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Time-Dependent Dilatonic Domain Walls

arXiv:hep-th/9212041v1 6 Dec 1992CTP-TAMU-78/92hep-th/9212041Time-Dependent Dilatonic Domain WallsHoSeong La∗Center for Theoretical PhysicsTexas A&M UniversityCollege Station, TX 77843-4242, USATime-dependent domain wall solutions with infinitesimal thickness are obtained in thetheory of a scalar field coupled to gravity with the dilaton, i.e. the Jordan-Brans-Dickegravity.

The value of the dilaton is determined in terms of the Brans-Dicke parameter ω.In particular, the solutions exist for any ω > 0 and as ω →∞we obtain new solutions ingeneral relativity. They have horizons whose sizes depend on ω.11/92∗e-mail address: hsla@phys.tamu.edu, hsla@tamphys.bitnet

In the cosmological models for the early universe based on particle physics and generalrelativity (GR), it is known that these models admit topological defects as their classicalsolutions[1][2]. These topological defects in principle can be formed when the universegoes through phase transitions, although the chances of actual existence are not necessar-ily high.

We however anticipate that GR may not be the ultimate theory to describe thevery early universe for which we need a quantum gravity. Thus we should look for otherpossibilities how these topological defects can be formed in other gravitational theories.Besides, the ultimate theory should not only be able to predict the existence of the exist-ing objects but also be able to disprove the nonexistence of the nonexisting objects.

Instring theory context there have been various approaches, but the complete results are stillelusive[3][4][5].In this paper we shall take another possible gravitational theory, namely, the Jordan-Brans-Dicke (JBD) theory, which is a gravitational theory with the dilaton in the four-dimensional space-time. From the string theory’s point of view, the JBD theory with aspecific Brans-Dicke (BD) parameter is a natural effective gravitational theory before thedilaton freezes up.Furthermore, a cosmological model can also be built based on theJBD theory[6].

Thus it is worth while to investigate the existence of topological defects inthis context. Cosmic string solutions in this theory were previously studied in ref.

[7] andstatic thin domain wall solutions have been found in ref.[8]. In this paper we shall look fortime-dependent thin domain wall solutions in the JBD theory.One subtle point in getting a domain wall solution is that solving the equations asymp-totically is not good enough.

Unless it satisfies proper matching conditions across the wall,it is not a domain wall solution of the system. As noted in ref.

[8], among many asymptoticsolutions only one with vanishing BD parameter survives as a true solution in the staticcase. Here we shall adopt the same prescription for the matching condition at the walland look for time-dependent solutions.Let us consider the action for the JBD theoryS =Zd4xp−ege−2eφ eR −4ω∂µeφ∂µeφ+ SM[egµν],(1)where ω is the BD parameter of the theory.

In particular the ω = −1 case corresponds tothe action of the dilaton gravity from string theory[9]. For our purpose we take SM as theaction for the real scalar field with a double well potential:SM = −12Zd4xp−eg∂µΦ∂µΦ + λ(Φ2 −η2)2.

(2)1

SM in particular has a ZZ2 discrete symmetry Φ →−Φ. Note that in the static case, thisdiscrete symmetry is equivalent to the time reversal symmetry.

This is why the domainwall structure is related to the CP phases[10][11]. Here we shall take Φ to be static, butthe metric tensor can depend on time.

In principle one can generalize to the case in whichΦ depends on time. The result however should be the same in the thin wall case if weassume the position of the wall does not change.For convenience we redefine the variables asω =14β2 −32,e−2eφ =116πGe−2βφ,egµν = e2βφgµν,(3)then the action now becomesS =116πGZd4x√−g(R −∂µφ∂µφ) + SM[e2βφgµν].

(4)Setting φ = 0, this action reduces to that of a real scalar field coupled to the Einsteingravity, in which case a time-dependent thin domain wall solution is known to exist[12].In our case although egµν is a physical metric of the space-time, eq. (4) is very convenientfor practical purposes.

Although eφ is the actual dilaton, we will call φ a dilaton too sincethey are related by a simple field redefinition. We shall also assume that φ does not dependon time.By varying eq.

