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arXiv:gr-qc/0703091v6 27 Feb 2008

Black string and velocity frame dragging

Jungjai Lee∗

Department of Physics, Daejin University, Pocheon, 487-711, Korea.

Hyeong-Chan Kim†

Department of Physics, Yonsei University, Seoul 120-749, Republic of Korea.

(Dated: February 27, 2008)

We investigate velocity frame dragging with the boosted Schwarzschild black string

solution and the boosted Kaluza-Klein bubble solution, in which a translational sym-

metry along the boosted z-coordinate is implemented. The velocity frame dragging

effect can be nullified by the motion of an observer using the boost symmetry along

the z−coordinate if it is not compact. However, in spacetime with the compact

z−coordinate, we show that the effect cannot be removed since the compactification

breaks the global Lorentz boost symmetry. As a result, the comoving velocity is

dependent on r and the momentum parameter along the z−coordinate becomes an

observer independent characteristic quantity of the black string and bubble solutions.

The dragging induces a spherical ergo-region around the black string.

PACS numbers: 04.70.-s, 04.50.+h, 04.90.+e, 11.30.Cp

After the discovery of Schwarzschild black hole solution [1] in general relativity, there

has been an enormous increase of interest in black objects such as several kinds of black

hole, black string, and black p−brane. The black holes in (3+1)-dimensional Einstein-

Maxwell theory are classified by a few parameters (so called hairs) such as mass M, angular

momentum J, and charge Q. By the three parameters, the solutions are classified into

four specific families, the Schwarzschild metric [1], the Reissner-Nordstr¨ om metric [2, 3],

the Kerr metric [4], and the Kerr-Neumann metric [5]. In the presence of matters other

than electromagnetic field, several solutions are possible with different hairs, non-abelian

hairs [6, 7, 8, 9, 10], dilatonic hair [11, 12, 13, 14], quantum hair [15, 16], and so forth. In

addition to these, there are another kind of solutions with different asymptotic spacetime

structure such as the BTZ black hole [17, 18].

These parameters plays a role in changing the structure of black objects. The mass

M develops event horizon and the charge Q allows a black object to interact with gauge

field. The angular momentum (J ?= 0) induces a strong frame dragging and develops an

ergo-region around its event horizon. The rotational frame dragging effect was first derived

from the theory of general relativity in 1918 by Lense and Thirring, and is also known as

the Lense-Thirring effect [19, 20, 21]. The rotational frame dragging has been measured in

recent experiments [22, 23, 24, 25, 26], which shows that the general relativity is correct.

At first look, it seems that the momentum parameter P does not play a role in changing

the spacetime structure and can be gauged away by the simple coordinate transform like

∗Electronic address: jjlee@daejin.ac.kr

†Electronic address: hckim@phya.yonsei.ac.kr

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the boost. Contrary to this naive expectation, in this letter, we discuss that the boosted

spacetime with non-zero momentum P is not equivalent to the static one if the boosted coor-

dinate is compact. Let us argue this using the boosted black string [27, 28] with the compact

fifth z−coordinate with z = 0 being identified with z = L. An analogy of twin paradox

helps us understand the situation. Let one of the twin circumnavigate along the compact

z-coordinate with constant velocity and meet the other staying at z = 0. If the relativity of

velocity holds, the twin paradox reemerges since none of the twin experience acceleration.

For this paradox being resolved, the relativity of velocity should be broken for compact co-

ordinate. This manifestly shows that the rigid Lorentz boost along the compact coordinate

is not a symmetry of the spacetime any more. In fact, the rigid Lorentz boost spoils the

space-like feature of the transformed space coordinate. Therefore, the rigid Lorentz boost

along the z−coordinate is excluded in the diffeomorphism group of the compact spacetime

any more.

Actually, the velocity frame dragging effect induced by P is not well known and con-

troversial. If the momentum P do the role in changing the spacetime structure, the ve-

locity frame dragging effect will be observable in gravitating objects, such as black ob-

jects [29, 30, 31, 32, 33, 34, 35] in string theory and other higher dimensional gravity

theory [36, 37, 38]. If there is a velocity frame dragging in these objects, the momentum

flow (P ?= 0) can induce an ergo-region around its event horizon to alter the spacetime

structure. In particular, the black string with a compact coordinate or the Kaluza-Klein

bubble solution may exhibit this velocity frame dragging effect. This field is not touched

at all so far. In this paper, we investigate the frame dragging effect in these solutions by

comparing it with that in the Kerr black hole. We study how the frame dragging effect

changes when the spacetime has a compact fifth coordinate.

