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Published by Institute of Physics Publishing for SISSA/ISAS

Received: June 20, 2003

Revised: August 7, 2003 Accepted: August 22, 2003 Black strings in asymptotically plane wave geometries Eric G. Gimon, Akikazu Hashimoto and Oleg Lunin

J H E P 0 8 ( 2 0 0 3 ) 0 3 5

2.2 Properties of the asymptotically P10 black string solution

1. Introduction
Maximally symmetric plane waves have emerged as an important background space-time
of string theory [

Schwarzschild black holes provide important insights into the nature of gravity. While
black objects in other maximally symmetric spaces are well known, analogous solutions
in plane wave geometries have yet to be constructed. Nonetheless, it was shown in [

{ 1 { J

In this article, we present a systematic solution generating technique which can be
used to construct a general class of asymptotically plane wave solutions. Our method
is based on the observation of [

Since the asymptotically plane wave black string solutions we ¯nd in this paper have a regular horizon, it is straightforward to compute its area. We ¯nd that this area is identical to the area of the black string before the null Melvin twist. We will provide a proof that the null Melvin twist is a procedure which keeps the area of the horizon invariant.

The null Melvin twist can be used to generate a plane wave background with the same
metric as the Penrose limit of AdS5 £ S5 in type IIB supergravity which was identi¯ed in [

The organization of the paper is as follows. In section 2, we construct the black string
solution in ten dimensional asymptotically plane wave background and describe some of
its properties. In section 3, we describe various generalizations including adding charges
and angular momenta. In section 4, we describe the analogous construction for plane wave
black strings in other dimensions. Concluding remarks are presented in section 5.
2. Asymptotically plane wave black string in 10 dimensions
In this section, we describe the black string in an asymptotically 10 dimensional plane wave
geometry. We will denote the maximally symmetric plane wave geometry in d dimensions
by Pd. The metric of P10 is
If this metric is supported by a self dual 5-form ¯eld strength as in [

The P10 solution with NS-NS ¯elds above is particularly well suited as a background for introducing a black string for two reasons. First, the action of the isometry group U(4) is transitive on the transverse seven-sphere (the orbits are U(4)= U(3)), which means that we can expect to modify the metric with functions of the transverse distance only and thus can use brute force to ¯nd the black string solution. Second, and more importantly, we can ¯nd the black string by using the fact that Minkowski space is related this P10 by a null Melvin twist.

In this subsection, we will describe the sequence of solution generating manipulations which
we call the null Melvin twist. We start by constructing a neutral black string in P10 as
an example. As was shown in [

1. Consider a Schwarzschild black hole solution in 8+1 dimensions [

{ 3 { dss2tr = ¡f (r) dt2 + dy2 + 3. T-dualize along y. This gives a solution of IIA supergravity with fundamental string charge QF 1 = M sinh ° cosh °. Translation along y and SO(8) rotations along the transverse S7 remain isometries of this geometry. 4. Twist the rotation of S7 along y. By twisting, we mean parameterizing the plane transverse to the string in cartesian coordinates xi, and making the following change of coordinates (2.2) (2.3) (2.4)

J
H
E
P
0
8
(
2
0
0
3
)
0
3
5
where the parameter ® controls the amount of twisting. For the sake of simplicity,
we twist in all four planes of rotation by the same amount. This has the e®ect of
replacing the metric on the 7-sphere according to
d72 ! d72 + ® ¾ dy + ®2 dy2 ;
(2.5)
where
5. T-dualize along y. This geometry now corresponds to a black string in type IIB
supergravity with momentum Py = M sinh ° cosh ° sitting at the origin of the Melvin
universe [

¯ = 12 ® e° = ¯xed : The end result is the black string in an asymptotically P10 spacetime. dt dy + µ1 ¡ ¯k2(rr)2 ¶ dy2 +

Steps 3 through 5 are the manipulations involved in generating an ordinary Melvin
°ux tube solution by twisting along a spatial isometry direction y which was originally
described in [

{ 4 { (2.7) (2.8) (2.9)