(4) with respect to the new metric gµν, we obtainRµν = ∂µφ∂νφ + 8πG(Tµν −12gµνT),(5)where the “energy-momentum” tensor is given byTµν = −2√−gδSMδgµν = −12e2βφgµνgαβ∂αΦ∂βΦ −2e2βφ∂µΦ∂νΦ + gµνe4βφλ(Φ2 −η2)2. (6)Note that Tµν is not covariantly conserved due to the dilatonic contribution.

The physicalenergy-momentum tensor eT matterµν+ eT eφµν satisfies the gravitational equation of motion forJBD theory[13],eRµν −12egµν eR = 8πe2eφ eT matterµν+ eT eφµν,(7)where ∇µ eT matterµν= 0 andeT eφµν = 18π e−2eφ 2∂µ∂ν eφ + 2eΓαµν∂α eφ + 12e2eφegµν e 2e−2eφ−ω2π e−2eφ ∂µ eφ∂ν eφ −12egµν∂αeφ∂α eφ. (8)2

The second term of eq. (8) is proportional to ω so that it vanishes if ω = 0, but the firstterm is independent of ω.The field equations for the dilaton φ is2φ =1√−g ∂µ(√−ggµν∂ν)φ = −8πGβT(9)and the matter scalar field satisfies∂µ(√−ge2βφgµν∂ν)Φ −2λ√−ge4βφΦ(Φ2 −η2) = 0.

(10)In general, domain wall solutions are obtained in theories where a discrete symmetryis spontaneously broken. Note that the action eq.

(2) for the matter field Φ has a discretesymmetry Φ →−Φ so that we can look for domain walls, when this symmetry is spon-taneously broken. In the case where domain walls have infinitesimal thickness, we canapproximate the wanted scalar field to behave asΦ(z) =ηif z > 0;−ηif z < 0.Then we are interested in the the domain wall which separates a space of the Φ = η phasefrom a space of the Φ = −η phase.

Such an approximation is in fact reasonable for thecases where the Compton wavelength of the test particle is much longer than the thicknessof the wall.For the static cases we used the following ansatz for domain wall solutions[14][15][8]:ds2 = A(|z|)(−dt2 + dz2) + B(|z|)(dx2 + dy2).Note that we have required the reflection symmetry between each side of the wall, whichis an infinite plane perpendicular to the z-direction at z = 0. Now for time-dependentdomain wall solutions we shall tryds2 = −A(|z|)dt2 + C(|z|)dz2 + B(|z|)U(t)(dx2 + dy2).

(11)Again we have the reflection symmetry. The time dependent part of the metric is addedin gxx = gyy.

This ansatz in fact is a generalized version of the time-dependent domainwall solutions in GR, in which case C = 1[12].Now we need a prescription to take care of the matching conditions across the wall.Strictly speaking |z| is not analytic at z = 0. However from physics’ point of view this3

merely is an approximation due to the thin wall assumption. One should treat it as a limitcase of analytic function.

One way to introduce a reasonable analytic property for |z| isto use a step function. Thus we can use the following prescription for |z| to avoid such adifficulty[8]:|z| = z[θ(z) −θ(−z)],(12)where θ(z) is a step function defined by θ(z) = 1 for z ≥0, θ(z) = 0 for z < 0.

Then∂z|z| = [θ(z) −θ(−z)] + 2zδ(z). If we promise that ∂z|z| shall be multiplied with somefunction of z that does not have a pole at z = 0, we can safely use an identification∂z|z| ≡θ(z) −θ(−z).

Similarly, ∂2z|z| ≡2δ(z). The reason we try to be careful aboutsuch analyticity is to check the consistency of the solutions at the wall, which turns out tobe important to provide interesting constraints on the solutions.

Again we would like toemphasize that this should be regarded as an approximation.Using this ansatz, eq. (11), we findRtz = 12˙UUA′A −B′B,(13)where the prime and the dot denote ∂z and ∂t respectively.The corresponding fieldequation Rtz = 0 leads toA = B.

