property of this new solution is that the type of the singularity is null rather than spacelike. This can be jus-tified as follows

The singular 共or horizon兲 surface is described by

S共␶兲⫽1⫺␶ 共35兲

共in which we have assumed an equatorial plane, namely, ␴

⫽0, in the transformed BHs space-time兲. We compute the normal vector to this surface

共ⵜS兲2_⫽g␶␶_S ␶ 2_⫽共1⫺␶ 2_兲e⌫ X ⫽ ␣2_␶2(k₁2⫹k₂2)_共1⫺_␶2_兲1⫺k₁2 共1⫺p␣␶兲2 . 共36兲 Since we are interested in the limit as ␶→1, for k₁2 ⬍1, (ⵜS)2_{⫽0 which indicates a null property. The blow-up}

of the curvature scalar provides a generic null singularity on this surface. The existing isometry can provide us with a null singularity in the corresponding BH problem.

Using the transformations given in Eqs. 共4兲 and 共5兲 the metric 共33兲 transforms into

␣2_m2_ds2_⫽共1⫺_␳⫺2_␻_兲dt2

⫺sin2_␪_关⌬⫹_␻_共1⫹a2_␳⫺2_sin2_␪_兲兴d_␾2

⫺2a␻␳⫺2_sin2_␪_{dt d}_␾

⫺␳2_e⫺⌫_共⌬⫺1_dr2_⫹d_␪2_兲, _共37兲

where␳, ␻, and⌬ are given in Eq. 共6兲. This metric repre-sents the KN BH coupled with the particular scalar field which reads in the equatorial plane

␾⫽k1

2 ln

冉

r_⫹⫺r

r⫺r_⫺

冊

. 共38兲

In the same plane we have

e⌫⫽关共r_⫹⫺r兲共r⫺r_⫺兲兴⫺k1

2

冉

m⫺r

冑

m2⫺Q2

冊

2(k₁2⫹k₂2)

. 共39兲

Similar analysis on the singular surface

S共r兲⫽r⫺r_⫺ 共40兲

results in (ⵜS)2⫽0 as r→r_⫺. This result retains the null singularity formation on the corresponding BH problem.

Note that, the scalar field in Eq.共31兲 is singular as␶→1. Let us now choose the following scalar field which is regular, in the limit ␶→1,

␾⫽␣0␶␴, 共41兲

where␣₀is any constant parameter. The metric function⌫ is obtained as

e⌫⫽e␣0[␶2⫹␴2(1⫺␶2)]_. 共42兲

The curvature scalar due to this scalar field is

4␲T_␮␮⫽␣0

2_e_⌫

X 共␶

2_␦_⫺_␴2_⌬兲. _共43兲

The metric function ⌫ and curvature scalar remains regular as the horizon is approached. The Weyl scalars are also checked and found that as ␶→1, the space-time remains regular. This is another justification of the indeterminate character of CHs.

VI. DISCUSSION

It is known that the relations between the mathematical theory of BHs and that of CPW space-time requires that the interior of every standard BH solution is locally isometric to the interaction region of CPW space-time.

In this paper, we have used this fact to investigate the singularity structure of the charged spinning BH namely the KN. In our analysis, we have coupled matter共null shells兲 and scalar fields to the geometry of CPW space-time and using the isometry, we transformed the resulting metric to the BH geometry and investigated the effect of these fields on the CH. Since our analysis is based on a completely analytic exact solutions, we believe that it reflects the real character of the Killing CHs of CPW and BH space-times.

First, a pair of test null dust is inserted in the CX colliding wave space-time. Since the energy-momentum scalar di-verges then we conclude that the CH is unstable and an exact back reaction solution must yield a scalar curvature singular-ity. The exact solution which we present involves a regular conformal factor as a source so that all Weyl scalars remain finite while some Ricci’s 共and scalar curvature兲 diverge on the horizon. Inclusion of the conformal factor is equivalent energetically to the de Sitter background and through the isometry it makes it possible for us to embed a BH in a de Sitter 共or anti–de Sitter兲 background. Although we have in-serted a matter field as a regular conformal factor, the slight-est effect of the source共through the scalar curvature or cos-mological constant兲 of such a background suffices to destroy the inner horizon and converts it into a spacelike curvature singularity. Second, we have inserted scalar fields to the ge-ometry of CX CPW solution. Inclusion of particular scalar fields was shown to create null singularities rather than spacelike ones both in the space of colliding waves and that of corresponding BH space-times. With these particular exact solutions, we may conclude that the Killing CH of CPW space-times and the inner horizon of BHs have dual charac-ter when they are subjected to the inclusion of matcharac-ter and scalar fields. It remains an open question whether this null singularity is an intermediate stage between the horizon and spacelike curvature singularity.

