Colliding wave solutions from five-dimensional black holes and black pp-branes

E. Halilsoy, M. Halilsoy, and O. Unver

Citation: Journal of Mathematical Physics 47, 012502 (2006); doi: 10.1063/1.2157051 View online: http://dx.doi.org/10.1063/1.2157051

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/47/1?ver=pdfcov Published by the AIP Publishing

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Colliding wave solutions from five-dimensional black

holes and black p-branes

E. Halilsoy,a兲M. Halilsoy,b兲 and O. Unverc兲

Physics Department, Eastern Mediterranean University, G.Magosa (N. Cyprus), Mersin 10, Turkey

共Received 22 July 2005; accepted 9 November 2005; published online 13 January 2006兲 We consider both the five-dimensional Myers-Perry and Reissner-Nordstrom black holes共BHs兲 and black p-branes in 共4+p兲-dimensions. By employing the isometry with the colliding plane waves共CPWs兲 we generate Cauchy-Horizon 共CH兲 forming CPW solutions. From the five-dimensional vacuum solution through the Kaluza-Klein reduction the corresponding Einstein-Maxwell-dilaton solution is obtained. This CH forming cross polarized solution with the dilaton turns out to be a rather complicated nontype D metric. Since we restrict ourselves to the five-dimensional BHs we obtain exact solutions for colliding 2- and 3-form fields in 共p+4兲-dimensions for p艌1. By dualizing these forms we can obtain also colliding 共p+1兲- and 共p+2兲-forms which are important processes in the low energy limit of the string theory. All solutions obtained are CH forming, implying that an analytic extension beyond is possible. © 2006 American Institute of Physics.

关DOI:10.1063/1.2157051兴

I. INTRODUCTION

Black holes 共BHs兲 are known to have region isometric to the space of colliding plane waves 共CPW兲.1,2

This may be either in between the two horizons共i.e., inner and outer兲 or given the case with single horizon the inner region of the event horizon. Such an isometry renders it possible to generate CPW solutions from known solutions of BHs. In a recent paper3we gave a prescription for generating CPW solutions from an Einstein-Maxwell-Dilaton-Axion共EMDA兲 theory. In this theory the dilaton was linear and the BH was not asymptotically flat. In this solution the axion arises as the cross polarizing agent for the CPWs. This means that the limit of linear polarization removes the axion leaving behind only the Einstein-Maxwell-Dilaton 共EMD兲 theory. Another solution with similar features but valid only in the zero dilaton limit was obtained previously.4 Interesting physical property shared by both of these solutions is that the space-time subsequent to the collision of waves emerges free of physical singularities. Horizon forming CPW1,2,5–7solutions in the EMDA theory are naturally of utmost important to the string theory. Since the idea of higher dimensions has already gained enough momentum it is important to investigate the collision of waves in higher dimensions.8,9 It is known already that the four-dimensional EMDA theory is equivalent to the six-dimensional Ricci flat, vacuum solution.10,11

In this paper we restrict ourselves to the five- 共and four-兲 dimensional space-times and their extension through the brane world. We consider first the five-dimensional collision of gravitational 共impulse and shock兲 waves obtained from the isometry with the Myer-Perry black hole 共MPBH兲.12,13

This particular BH contains two rotation parameters in addition to the mass. For simplicity we make the special choice in which the two angular momenta are equal. Then we identify the共r,␪兲 sector of the BH at hand with the null coordinates sector 共u,v兲 of the colliding

a兲_{Electronic mail: elif.halilsoy@emu.edu.tr} b兲_{Electronic mail: mustafa.halilsoy@emu.edu.tr} c兲_{Electronic mail: ozlem.unver@emu.ed}

47, 012502-1

waves and accompany this with the necessary coordinate transformation. Inclusion of the Heavi-side step functions along with the null coordinates must guarantee that no additional current sheets are created at the boundaries. In the standard Einstein and Einstein-Maxwell共EM兲 theories these are summarized in the O’Brian-Synge14 boundary conditions, respectively. Similar arguments rightly follow in the higher dimensional space-times as well.

The static MPBH is transferred to the linearly polarized CPWs, which turns out to be a nonsingular type-D solution. The rotating MPBH transforms through the Kaluza-Klein共KK兲 re-duction procedure to the CPW space-time with cross polarization in the four-dimensional EMD theory. This space-time is also regular but it does not belong to the type-D class. As a matter of fact the dilaton involved cross polarization共instead of axion, as it arises in the above-mentioned solutions兲 makes the space-time structure rather involved.

As a second example we consider the Reissner-Nordstrom共RN兲 BH15in five dimensions from which we obtain CPW solution in the five-dimensional EM theory. Similar to the CPW solution obtained from the MPBH this one also is singularity free. Our examples of five-dimensional BHs exclude the extremal limits because in such a limit which removes the isometric region of BHs with CPWs the equivalence fails to work. Under such circumstances an alternative transformation, analogous to the RN-Bertotti-Robinson equivalence, must be pursed which is out of our scope in this paper.

As a third example we consider the black-branes in the共d+p兲-dimensional brane world. We find regular CPW solutions to colliding 3-form fields in higher dimensions. Another solution that we obtain from the same black-brane metric is colliding 共EM兲 共2-form兲 fields in higher dimen-sions.

Our study may lay the foundation for promoting the string theory in approximation in low energies from single plane wave background to the more realistic CPW background. The regular initial data of CPWs provides a natural choice among the nonunique Penrose limits of space-times.16–18It is known that the incoming region of a CPW space-time admits automatically a Penrose limit of the interaction region. The advantage is that we have the double Penrose limits which are both well-defined initial data. Any Penrose limit does not qualify as an initial data toward construction of the interaction region.

