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Topological Defects in Gravitational Theories

arXiv:gr-qc/9301024v1 1 Feb 1993Topological Defects in Gravitational Theorieswith Non Linear LagrangiansJ. Audretsch, A. Economou and C.O.

LoustoFakult¨at f¨ur Physik der Universit¨at Konstanz, Postfach 5560, D - 7750 Konstanz, GermanyThe gravitational field of monopoles, cosmic strings and domain walls is studied in the quadraticgravitational theory R + αR2 with α|R| ≪1, and is compared with the result in Einstein’s theory.The metric aquires modifications which correspond to a short range ‘Newtonian’ potential for gaugecosmic strings, gauge monopoles and domain walls and to a long range one for global monopoles andglobal cosmic strings. In this theory the corrections turn out to be attractive for all the defects.

Weexplain, however, that the sign of these corrections in general depends on the particular higher orderderivative theory and topological defect under consideration. The possible relevance of our resultsto the study of the evolution of topological defects in the early universe is pointed out.I.

INTRODUCTIONAfter the paradigm of the Hilbert’s Lagrangian formulation of Einstein’s theory of gravity it was clearhow one could consistently formulate other, higher derivative gravitational theories (that is theories inwhich the field equations have higher than second metric derivatives). And such theories where indeedproposed and used as alternatives to Einstein’s theory in attempts to unify other fields with gravity [1]and to remedy some of its seemingly undesirable consequences as, for example, at the classical level,the unavoidance of cosmological singularities [2] and, at the quantum level, the non renormalizabilityof the quantized version of general relativity [3].One of the main motivations for studying higher derivative theories comes from the semiclassicalgeneral relativity.

There, it seems to be a matter of self-consistency to consider higher derivative termsin the gravitational Lagrangian since such terms arise generically in one-loop calculations [4]. Certainlythis notion of self-consistency is a delicate issue and, as Simon has recently suggested [5], it needs tobe reconsidered if one wants to avoid undesirable semiclassical predictions such as unstable Minkowskispacetime.

Another recent motivation for considering higher derivative gravitational theories is thatsuch theories have arisen as low energy limit of several superstring theories [6].Higher derivative theories are of interest to cosmology mainly because, even vacuum theories admitcosmological models which give rise to the, so called, Starobinsky inflation [7] (see however Ref. [5] fora critisism on its consistency in the semiclassical limit), without fine tuning of the initial conditions [8].In this paper we want to look at another topic of cosmological relevance namely, the effects of higherderivative theories on the gravitational field of topological defects as monopoles, cosmic strings anddomain walls.

These are objects that may have formed during phase transitions in the cooling downof the Early-Universe and may have played a key role in the formation of the large scale structureof the Universe mainly through their gravitational interactions [9,10].Since their main interactionis gravitational, it is important to have an idea of what modifications one should expect in theirgravitational field when the relevant gravitational theory has higher derivatives. Some work has recentlybeen done in this direction, but only for gauge cosmic strings [11].

This work shows that in the weak fieldlimit only short range corrections to the Einstein theory arise which are associated with the presenceof additional massive fields in the spectrum of higher derivative theories. However, this is not expectedto be in general true, especially for global topological defects which are extended field configurationsand not localized as the gauge cosmic strings.In this work we have in mind theories that can be separated in a part LG for the gravitational field gµνconstructed with geometrical scalars of the Ricci tensor Rab, and another part LM containing matterfields with standard coupling to the gravitational field gµν1

L = 12κLG(Rab) + LM(gab). (1)Hereafter κ := 8πG where G is the gravitational constant.

For theories of this type it has been notedthat they can be recasted into an equivalent theory of Einstein gravity interacting with additionalmatter fields [12–14]. However, as it was stressed by Brans [15] and we shall explain in the next section,this equivalence is in general only at a mathematical level and not at a physical one.

Nevertheless, basedon such an equivalent system, Whitt [12] was able to show that the black holes of general relativity arethe only black hole solutions of R + R2 theories (no hair theorem).For the discussion of this paper we will deal with theories that have as gravitational part the following,often appearing in the literature, LagrangiansLG = √−g(R + αR2 + βRµνRµν)(2)andLG = √−gF(R),(3)where α, β are some coupling constants, g := detgµν, and R = gµνRµν. Finally the F in Eq.