The metric of the Schwarzschild black string solution in (4+1) dimensions is

ds2= −

?

1 −2M

r

?

dt2+ dz2+

1

1 − 2M/rdr2+ r2?dθ2+ sin2θdφ2?.(1)

The Kaluza-Klein bubble solution [39] is given by the double-Wick rotation t → iz and

z → it of Eq. (1):

?

We see that there is a minimal 2-sphere of radius 2M located at r = 2M for this bubble

solution. To avoid a conical singularity at r = 2M we need that z is a periodic coordinate

with period L = 8πM. Clearly, the solution asymptotes to M4× S1for r → ∞.

Let us check the velocity frame dragging effect in these two solutions (1) and (2). Consider

an observer moving with the velocity v = tanhξ with the coordinates transform,

ds2= −dt2+1 −2M

r

?

dz2+

1

1 − 2M/rdr2+ r2?dθ2+ sin2θdφ2?. (2)

?t′0

z′4

?

=

?coshξ− sinhξ

−sinhξ coshξ

??t

z

?

,(3)

with respect to the static metrics (1) and (2). Then, a stationary metric can be obtained to

be

ds2= gµνdxµdxν= −F(r)dt2+ 2X(r)dtdz + H(r)dz2

+

1 − 2M/rdr2+ r2?dθ2+ sin2θdφ2?,

(4)

1

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where the functions F(r), H(r), and X(r) are,

F(r) = 1 −2M cosh2ξ

r

,H(r) = 1 +2M sinh2ξ

r

X(r) =M sinh2ξ

r

,

for the boosted Schwarzschild solution and

F(r) = 1 +2M sinh2ξ

r

,H(r) = 1 −2M cosh2ξ

r

X(r) = −M sinh2ξ

r

,

for the Kaluza-Klein bubble solution, respectively. The mass, tension, and the momentum

flow along z− coordinate, MADM, τ, and P are

?

τ =

MADM =1 ±cosh2ξ

?

P = ±M sinh2ξ

3

?3M

?3M

4

,(5)

1 ∓cosh2ξ

34

,

4

,

where the upper/down sign in Eq. (5) is for the boosted Schwarzschild/bubble solution. In

the boosted Schwarzschild solution, the velocity frame dragging is induced by the momen-

tum P. However, an observer related with the coordinates transform (3) does not see any

dragging effect and see a static metric. This is one of main difference of the momentum

driven metric from that of the well-known Kerr solution.

Note that the bubble solution with L = 8πM becomes naked (conical) singular at r = 2M

if it is the boosted [27]. Therefore, the boost along the z-coordinate alters the physical

properties of the r = 2M surface. Instead, we could choose the compactification length

Lboost= 8πM coshξ for the boosted bubble solution to avoid the naked singularity. We can

conclude that a given static bubble solution is inequivalent to the boosted bubble solution.

Before we study the black string solution with compact z−coordinate, let us consider the

Kerr blackhole with metric,

ds2= −

?

r2+ a2+2Mrα2

1 −2Mr

ρ2

?

dt2+ρ2

Λ2dr2+ ρ2dθ2

?

(6)

+

?

ρ2

sin2θsin2θdφ2−4Mrα

ρ2

sin2θdtdφ,

where α = J/M, ρ2= r2+ α2cos2θ, and Λ2= r2− 2Mr + α2. In the non-relativistic limit

where M goes to zero, the Kerr metric becomes the orthogonal metric for oblate spheroidal

coordinate,

ds2= −dt2+

ρ2

r2+ α2dr2+ ρ2dθ2+ (r2+ α2)sin2θdφ2.

We may re-write the Kerr metric (6) in the following form:

ds2= −

?

gtt−g2

tφ

gφφ

?

dt2+ grrdr2+ gθθdθ2+ gφφ

?

dφ +gtφ

gφφdt

?2

.

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This metric is equivalent to a co-rotating reference frame that is rotating with angular speed

Ω = ω(r,θ) that depends on both the radius r and the colatitude θ,

ω(r,θ) = −gtφ

gφφ

=

2Mαr

ρ2(r2+ α2) + 2Mα2rsin2θ.

(7)

Thus, an inertial reference frame is drawn by the rotating central mass, the frame dragging.

An extreme version of frame dragging occurs within the ergosphere of a rotating black

hole. The ergosphere of the Kerr metric is bounded by two surfaces on which it appears

to be (coordinate) singular. The inner surface corresponds to a spherical event horizon at

rH= M +√M2− α2, where the purely radial component grrof the metric goes to infinity.