Finally we wish to express the view that our technique can be extrapolated to higher dimen-sions provided some associated difficulties are overcome. The most important problem is the analytic integration of the radial coordinate共r兲 of BHs in terms of the prolate-type coordinate 共␶兲 of CPWs. And as the second major difficulty we cite of the necessary proper representation of the higher dimensional spherical line element suitable for the ideals of the geometry of CPWs.

The organization of the paper is as follows. In Sec. II we obtain CPW solutions from the five-dimensional BHs, whose details are tabulated in Appendixes A, B, and C. Section III inves-tigates the physical properties of the metrics obtained in Sec. II. Section IV contains solutions for colliding 3-form fields in higher dimensions and their KK reductions. Section V focuses attention on a class of by-product solutions of colliding EM shock waves in higher dimensions. We dualize our 2-form fields of Sec. V in Sec. VI to obtain colliding共p+2兲-form fields in 共p+4兲-dimensions. Our conclusion and discussion is in Sec. VII.

II. CPW SOLUTIONS FROM FIVE-DIMENSIONAL BHs

In this section we concentrate on the two well-known types of BHs in five-dimensions. First we consider the MPBH and then RNBH. Our analysis applies, however, to any five-dimensional BHs that possesses two nonoverlapping horizons albeit the technical difficulties. We comment on this point at the end of the section in concentration with the Schwarzschild-de共-anti兲 Sitter BH.

共A兲 The MPBH in five dimensions with two equal angular momenta is given ds52= g˜ABdxAdxB 共A,B, ... = 0, ... 4兲,

ds₅2= dt2− ␮ ␳2

冋

dt + a ¯ 2共d␸− cos␪d␩兲

册

2 − ␳ 4_d␳2 ␳4_−␮␳2_+␮¯a2−␳ 2_d⍀ 3 2 , 共1兲

where␮and a¯ are, respectively, proportional to the mass and angular momentum of the BH. We

note that five-dimensional suffices are denoted by capital italic letters while four dimensional ones by greek letters. A tilde over specifies also the five-dimensional geometrical object. For the three-dimensional metric of S3we choose the representation

d⍀_共3兲2

=1₄共d␪2+ d␩2+ d␸2− 2 cos␪d␩d␸兲, 共2兲 where 0⬍␪⬍␲, and the angles␩and␸are defined modulo 2␲. The static MPBH corresponds to

a = 0, while the extreme case is defined by␮= 4a¯2_{. The CPW form in five dimensions is obtained} by imposing the identification of the 共␳,␪兲 sector in the above metric with the 共␶,␴兲 sector of CPW as follows:3

冉

4␳2d␳2 ␮␳2_−␳4_−␮¯a2− d␪ 2

冊

₌

冉

d␶ 2 1 −␶2− d␴2 1 −␴2

冊

. 共3兲

In the sequel, for simplicity we choose␮= 1 leading us to the solution 2␳2= 1 +

冑

1 − 4a¯2_␶,

共4兲 cos␪=␴

implying further that we impose 兩a¯兩 ⬍1₂. Supplementing this transformation with the identifica-tions t→ x, ␸→ y, 共5兲 ␩→ z, ao 2 = 2a¯2

followed by an appropriate rescaling of coordinates we obtain the five-dimensional vacuum metric apt for CPWs: ds₅2= F

冉

d␶ 2 ⌬ − d␴2 ␦

冊

− 1 F关Zo共dy − 2␴dz兲dy + Z dz 2_{+ 4a} o共dy −␴dz兲dx − 共1 − k␶兲dx2兴. 共6兲

Our abbreviations in this metric stand as follows: ⌬ = 1 −␶2_, ␦= 1 −␴2, F = 1 + k␶, Z = F2+ 2ao2␴2, Zo= F2+ 2ao 2_,

k =

冑

1 − 2ao

2 ,

0⬍ k 艋 1, 共7兲

where the coordinates共␶,␴兲 are related to the null coordinates 共u,v兲 through ␶= sin共au + bv兲,

␴= sin共au − bv兲, 共8兲

共a,b:constants兲.

Now, the crucial point toward the interpretation of this metric as a CPW metric is by making the substitutions

u→ u␪共u兲,

共9兲

v→ v␪共v兲

in the metric functions, where␪stands for the Heaviside unit step function. This process must not create currents 共sources兲 on the null boundaries u=0=v. Alternatively this implies that the five-dimensional Ricci terms all vanish globally

R ˜ AB= 0 共10兲 共xA :u,v,x,y,z兲.

The Riemann tensor components R˜ABCD, however, involve Dirac delta functions, indicative of

impulsive gravitational waves in addition to the shock waves required commonly by the step functions. In Appendix A we tabulate all components exhaustively from which we can easily identify the nonvanishing ones in the incoming regions. By setting v⬍0共u⬍0兲 we restrict

our-selves to the incoming region II 共III兲, comprising of five-dimensional gravitational plane waves alone. Obviously, for both u⬍0 and v⬍0 we obtain the region I which is a five-dimensional flat space-time given by

ds₅2= 4ab du dv −共1 + ao2兲dy2− dz2− 4aodx dy − dx2. 共11兲

The Kretschmann scalar in the interaction region共region IV, u⬎0,v⬎0兲 turns out to be

K = R˜ABCDR˜ABCD=

6

共1 + k␶兲6关4k2共k +␶兲2−共1 + k␶兲2兴 共12兲 which is free of singularities.