(14)With this identification the field equations now becomeRtt = 14A′2AC −14A′C′C2+ 12A′′C + 12˙U 2U 2 −¨UU = −4πGλAe4βφ Φ2 −η22 ,(15a)Rzz = 34A′2A2 + 34A′C′AC −32A′′A =φ′2+8πGe2βφΦ′2+ 12λCe4βφ(Φ2−η2)2,(15b)Rxx = Ryy= U −14A′2AC + 14A′C′C2−12A′′C + 12¨UU!= 4πGλAUe4βφ Φ2 −η22 ,(15c)1A3/2C1/2A3/2C1/2 φ′′= 8πGβe2βφ 1C Φ′2 + 2λe4βφ(Φ2 −η2)2,(15d)A3/2C1/2 e2βφΦ′′= 2λA3/2C1/2e4βφΦ(Φ2 −η2). (15e)For thin walls we have Φ2 = η2 and Φ′ = 0 away from z = 0 so that we shall firstsolve the above equations away from z = 0, then shall check the consistency at the wall.Solving eqs.

(15a, c) for z ̸= 0, we obtainU(t) = eκt,(16)4

and from eq. (15d) we obtainφ′ = αC1/2A3/2 ,(17)where κ, α are constants yet to be determined.Using these in eqs.

(15a −c), we can determineC =AA′2κ2A2 + 23α2 . (18)With such C eq.

(17) now can be integrated asφ = 12q32 lnq1 + 3κ22α2 A2 −1q1 + 3κ22α2 A2 + 1,(19)where we have taken A(0) = 1, C(0) = 1, and accordingly φ(0).Since this eq. (18) satisfies eqs.

(15a −c) identically away from z = 0, there is no otherconstraint on the function A so that for any A we can obtain an asymptotic solution. Thisis rather intriguing, although some of them might fail to satisfy the matching conditionsacross the wall.

Also some proper asymptotic boundary conditions will distinguish them.For practical purposes, it is more convenient to choose C and solve for A to check thematching conditions across the wall.If we assume C = 1, then for α = 0 we reproduce the known solutions in the generalrelativity case[12], which can be shown easily from eq(18). For α ̸= 0 there are othersolutions but the indefinite integral to obtain A cannot be performed analytically.From now on we shall concentrate on one particularly interesting case of C = A andleave the rest as future exercises.

For C = A eq. (15a) is precisely the same as eq.

(15c).Now eq. (18) can be integrated to yieldA = peκz + qe−κz,(20)where4pqκ2 + 23α2 = 0.

(21)With the required boundary condition, A(0) = 1, i.e. p+q = 1, we can explicitly determinep, q asp = 12 1 −r1 + 2α23κ2!,q = 12 1 +r1 + 2α23κ2!.

(22)5

Thus we haveC = A = B = coshκ|z| −r1 + 2α23κ2 sinhκ|z|,(23)and from eq. (19), we getφ =q32 ln1 −q32κα1 −q1 + 2α23κ2eκ|z|1 +q32κα1 −q1 + 2α23κ2eκ|z|.

(24)κ shall be determined in terms of the energy density so that we are left with a free parameterα. Thus these are one parameter family of asymptotic solutions.

But as we mentionedbefore, not all of them are true domain wall solutions.Now let us check the consistency of the solution at the wall.Using the analyticproperty eq. (12) we prescribed, For small κ eq.

(15a, c) reduces toκr1 + 2α23κ2 δ(z) = 4πGλΦ2 −η22 ,(25)which leads to κ > 0. Later we shall find out that this small κ assumption is indeed relatedto the weak gravitational field approximation in the sense that the energy density is small.Similarly, eq.

(15b) leads toκr1 + 2α23κ2 δ(z) = 4πG(Φ′)2. (26)and eq.

(15d) reduces to αβ −2κr1 + 2α23κ2!δ(z) = 4πG(Φ′)2. (27)The consistency of eqs.

(26)(27) identifies the coefficients of the delta-function so thatwe can determine α in terms of β asα =3βκp1 −6β2 . (28)Thus the solution exists only for β2 < 1/6, i.e.

