ISSN 0021�3640, JETP Letters, 2010, Vol. 92, No. 5, pp. 276–280. © Pleiades Publishing, Inc., 2010.

276

1. INTRODUCTION

Brane�world cosmological scenarios [1, 2] leaveopen the possibility of presence of matter in the bulk,at least in the form of black holes [3]. Interaction ofbulk black holes with the three�brane is importantboth for the fate of the brane itself [4], as well as for thefate of black holes presumably created at colliders inTeV�scale gravity [5]. This problem was extensivelystudied recently using the Dirac–Nambu–Goto(DNG) equation on different black hole backgrounds[6] leading to variety of interesting predictions. Herewe would like to investigate collision between the massand the three�brane as genuine two�body problem,treating both of them on equal footing. Clearly, rela�tivistic two�body problem with field mediated interac�tion has no exact solutions already in the case of twopoint particles in fiat space. The standard approach isperturbation theory in terms of the coupling constant.We therefore adopt the same strategy. In order to max�imally simplify the problem, we will treat the mass aspoint�like, gravity—within the Fierz–Pauli theory,and we restrict by an iterative solution in terms of thecoupling constant κ, related to the five�dimensionalgravitational constant G5 as κ2 = 16πG5.

Gravitational constant in five dimensions hasdimension of length3. Combining it with the particlemass m (dimension of length–1) and the brane tensionμ (dimension of length–4) we have two length parame�

ters: l = 1/κ2μ, rg = κ , the first corresponding tothe curvature radius of the bulk in the RS II setup, andthe second—to gravitational radius of the mass m. Tokeep contact with the RS II model we have to considerdistances small with respect to l, while to justify thelinearization of the metric for the mass m we have toconsider distances large with respect to rg. So to applythe linearized theory to both objects we have to assumel � rg, or m μ2κ6 � 1. Clearly, in this approach we will

¶The article is published in the original.

m

not be able to capture such essentially non�linear phe�nomena as formation of a hole in the three branebounded by two�brane, and an associated Chamblin–Eardley instability [7]. Treating the mass as point�like,we exclude formation of a hole, but instead we will beable to describe another interesting manifestation ofthe brane perforation: an excitation of the quasi�freeNambu–Goldstone (NG) field.

Our model of collision is therefore quite simple: wetreat bulk gravity at the linearized level on Minkowskibackground, with both the brane and the particlebeing described by geometrical actions. Gravitationalinteraction between them is repulsive in the co�dimension one, and the force does not depend on dis�tance. When the mass pierces the brane reappearingon the other side of it, the sign of this force changes,and the brane gets shaken. We show that this shakegives rise to spherical NG wave which then expandswithin the brane with the velocity of light indepen�dently on further particle motion. Though we do notdiscuss here secondary effects due to interaction of theNG wave with matter fields on the brane [8, 9], it isclear that this interaction will transfer energy to thebrane matter modes. Thus, for an observer living onthe brane the act of brane perforation by a massivebody from the bulk will look like a sudden explosiveevent not motivated by any visible reasons.

2. SET UP

Consider the three�brane propagating in the five�dimensional spacetime (�5, gMN), whose world�vol�ume �4 is given by the embedding equations xM =XM(σμ), M = 0, 1, 2, 3, 4, parameterized by arbitrarycoordinates σμ, μ = 0, 1, 2, 3 on �4. The DNG equa�tion reads

(1)∂μ Xν

NgMNγμν γ–( ) 1

2��gNP M, Xμ

NXν

Pγμν γ– ,=

Nambu–Goldstone Explosion under Brane Perforation¶

D. V. Gal’tsov, E. Yu. Melkumova, and S. Zamani�MogaddamDepartment of Physics, Moscow State University, Moscow, 119899 Russia

e�mail: galtsov@physics.msu.ruReceived July 14, 2010

We show that perforation of the three�brane by mass impinging upon it from the five�dimensional bulk excitesNambu–Goldstone spherical wave propagating outwards with the velocity of light. It is speculated that suchan effect can give rise to “unmotivated” energy release events in the brane�world cosmological models.