We wish now to apply the KK reduction procedure to our five-dimensional metric共6兲 in order to obtain CPWs of the EMD theory in four dimensions. We follow the KK reduction procedure through the identification

g

˜AB=␾−1/3

冉

g_␮␯+ 4␾A_␮A_␯ 2␾A_␮

2␾A_␯ −␾

冊

. 共13兲

This makes it possible to read both the four-dimensional metric g_␮␯, as well as the dilaton␾, and the EM potential A_␮. The results are as follows:

ds42=共FZ兲 1 2

冋

d␶2 ⌬ − d␴2 ␦ − 1 Z

冉

L Fdx 2_+␦Z ody2− 4ao␦dx dy

冊

册

,

␾=

冉

Z F

冊

3 2 , A_␮=␴ Z

冉

0,0,ao, 1 2Zo

冊

, 共14兲

where the notations are as in 共7兲, and in addition we have labelled L=2F−Z. The action of the resulting four-dimensional EMD theory is

S = 1 16␲

冕

兩g兩 1 2dx4

冋

− R −␾F_␮␯F␮␯+共ⵜ␾兲 2 6␾2

册

共15兲

so that the dilaton and Maxwell equations take the respective forms

䊐共ln␾兲 = − 3␾F_␮␯F␮␯, 共16兲

ⵜ␣共␾F␮␣兲 = 0, 共17兲

in which䊐 stands for the covariant Laplacian. To complete the set of Einstein equations we need also the Ricci tensor which is given by

R_␮␯=1 6 ␾,␮␾,␯ ␾2 − 2␾

冉

F␮␣F␯ ␣₋1 4g␮␯F␣␤F ␣␤

冊

_{. 共18兲}

We note that this representation of dilaton is different from the standard one expressed as an exponential function in the action. This more familiar latter form is related to the present one by the substitution

␾= e−2a␴ 共19兲

which casts the action into

S = 1

16␲

冕

兩g兩 1

2dx4关− R + 2共ⵜ␴兲2− e−2a␴F_␮␯F␮␯兴, 共20兲 where the dilatonic parameter is

冑

3. The physical properties of the EMD space-time obtained hitherto will be studied in the next section.

共B兲 The five-dimensional RNBH is given by ds₅2=

冉

1 −m r2+ q2 r4

冊

dt 2₋

冉

_{1 −}m r2+ q2 r4

冊

−1 dr2− r2d⍀_共3兲2 , 共21兲 where m and q are, respectively, related to the mass and charge of the BH. The EM vector potential one-form is given by

A = A_␮dx␮=

冑

3q

2r2dt. 共22兲

Here also, similar to the MP case we choose the S3line element as in共2兲. The transition to the CPW metric is accomplished here by the identification

4 dr2 m − r2−q 2 r2 − d␪2₌ d␶ 2 1 −␶2− d␴2 1 −␴2. 共23兲

r2=m 2共1 + l␶兲, 共24兲 where l =

冑

1 −4q 2 m2 ⬎ 0.

By choosing m = 1 in addition to the identifications

␴= cos␪= sin共au − bv兲,

t→ x,␸→ y,␩→ z,

␶= sin共au + bv兲,

共a,b = constants兲 共25兲

we obtain the metric共after rescaling of coordinates兲

ds₅2=共1 + l␶兲共4ab du dv − dy2− dz2− 2 sin共au − bv兲dy dz兲 − ⌬ 共1 + l␶兲2dx

2_{. 共26兲}

The EM vector potential one-form takes the form under the above transformation

A =

冑

3q

2

冑

2l共1 + l␶兲dx. 共27兲

The interpretation of this metric as a CPW is completed by inserting the step functions into the null coordinates u andv. This metric represents collision of EM plane waves in five dimensions.

For l = 1共or q=0兲 it reduces to the CPW metric of the five-dimensional pure gravity and coincides with the ao= 0 case of the metric共6兲. Thus, 共26兲 is the EM extension while 共6兲 was the rotational

extension of the same CPW metric obtained from the five-dimensional Schwarzschild metric. The Riemann components of the metric 共26兲 are given in Appendix B from which we compute the Kretschmann scalar to find共for u⬎0 and v⬎0兲.

K =127l

4_{+ 180l}3_{␶− 2l}2_{− 72⌬l}2_{+ 19 − 36l␶}

4共1 + l␶兲4 共28兲

which is also regular to the future of the collision point u = 0 =v.

Finally we wish to comment on other BHs and corresponding CPW solutions in five dimen-sions. Although our method applies to any such BH that admits inner and outer horizons such that the region in between possesses two spacelike Killing vectors technically some cases are not tractable. As an example we cite the Schwarzschild-de共-anti兲 Sitter BH given by the line element ds₅2= h共r兲dt2− h共r兲−1dr2− r2d⍀_共3兲2 , 共29兲 where h共r兲=k−共m/r2_兲±共r2_{/ l}2_{兲, in which k= ±1, m is related to mass and l to the cosmological} constant. To obtain the associated CPW solution we demand now that

冕

d␳

冑

␳

冑

m − k␳±␳ 2

l2

= ± sin−1␶, 共30兲

where we have used ␳= r2_{. The inversion of such an elliptical integral seems to be beyond} analytical calculation which must therefore be handled within the scope of numerical analysis.