J
H
E
P
0
8
(
2
0
0
3
)
0
3
5
2.2 Properties of the asymptotically P10 black string solution
The solution (2.8) is very simple. By inspection, if we set M to zero the solution reduces
to the P10 geometry described at the beginning of this section. On the other hand, setting
¯ = 0 will reduce the solution to the black string solution (2.2). There is a regular horizon at
rH = M 1=6
(2.10)
which persists for ¯nite values of ¯. One can therefore interpret (2.8) as the black string
deformation of P10. Furthermore, since both f (r) and k(r) asymptote to 1 as r is taken to
be large, the e®ect of M decays at large r. Unlike the six dimensional solution described
in [

Because both the dilaton and the metric asymptote to P10 in (2.8), one can unambiguously de¯ne the area of the horizon in Einstein frame with the same asymptotics. The area of the ¯xed (r,t)-surface in Einstein frame for (2.8) is

A = L pk(r) ¡ ¯2r2 r7 7 ; where 7 = ¼4=6 is volume of a unit S7 and L is the length of the translationally invariant y-direction. At the horizon, this evaluates to

AH = L M 7=6 7 :

S =

L M 7=6 7

as an entropy of some sort. A remarkable fact is that this quantity is independent of the parameter ¯.

It would also be interesting to compute the temperature associated to this black string. Ordinarily one computes the temperature of a black object in terms of the surface gravity where »a is the null generator of the horizon. The temperature is then given in terms of the surface gravity as TH = ·=2¼.

For the solution (2.8), the null generator of the horizon is simply the Killing vector ·2 = ¡ 1 ³ra»b´ (ra»b) ;

2 Special care is necessary with regards to the normalization of this Killing vector in the computation of the temperature. In asymptotically °at space, for example, it is natural to normalize the Killing vector so that it is of unit norm asymptotically. It is a priori not clear what constitutes a natural normalization of the Killing vector in an asymptotically plane wave geometry. Let us therefore use a normalization such that the Killing vector { 5 { (2.11) (2.12) (2.13) (2.14) (2.15)

J H E P 0 8 ( 2 0 0 3 ) 0 3 5 takes precisely the form (2.15) and interpret the temperature as being measured in units of inverse coordinate time t. With this caveat in mind, the temperature at the horizon of the solution (2.8) is found to be

TH = 23¼ M ¡1=6 : (2.16) What is remarkable about these results is the fact that both the temperature and entropy are independent of the parameter ¯. We will in fact show in the appendix that the area of the horizon is invariant under a null Melvin twist for a general class of spacetimes. This would appear to indicate that the parameter M has a natural interpretation as the mass density of the black string. This is a rather non-trivial statement since a proper notion of mass analogous to the ADM mass for an asymptotically plane wave geometry has not yet been de¯ned.

The solution (2.8) is free of closed time-like curves so long as the y coordinate is decompacti¯ed at the end of chain of dualities. If the y coordinate is compact, closed time-like curves can appear just as in the case of ordinary plane-waves. 3. Generalizations The null Melvin twist construction is extremely simple and can be applied to generate a wide variety of asymptotically plane wave geometries. In this section we will describe a few examples.

One simple generalization of (2.8) is to add angular momentum. Consider a rotating black
string in type IIB supergravity [

dss2tr = ¡dt2 + (1 ¡ f (r)) µdt + ` ¾¶2 2 + dy2 +

1 dr2 + r2d72 ; h(r) where In general, rotating black strings in 10-dimensions admit four independent angular momentum charges. We have taken all four angular momenta to equal ` for simplicity.