The outer surface is the stationary limit,

rstationary= M +√M2− α2cos2θ,

which touches the event horizon at the poles of the rotation axis, where the colatitude θ

equals 0 or π.

What is the difference of the frame dragging effect in the present stationary solution (4)

from that in the Kerr blackhole? We have asymptotic observer at r = ∞ in black string

metric in which frame dragging may exist. However, the frame dragging is the same at

all points of the spacetime if it is observed in the coordinates of the asymptotic observer.

Therefore, the dragging effect can be nullified by the motion of the asymptotic observer with

velocity v = tanhξ.

An interesting question is whether we can identify the presence of frame dragging effect

or not, if we are restricted to a circle with fixed radius and polar angle in Kerr spacetime.

To begin with, let us review a well known thought experiment, which defines a locally non-

rotating observer, in the Kerr spacetime (See Exercise 33.3 of Ref. [40]). Place a rigid, circular

mirror (“ring mirror”) at fixed (r,θ) around a black hole. Let observer at (r,θ) with angular

velocity Ω emit a flash of light. Some of the photons will get caught by the mirror and will

skim along its surface, circumnavigating the blackhole in the positive-φ direction. Others

will get caught and will skim along in the negative-φ direction. Then, only the observer

with his angular velocity Ω = ω(r,θ) in Eq. (7) will receive back the photons from both

directions simultaneously. Only the observer regard the +φ and −φ directions as equivalent

in terms of local geometry. Therefore, there is a preferred “locally nonrotating” observer in

this situation. The metric inside the ring with respect to the locally nonrotating observer

takes the form:

ds2

nr= −dt2+ R2dφ2,(8)

where we have ignored the θ, r coordinates and rescaled the time coordinates since the

observer are restricted to the ring. The coefficient R is independent of t and φ. This

metric is flat with respect to the coordinate (t,Rφ). However, there is no Lorentz boost-like

symmetry which mixes t and φ, since there is a preferred observer: the locally nonrotating

observer. Therefore, the observer restricted to the ring can determine he is rotating or not

with respect to the locally nonrotating frame. However, the locally nonrotating observer

cannot determine whether he is rotating or not with respect to an asymptotic infinity without

comparing his coordinates with respect to the asymptotic infinity. Only after he comparing

his coordinates with the asymptotic one, he can determine he is rotating or not.

It is interesting to ask what takes away the apparent boost-like symmetry along φ in the

metric (8). At first glance, one of the two is responsible for the breaking of the symmetry, the

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work done by the mirror on the light and the compactness of the angle φ. To examine these

possibilities, we consider three limiting thought experiments. First, we take the vacuum

(M = J = 0) limit of the blackhole. In this case, we have ω(r,θ) = 0 and therefore,

the locally nonrotating frame selects the static frame. The light bounces by the mirror to

circulate the ring. However, it should be noted that the force given by the mirror to the

light is orthogonal to the velocity of the light so that it does not affect to the angular motion

of light. Second, we consider the r → ∞ limit. In this case, the experiment cannot select

any observer since no light can circumnavigate to return to the observer within finite time.

Therefore, there disappears the preferred locally nonrotating observer and the boost-like

symmetry along φ will be restored. Finally, let r be placed at the last unstable circular orbit

of photon (the photon sphere). In this case, we do not need the mirror which restricts the

path of the light. The observer may simply send light for both side of the φ directions and

then waits until the light to arrive him after a full circulation of the geodesic path. There

are no artificial work done by the mirror on the light, however, the light select the locally

nonrotating observer. This observation indicates that the breaking of the Lorentz boost-like

symmetry is not due to the work given by the mirror. Since in this case, the light simply

follows the geodesics, we may accept the metric (8) as a 2−dimensional spacetime metric

with symmetry φ = φ + 2π without the r and θ coordinates. In conclusion, the breaking

of the boost-like symmetry is solely due to the compactness of the φ coordinate, not due to

the work given by the mirror.

Similarly, the boosted black string solution with the compact z-coordinate is independent

of the static one. Now consider spacetime given by the metric (4) with the compact z-

coordinate with period L. We examine the frame dragging effect by analyzing the metric

from the point of view of moving observer along the z−direction with velocity q at r → ∞.

The metric seen by the moving observer can be obtained by using the remaining translational

symmetry[42]:

z′= γ(z + qt),t′= γ−1t,(9)

where γ =

1

√

1−q2. The metric (4), from the point of view of moving observer with velocity

q at r → ∞, becomes

ds2= −γ2?F + 2qX − q2H?dt′2+ 2(X − qH)dz′dt′+ γ−2Hdz′2.