DOI: 10.1134/S0021364010170029

JETP LETTERS Vol. 92 No. 5 2010

NAMBU–GOLDSTONE EXPLOSION 277

where = ∂XM/∂σμ are the tangent vectors, γμν is the

inverse induced metric on �4, γμνγνλ = ,

(2)

and γ = detγμν.

The energy–momentum tensor of the brane is

(3)

where μ is the brane tension.

Bulk gravitational field will be described within thelinearized theory expanding the metric as

(4)

with ηMN = diag(1, –1, –1, –1, –1), and the metricdeviation hMN being considered as the Minkowski ten�sor. The linearized Einstein equation in the harmonic

gauge ∂MhMN = ∂Nh/2, h = reads:

(5)

where �5 = –∂M∂M is the bulk D’Alembertian, and themetric potentials satisfy the superposition principlebeing the sum of the brane potentials and the mass

potentials: hMN = + . For consistency, therespective energy–momentum tensors at the right

hand side must be fiat space divergenceless ∂N =0, so, building them, we should not take into accountgravitational.

In zero order in κ the brane is assumed to be un�

excited = , where are four constant bulkMinkowski vectors which will be normalized as

= ημν. Obviously, this is a solution to theEq. (1) for κ = 0, and the corresponding induced met�ric is four�dimensional Minkowski

(6)

One can further identify the coordinates σμ on theunperturbed brane with the bilk coordinates σμ = xμ,

in which case = . Denoting the fifth coordinate

as z = x4 we obtain the solution of (5):

(7)

Xμ

M

δλ

μ

γμν Xμ

MXν

NgMN x X= ,=

TbMN μ Xμ

MXν

Nγμνδ5 x X σ( )–( )

g–������������������������� γ– d

4σ,∫=

gMN ηMN κhMN,+=

hPP

�5hMN κ TMN13��ηMNT–⎝ ⎠

⎛ ⎞ ,=

hMNb hMN

m

TMNb m,

X0M Σμ

Mσμ Σμ

M

Σμ

MΣν

NηMN

TbMN μ Σμ

MΣν

Nημνδ5 x X0 σ( )–( )d4σ, T∫ TPP.= =

Σμ

M δμ

M

hMNb κμ

2����� ΣM

μ ΣNμ43��ηMN–⎝ ⎠

⎛ ⎞ z=

= κμ6

�����diag 1– 1 1 1 4, , , ,( ) z ,

this is a potential linearly growing on both sides of thebrane. Comparing this with the RS II metric

(8)

at the distance z from the brane small compared withthe curvature radius of the AdS bulk, kz � 1, so that

� 1 – 2k see that they differ by coordinatetransformation. Indeed, the gauge for the RS solutionis non�harmonic. To pass to the harmonic gauge onehas to transform the fifth coordinate as

(9)

This reproduces our solution (7). Note that this trans�formation is non�singular on the brane: ∂zRS/∂z = 1 atz = 0.

Consider now motion of the point mass in the bulkalong the world line xM(τ) = (t(τ), 0, 0, 0, z(τ)):

(10)

We assume the mass moves in the positive z directionand hits the brane at t = 0 having the velocity v, and

correspondingly, = γ, = γv, γ = 1/ . Thenwe find from (10) that just before and after collision

(11)

Particle energy and momentum , have no dis�continuity at the location of the brane z = 0, but theirderivative have. It can therefore perforate the branewithout loosing the energy–momentum, but with sud�den change of acceleration. The sign in (11) corre�sponds to gravitational repulsion, as could be expectedin the co�dimension one case [10].

