Nonsingular colliding wave solutions in Einstein-Maxwell-dilaton-axion theory E. Halilsoy and M. Ha

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Nonsingular colliding wave solutions in Einstein-Maxwell-dilaton-axion theory

E. Halilsoy*and M. Halilsoy†

Physics Department, Eastern Mediterranean University, G. Magosa, Mersin 10 (N. Cyprus), Turkey 共Received 25 December 2003; published 25 June 2004兲

The local isometry between black holes and colliding plane waves is employed to derive new colliding wave solutions in the Einstein-Maxwell-dilaton-axion theory. The technique is applied to the asymptotically nonflat linear dilaton black holes. We obtain two new metrics which we label共from the language of black holes兲 as Kerr and Newman-Unti-Tamburino共NUT兲 types. The NUT type turns out to be type D while the Kerr type belongs to the general class. Both types share the common feature that, instead of an all encompassing generic singularity, Cauchy horizons develop in the process of collision.

DOI: 10.1103/PhysRevD.69.124021 PACS number共s兲: 04.20.Jb

I. INTRODUCTION

Chandrasekhar and Xanthopoulos共CX兲 first observed that a particular metric of colliding plane waves 共CPWs兲 trans-forms into the trapped region between horizons of the Kerr black hole 共BH兲关1兴. The reason for this local isometry is simple: in that region the Kerr black hole admits two space-like Killing vectors, the same as required by the space of CPWs. A coordinate transformation maps the one problem into the other provided the boundary conditions are satisfied. By this it is meant that continuous matching of the different wave regions holds, such that no source currents arise at the boundaries. A special case covers naturally the Schwarz-schild BH where for r⬍2m 共i.e., inside the horizon兲 it ad-mits two spacelike Killing vectors and the corresponding CPW spacetime can easily be derived 关2兴. Extension of the Kerr BH to the Kerr-Newman case and the associated CPW metric in Einstein-Maxwell 共EM兲 theory was also given by CX关3兴. The same idea of local isometry has also been used to obtain CPW solutions in Einstein-dilaton-axion 共EDA兲 theory 关4兴. More recently, we have given an example of the CPW metric in the Einstein-Maxwell-dilaton-axion共EMDA兲 theory in the limit of zero dilaton field, which also employs an isometry between the throat region of extremal BHs and CPWs 关5兴. This example suggests that the local isometry in question has a larger scope than envisaged. In a separate work we showed the exact equivalence of the near horizon geometry of extremal BHs and CPWs in the EM theory关6兴. All BH solutions alluded to so far share the common feature that they are asymptotically flat. A new type of BH in the linear dilaton gravity has been introduced, on the other hand, which fails to satisfy asymptotic flatness 关7–11兴. Since the space of CPWs also shares this latter condition, the local isometry between such BHs and CPW spacetimes must be expected in a more natural way. In addition to the linear dilaton, these asymptotically non-flat BHs admit electromag-netic共em兲 and axion fields, which enable us in this paper to obtain new CPW metrics in the EMDA theory. From the physics standpoint our solutions are important since they are free of physical singularities. Singularities, which used

mostly to doom the interaction region of CPWs, are replaced here by extendable Cauchy horizons. By standard solution generation techniques, new solutions in the EMDA theory can be obtained, but singularity-free solutions are not guar-anteed关12,13兴. Finally, we wish to comment that we can add massless scalar fields to the already existing dilaton, axion, and em fields by using a method which we have developed recently 关14,15兴.

The organization of the paper is as follows. In Sec. II we review the linear dilatonic BH and its extension to stationary form. Section III covers the derivation of our CPW metrics whose details are tabulated in Appendixes B and C. We con-clude the paper in Section IV with a conclusion and discus-sion.