The metric component gtz′ is

(10)

gt′z′ =

−q +M sinh2ξ(1 − q tanhξ)

−q −M sinh2ξ(−q + tanhξ)

r

,Schwarzschild ,

r

,Kaluza-Klein bubble,

whose asymptotic value is −q. As a result, the metric (10) with compact z−coordinate

is described by 3−parameters (M,ξ;q), where the set (M,ξ) denotes observer independent

geometric properties and q, the velocity of the observer.

The comoving velocity along z-coordinate at r is

v(r) = −gt′z′

gz′z′= γ2

?

q −X

H

?

,(11)

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which asymptotically approaches to q at r → ∞ as it must. For the boosted Schwarzschild

solution the comoving velocity in Eq. (11) is

vS(r) = γ2

?

q −

M sinh2ξ

r

1 +2M sinh2ξ

r

?

. (12)

Note that the comoving velocity is dependent on r. Therefore, to an observer moving with

the velocity q, only the geometry at rq =

there. For different r, the geometry is stationary and therefore there exists frame dragging.

At the horizon the velocity (12) becomes vS(2M) = γ2(q − tanhξ), therefore the moving

observers with velocity q = tanhξ see the horizon static, which is intuitively correct.

For the boosted Kaluza-Klein bubble solution, we have the comoving velocity,

sinh2ξ M

q

(1 − q tanhξ) is static since vS(rq) = 0

vKK(r) = γ2

?

q +

M sinh2ξ

r

1 −2M cosh2ξ

r

?

. (13)

To an observer moving with velocity q, only the geometry at

rq= 2M cosh2ξ(1 −tanhξ

q

)(14)

is static. For different r, the geometry is stationary and therefore there exists frame dragging

effect. If an observer at infinity want to see a geometry at a given rqstatic, he should move

with the velocity q satisfying Eq. (14). This velocity, however, becomes the light velocity at

r = 2M cosh2ξ(1±tanhξ) and even larger than the light velocity in range between the two

values of r. Therefore, it is impossible that an observer at infinity see this region static.

At r = 2M, the comoving velocity becomes vKK(2M) = γ2(q−cothξ). If we restrict the

velocity |q| < 1, since the motion of observer at infinity is restricted, we cannot make the

velocity (13) at r = 2M surface to zero.

The static limit is present at re(q,ξ) satisfying

F + 2qX − q2H = 0.(15)

The static limit changes according to the velocity q of the observer at infinity. For the

Schwarzschild case, Eq. (15) gives

re =2M cosh2ξ(1 − q tanhξ)2

1 − q2

≥ 2M,

where retakes its minimum value 2M at q = tanhξ. The ergo-region is located at 2M < r ≤

re. To an observer moving with velocity q = tanhξ, there is no ergo-region since re= 2M

overlaps with the horizon. For the Kaluza-Klein bubble case, Eq. (15) gives

re = −2M cosh2ξ(q − tanhξ)2

1 − q2

.

Since we have restricted to r ≥ 2M for the bubble solution and the size of the velocity

q must be smaller than one, we have re < 0 always. Therefore, there does not exist the

ergo-region for the boosted bubble solution.

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In discussion, the boosted black string solution in spacetime with the compact

z−coordinate is not equivalent to the static solution and has a quantity P, the momentum

with respect to an asymptotically static observer, as a characteristic physical parameter. As

a result, the velocity frame dragging effect becomes a physical observable.

The momentum parameter also becomes a physical observable for the boosted Kaluza-

Klein bubble solution which has an innate compactified length. This compactness destroys

the boost symmetry. So, in the boosted Kaluza-Klein bubble solution, the momentum pa-

rameter along compact direction becomes a physical quantity which cannot be gauged away

by boost transformation. In this direction, it has been studied recently the thermodynamics

of boosted non-rotating black holes with the momentum along the compact dimension [41].

How does the velocity frame dragging effect appear from the point of view of an observer

in four dimensions? The most distinguished fact is that there appears a spherical ergo-region

around the black string. Therefore, if one find an object with spherically symmetric static

limit, it must be a black string solution with a compact z−coordinate which has the velocity

frame dragging effect. This is different from the ergo-region around the Kerr blackhole,

which is not spherically symmetric.

It would be interesting to investigate the velocity frame dragging effect for other black

objects with compact dimensions and other gravity objects such as stationary vacuum black

string solution which is not equivalent to static solution [43].

Acknowledgments

We are very grateful to Gungwon Kang for helpful discussions. This work was supported

by the Daejin University Special Research Grants in 2007.

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