Now compute the gravitational potentials of themass constructing the source energy–momentum ten�sor in zero order in κ. With gravitational interactionbeing neglected, the mass moves freely with the veloc�ity v, so

(12)

Substituting the corresponding energy–momentumtensor

(13)

into the Eq. (5), we obtain

(14)

dsRS2 e

2k zRS–ημνdxμdxν dzRS

2, k– κ2μ

12�������,= =

e2k zRS–

zRS

zRS z 2kz2 z( ).sgn–=

ddτ���� x· NgMN( ) 1

2��gPQ M, x· Px· Q

.=

t· z· 1 v2–

t·· κ2μ6

�������γ2v z( ), z··sgn κ2μ

12�������γ2 1 4v2+( ) z( ).sgn= =

mt· mz·

xM τ( ) uMτ γ 1 0 0 0 v, , , ,( )τ, uMconst.= = =

TmMN m uMuNδ x x τ( )–( ) τd∫=

hmMN κm

2π( )2�����������

uMuN13��ηMN–⎝ ⎠

⎛ ⎞

γ2 z vt–( )2 r2+�������������������������������,–=

278

JETP LETTERS Vol. 92 No. 5 2010

GAL’TSOV et al.

where r2 = (x1)2 + (x2)2 + (x3)2. This is nothing but theLorentz�contracted five�dimensional Coulomb gravi�tational field of a moving point particle.

3. BRANE DEFORMATION

The transverse coordinates of the brane can beviewed as Nambu–Goldstone bosons (branons) whichappear as a result of spontaneous breaking of the trans�lational symmetry [8]. These are coupled to gravityand matter on the brane via the induced metric (for arecent discussion see [9]). In our case of co�dimensionone there is one such branon, which is a massless fieldcoupled to bulk gravity.

To derive the NG wave equation we consider the

deformation of the brane XM = + δXM caused by

gravitational field due to matter in the bulk, com�pute the induced metric and linearize the resultingequation with respect to δXM. Only transverse to thebrane perturbation Φ(xμ) = δX4 is physical, for whichwe obtain the wave equation

(15)

where �4 = –∂μ∂μ is the flat D’Alembertian on �4, andthe source term J = J4, where

(16)

with hPQ = . Substituting here the potentials (14),we obtain explicitly:

(17)

The retarded solution of the Eq. (15) reads:

(18)

where kμ = (ω, k) and J(k) is the Fourier transform ofthe source:

(19)

Evaluating the integral we find the following solutionconsisting of two terms:

(20)

X0M

hMNm

�Φ x( ) J x( ),=

JN κ 12��∂NhPQ ∂PhQ

N–⎝ ⎠⎛ ⎞

z 0=

Σμ

PΣν

Qημν,=

hPQm

J x( ) λvt

r2 γ2v

2t2+( )2

��������������������������, λ mκ2γ2

2π( )2������������ γ2

v2 1

3��+⎝ ⎠

⎛ ⎞ .= =

Φ Gret x x'–( )J x '( )d4x'∫=

= 1

2π( )4����������� e ikx–

k2 2i�ω+�������������������J k( )d4k,∫

J k( ) eikxJ x( )d4x∫2π2λγ

���������� iω

ω2 γ2v

2k2+����������������������� .= =

Φ λ

2γ3������

F0 r t,( )r

��������������F1 r t,( )

r��������������–⎝ ⎠

⎛ ⎞ ,=

where

(21)

The first part F0, proportional to the Heaviside stepfunction of time θ(t), is zero until the moment t = 0 ofperforation. It describes an expanding wave caused bythe perforation shake discussed above (the first term inF0 looks contracting, but actually it is a constant,θ(t) � (r + t) = 1). This wave, propagating with thevelocity of light, does not correlate with furthermotion of the particle.

The second part F1 is non�zero and smooth bothbefore (t < 0) and after (t > 0) the perforation. It doescorrelate with position of the mass describing a con�tinuous deformation of the brane caused by its gravita�tional field. It is small when the particle is far awayfrom the brane, and grows up to the maximal absolutevalue equal to π/2 when it approaches the brane.