(3) is inprinciple an arbitrary function of the curvature scalar R. However, later on we will take F to differ onlyslightly from the Einstein value R, that is F = R + αR2 with α|R| ≪1. See Ref.

[12] for a treatment ofthe F = R+αR2 theory in vacuum and the Ref. [16] together with references therein for generalizationsto arbitrary F(R) in the presence of particular forms of matter.The structure of the paper is as follows.Section II contains a brief review of higher derivarivetheories to the extent needed in this paper.

The field equations for the theories in (1),(2) and (1),(3),are written down, and their spectrum is explained. With the procedure that enables the recasting ofthese field equations into Einstein type ones we obtain the basic result that is used in the Sec.

IIIfor the comparison of the gravitational field of global monopoles, cosmic strings and domain walls inEnstein’s theory, and in the quadratic R + αR2 theory with α|R| ≪1. Section III also contains at thebeginning a brief introduction to the topological defects.

Finally in Sec. IV we conclude with a briefsummary and comments.Throughout this paper we use the conventions ¯h = c = 1, metric signature (−+++), Riemann tensorRabcd := −∂dΓabc + .

. ., and Ricci tensor Rab := Rcacb.II.

THEORIES WITH HIGHER DERIVATIVESA. Field equationsWe shall give now the gravitational field equations for the higher derivative theories given by the Eqs.

(1), (2) and Eqs. (1), (3).

The field equations for gµν are obtained by varying the action correspondingto Eq. (1) with respect to gµν and contain derivatives of the metric up to the fourth order.

For the caseof LG of Eq. (2) they read(1 + 2αR)(Rµν −12gµνR) + α2 R2gµν+ (2α + β2 )gµνR;p;p −(2α + β)R;µν+ βRµν;p;p −β2 RpqRpqgµν + 2βRpqRµpν q= −2κ√−gδSMδ(gµν) := κT (M)µν .

(4)Notice that the trace of this equation is an inhomogeneous massive Klein-Gordon equation for thecurvature scalar R(6α + 2β)R;p;p −R = κT (M). (5)Finally, the field equations for the theory (1) and (3) can be written as2

F ′Gµν = κT (M)µν+ 12gµν(F −F ′R −2F ′;p;p) + F ′;µν,(6)where F ′ = ∂F/∂R and Gµν = Rµν −12gµνR is the Einstein tensor. The trace of this equation is3F ′;pp + F ′R −2F = κT (M).(7)B.

Spectrum of quadratic theories -Weak gravitational limitWe would like to stress here the fact that quadratic theories do not contain only the usual massless(long-range) spin-2 graviton field but also, in general, two massive (short-range) fields with spin-0 andspin-2.This spectrum can be easily recognized in the case of LG of Eq. (2) when one writes the fieldequations in the linearized weak field limit using a convenient gauge (coordinate system).Indeedfollowing Teyssandier [17] we have that gµν can be decomposed in the weak gravitational limit (wheregµν = ηµν + hµν with |hµν| << 1) asgµν = ηµν + h(E)µν + χηµν + ψµν,(8)with the field equations✷h(E)µν = −2κ(Tµν −12T ηµν),(✷−m20)χ = −13κT,m−20:= 6α + 2β,(✷−m21)ψµν = 2κ(Tµν −13T ηµν),m−21:= −β,(9)and the gauge conditions∂a(h(E)µa −12h(E)λληµa) = 0,(ψab −ψλληab),ab = 0.

(10)Here indices are raised and lowered with the Minkowski metric tensor and the operator ✷is theMinkowskian one. One recognizes in Eqs.

(8)-(10) the usual Einstein contribution h(E)µν , that is thegraviton field which has 2 degrees of freedom. Then, a scalar field χ with mass m0, which obviously hasone degree of freedom and appears as an overall conformal factor (in the considered approximation).Finally the massive tensorial field ψµν with mass m1 which turns out to have five degrees of freedom(note that in contrast to h(E)µν its components satisfy only one gauge condition) and thus it posseses thestucture of a massive spin-2 field.

In order to keep the “mass” parameters m0, m1 real we shall demandthe no-tachyon constraint3α + β ≥0,β ≤0. (11)We leave for the next subsection the case of the Lagrangian in Eq.