III. PROPERTIES OF THE COLLIDING EMD SPACE-TIME

The linearly polarized CPW metric 共14兲 is rather transparent so we restrict ourselves to the case ao= 0, first. Upon rescaling of x and y we have the metric

ds2=共1 +␶兲32

冉

2 du dv − 1 −␶ 共1 +␶兲2dx

2_−␦dy2

冊

_共31兲 in which ␶and␴ are implied with the step functions. By the choice of Newman-Penrose共NP兲 null-tetrad basis one-forms,

l =共1 +␶兲34du,

n =共1 +␶兲34dv,

冑

2m =共1 −␶兲12共1 +␶兲−1/4dx + i

冑

␦共1 +␶兲 3

4dy , 共32兲

we obtain all Ricci and Weyl components as tabulated in the Appendix C. It is observed by studying the Weyl scalars ⌿o, ⌿2, and ⌿4 that the space-time is regular everywhere for 共u ⬎0,v⬎0兲. On the boundaries, however, both ⌿o and ⌿4 suffer from singularities at 共u=0,bv =␲/ 2兲 and 共v=0,au=␲/ 2兲, respectively. These are the typical null singularities inherited from the problem of colliding EM shock waves, therefore such singularities in the present problem is not unexpected at all. From the metric共31兲 we observe that ␶= 1 and ␴= 1 are spurious, removable coordinate singularities since they do not show up in the Weyl scalars. In particular,␶= 1 is the location of the horizon in the interaction region beyond which the metric can be extended ana-lytically. The other coordinate singularity ␴= 1 is out of question since it does not belong to the interaction region. The incoming EMD waves prior to the collision can also be easily identified from Appendix C. In the region II we have

⌿2= − 3 8a 2_␪共u兲 共5 + sin共au兲兲 共1 + sin共au兲兲52 , 共33兲 ⌽22= 1 16a 2_␪共u兲 共7 + sin共au兲兲 共1 + sin共au兲兲52 ,

while in the region III we must replace au↔bv to obtain ⌿o and⌽oo. The incoming waves that

comprise Ricci components⌽₂₂共⌽oo兲 is obviously constructed from both the EM and the dilaton

parts. Inside the collision region we observe also that the condition 9⌿2

2

=⌿0⌿4 共34兲

holds among the Weyl scalars, showing its type-D character. Direct choice of a Kinnersley type tetrad eliminates both⌿oand⌿4components of the Weyl scalars.

19

Such a tetrad is given by the basis one-forms

l =

冑

1 +␶ 1 −␶ d␶− dx, 2n = d␶+

_冑

1 −␶ 1 +␶dx,

冑

2m =共1 +␶兲34

冉

d␴

冑

␴− i

冑

␦dy

冊

. 共35兲

This choice gives now the only nonzero component ␺2= − 1 8 共5 +␶兲 共1 +␶兲52 共36兲 verifying its manifestly type-D character.

Now returning to the general CPW metric共14兲 with ao⫽0, we can discuss again the

interac-tion region alone. For this reason we omit all step funcinterac-tions and consider an NP tetrad basis one-form

冑

2l =共FZ兲14

冉

冑

d␶ ⌬+ d␴

冑

␦

冊

,

冑

2n =共FZ兲14

冉

d␶

冑

⌬− d␴

冑

␦

冊

,

冑

2m =

冑

L 共FZ兲14

冋

dx +

冉

ao␦F L + i

冑

␦FZo L

冊

dy

册

. 共37兲

The results are rather tedious so we shall refrain from tabulating the Ricci and Weyl compo-nents. Instead, relying on a numerical computation we have verified that the condition共34兲 fails to hold in the present case. Thus, we have seen numerically at least that our space-time is not

type-D. By the same numerical analysis we conclude that our space-time is not singular. Another

approach to study this space-time is to search for a possible Kinnersley tetrad that serves to generalize 共35兲. To attain this goal we define the null vector l␮ out of the geodesics equation. Unfortunately, in contrast to the BH case the choice␪=␪o= constant which used to simplify the

problem significantly, remains ineffective. By this analysis we obtain a set of Kinnersley-type tetrad as follows: l␮=

冑

F ⌬兵k⌬S,0,F,− ao其, 2n␮= 1 k2F

冑

ZS2兵k⌬S,0,− F,ao其,

冑

2m␮= F 1 4

冑

␦Z34S

再

0,−␦S

冑

Z F,− iao␦,i

冎

, 共38兲 where S = Z−1/2_共F−a o 2_␦兲1 2.

In the limit ao= 0, this tetrad reduces to共35兲, as it should. This fact that our space-time is not

type-D reflects in the computation of the spin coefficients since none turn out to vanish. In conclusion, we state that colliding cross-polarized EMD space-time obtained from the five-dimensional MPBH through the KK reduction procedure turns out to be nonsingular in spite of all its complication. This reflects the highly transcendental coupling between the dilaton and the other fields. Linear polarization limit removes all complication and we obtain a much simpler space-time structure.

IV. CPW SOLUTIONS FROM BLACK p-BRANES

A class of black p-brane solutions in d-dimensions of the action

S =

冕

d共d+p兲x

冑

− g

冉

R − 2 共d − p兲!F共d−2兲2

冊

共39兲 with F_共d−2兲= 1 共d − 2兲!F␮1¯␮共d−2兲dx ␮1∧ ¯ ∧ dx␮d−2

is given by the metric.20

ds_共d+p兲2 = AdBd共1−p兲/共1+p兲dt2−共AdBd兲−1dr2− Bd2/共p+1兲dy dy − r2d⍀_共d−2兲2 共40兲 in which Ad= 1 −

冉

r₊ r

冊

d−3 , Bd= 1 −

冉

r− r

冊

d−3 .