It is easy to verify that for all t ≠ 0, r ≠ 0 this term sat�isfies the inhomogeneous wave equation:

(22)

reproducing the right hand side of the Eq. (15). How�ever, it has finite and unequal limits as t ±0 for allr ≠ 0:

(23)

As was already explained, change of sign is due tochange of direction of the gravitational force betweenthe brane and the particle. The repulsive nature of thisforce manifests itself in signs: when the mass approachthe brane (t < 0) the repulsive deformation is directedalong z, which corresponds to Φ > 0, when the particlereappears on the other side of the brane it repels thebrane in the negative z direction, thus Φ < 0. Since theforce does not vanish and instantaneously changessign at the moment of perforation, the second part ofthe solution has singular t�derivatives. Indeed, apply�ing the box operator in the sense of distributions we getan additional delta�derivative term at the right handside of the Eq. (22) which was obtained for t ≠ 0:

(24)

The delta�derivative term describes the instantaneousshaking force exerted upon the brane. It excites theNG shock wave described by the first term in (21).Indeed, acting by D’Alembert operator on the F0 part

F0 r t,( ) π2��θ t( ) � r t+( ) � r t–( )+[ ],=

F1 r t,( ) rγvt������.arctan=

�F1 r t,( )

r�������������� 1

r�� ∂t

2 ∂r2–( )F1 r t,( ) 2γ3

vt

r2 γ2v

2t2+( )2

��������������������������,= =

rγvt������arctan

t 0±→lim π

2��� t( ).=

�F1 r t,( )

r�������������� 2γ3

vt

r2 γ2v

2t2+( )2

�������������������������� πδ ' t( )r

������������.+=

JETP LETTERS Vol. 92 No. 5 2010

NAMBU–GOLDSTONE EXPLOSION 279

of the solution, we find exactly the same delta�deriva�tive term:

(25)

Actually, the right�hand side arises as the second timederivative of the Heaviside function θ(t) entering F0,while the remaining factor [�(r + t) + �(r – t)]/rdescribes spherical shock waves satisfying the homo�geneous wave equation:

(26)

The sum of the two terms in (20) therefore has no dis�continuity at t = 0, while the discontinuity at t = r cor�responds to the expanding shock wave, satisfying thehomogeneous wave equation. One could expect thatbrane deformation caused by the gravitational field ofthe mass could be tight to the mass motion, i.e. be ofthe F1 type only. However, as we have shown, non�smoothness of gravitational interaction at the momentof piercing gives rise to NG explosive wave F0 whichthen propagates freely along the brane.

Even more surprising is that in the limit of zeromass velocity our solution remains non�zero:

(27)

Moreover, acting on this expression by box operator,one obtains zero, except for the point r = 0, at whichone has to perform calculations in terms of distribu�

tions. By virtue of the identity = –4πδ3(r) we then

find an extra term:

(28)

This might seem paradoxical, since the source term(17) in the NG equation (15) looks to be zero for v =0. The paradox is solved if we regard the source J(x) asdistribution. It is easy to see, that in the limiting casest 0 or v 0, J(x) exhibits properties of the three�dimensional delta�function. Denoting α = γvt, wehave:

(29)

Since the box operator in (15) contains the three�

dimensional Laplace operator, � = Δ – , it is rea�sonable to consider J(x) in the sense of distributions on

�F0 r t,( )

r�������������� πδ ' t( )

r������������.=

�� r t±( )

r�������������� 1

r�� ∂t

2 ∂r2–( )� r t±( ) 0.= =

Φ0 Φv 0→lim mκ2

48π��������� r t–( )

r��������������.= =

Δ1r��

�Φ0 QBδ3 r( ), QB

mκ2

12��������� t( ).= =

α

r2 α2+( )2

�������������������α 0±→lim

0, if r 0,≠

∞, if r± 0.=⎩⎨⎧

=

∂t2

R3. Now, the integral of J(x) over the three�dimen�sional space is α�independent up to the sign and finite,

(30)

where �(α) = α/ so, taking into account that�(α) = �(t), we obtain:

(31)

which for v = 0 reproduces the right hand side of (28).Remarkably, this limiting value is the same if we con�sider time in the close vicinity of the perforationmoment t 0 for any velocity v of the mass m, or ifwe consider the limit of small velocity v 0. In thelatter case of quasi�static perforation this limit holdsfor any t, and since the coefficient λ remains finite asv 0, the point�like source in (15) might be attrib�uted to some NG “charge” (shake charge).