Applying the null Melvin twist procedure of the previous section leads to the following solution of type IIB supergravity e' = B =

dr2 + h(r) 1 pk(r) ¯r2 µ 2k(r) f (r) + ¯2r2 h(r) dt2 ¡ 2¯2r2h(r) k(r) k(r)

dt dy + The inner and outer horizons are located at the roots of h(r). Just as in the non-rotating case, this geometry asymptotes to plane wave for large r or small M , but reduces to a rotating black string in the small ¯ limit. The horizon area and the surface gravity are independent of ¯ and agree with the results for a rotating black string in asymptotically °at space. The outer horizon of the ` 6= 0 solution carries non-vanishing angular velocity

It is also easy to add charges to (2.8). Simply start with the non-extremal fundamental string solution and apply the null Melvin twist. This gives

(f (r)dt + dy) ^ ¾ k±(r) ´ 1 +

M sinh2 ± r6 +

M ¯2
r4 :
A non-extremal D-string solution can be obtained in the same way if we perform the null
Melvin twist starting from the nonextremal D-string instead of the fundamental string.
The extremal limit of this solution is closely related to the solutions described in [

So far, we have considered the case where one twists the four complex planes transverse to the black string equally. One can readily generalize this to independent twists in each of the four complex planes x1 + ix2 ! e2i® v1 y(x1 + ix2) x3 + ix4 ! e2i® v2 y(x3 + ix4) x5 + ix6 ! e2i® v3 y(x5 + ix6) x7 + ix8 ! e2i® v4 y(x7 + ix8) : { 7 { (3.4) (3.5) (3.6) (3.7) (3.8)

J H E P 0 8 ( 2 0 0 3 ) 0 3 5 e' = B = (f (r) dt + dy) ^ ¾v kv(r; 7) = 1 + ¯2M j¾vj2 ;

r4 r2¾v = v1 (x1dx2 ¡ x2dx1) + v2 (x3dx4 ¡ x4dx3) + 2

+ v3 (x5dx6 ¡ x6dx5) + v4 (x7dx8 ¡ x8dx7) ; j¾vj2 = r12 ¡v12(x12 + x22) + v22(x32 + x42) + v32(x52 + x62) + v42(x72 + x82)¢ : Note that since the U(4) isometry is broken, kv(r; 7) can now depend non-trivially on the coordinates of S7. One can explicitly check, though, that the horizon area and the surface gravity remain independent of ¯ and the vi's.

To be completely general, one can construct a solution with 13 independent parameters M , ¯, Py, QF 1, QD1, `1, `2, `3, `4, v1, v2, v3, and v4. We will not write this most general solution explicitly. 4. Asymptotically plane wave black string in other dimensions Another natural generalization of our procedure is to apply the null Melvin twist to black branes smeared in more dimensions. A black p-brane in 10 dimensions, when compacti¯ed along p¡1 of the translationally invariant directions, will look like a black string in d = 11¡p dimensions. The p ¡ 1 extra dimensions play a spectator role, and so by applying the null Melvin twist on the e®ective d dimensional black string, one can construct black string solutions which are asymptotically Pd.

dss2tr = ¡fd(r) dt2 + dy2 +

M fd(r) = 1 ¡ rd¡4 :

1 fd(r)

10¡d dr2 + r2 dd¡3 + X dzi2 ; i=1 To simplify the discussion, let us restrict to even values of d. Then there are (d ¡ 2)=2 independent null Melvin twist parameters that one can independently adjust. Let us further { 8 {

(3.9) (3.10) (3.11) (4.1)

J H E P 0 8 ( 2 0 0 3 ) 0 3 5 e' = take all the (d ¡ 2)=2 twist parameters to be equal for simplicity. Then, we ¯nd that the supergravity solution for the neutral black string in Pd takes the form with For d > 6, (4.2) asymptotes to Pd and closely resembles P10 in many ways.

The d = 6, is special in that k6(r) does not asymptote to 1. This is similar to conical
de¯cits which arise as a result of mass deformation in 2+1 dimensions and cause the
background to be deformed by a ¯nite amount even for large r. Nonetheless, (4.2) for
d = 6 can be considered as a black string deformation of P6 in the sense that in the
small M limit, the solution reduces to P6. The black string deformation of P6 was also
constructed in [