We consider here only the nonextremal case r+⬎r−, where the region r−⬍r⬍r+enables us to construct nonsingular CPW solutions. As examples we shall present solutions for p = 1, p = 6, and

p→⬁, however, our procedure applies for any p艌1. In particular, the six-dimensional

magneti-cally charged metric共40兲 becomes

ds₆2= A5dt2−共A5B5兲−1dr2− B5dy2− r2d⍀共3兲2 ,

F_共3兲= Q⑀3, 共41兲

where⑀₃is the volume form on the 3-sphere and Q2_{= 2共r}

+r−兲2. By the KK reduction procedure the six-dimensional action is reduced to the five-dimensional one

S =

冕

d5x

冑

− g

冋

R − 2共ⵜ␾兲2−1 3e −2c␾_F 共3兲 2

册

, 共42兲

where c =

冑

2₃ and the metric, dilaton and 3-form fields are ds₅2= B₅ 1 3关A 5dt2−共A5B5兲−1dr2− B5dy2− r2d⍀共3兲2 兴, ec␾= B₅− 1 3_,

F_共3兲=

冑

2共r+r−兲⑀3. 共43兲 The corresponding CPW solutions are obtained by the transformation

r2=12共ao+ bo␶兲,

cos␪=␴,

t = 2x,

␸= z,

␩= w, 共44兲

in which we have adopted the representation for d⍀_共3兲2

and introduced the abbreviations

ao= r+2+ r−2,

共45兲

bo= r+2− r−2. The resulting CPW metric, dilaton and the 3-form fields are

ds₅2=

冉

1 +␶ k +␶

冊

1/3

再

共k +␶兲

冋

2 du dv − dz2_{− dw}2_{+ 2␴dz dw −}1 −␶ k +␶dx 2

册

冎

_, ec␾=

冉

1 +␶ k +␶

冊

−1/3 , 共46兲 F_共3兲uzw= Qa␪共u兲

冑

␦, F_共3兲vzw= − Qb␪共v兲

冑

␦, where Q =共1/

冑

2兲共k2_{− 1兲 and k=a}

o/ bo⬎1. Our notation for ␶,␴, and␦ are as in the preceding

sections and in transforming共43兲 into 共46兲 we used the freedom of rescaling of x and ds2_{. This} metric has the scalar curvature共for u⬎0, v⬎0兲

R =4

3

ab共k − 1兲

共1 +␶兲4/3_{共k +␶兲}5/3 共47兲

which is regular in the interaction region. The colliding 3-form metric corresponding to 共41兲 becomes

ds₆2=共k +␶兲共2 du dv − dz2− dw2+ 2␴dz dw兲 − 1

k +␶关共1 −␶兲dx

2_{+共1 +␶兲dy}2_{兴, 共48兲}

whereas the 3-form field preserves its form. This metric represents the collision of 2-form fields in flat background. The 3-form field is obviously obtained from the 2-form potential by

F_共3兲= dA_共2兲, 共49兲

where

Azw= Q sin共au␪共u兲 − bv␪共v兲兲.

By a similar analysis we obtain the collision of these 3-form fields in 11-dimensional space. The result is ds₁₁2 =共k +␶兲共2 du dv − dz2_{− dw}2_{+ 2␴dz dw兲 −}

冉

1 +␶ k +␶

冊

2 7

兺

i=1 6 共dyi_兲2₋

冉

1 −␶ k +␶

冊冉

1 +␶ k +␶

冊

−5/7 dx2 共50兲 which has the regular scalar curvature,

R = −5

7

ab共k2− 1兲

共k +␶兲3 . 共51兲

The KK reduction of this 11-dimensional metric to the fifth dimension is expressed by

ds2=

冉

1 +␶ k +␶

冊

4 7

冦

共k +␶兲关2 du dv − dz2_{− dw}2_{+ 2␴dz dw兴} −

冉

1 −␶ k +␶

冊冉

1 +␶ k +␶

冊

−5 7 dx2

冧

, 共52兲 ec␾₌

冉

1 +␶ k +␶

冊

−4/7 , and F_共3兲components are as in共49兲.

V. COLLIDING EM WAVE SOLUTION IN ANY HIGHER DIMENSION

The action for the共4+p兲-branes is given by

S =

冕

d共4+p兲x

冑

− g共R − F_共2兲2 兲, 共53兲

in which F_共2兲stands for the EM 2-form. Solution is given by20 ds_4+p2 = AB共1−p兲/共1+p兲dt2− B2/共p+1兲dy dy −共AB兲−1dr2− r2d⍀_共2兲2 , 共54兲 where A = 1 −r+ r, B = 1 −r− r, and F = Q⑀2. Now, the transformation共with r+= ao+ boand r−= ao− bo兲

r = ao+ bo␶,

␸= z,

t = x, 共55兲

yields the CPW metric,

ds_4+p2 =共k +␶兲2共2 du dv −␦dz2兲 −

冉

1 +␶ k +␶

冊

2/共p+1兲

兺

i=1 p 共dyi_兲2₋

冉

1 −␶ k +␶

冊冉

1 +␶ k +␶

冊

共1−p兲/共1+p兲 dx2 共56兲 in which we have introduced k = ao/ bo⬎1, and rescaled the coordinates. The EM potential 1-form

is given by

A = − Q sin共au␪共u兲 − bv␪共v兲兲dz.

So that the nonzero field 2-form components are

Fuz= − Qa␪共u兲

冑

␦,

共57兲

Fvz= Qb␪共v兲

冑

␦.