This notion allows us to make distinction betweentwo static situations. The first is that of the point masssitting on the brane all the time. In this case, comingback to the Eq. (16) for brane perturbations, we findthat the source term at the right hand side will be zero.This “eternally” sitting on the brane mass is not in factthe bulk particle. On the contrary, the bulk particlearriving at the brane with zero velocity interacts gravi�tationally with the brane in such a way that a non�zeroNG charge as a source term in the branon wave equa�tion is produced. Therefore, it does generate the NGwave (28). This NG “charge” is a manifestly non�con�served quantity, changing sign at the moment of perfo�ration. For an observer on the brane the perforation isfelt as a sudden shake, and the corresponding NG fieldwill be not a static Coulomb field, but an expandingwave.

4. CONCLUSIONS

We have considered a simple model of collisionbetween an infinitely thin three�brane and a point�likebulk particle interacting gravitationally in five�dimen�sional spacetime within the linearized Einstein theory.In this setup the particle impinging normally on thebrane freely passes through it to reappear on the otherside. Since the particle has no size, no hole is createdin the three�brane and no two�brane appears sur�rounding the hole. These phenomena which have toarise in the case of an impinging mass of finite radius(a black hole) are beyond the scope of the linearizedgravity theory we adopt here. But as we have shown,perforation has another important effect which is wellcaptured by the linear gravity approximation: detona�tion of a shock NG expanding wave at the moment ofpiercing. This might seem surprising, since no directnon�gravitational force between the mass and thebrane exists in our model, and the particle passes

J x( )d3x∫4πλγ

�������� αr2

r2 α2+( )2

������������������� rd

0

∞

∫π2λγ

�������� α( ),= =

α

J x( )α 0±→lim π2λ

γ�������� t( )δ3 r( ),=

280

JETP LETTERS Vol. 92 No. 5 2010

GAL’TSOV et al.

through the brane feeling only its gravitational field.Similarly, the brane is not hit by the mass in themechanical sense, but only reacts on its gravitationalfield. But the potential energy of the point mass in thefield of the brane immersed in spacetime with co�dimension one is linearly growing, so the force isrepulsive and distance�independent. When the masspierces the brane, this finite repulsive force instanta�neously changes sign, so its time derivative behaves asdelta�function of time.

Apart from this shock NG wave expanding with thevelocity of light, the retarded solution contains theprecursor/tail part which is non zero both before andafter the perforation. This component is in direct cor�relation with position of the particle, so its measure�ment may serve as a tool to see invisible matter in thebulk.

Our model was inspired by the RS II setup. Indeed,using the linearized Einstein theory we were able toreproduce the RS II solution at distances small withrespect the curvature radius of the bulk. But differentlyfrom the RS approach, we have considered both thebrane and the particle on equal footing as test interact�ing objects in the Minkowski background. Thisallowed us to tackle the problem relativistically (in thespecial relativity sense) and to reveal existence ofexplosive NG wave triggered at the moment of perfo�ration. One can speculate that perforation of thethree�brane in the brane�world models may give rise to“unmotivated” explosive events in the observed Uni�verse. Indeed, the NG field universally interacts withmatter on the brane via the induced metric [8, 9], and,consequently, the NG explosion will transform intothe matter explosion. In absence of matter, gravita�tional radiation will be excited, to see this it is enoughto pass to the second postlinear order of Einstein the�ory. The Nambu–Goldstone explosion can beexpected to hold in the full non�linear treatment aswell. Indeed, excitation of the brane oscillations in theblack hole case is likely to have been observed innumerical experiments [4].