(3) where we will go beyond theweak gravitational limit and we will see that the spectrum of this theory consists of a graviton and amassive interacting scalar field.C. Reformulation of quadratic theoriesInterestingly enough, besides gµν there is an alternative canditate for the metric field of the spacetime[13,14], namely the γµν which is the inverse of γµν where√−γγµν := ∂LG∂Rµν,(12)3

and γ := detγµν.In particular for the LG of Eq. (2) we have√−γγµν := √−gh(1 + 2αR)gµν + 2βRαβgαµgβνi.

(13)Expressing the theory in terms of γµν via a Legendre transformation, one can reduce the order of thederivatives that appear in the field equations from fourth to second. But what is also important is,that the resulting theory takes the form of Einstein gravity for the metric γµν plus some additionalmassive fields interpreted as matter fields.

Thus the equation (13) can be considered as a non lineardecomposition of gµν in the physical spectum of the full theory: a spin-2 massless field γµν, a scalarfield that appears as a conformal factor and is a linear function of R, and finally, a tensor field relatedto Rab with 5 degrees of freedom (Rab is symmetric satisfying the 4 contracted Bianchi equations andits trace is essentially the previously mentioned scalar field).We can be more explicit in the interesting case of the theory in Eq. (1) where LG is given by Eq.

(3).Here, only an additional scalar field appears since γµν and gµν are conformally related. Indeed Eq.

(12)gives√−γγµν := √−gF ′gµν,(14)which implies thatγµν = F ′−1gµν. (15)Defining a scalar field ψ (not to be confused with the tensorial field ψµν of Eqs.

(8), (9) viaF ′ = exp r2κ3 ψ!,(16)the field equations (6) are written in the system γµν asbGµν = κF ′ T (M)µν (gab) + κT (ψ)µν(17)where, hereafter, hats denote quantities with respect to the metric γµν andT (ψ)µν = b∇µψ b∇νψ −12γµν b∇λψ b∇λψ −12γµνU(ψ). (18)The potential U(ψ) is given byU(ψ) = 12κF ′−2(RF ′ −F),(19)which can be written as a function of ψ alone in regions where Eq.

(16) is invertible. Finally the scalarfield ψ satisfies the equationb✷ψ =κ61/2F ′−22F −RF ′κ+ gµνT (M)µν (gab),(20)which can be checked to be equivalent to the trace (7) of the initial field equations (6).These field equations (17) and (20) follow from the LagrangianL′ =12κ√−γ bR(γab)+ √−γ[−12γµνψ,µψ,ν −U(ψ)] + LM(gab),(21)which shows that the quadratic theory (1) is, loosely speaking, equivalent to “Einstein’s” gravitationaltheory for the metric γab plus an interacting massive scalar field ψ plus the “peculiar” (not anymoreusual) matter fields of LM(gab).

They are indeed peculiar if γab is considered as the metric of spacetimesince then the dependence of LM on gab implies, via Eq. (15), a non standard interaction of the metric4

field γab and of the field ψ with the matter fields of LM. More on this issue of equivalence will be saidat the end of this section.Case: F(R) ≈R + αR2 with α|R| ≪1.We will now consider the interesting case where F(R) can be expanded as a Taylor series around R = 0and deviates only slightly from Einstein’s theoryF(r) = R + αR2 + O(R3),α|R| ≪1,α := F ′′2R=0.

(22)Assuming that the O(R3) terms can be ignored in this expression then the field equations (17) and (20)simplify considerably. Indeed, in this approximation Eqs.

(16), (22) implyψ ≈ 6κ1/2αR,(23)while the leading term in the potential U(ψ) of Eq. (19) is, assuming the no-tachyon constraint α > 0,a mass termU(ψ) = 12m20ψ2 + O(ψ3),m20 = 16α.

(24)The metric γµν and the field ψ are obtained from the field equations (17) and (20) which, to lowestorder in the approximation (22), readbGµν ≈κT (M)µν (γab),(b✷−m20)ψ ≈κ61/2T (M)(γab). (25)Making use of Eq.

(15) one can finally obtain the metric gµν. Notice that, to the considered approxi-mation, it does not matter which metric we actually use in T (M)µν .