It is observed now, that it is a simple matter to obtain the CPW metrics for an arbitrary p 艌1. In particular, for p=1 and p=7 we have

ds₅2=共k +␶兲2共2 du dv −␦dz2兲 − 1 k +␶关共1 +␶兲dy 2_{+共1 −␶兲dx}2_{兴 共58兲} and ds₁₁2 =共k +␶兲2共2 du dv −␦dz2兲 −

冉

1 +␶ k +␶

冊

1 4

兺

i=1 7 共dyi_兲2₋

冉

1 −␶ k +␶

冊冉

1 +␶ k +␶

冊

−3/4 dx2, 共59兲 respectively. By letting p→⬁, we can easily obtain also the colliding EM wave solutions in the ⬁-brane world. The KK reduction to the fourth dimension for an arbitrary p-brane is

ds₄2=

冉

1 +␶ k +␶

冊

p/共p+1兲

再

共k +␶兲2_{共2 du dv −␦dz}2_{兲 −}

冉

1 −␶ k +␶

冊冉

1 +␶ k +␶

冊

共1−p兲/共1+p兲 dx2

冎

共60兲 with ec␾=

冉

1 +␶ k +␶

冊

−p/关2共p+1兲兴 , c =

冑

p p + 2, Fuz= − Qa␪共u兲

冑

␦,

Fvz= Qb␪共v兲

冑

␦,

Q2=1 2共k

2_{− 1兲}

冉

p + 2

p + 1

冊

.

This metric describes the collision of dilaton coupled EM waves in four dimensions.

VI. COLLIDING_{„p+2…-FORMS IN „p+4…-DIMENSION}

In Sec. V we have constructed CPW solutions for the 2-form fields in共p+4兲-dimension. By applying the duality principle we can obtain共p+2兲-form fields and consider their collision at equal ease. We define the duality

F ˜␮1¯␮k₌ 兩g兩 −1/2 共n − k兲!⑀␮1¯␮k␮k+1¯␮nF␮k+1¯␮n, 共61兲 where共n⬎k兲 and F_␮ k+1¯␮n is assumed known.

The permutation symbol⑀␮1¯␮n_satisfies21

⑀␮1¯␮n

⑀␮1¯␮n=共− 1兲

l

n ! , 共62兲

where l⫽number of minus signs in g␮␯. Since we have readily available 2-form at hand we define its dual F ˜␮1¯␮p+2₌ 1 2!兩gp+4兩 −1/2_⑀␮1¯␮p+4_F ␮p+3␮p+4 共63兲

in共p+4兲-dimension. The action of 共gravity +F˜p+2兲 is taken as

S =

冕

d4+px

冑

g

冉

R − 2

共p + 2兲!F˜p+2

2

冊

_共64兲

with the field equations

R_␮␯= 2 共p + 2兲!

冉

F˜␮␮1¯␮p+1F ˜ ␯ ␮p¯␮p+1₋ 共p + 1兲 共p + 2兲2g␮␯F˜ 2

冊

共65兲 ⳵␮共兩gp+4兩1/2F˜␮␮1¯␮p+1兲 = 0, where F˜2_{= F}˜ ␮1¯␮p+2F ˜␮p¯␮p+2_.

We proceed with two particular examples, p = 1 and p = 6.

共i兲 p=1 case. The 3-form field F˜3 is from the metric 共58兲 and Fuz= −Qa␪共u兲

冑

␦, Fvz

= −Qb␪共v兲

冑

␦. It is given by F ˜ 3= Q

冑

⌬ 共k +␶兲2共a␪共u兲du + b␪共v兲dv兲 ∧ dx ∧ dy 共66兲 which can be associated through F˜3= dA˜2to the 2-form potential

A ˜

2= −

Q

共k +␶兲dx∧ dy. 共67兲

The incoming region 共II兲 metric and 3-form fields can also be expressed in the Brinkmann form since they are given here in the Rosen form. For this we define new coordinates 共U,V,X,Y ,Z兲 as follows:

U =

冕

共k + sin au兲2du,

X = A共u兲x, Y = B共u兲y, Z = C共u兲z,

V =v +x 2 2AAu+ y2 2BBu+ z2 2CCu, where A2共u兲 =1 − sin au␪ k + sin au␪,

B2共u兲 =1 + sin au␪ k + sin au␪,

C =共k + sin au␪兲cos au␪. 共68兲

The relation between U and u can be chosen as

U =3 2u + 2 a共1 − cos au兲 − 1 4asin 2au, 共69兲

so that u = 0 and U = 0 coincide. Further, the graph of U共u兲 reveals that in the interval 0⬍au ⬍␲/ 2, u⬎0 implies that U⬎0. However, as it is observed we cannot invert u in terms of U, and this enforces us to keep the Brinkman form in an implicit form. We have ultimately

ds2= 2 dU dV − dX2− dY2− dZ2− 2H共u共U兲,X,Y,Z兲dU2, where H共u共U兲,X,Y,Z兲 =a 2␦共U兲

冋

Y 2_{− X}2₊1 k共2Z 2_{− X}2_{− Y}2_兲

册

₊ a 2_␪共U兲

4共k + sin au兲2关共k + 1兲共3 − k + 2 sin au兲X 2

−共k − 1兲共3 + k − 2 sin au兲Y2−共k + sin au兲共k + 4 sin au兲Z2兴. 共70兲 We recall that a general class of metrics given by

ds2= 2 dU dV −

冉

兺

i,j

Aij共U兲XiXj

冊

du2−

兺

i

共dXi_兲2_, where Aij= constant, is known as Cahen-Wallach space.22

It is clear that we haveⵜ2_{H⫽0 in our case, indicating the presence of energy momentum for} the 3-form field as it should. The 3-form field, in the Brinkman form for region II is

F ˜

3=

aQ␪共U兲

共k + sin au兲3dU∧ dX ∧ dY, 共71兲

where the inversion of the expression共69兲 is implied.