For d = 6, (4.2) becomes (1 ¡ M=r2)(1 + ¯2 r2) 1 + ¯2M dt2 ¡ 2¯2r2(1 ¡ M=r2) 1 + ¯2M dt dy + µ1 ¡ 1 +¯2¯r22M ¶ dy2 + dr2 r2 µ + 1 ¡ M=r2 + r2 d32 ¡ 4 e' = µµ1 ¡ Mr2 ¶ dt + dy¶ The solution presented in [12, equation (26)] reads ds2 = µ1 ¡ 2rm2 ¶

dt~2 + dy~2 ¡ 2jr2¾6(dt~¡ dy~) ¡ 2j2m r2¾62 +
+ µ1 ¡
2m(1 ¡ 8j2m) ¶¡1
r2
dr2 + r2d32 :
Here, we relabeled the coordinates (t; y) of [

{ 9 { (4.2) (4.3) (4.4) (4.5)

J H E P 0 8 ( 2 0 0 3 ) 0 3 5

To facilitate the comparison one must rewrite the metric (4.4) in the coordinate chart in which (4.5) is written. This is achieved by ¯rst twisting and then performing a change of variables so that the metric becomes ¾6 ! ¾6 + 2¯(dt + dy) ; t = (1 + ¯2M ) t~ y = ¡¯2M t~¡ y~

M (1 +r2¯2M ) ¶ dt~2 + dy~2 + µ Now, (4.8) and (4.5) have the same form, and they can be seen to be identical if we identify the parameters according to (4.6) (4.7) (4.8) (4.9) (4.10)

J H E P 0 8 ( 2 0 0 3 ) 0 3 5 2m = M (1 + ¯2M ) 2j = (1 + ¯¯2M ) : 8j2m =

¯2M 1 + ¯2M : So the dimensionless combination 8j2m is related to the dimensionless combination ¯ 2M by The critical value 8j2m = 1, which causes the solution (4.5) to degenerate, corresponds to ¯2M being in¯nite.

The fact that (4.4) do not asymptote to plane wave geometry makes the identi¯cation of quantities such as mass, entropy, and temperature even more subtle. One can nonetheless map (4.4) to Einstein frame and compute the horizon area and the surface gravity using the killing vector (2.15) for these coordinates. This yields a ¯ independent answer AH = LM 3=23VT 4 ; ·2 = 1 :

M (4.11) This was essentially guaranteed based on the arguments presented in the appendix, and suggests that the parameter M can be interpreted as a physical mass. 5. Discussion In this article, we presented a powerful technique for generating a large class of asymptotically plane wave geometries. Using this technique, we have succeeded in constructing a supergravity solution which contains an event horizon while asymptoting to the plane wave geometry.

With explicit solutions at hand, one can explore various thermodynamic interpretations of the black string geometry. We ¯nd that the null Melvin twist leaves the area of the event horizon invariant, suggesting that the entropy of the black string in the plane wave geometry is identical to the entropy of the black string before taking the null Melvin twist. Furthermore, with certain assumptions regarding the de¯nition of temperature in asymptotic plane wave geometries, we found that it too is left unchanged under the null Melvin twist. This suggests that the mass parameter of the black string solution corresponds to a physical measure of mass in some canonical way. It would be interesting to make these ideas more precise by formulating a satisfactory de¯nition of mass and temperature in an asymptotically plane wave geometry.

We also constructed a black string solution in an asymptotically six dimensional plane
wave background using the null Melvin twist technique. We showed that this solution is
equivalent to the solution presented in [

Clearly, this null Melvin twist can be used to generate a more general class of solutions than the ones considered in this article. Unfortunately, this technique can only be used to construct certain subset of asymptotically plane wave geometries. A plane wave geometry of particular interest which can not be constructed using this technique is the one which arises as a Penrose limit of AdS5 £ S5. It would be extremely interesting to ¯nd the analogous black string solution in this asymptotic geometry.

With very little e®ort, one can dualize the Schwarzschild black string solution in Pd to a black hole solution in (d ¡ 1)-dimensional GÄodel universe Gd¡1. The horizon and the velocity of light surface meet only when ¯2M is in¯nite.