It is more convenient, however, fromthe technical point of view to use γµν.Concluding we arrive at the following result:For a given matter source T (M)µν (gab), the metric gµν in the quadratic theory (1) and (3) with F(R) ≈R + αR2 and α|R| ≪1 is given bygµν = [1 + χ]γµν,(26)where γµν is the metric in the Einstein’s theory with source T (M)µν (γab) while the field χ satisfies theequation(✷−m20)χ = −κ3 T (M)(γab),m20 := 16α(27)with the ✷operator taken with respect to the γab metric.This result follows directly from Eqs. (25), (15) using the variable χ related to the field ψ via χ :=−2[κ/6]1/2ψ.

Note that from (23) follows αR = −χ/2 and therefore the condition α|R| ≪1 for ourapproximation is equivalent to |χ/2| ≪1. Finally, let us notice that in the weak gravitational limit theEqs.

(26) and (27) are consistent with the β = 0 limit of the linearized equations (8) and (9).D. Some remarksBased on the decomposition (12) several authors [13,18,14,19,16] have dealt with the question ofwhether quadratic theories are equivalent to Einstein’s theory plus some additional fields.

It seemsthat this may well be true for vacuum theories. However, as was pointed out by Brans [15], (see alsoRefs.

[20,21]), a subtlety appears in the case where usual matter is present. The problem is that theequivalence principle, a basic guide that one may use in constructing theories coupled to gravity andin particular to Einstein’s theory, cannot be valid in both the original and the reformulated theories.

Ifit is valid in the original theory, then a test matter in LM of the Lagrangian (1) will follow geodesics5

of the spacetime with metric gµν but, in general, will fail to do the same in the spacetime with themetric γµν. In this sense we are not entitled to consider the reformulated theory as Einstein’s theory inthe presence of some interacting fields.

In the case that one is philosophically inclined to consider theγµν as the physical metric, while the gµν as some sort of unifying field, then the equivalence principleshould be implemented in the matter part LM of Eq. (1) using the metric γµν in the place of gµν.

Ofcourse then, this LM will be non standard with respect to gµν.Whether or not nature chooses to couple usual matter universally only to a spin-2 field (as the γµν)and not to a more composite one (as the gµν) is far from being experimentally testable. Trying to findan answer one may, however, employ some criteria of principle, as positivity of energy [20].

In any case,the use of new variables, as those of Eq. (13) and Eq.

(15), which turn out to simplify technically aphysical problem, is undoubtfully very useful even if it is not clear whether one can attribute to thesevariables a foundamental character.III. TOPOLOGICAL DEFECTSIN HIGHER DERIVATIVE THEORIESCosmological defects are formed during phase transitions in the evolving Early Universe wheneverthe symmetry group G of the relevant field theory breaks down to a subgroup H so that the vacuummanifold M = G/H has some non trivial homotopy group [22].

Such a symmetry breakdown at anenergy scale η can be realized, e.g., with an n-component scalar field φ(i) having a Mexican-hat type ofpotentialV (φ) = −λ4 (nXi=1φ(i)φ(i) −η2)2. (28)The homotopic structure of the vacuum manifold depends on the number n of components of the scalarfield and, thus, we may have the formation of domain walls for n = 1, cosmic strings for n = 2, monopolesfor n = 3.

These defects are respectively surface-, line-, and point-like configurations. Sufficiently awayfrom these configurations, at distances d ≫δ, the scalar field φ(i) approaches quickly its vacuum valuePi φ(i)φ(i) ≈η2.

Here δ is the width of the core of these defects, of the order of m−1φwhere mφ = η√λis the mass of the scalar field’s massive mode. Typically, for symmetry breaking at grand unificationscale, δ ≈10−30cm and κη2 ≈10−6.Depending on whether the symmetry that breaks down is a gauge (local) or a global one we haverespectively the formation of gauge or global topological defects.

In the case of gauge symmetry thereexists a well defined core, with width δ, where most of the energy of the topological defect configurationis localized. On the other hand, for global topological defects the components of the respective stress-energy-momentum tensor have, outside the “core”, a relatively slow fall offdue to the gradients of theGoldstone modes of the scalar field φ(i).

Thus, global defects are extended configurations. The reasonfor this difference between gauge and global defects is that in the case of gauge symmetry the presenceof gauge fields can compensate the gradients of the scalar field.