This work was supported by the Russian Founda�tion of Basic Research, project no. 08�02�01398�a.

REFERENCES

1. K. Akama, Lect. Notes Phys. 176, 267 (1982)[arXiv:hep�th/0001113]; V. A. Rubakov and M. E. Sha�poshnikov, Phys. Lett. B 125, 136, 139 (1983); M. Vis�ser, Phys. Lett. B 159, 22 (1985) [arXiv:hep�th/9910093]; G. W. Gibbons and D. L. Wiltshire, Nucl.Phys. B 287, 717 (1987) [arXiv:hep�th/0109093].

2. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999) [arXiv:hep�ph/9905221v1]; Phys. Rev. Lett. 83,4690 (1999) [arXiv:hep�th/9906064v1].

3. V. A. Rubakov, Usp. Fiz. Nauk 171, 913 (2001) [Phys.Usp. 44, 871 (2001)] [arXiv:hep�ph/0104152]; D. Lan�glois, Prog. Theor. Phys. Suppl. 148, 181 (2003)[arXiv:hep�th/0209261].

4. A. Flachi and T. Tanaka, Phys. Rev. D 76, 025007(2007) [arXiv:hep�th/0703019]; K. Hioki, U. Miya�moto, and M. Nozawa, Phys. Rev. D 80, 084011 (2009)[arXiv:0908.1019 [hep�th]].

5. A. Flachi, O. Pujolas, M. Sasaki, and T. Tanaka, Phys.Rev. D 74, 045013 (2006) [arXiv:hep�th/0604139];A. Flachi, O. Pujolas, M. Sasaki, and T. Tanaka, Phys.Rev. D 73, 125017 (2006) [arXiv:hep�th/0601174];A. Flachi and T. Tanaka, Phys. Rev. Lett. 95, 161302(2005) [arXiv:hep�th/0506145].

6. V. P. Frolov, D. V. Fursaev, and D. Stojkovic, Class.Quant. Grav. 21, 3483 (2004) [arXiv:gr�qc/0403054];J. High Energy Phys. 0406, 057 (2004) [arXiv:gr�qc/0403002]; V. P. Frolov, M. Snajdr, and D. Stojkovic,Phys. Rev. D 68, 044002 (2003) [arXiv:gr�qc/0304083];V. P. Frolov, Phys. Rev. D 74, 044006 (2006) [arXiv:gr�qc/0604114]; D. Kubiznak and V. P. Frolov, J. HighEnergy Phys. 0802, 007 (2008) [arXiv:0711.2300 [hep�th]].

7. A. Chamblin and D. M. Eardley, Phys. Lett. B 475, 46(2000) [arXiv:hep�th/9912166]; D. Stojkovic, K. Freese,and G. D. Starkman, Phys. Rev. D 72, 045012 (2005)[arXiv:hep�ph/0505026].

8. T. Kugo and K. Yoshioka, Nucl. Phys. B 594, 301(2001) [arXiv:hep�ph/9912496].

9. Y. Burnier and K. Zuleta, J. High Energy Phys. 0905,065 (2009) [arXiv:0812.2227 [hep�th]]; J. Alexandreand D. Yawitch, New J. Phys. 12, 043027 (2010)[arXiv:0910.5150 [hep�th]].

10. V. A. Rubakov and S. M. Sibiryakov, Class. Quant.Grav. 17, 4437 (2000) [arXiv:hep�th/0003109v1];W. Mueck, K. S. Viswanathan, and I. V. Volovich, Nucl.Phys. B 590, 273 (2000) [arXiv:hep�th/0002132];S. L. Dubovsky, V. A. Rubakov, and P. G. Tinyakov,Phys. Rev. D 62, 105011 (2000).