So far, we have only considered black string solutions. It would be extremely
interesting to ¯nd the solution corresponding to black holes. Unfortunately, the null Melvin
twist technique involves T-duality, and can not be applied at the level of supergravity
to backgrounds which do not contain at least one translational isometry direction.
However, since certain plane wave backgrounds can be constructed using only ¯elds in the
NS-NS sector, it may be possible to extract such a geometry from the consideration of
the sigma model for the string world sheet in this background. A related discussion in
the context of duality between Kaluza-Klein monopole and the NS5-brane can be found
in [

Acknowledgments We would like to thank D. Berenstein, E. Boyda, S. Ganguli, O. Ganor, P. Ho·rava, G. Horowitz, and S. Ross for discussions. The work of E.G.G. is supported by NSF grant PHY-0070928 and by Frank and Peggy Taplin. A.H. is supported in part by DOE grant DE-FG02-90ER40542 and the Marvin L. Goldberger Fund. V.E.H. is supported by NSF H E P 0 8 ( 2 0 0 3 ) 0 3 5 Grant PHY-9870115. O.L. is supported by NSF grant PHY-0070928. M.R. acknowledges support from the Berkeley Center for Theoretical Physics and also partial support from the DOE grant DE-AC03-76SF00098 and the NSF grant PHY-0098840.

A. Null Melvin twist and the area of the horizon In this appendix, we will show that the area of the horizon does not change under rather general set of manipulations for which the null Melvin twist is a special case. The proof will be based on an assumption that the B¹º ¯eld for the initial con¯guration is regular at the horizon.

Consider starting from a space-time with a horizon and a space-like translation symmetry along a coordinate y in type II supergravity theory. We will assume that the metric1 is written in the Boyer-Lindquist coordinates so that there is a coordinate r which does not mix with other coordinates, allowing one to write the metric in the form 0 grr 0 1

CC :

M¹º CA

A¹º = g¹º det M : One of the coordinates of M¹º will be t. By horizon, we will mean a ¯xed (t; r) surface where grr diverges and det M vanishes. Let A¹º denote the cofactor of M¹º . The coordinates are regular away from the horizon, then one can write The area of the horizon is the square root of Att evaluated at the horizon.

Consider starting from a generic space-time with the properties described above, and applying the following sequence of manipulations:

x¹ ! x¹ + ®¹y

These manipulations give rise to a new geometry again in Boyer-Lindquist coordinates, whose horizon area is given by the square root of

Att + 2 sinh2 ° ®¹B¹º Ayº + sinh2 ° ®¹®º (B¹½Bº¾ + g¹½gº¾ )A½¾ : (A.4) 1In this appendix, we always write the metric in the Einstein frame. (A.1) (A.2) (A.3)

J P 0 8 ( 2 0 0 3 ) 0 3 5 Using the relation (A.2) away from the horizon, we can rewrite the last term in this expression as sinh2 ° ®¹®º (B¹½Bº¾ + g¹½gº¾)A½¾ = sinh2 ° detM ®¹®º (B¹½Bº ½ + g¹º ) : (A.5) This is a scalar multiplied by detM , so it has to vanish at the horizon. The second term is an expression proportional to if we set V¹ = ®º B¹º which we assume to be regular at the horizon. To prove the invariance of the area, it is su±cient to prove that for any regular vector V¹, (A.6) goes to zero as we approach the horizon.

To show this we introduce a vielbein e¹a for the D ¡ 1 dimensional \metric" M¹º : V¹A¹º = V º detM

M¹º = ´abe¹aeº b : Since there is no curvature singularity at the horizon, all components of the vector V a must be regular there. In terms of the vielbein e¹a and its cofactor E¹a, we can write V¹A¹º = ¡eº aV a(det e)2 = ¡Eº aV adet e : (A.8) Since all elements of M¹º are ¯nite at the horizon, we can choose all eº a to be ¯nite as well. This in turn leads to ¯nite minors E º a. On the other hand, since detM vanishes at the horizon, so does det e, proving that V¹A¹º vanishes at the horizon for any value of º.

This shows that the area of the horizon is invariant with respect to sequence of manipulations described in this appendix. In particular, by taking ®¹ to rotate the four independent transverse rotations of S7 and scaling the twist with respect to the boost, we show that the area of the horizon is invariant under null Melvin twist. (A.6) (A.7)

J P 0 8 ( 2 0 0 3 ) 0 3 5 J H E 0 (