Finally, in the case of discrete symmetrybreaking, which gives rise to domain walls, there are no Goldstone modes and thus domain walls arelocalized configurations.Based on the above properties, we will make in what follows the following approximations:(i) Gauge topological defects and domain walls will be considered in the zero core-thickness approx-imation and thus their stress-energy-momentum tensors will have components with appropriateDirac δ-fuctions. (ii) For global defects, we will make the σ-model approximation where the scalar field is fixed toits asymptotic vacuum value everywhere outside the defect.

This is a sensible approximation atdistances from the defect sufficiently larger than the “core” width δ.In the following subsections we will obtain the gravitational field of cosmic strings, monopoles anddomain walls in the quadratic theory R + αR2 with α|R| ≪1. For this we will make use of the resultof the previous section (see Eqs.

(26), (27)), stating that the metric in the quadratic theory, ds2(Q), isconformally related with the metric in Einstein’s theory, ds2(E),ds2(Q) = (1 + χ)ds2(E),|χ/2| ≪1,(29)6

with χ satisfying the massive Klein-Gordon equation (27) in the ds2(E) metric. A consequence of Eq.

(29) is that there will be a modification of the “Newtonian” potential equal to χ/2. We will have belowthe oportunity to study its nature and its range in the case of topological defects, be them localized orextended sources.In general we shall restrict our attention to sufficiently large distances, d, away from the core, (d ≫δ),but we will keep in mind that a proper treatment at short distances requires a proper model for thecore of the defect itself.

In this way we will be able to use the existing results in General Relativity forthe gravitational field of cosmic strings, monopoles and domain walls which were obtained by makinguse of the above approximations in model Lagrangians with symmetry breaking potential of the form(28).A. Global monopolesThe stress-energy-momentum tensor of a global monopole configuration, in regions far away from thecore, can be approximated by [23,24]T tt = T rr ≈−η2r2 ,T θθ = T ϕϕ = 0,(30)while the respective metric in Einstein’s theory of gravity is (approximately) given by [23,24]ds2(E) = −(1 −∆)dt2 + (1 −∆)−1dr2 + dΩ2dΩ2 := r2(dθ2 + sin2 θdϕ2),∆:= 8πGη2 = κη2.

(31)This metric corresponds to a spacetime with a solid deficit angle: test particles are deflected by an angleπ∆/2 irrespective of their velocity and their impact parameter. Here it should be added that a morecareful treatment [24] that takes into account the actual behaviour of the field at the monopole core,shows that the metric (31) gets modified by terms which at distances r ≫δ = (√λη)−1 correspondeffectively to a negative mass term Meff, that is e.g.

gtt ≈(1 −∆−2GMeff/r). According to numericalanalysis [24] Meff≈−6π√λη.

Thus, besides the topological deflection caused by the solid deficit angle,test particles experience also a repulsive radial force −GMeff/r2 away from the monopole.The metric in the quadratic theory is given by Eq. (29) with χ satisfying Eq.

(27). Looking forspherically symmetric solutions we find that this equation for χ = χ(r) reads 1r2ddrr2 ddr−bm2χ(r) =2∆3(1 −∆)r2 ,bm2 := m20/(1 −∆).

(32)Making use of the Green function for this equation,G(r, r′) = −1bmrr′ [e−bmr′sinh( bmr)Θ(r′ −r)+e−bmr sinh( bmr′)Θ(r −r′)],(33)where the step function Θ(z) := {0, 1, 1/2} for {z < 0, z > 0, z = 0} respectively, we can write down thesolution for χ(r) in terms of the Exponential-Integral (Ei) and Hyperbolic-Sine-Integral (shi) functions[25] asχ(r) =2∆3(1 −∆)1bmr [Ei(−bmr) sinh( bmr) −e−bmrshi( bmr)]. (34)Checking numerically the behavior of this function we find that its contribution to the ‘Newtonian’potential χ/2 is an attractive one.In particular, using the asymptotic behavior of the Ei and shifunctions [25] we find that at large radial distances r →∞χ(r) ≈−23∆(m0r)2 ,(35)7

which implies a long range potential, exerting on test particles an attractive force −(2∆/3m20)r−3.Comparing this force to the repulsive force due to the core of the monopole we see that the former fallsofffaster by one power of r and thus is negligible at very large distances. It overcomes, however, theeffect of the latter at a distance r ≈m−20 /(λδ) and, thus, it can be the dominant force within the regionδ ≪r ≪m−20 /(λδ) which will exist provided that m−10≫δ.Finally, let us note that the expression (34) diverges as r →0.This is due to the form of theenergy-momentum-tensor in Eq.