共ii兲 p=6 case. The 8-form field F˜8is found from the metric 共56兲,

F ˜

8= F˜uxy1_¯y6du∧ dx ∧ dy1∧ ¯ ∧ dy6+ F˜_vxy1_¯y6dv∧ dx ∧ dy1∧ ¯ ∧ dy6, 共72兲

F ˜uxy1¯y6 =Qb␪共v兲

冑

␦ 兩g10兩1/2 , F

˜vxy1¯y6₌Qa␪共u兲

冑

␦ 兩g10兩1/2

and we have chosen⑀uvzxy1¯y6_{= + 1. The corresponding 7-form potential is}

A ˜ 7= − Q k +␶dx∧ dy 1_{∧ ¯ ∧ dy}6 _共73兲

which derives F˜8 through F˜8= dA˜7.

It is seen that the collision problem of these 8-form fields is automatically solved with well-defined incoming states. The solutions, as we stated earlier are regular but our procedure does not allow at the moment to obtain the collision problem of arbitrary n-form fields. Our procedure limits itself only with the 2共3兲-form fields and their duals. Different authors addressed themselves to the more general problem but they obtained only perturbative and singular solutions.23,24

VII. CONCLUSIONS

In this paper we have concentrated first on two five-dimensional BHs, namely Myers-Perry 共MP兲 and Reissner-Nordstrom 共RN兲. These are both extensions of the five-dimensional Schwarzs-child BH, MP with rotation while RN with electric charge. The inherent isometry between the BHs and colliding plane waves 共CPWs兲 yields regular, horizon forming solutions to the latter. By regular, throughout the paper we imply a Cauchy-Horizon 共CH兲 forming space-time with finite curvature invariants. We have not attempted to extend our space-time beyond CH. Once this is done by Chandrasekhar and Xanthopoulos,1we may face various singularities ranging from time-like to spacetime-like ones or no singularities at all. Another issue that we have not addressed ourselves in the paper is the stability of the CH formed in the collision. There are strong arguments that under certain perturbations the CHs of the CPWs transform into curvature singularities.25 Defi-nitely this matter is far from being conclusive and requires further investigation. We note also that beside the BHs the more general Weyl solutions can be employed in the generating of CPWs.26

Our particular cross-polarized dilatonic non-type-D metric with CH provides an example to be taken into account other than the singular ones used in string theory.27,28Our procedure is extend-ible to higher dimensional BHs provided technical matters are overcome. One such problem is to find representation for the n-dimensional spherical line element which admits共n−1兲-dimensional Abelian subspace. Equation共2兲 performs just this for the three-dimensional sphere. Although we obtain colliding 2共3兲-form fields by our procedure through, employing five-dimensional BHs, we can dualize our forms and obtain colliding 共p+1兲- and 共p+2兲-form fields in 共p+4兲-dimensions. Extension of our work to arbitrary form fields will be the next stage of our study. Presumably all these metrics will find application in higher dimensional space-times and low energy limit of the string theory.

APPENDIX A

The nonzero components of the metric共6兲 are given below

R ˜

uvuv= − 2a2b2k2␪共u兲␪共v兲,

R ˜

R ˜

uvzx= 2abkao␪共u兲␪共v兲,

R ˜ uyvy= abk2ao 2_{␪共u兲␪共v兲,} R ˜ uyvx= abk2ao 2_{␪共u兲␪共v兲 = R˜} uxvy, R ˜

uzvy= abkao2␪共u兲␪共v兲 = − R˜uyvz,

R ˜ uzvz= − abao 2_{␪共u兲␪共v兲,} R ˜

uzvx= abkao␪共u兲␪共v兲 = − R˜uxvz,

R ˜

uxvx= abk2␪共u兲␪共v兲,

R ˜ yzyz= −

共

1 2+ 2ao 4

兲

_{␪共u兲␪共v兲,} R ˜ yzzx= 2ao 3_{␪共u兲␪共v兲,} R ˜ yxyx= 1 2k 2_{␪共u兲␪共v兲,} R ˜ zxzx=

共

k2− 1 2

兲

␪共u兲␪共v兲, R ˜

uyuy= − a2␪共u兲Y1+ a␦共u兲cos共bv␪共v兲兲Y2,

R ˜

vyvy= − b2␪共v兲Y1+ b␦共v兲cos共au␪共u兲兲Y2,

R ˜ uxux= − 2a2␪共u兲Y3− 2ak D2␦共u兲cos共bv␪共v兲兲, R ˜ vxvx= − 2b2␪共v兲Y3− 2bk D2␦共v兲cos共au␪共u兲兲, R ˜ uyux= − a2ao␪共u兲Y4− 2aaok D2 ␦共u兲cos共bv␪共v兲兲, R ˜ vyvx= − b2ao␪共v兲Y4− 2baok D2 ␦共v兲cos共au␪共u兲兲, R ˜_{uzuz= a}2_␪共u兲Y 5+ a␦共u兲cos共bv␪共v兲兲Y6, R ˜ vzvz= b2␪共v兲Y5+ b␦共v兲cos共au␪共u兲兲Y6,