II. LINEAR DILATON BLACK HOLES

The field equations in the EMDA theory can be generated from the action

S⫽ 1 16␲

冕

d 4_x_兩g兩1/2

冋

_{⫺R⫹2共ⵜ}_␾_兲2_⫹1 2e 4␾_共ⵜ_␬_兲2 ⫺e⫺2␾_F ␮␯F␮␯⫺␬F␮␯F˜␮␯

册

共1兲

where ␾ is the dilaton, ␬ is the 共pseudoscalar兲 axion, and F_␮␯ stands for the em field tensor. The dual field tensor is defined by F˜␮␯⫽1

2兩g兩⫺1/2⑀␮␯␣␤F␣␤ in which we choose

⑀0123_{⫽⫹1. In addition to the Einstein equations}

G_␮␯⫽⫺8␲T_␮␯ 共2兲

the remaining EMDA field equations are

⳵␮关兩g兩1/2共e⫺2␾F␮␯⫹␬F˜␮␯兲兴⫽0, 共3兲

2䊐␾⫽e4␾_共ⵜ_␬_兲2_⫹e⫺2␾_F

␮␯F␮␯, 兩g兩⫺1/2_⳵

␮共兩g兩1/2e4␾g␮␯␬,␯兲⫽⫺F␮␣F˜␮␣,

in which䊐 stands for the covariant Laplacian. In Appendix A we give the total energy-momentum tensor T_␮␯in terms of both the fields and the tetrad scalars. The following diagonal metric solves the EMDA equations without the axion关9兴: *Electronic address: elif.halilsoy@emu.edu.tr

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共d␪ 2_⫹sin2_␪_d_␾2_兲. _共4兲 The dilaton and Maxwell two-form (F⫽dA) are

e2␾⫽e2␾⬁

冉

1⫺r⫺ r

冊

, F⫽Qe

2␾_⬁

r2 dr⵩dt, 共5兲

respectively, where␾_⬁is the asymptotic value of the dilaton and the mass共M兲 and electric charge 共Q兲 of the BH are

M⫽1 2r⫹,

Q⫽e⫺␾⬁

冑

r⫹r⫺

2 . 共6兲

In the string frame the dilaton is a linear function of distance, and in the near horizon limit this solution of EMD theory transforms into ds2_⫽r⫺b r0 dt2_⫺ r0 r⫺bdr 2_⫺r 0r共d␪2⫹sin2␪d␾2兲共7兲 with e2␾⫽r r₀, F⫽ 1

冑

2r0 dr⵩dt.

The new constants b and r0 that arise in the near horizon geometry are related to the mass (b⫽2M) and the electric charge (Q⫽r0/

冑

Thus the six potentials, namely, f,␹, u,v, ␾, and␬ param-etrize overall the target space apt for the EMDA theory. The Kerr Newman-Unti-Tamburino共NUT兲 extension of the static metric 共7兲 is obtained accordingly as

ds2⫽⌬˜⫺a 2_sin2_␪ ⌫ 共dt⫺wd␾兲2 ⫺⌫

冉

dr 2 ⌬˜ ⫹d␪ 2_⫹ ⌬˜ sin 2_␪ ⌬˜⫺a2_sin2_␪d␾ 2

冊

_共11兲

共we note that we put a tilde over ⌬ in order to avoid any

confusion with the⌬ that we shall be using in the next sec-tion兲. The dilaton, axion, and (u,v) potentials are

e2␾⫽r 2_{⫹共N⫹a cos}_␪_兲2 ⌫ , ␬⫽r0 M N共r⫺M 兲⫺aM cos␪ r2⫹共N⫹a cos兲2 , v⫽r 2_{⫹共N⫹a cos}_␪_兲2 ⌫ , u⫽r0 M N共r⫺M 兲⫺aM cos␪ ⌫ , 共12兲

where a and N are the Kerr共rotation兲 and NUT parameters, respectively, while other abbreviations are as follows:

⌬˜⫽r2_⫺2Mr⫹a2_⫺N2_, ⌫⫽r0 M共Mr⫹N 2_{⫹aN cos}_␪_兲, w⫽⫺r0 M

N⌬˜ cos␪⫹a共Mr⫹N2兲sin2␪

⌬˜⫺a2_sin2_␪ .