(30) which is not valid at distances comparable to the core of themonopole.B. Gauge MonopolesA gauge monopole is a spherically symmetric configuration with mass M and a magnetic charge g.Its stress energy momentum tensor can be approximated byT tt = −M4πδ(r)r2−(g/4π)2r4,T rr = −T θθ = −T ϕϕ = −(g/4π)2r4.

(36)We consider the case where the metric outside the core of the monopole matches to a Reissner-Nordstromone, (see Ref. [26] for a recent review and new results on the gravitational field of monopoles)ds2 = −1 −2GMr+ Gg24πr2dt2+1 −2GMr+ Gg24πr2−1dr2 + dΩ2.

(37)Since the source for the χ field is the trace of the stress-energy-momentum tensor, only the massterm in Eq. (36) will contribute.

Furhermore, if we consider distances sufficiently far from the monopoler ≫δ ≫GM, the equation for χ approximately reads 1r2ddrr2 ddr−m20χ(r) = κM12πδ(r)r2 . (38)Demanding finiteness at radial infinity, this equation has as solution the Yukawa fuctionχ(r) = −κM12πe−m0rr.

(39)Notice that the Newtonian potential of the monopole will be modified by the ammount χ/2 correspond-ing to an attractive potential exponentially decreasing with an e-folding term characteristic of a massivescalar field with mass m0.It worths remarking that the short range corrections of Eq. (39) apply also to the external metricof spherically symmetric mass distributions [17] such as neutron strars, giving thus rise to “fifth force”terms.

However, when one deals with black holes, the no-hair theorem for R + R2 theories [12] impliesthat corrections of the type (39) are absent.C. Global Cosmic StringsAs we explained in the introductory part of this section global cosmic strings are extended lineconfigurations.

The stress-energy-momentum tensor for a straight, static, cylindrically symmetric globalstring lying along the z-axis is approximately given for r ≫δ byT tt = T zz = T rr = −T θθ ≈−η22r2 . (40)The respective exact solution for the metric in Einstein’s theory has been found in [27].

However, itis quite complicated for the purpose of solving the equation (27) for the field χ. Furthermore, besides8

this technical problem, the spacetime of a global string has true spacetime singularities [28,29], a factthat demands carefull checking of the range of validity of the approximation (α|R| ≪1) on which ourtreatment is based. Instead, we prefer to work here in the weak field limit where the equation for theχ field is in Minkowski background metric.In the weak field limit of general relativity the metric of the global string reads [28]ds2(E) =1 −4Gµ ln(rδ )(−dt2 + dz2) + dr2+r21 −8Gµ ln(rδ + c)dθ2,(41)Here µ := πη2, δ is the core width and c is a constant of order unity that may partially take into accounta global effect of the string core.

Studying the motion of test particles it is seen that the static globalstring exerts a repulsive force 2Gµ/r [28]. It is interesting to explore how this force is modified in thequadratic theory that we are currently considering.The equation that χ statisfies in the weak field limit is1rddrr ddr−m20χ(r) =κµ3πr2 .

(42)The Green function for the homogeneous part of this differential equation, with the boundary conditionsof finitness at the origin and at infinity, is easily found to beG(r, r′) = −K0(m0r)I0(m0r′)Θ(r −r′)−I0(m0r)K0(m0r′)Θ(r′ −r)(43)where Θ is the step function. Thus the solution of Eq.

(42) can be written asχ(r) = −κµ3πhK0(m0r)Z rδI0(m0r′)dr′r′+ I0(m0r)Z ∞rK0(m0r′)dr′r′i,(44)where we have introduced a lower cutoffat r = δ to cope effectively with the divergence that appearsin the first integral if we let r →0.This divergence is only due to the approximate form of thestress-energy-momentum tensor which as we have already stressed is not valid near the core of thestring.The leading term in an asymptotic expansion of Eq. (44) at large radial distances isχ(r) ≈−κµ3πm20r2 ,r →∞.