ω > 0.Note that κ and η have mass dimensions and the Newton’s gravitational constant Ghas inverse mass square dimension, while λ is a dimensionless coupling constant. Using adimensional analysis for a possible thick wall, if we have Gλη4 ≫κ η√λ and Φ(z = 0) = 0,6

the LHS of eqs. (25)(26) are effectively comparable to the RHS by smearing out the deltafunction Thus this is a good approximate solution and the condition in fact correspondsto the weak gravitational field limit.Finally, κ can be determined from the “energy” density as follows: Using eqs.

(25)-(27)we can compute the “energy-momentum” tensor eq. (6) asTµν =κ4πGp1 −6β2 δ(z)diag(1, −1, −1, 0),κ > 0.

(29)As we promised, the small κ implies the small energy density. The corresponding physicalenergy-momentum tensor can be computed from eq.

(7).Here we would like to call the readers attention to the fact that we have differentialequations with the Dirac delta-function.Some may find that this is unreasonable be-cause after all the Dirac delta-function is not a function but a distribution. But this is notcompletely unreasonable in field theory when we often need to be careful about the analyt-icity.

The main intention is not to solve the differential equations in question but to checkthe consistency between equations. In this sense this is a sufficiently good approxima-tion.

In fact one can be more careful about this situation and can introduce distributionalenergy-momentum tensor in terms of delta-function from the beginning[16]. The resulthowever is more or less equivalent because we also have derived the distributional “energy-momentum” tensor using our prescription eq.(12).

We can also further clarify the resultby introducing infinitesimal thickness of the wall and taking approximation around z = 0,although we cannot determine the shape of the solution within this thickness exactly.Now the physical metric can be obtained by multiplying the conformal factor (seeeq. (3)) as des2 = e2βφds2 so that we obtaindes2 = e2βφA(|z|)−dt2 + dz2 + eκt(dx2 + dy2),(30)wheree2βφ =√6β −1 −p1 −6β2eκ|z|√6β +1 −p1 −6β2eκ|z|√6β,A(|z|) = coshκ|z| −1p1 −6β2 sinhκ|z|!.The value of the dilaton eφ can be obtained in terms of β, which in turn related to the BDparameter ω in eq.

(3), aseφ = βφ + 12 ln 16πG. (31)7

In particular, at the wall we haveeφ(0) =q32β lns1 −√6β1 +√6β + 12 ln 16πG.Note that as ω →∞, i.e. β →0, we have new solutions in general relativityA(|z|) = e−κ|z|.

(32)For a given BD parameter ω > 0 we have found time-dependent thin domain wallsolutions in the JBD theory. For ω = 0, it is known that stable static thin domain wallsolution exists as proven by the author[8].

This in particular also proves that there is nothin domain wall solutions for ω < 0, i.e. in the 4-d analogue of the dilaton gravity.As in the GR case, the time-dependent domain wall we have obtained also has ahorizon at |z| = 1κ tanh−1 p1 −6β2.

As ω increases (i.e. β →0.

), the size of the horizonalso increases. As ω →0 (i.e.

β2 →1/6. ), the horizon shrinks and eventually disappears.Note that the signature of the metric changes behond the horizon.

Thus we should regardthe region within thee horizon also as where our choice of the coordinate system is valid.This however is not unusual.As is known eeeeveen in GR case, the metric of time-dependent thin walls become singular as |z| →∞.Although we have shown only one case explicitly, there are enormous amount of possi-ble solutions according to eeeq.(18). At this moment we do not understand why there areso many solutions.

We however expect that there should be some classification scheme, orsome arguments to exclude some of them. Also it will be interesting to search for solutionswhich behave well outside the horizon.Much work is needed to understand the structure of all other possible solutions andwhat kind of cosmological implications these dilatonic domain wall solutions have.AcknowledgementsThe author thanks S. Fulling for allowing him to access to the MathTensor.

This workwas supported in part by NSF grant PHY89-07887 and World Laboratory.8

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