共13兲

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III. CPW SOLUTIONS FROM LINEAR DILATON BH The general metric for CPWs is represented by关1兴

ds2⫽X

冉

d␶ 2

⌬ ⫺

d␴2

␦

冊

⫺共Ydx2⫹Zdy2⫾2Wdxdy兲共14兲 where ⌬⫽1⫺␶2, ␦⫽1⫺␴2, and the metric functions de-pend only on the variables (␶,␴). Next, by introducing null coordinates (u,v) through

␶⫽sin共au⫹bv兲,

␴⫽sin共au⫺bv兲

共a,b⫽const兲, 共15兲

we observe that the line element is cast into the standard form suitable for CPWs. The colliding wave formulation of the problem follows by the substitutions u→u␪(u), v →v␪(v), where ␪ is the Heaviside unit step function. The problem of local isometry requires that the Kerr-NUT metric

共11兲 and 共12兲 be transformed into the form of CPWs such

that X⬎0, Y⭓0, and Z⭓0 necessarily. Vanishing of metric functions signals singularities of the coordinate type or ge-neric curvature singularities. We observe that the (r,␪) sec-tor of the BH metric共11兲 can consistently be mapped into the (␶,␴) form provided ⌫共r兲

冉

y⫽␸. 共19兲

By imposing appropriate scaling of the coordinates共adapting M⫽r0⫽1) and defining q⫽a 共or q⫽N) with a related pa-rameter p⭓1 such that

_dy_⫺ qdx p⫹sin au

冊

2

册

⫺ cos 2_au p⫹sin audx 2 _共23兲

共and a similar metric with dsIII

2

with au↔bv for region III兲. For u⬍0, v⬍0, we get the flat metric 共region I兲

ds_I2⫽4abpdudv⫺1 pdx 2_⫺p

冉

_dy_⫺q pdx

冊

2 共24兲

expressed in a scaled coordinate system. The dilaton, axion, and em fields can also be easily obtained in the incoming regions. By inverting the problem, this information consti-tutes our initial data which naturally all vanish, as it should in the flat region u⬍0, v⬍0. In Appendix B we give the nonzero curvature and Ricci components of this spacetime. The interesting property is that it is not singular. All Weyl scalars are regular and the singularities at ␶⫽1(␴⫽1) are spurious coordinate singularities. Another interesting prop-erty is that in contrast to the Kerr metric our Kerr-type metric

共21兲 is not type D. This becomes evident after we compute ⌿0⌿4⫺9⌿2

2_⫽ q 2␦_⌬

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In the limit q→0 (p→1), which corresponds to the CPW generated from the static dilaton metric 共7兲, it becomes type D.

共2兲 The NUT-type CPW metric (a⫽0,q⫽N⫽0)

ds2⫽共p⫹␶兲

冋

d␶ 2 ⌬ ⫺ d␴2 ␦ ⫺␦d y2

册

⫺ ⌬ p⫹␶共dx⫹q␴d y兲 2_. 共26兲

The dilaton, axion, and Maxwell potential one-form are as follows: e2␾⫽p共1⫹␶ 2_兲⫹2_␶ ␶⫹p , ␬⫽ q␶ p共1⫹␶2兲⫹2␶, A⫽ 1

冑

2 p

冋

p共1⫹␶2兲⫹2␶ ␶⫹p dx⫺ pq␴⌬ ␶⫹p d y

册

. 共27兲

In the incoming region (v⬍0) 共region II兲 our metric takes

the form 关with u⫽u␪(u)]

ds_II2⫽共p⫹sin au兲关4abdudv⫺cos2aud y2兴

⫺ cos

2_au

p⫹sin au 共dx⫹q sin audy兲

2 _共28兲

and a similar form 共by au↔bv) follows for the region III incoming metric. For u⬍0, v⬍0 we obtain the flat metric

ds2⫽p共4abdudv⫺dy2兲⫺1 pdx

2_. _共29兲

The initial data for our incoming fields can also easily be found from Eq.共27兲. We present the details of this metric in Appendix C. The Weyl scalars suggest that, similar to the Kerr-type metric共21兲, the NUT-type metric 共26兲 is also regu-lar.␶⫽1 (␴⫽1) are coordinate singularities that can be re-moved. The significant difference between the NUT and Kerr types is that the NUT type turns out to be type D. The Weyl curvatures 共Appendix C兲 in the interaction region (u⬎0,v

⬎0) satisfy

9⌿₂2⫽⌿0⌿4. 共30兲

共3兲 The general static EMD metric was given in Eq. 共4兲.