(45)From this expression we conclude that the additional ‘Newtonian’ potential in the quadratic theoryimplies at large distances an attractive force −(κµ/3m20)r−3. Due to the slower fall offof the originalrepulsive force, the total force on test particles remains repulsive at large distances from the string.

Ataround r ∼m−10the total force is expected to change sign.D. Gauge Cosmic StringsGauge cosmic strings are, in contrast to global ones, localized line configurations.

The stress-energy-momentum tensor for a static, straight along the z-axis, gauge cosmic string with line energy density µisT tt = T zz = −µ2π(1 −κµ/2)δ(r)r ,T rr = T θθ = 0(46)with corresponding metric in Einstein’s theory [9]ds2(E) = −dt2 + dz2 + dr2 + (1 −κµ/2)2r2dθ2. (47)9

Here the polar coordinates r, θ have the usual range. This spacetime is everywhere flat except alongthe z-axis where the string is located.

As one goes around the string one notices an angle deficit. Thistopological property has the consequence that test particles which localy do not feel any gravitationalforces are, however, deflected by the string.Let us now turn our attention to the field χ.

It satisfies the equation1rddrr ddr−m20χ(r) =κµ3π(1 −κµ/2)δ(r)r . (48)which can be easily solved by demanding for the field χ finiteness at infinity and correct discontinuityat the origin.

The solution readsχ(r) = −κµ3π(1 −κµ/2)K0(m0r),(49)which can be easily checked that satisfies Eq. (48) using the small argument asymptotic behavior of themodified Bessel function K0(z) ≈−ln(z/2).

Finally notice that the field χ decays exponentially fastsince at large distances K0(m0r) ≈(π/2m0r)1/2 exp(−m0r). Very close to the string the expression(49) diverges logarithmically.

Again, as the physical cosmic string has a finite core this divergenceshould not appear in a more proper treatment near the core.From Eq. (49) follows that the ‘Newtonian’ potential of the cosmic string spacetime in an at-tractive, short range one.The respective force that the string will exert on test particles is−[κµm0/6π(1 −κµ/2)]K1(m0r).Because of the large distance exponential fall offbehavior ofK1(m0r) ∝(m0r)−1/2 exp(−m0r) it follows that this force is significant only close to the string upto distances r ∼m−10 .In closing this subsection let us remark that the result obtained here is in agreement with the recentresult of Linet and Teyssandier [11] in the weak field limit where κµ ≪1.

These authors have alsoobtained the cosmic string metric in the weak field limit of the quadratic theory (1) and (2) whichcontains also the massive tensorial field ψµν of Eqs. (8)-(10).

In particular they find that the effect ofthis field on the Newtonian potential is a repulsive one with a range set by the inverse mass m1 of thefield ψµν.E. Domain WallsThe energy content and the gravitational field of domain wall configurations in Einstein’s theory hasbeen studied extensively in the literature, see Ref.

[30] and references therein. We will consider hereVilenkin’s vacuum plain domain wall solution discussed in [31].

For a domain wall with surface energydensity σ, lying on the |z| = 0 plane, the stress-energy-momentum tensor isT tt = T xx = T yy = −σδ(z),T zz = 0,(50)while the respective domain wall spacetime is described by the metric [31]ds2(E) = (1 −ν|z|)2−dt2 + e2νt(dx2 + dy2)+dz2,ν := 2πGσ = κσ/4. (51)Note that some of the metric components are time dependent (no static solutions can be found).

Testparticles in this spacetime are repelled with a proper acceleration ν away from the wall [31], a propertythat we may deduce just by looking at the ‘Newtonian’ potential term in the gtt component of themetric (51). Finally we should mention that at |z| = ν−1 an event horizon appears [31].

In what followswe restrict our attention to spacetime regions with |z| ≤ν−1.Although the metric in (51) is time dependent we can find, however, static solutions to the equation(27) for the field χ depending only on |z|. For such solutions, equation (27) reduces to the ordinarydifferential equation(1 −ν|z|)−3 ddz(1 −ν|z|)3 ddz−m20χ(z) = 4νδ(z).

(52)10

The homogeneous part of this equation can be easily transformed into a Bessel differential equation forbχ where χ(|z|) := bχ(bz)/bz using the new variable bz = ν−1 −|z|. In this way we find that the solutionfor χ(z) is, in regions with z ̸= 0, a linear combination of the terms I1(m0bz)/bz and K1(m0bz)/bz whereK1, I1 denote modified Bessel functions.