R ˜_{uzux= − a}2_a o␪共u兲Y7− 2aao 共1 + 2k␶兲 D2 ␦共u兲cos共bv␪共v兲兲, R ˜ vzvx= b2ao␪共v兲Y7+ 2bao 共1 + 2k␶兲 D2 ␦共v兲cos共au␪共u兲兲, R ˜

uyuz= − a2␪共u兲Y8− a␦共u兲cos共bv␪共v兲兲Y9,

R ˜

vyvz= b2␪共v兲Y8+ b␦共v兲cos共au␪共u兲兲Y9, where we have used the following abbreviations:

D = 1 + k␶, Y1= 1 D3关4共1 + k␶− ao 2_{兲 − k}2_{⌬共3共1 + a} o 2_{兲 + k␶兲兴,} Y2= k2 D3关␶共k 2_{+ 2兲 + k共1 + 3␶}2_{+ k␶}3_兲兴, Y3= ao 2 − k2⌬ D3 , Y4= 1 D3关2共1 + k␶兲 − 3k 2_⌬兴, Y₅= 1 D3兵− 4共1 + k␶− ao2兲 + ⌬关ao2共11 + 12k2␶2兲 + k␶共7 − 6k2兲 − 3共1 − 2k2兲兴其, Y6= 1 D3关k +␶共3 − 10ao 2_{兲 + k␶}2_{共3 − 16a} o 2_{兲 + k}2_␶3_{共1 − 8a} o 2_兲兴, Y7= 1 D3关2共k +␶兲 − k⌬共5 + 6k␶兲兴, Y8= 1 D3关4共k +␶共1 − ao 2_{兲兲 + k⌬共k}2_{⌬ − 5 − a} o 2_{− 3k␶共1 + 2a} o 2_兲兲兴, Y₉= 1 D3关共2 − k 2_{兲共1 + 3k␶兲 + k}2_␶2_{共5 + k␶− 2k}2_兲兴. APPENDIX B

The nonzero Riemann components of the metric 共26兲 are with the step functions inserted as follows:

R ˜ uvuv= − 2la2b2 C 共l +␶兲␪共u兲␪共v兲, R ˜ uxux= − a2␪共u兲 C4 共3l 2_{+ 2l␶− 1兲 −}2a C3␦共u兲cos共bv␪共v兲兲共l +␶兲, R ˜ uxvx= ab C4␪共u兲␪共v兲共2l 2_{+ l␶− 1兲,} R ˜ uyuy= − a2␪共u兲 4C 共3l 2_{+ 4l␶+ 1兲 + a␦共u兲l cos共bv␪共v兲兲,} R ˜ uyuz= − a2␪共u兲␴ 4C 共3l 2_{+ 4l␶+ 1兲 + a␦共u兲cos共bv␪共v兲兲,} R ˜ uyvy= ab 4C共1 − l 2_{兲␪共u兲␪共v兲,} R ˜ uyvz= ab␴ 4C共1 − l 2_{兲␪共u兲␪共v兲,} R ˜ uzuz= − a2␪共u兲 4C 共3l 2_{+ 4l␶+ 1兲 + a␦共u兲l cos共bv␪共v兲兲,} R ˜ uzvz= ab 4C共1 − l 2_{兲␪共u兲␪共v兲,} R ˜ vxvx= − b2␪共v兲 C4 共3l 2_{+ 2l␶− 1兲 −}2b C3␦共v兲 cos共au␪共u兲兲共1 +␶兲, R ˜ vyvy= − b2_␪共v兲 4C 共3l 2_{+ 4l␶+ 1兲 + b␦共v兲l cos共au␪共u兲兲,} R ˜ vyvz= − b2␪共v兲␴ 4C 共3l 2_{+ 4l␶+ 1兲 − b␦共v兲cos共au␪共u兲兲,} R ˜ vzvz= − b2␪共v兲 4C 共3l 2_{+ 4l␶+ 1兲 + b␦共v兲lcos共au␪共u兲兲,} R ˜ xyxy= l⌬ 2C4␪共u兲␪共v兲共l +␶兲, R ˜ xyxz= l⌬␴ 2C4␪共u兲␪共v兲共l +␶兲,

R ˜ xzxz= l⌬ 2C4␪共u兲␪共v兲共l +␶兲, R ˜ yzyz= −␦␪共u兲␪共v兲 4C 共1 + 2l␶+ l 2_兲,

where we have used C = 1 + l␶.

APPENDIX C

The nonzero NP quantities for the metric共31兲 are ⌿2= 1 8ab␪共u兲␪共v兲 共5 +␶兲 共1 +␶兲52 , ⌿4= − 3 8a 2_␪共u兲 共5 +␶兲 共1 +␶兲52 +a 4 ␦共u兲 cos bv共␪共v兲兲 共3 +␶兲 共1 +␶兲3/2, ⌿0= − 3 8b 2_␪共v兲 共5 +␶兲 共1 +␶兲52 +b 4 ␦共v兲 cos au共␪共u兲兲 共3 +␶兲 共1 +␶兲3/2, ⌽22= 1 16a 2_␪共u兲 共7 +␶兲 共1 +␶兲52 , ⌽00= 1 16b 2_␪共v兲 共7 +␶兲 共1 +␶兲52 , ⌽02= − 1 4 ab␪共u兲␪共v兲 共1 +␶兲32 , ⌽11= 3 32ab␪共u兲␪共v兲 共1 −␶兲 共1 +␶兲52 = − 3⌳.

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