We wish now to obtain the corresponding CPW in this case as well. For this purpose we identify

dr2 共r⫺r⫺兲共r⫺r⫹兲 ⫺d␪ 2_⫽d␶ 2 ⌬ ⫺ d␴2 ␦ , 共31兲

which leads to the relation

2r⫽r_⫹⫹r_⫺⫹共r_⫹⫺r_⫺兲␶. 共32兲

The coordinate transformation toward our CPW metric is accomplished by this condition and ␴⫽cos␪, x⫽t, and y

⫽␸. The resulting metric is 共up to an overall constant rescaling兲 ds2⫽共1⫹␶兲共␣0⫹␤0␶兲

冋

d␶2 ⌬ ⫺ d␴2 ␦ ⫺␦dy2

册

⫺_␣1⫺␶ 0⫹␤0␶ dx2, 共33兲

where␣₀⫽r_⫹⫹r_⫺and␤₀⫽r_⫹⫺r_⫺. This is the CPW met-ric corresponding to a more general EMD theory without the axion. In the extremal case we choose␤0⫽0 and 共after res-caling the x coordinate兲 we obtain

ds2_⫽共1⫹_␶_兲

冋

d␶ 2

⌬ ⫺

d␴2

␦ ⫺␦dy2

册

⫺共1⫺␶兲dx2. 共34兲 This is precisely the limiting case ( p⫽1,q⫽0) of both the Kerr共21兲 and NUT 共26兲 type (a⫽0⫽N) metrics for CPWs. The metric共33兲 describes collision of waves in EMD theory, which is both regular and type D.

IV. CONCLUSION AND DISCUSSION

The local equivalence between the inner horizon region of BHs and the spacetime of CPWs has been fruitful in the generation of physically significant solutions in the colliding EMDA theory. For sample BHs we have chosen Kerr-NUT-type BHs in a linear dilaton background. As expected, the initial data for dilaton, axion, and em fields cannot be arbi-trary but are dictated by the original BH solution. The free-dom to eliminate the axion reduces the metric to diagonal and it leads to a regular CPW solution in the EMD theory. The incoming plane waves 共i.e., a holographic boundary in the string language兲 consisting of a mixture of dilaton, axion, and em waves extend smoothly into the interaction region. We realize once more共as in Ref. 关5兴兲 that the axion survives within the second polarization context of the colliding waves. We add, finally, that our technique applies also in higher dimensional BHs and colliding branes. One signaling problem in higher dimensions, however, is that the plane waves may propagate in lower dimensional backgrounds. It remains to be seen whether this feature may lead to the cre-ation of extra dimensions via colliding waves.

APPENDIX A

The total energy-momentum tensor is given by

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In terms of the null tetrad formalism of Newman and Pen-rose共NP兲, the energy-momentum is expressed as follows:

4␲T_␮␯⫽␾00n␮n␯⫹␾22l␮l␯⫹␾02m¯␮m¯␯⫹␾20m␮m␯

⫺␾01n␮m¯␯⫺␾10n␮m␯⫺␾12l␮m¯␯⫺␾21l␮m␯

⫹共␾11⫹3⌳兲共l␮n␯⫹n␮l␯兲

⫹共␾11⫺3⌳兲共m␮m¯␯⫹m¯␮m␯兲. 共A2兲 APPENDIX B

The null tetrad basis one-forms for the Kerr-type metric

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⌿2⫽ ab␪共u兲␪共v兲 12共␶⫹p兲3 关共1⫹p␶⫹iq兲 2_⫹共p_␶_{⫹1兲共1⫹iq兲兴,} ⌿4⫽bG2共u兲␦共v兲⫹ b2␪共v兲 4共p⫹␶兲3关⫺p 2_{⌬⫹3共1⫹p}_␶_{兲共1⫹iq兲兴,} ⌿0⫽aF2共v兲␦共u兲⫹ a2␪共u兲 4共p⫹␶兲3关⫺p 2_{⌬⫹3共1⫹p}_␶_{兲共1⫹iq兲兴,} 共C2兲

where the impulsive components are

G2⫽␭v and F2⫽⫺␴u,

in which the spin coefficients are

. 共C3兲

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