APPENDIX A: THE RICCI AND CURVATURE SCALARS The nonzero Ricci and curvature scalars of the collision of Einstein-Maxwell fields coupled with null shells 共in the same null tetrad employed by CX兲 are found as follows:

(6)

⌽22⫽共⌽22兲CX⫹␣0

eM

␾ 关␦共u兲⫹␪共u兲Mv兴, 共A1兲

⌽00⫽共⌽00兲CX⫹␤0

eM

␾ 关␦共v兲⫹␪共v兲Mu兴, 共A2兲

⌽02⫽共⌽02兲CX⫺

eM

2␾ 兵␣0关Vvcosh W⫺iWv兴␪共u兲

⫹␤0关Vucosh W⫺iWu兴␪共v兲其, 共A3兲

1⫺v2, ␾⫽1⫹␣0u␪共u兲⫹␤0v␪共v兲. 共A6兲

Since (1⫺u2⫺v2)⫽

冑

1⫺␶2

冑

1⫺␴2for␴⫽0 and␶→1 共on the horizon兲 the divergence of the scalar curvature ⌳ and ⌽11 is clearly manifest. A detailed calculation reveals that

⌽02is also divergent while⌽00and⌽22remain finite on the

inner horizon. Let us note that suppressing one of the incom-ing shells but retainincom-ing the other, still the above components diverge. This amounts to colliding Einstein-Maxwell wave from one side with an Einstein-Maxwell-null shell from the other side. However this form can not be interpreted as a de Sitter background in the corresponding black-hole problem.

APPENDIX B: ENERGY CONDITIONS

It should be noted that the inclusion of the matter field is acceptable if it satisfies some physical conditions such as the null and dominant energy conditions.

⌬ ,⫺ i␣

冑

␦共1⫺␣␶兲

冊

. 共B2兲 It is found that T_␮␯l␮l␯⫽⌽00, T_␮␯n␮n␯⫽⌽22, 共B3兲 where ⌽00⫽ ␣2_共1⫺_␣2_兲 2共1⫺␣␶兲4⫹ ␣eM ␾ 关␦共u兲⫹Mv␪共u兲兴, ⌽22⫽ ␣2_共1⫺_␣2_兲 2共1⫺␣␶兲4⫹ ␤eM ␾ 关␦共v兲⫹Mu␪共v兲兴, 共B4兲 which are positive and therefore satisfy NEC.

2. Dominant energy condition„DEC… The DEC is defined as

T00⭓兩Tab兩, 共B5兲 i.e., for each a,b, in the orthonormal basis the energy domi-nates the other component T_ab. The orthonormal vectors for the diagonal case of the new solution are

e₍₀₎␮ ⫽

冉

1

冑

X,0,0,0

冊

, 共B6兲

e₍₁₎␮ ⫽

冉

0,⫺ 1

(7)

e(2)␮ ⫽

冉

0,0,

1⫺␣␶

␣

冑

⌬ ,0

冊

, e₍₃₎␮ ⫽

冉

0,0,0,⫺ ␣

冑

␦共1⫺␣␶兲

冊

.

The nonzero energy-momentum tensors in orthonormal frames are T00⫽ 1 8␲关⌽00⫹⌽22⫹2共⌽11⫹3⌳兲兴, 共B7兲 T01⫽T10⫽ 1 8␲关⌽22⫺⌽00兴, T11⫽ 1 8␲关⌽00⫹⌽22⫺2共⌽11⫹3⌳兲兴, T22⫽ 1 4␲关⌽00⫹⌽11⫺3⌳兴, T33⫽ 1 4␲关⌽11⫺⌽02⫺3⌳兴,

where the expressions for⌽₀₀, ⌽₂₂, ⌽₁₁, ⌽₀₂, and⌳ are given in Appendix A.

关1兴 S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc.

Lon-don A415, 329共1988兲.

关2兴 U. Yurtsever, Phys. Rev. D 36, 1662 共1987兲.

关3兴 P. Bell and P. Szekeres, Gen. Relativ. Gravit. 5, 275 共1974兲. 关4兴 Y. Gursel, V.D. Sandberg, I.D. Novikov, and A.A. Starobinsky,

Phys. Rev. D 19, 413 共1979兲; S. Chandrasekhar and J.B. Hartle, Proc. R. Soc. London A284, 301共1982兲.

关5兴 A. Ori, Phys. Rev. Lett. 68, 2117 共1992兲. 关6兴 L. Burko, Phys. Rev. Lett. 79, 4958 共1997兲. 关7兴 L. Burko, Phys. Rev. D 59, 024011 共1998兲. 关8兴 U. Yurtsever, Class. Quantum Grav. 10, L17 共1993兲.

Lon-don A414, 1共1987兲.

Lon-don A418, 175共1986兲.

关11兴 T.M. Helliwell and D.A. Konkowski, Class. Quantum Grav.

16, 2709共1999兲.

关12兴 M. Halilsoy, J. Math. Phys. 41, 8351 共2000兲. 关13兴 A. Wang, J. Math. Phys. 33, 1065 共1992兲. 关14兴 O. Gurtug and M. Halilsoy, gr-qc/0103092. 关15兴 O. Gurtug and M. Halilsoy, gr-qc/0006038.