The coefficients of this solution are determined by demandingfiniteness at the horizon bz = 0, while on the domain wall, z = 0, the field χ should be continous andhave the appropriate discontinuity in its first derivative which, according to Eq. (52), is [ ddzχ]z=0 = 4ν.Thus we finally obtainχ(z) = −2hm0ν I2(m0ν )i−1 I1( m0ν [1 −ν|z|])1 −ν|z|.

(53)It is easy to check that this implies an attractive and short range contribution to the Newtonianpotential. This cannot overwhelm the original repulsive potential of the domain wall except very closeto the wall for |z| <∼m−10 .

This should be obvious in writing Eq. (53) in the sensible approximationm0/ν ≫1 and near the domain wall where χ takes its largest valuesχ(z) ≈−2νm0exp(−m0|z|).

(54)Away from the wall the field |χ| decreases exponentially and at the horizon χ attains the small value−1/I2( m0ν ).IV. CONCLUSIONS AND REMARKSWe have dealt in this paper mainly with the higher derivative theory (1), (3) with F = R+αR2 in theapproximation α|R| ≪1.

We showed that one can simplify the problem of solving the correspondingfourth order field equations using the conformal transformation (15) which leaves us with the systemof field equations (17), (20) having only second order derivatives. This is formally a system of Einsteintype equations plus the field equations for a massive scalar field with mass m20 = 1/(6α) interacting nonminimally with gravity.

We then found that in the approximation α|R| ≪1 the gravitational fields inthe R + αR2 theory and in the Einstein theory are conformally related according to the Eqs. (26), (27).Using this result we looked in Sec.

III for solutions representing the gravitational field of monopoles,cosmic strings and domain wallsFor localized topological defect configurations as gauge strings, gauge monopoles and domain wallswe have found short range, attractive corrections. Their range ∼1/m0 is characteristic of the pressenceof the massive field.For extended sources as global monopoles and global strings we have again found attractive correctionsbut with a long range.

Their fall offrate depends on the corresponding stress-energy-momentum of thesedefect configurations. In particular, for distances r ≫m−10is found that the attractive correction to theNewtonian potential is ≈κT/(6m20) where T is the trace of the corresponding stress-energy-momentumtensor.A more detailed investigation of the gravitational effects of topological defects in more general higherorder derivative theories is in progress and we hope to present the results elsewhere.

We can howeveralready here reestablish the above given conclusions in the R+αR2 theory and at the same time extendthem to the case of the theory (1),(2) which, for β ̸= 0, contains also a massive tensorial field. Thiswill be done by making use of the linearized field equations (8), (9).

First observe that the Newtonianpotential ΦN will have, besides the Einstein term Φ(E)N , also the contributions Φ(χ)Nand Φ(ψµν)Nfrom thescalar field χ and massive field ψµν respectivelyΦN = Φ(E)N+ Φ(χ)N + Φ(ψµν)N= 12[−h(E)00 + χ −ψ00]. (55)For static spacetimes we have from Eqs.

(8) and (9) that∇2Φ(E)N= κ2 (ρ + P1 + P2 + P3),(∇2 −m20)Φ(χ)N= κ6 (ρ −P1 −P2 −P3),11

(∇2 −m21)Φ(ψµν)N= −κ3 (2ρ + P1 + P2 + P3)m−20:= 6α + 2β,m−21:= −β,(56)where ρ denotes the mass density and P1, P2, P3 the principal pressures of the matter.Sign of forces: Look at the r.h.s of these equations. For the topological defect configurations discussedin the present paper ρ+Pi Pi is > 0 for gauge monopoles, 0 for gauge strings and global monopoles, and< 0 for global strings and domain walls.

On the other hand ρ−Pi Pi > 0, while −(2ρ+Pi Pi) ≤0 withequality holding for domain walls. Thus the Einstein contribution in Eq.

(55) is attractive for gaugemonopoles, zero for gauge strings and global monopoles, and repulsive for global strings and domainwalls. The χ-contribution is always attractive, while the ψµν-contribution is in general repulsive exceptfor domain walls where it is zero.Range of forces: From Eqs.

(56) it is clear that the Einstein term provides in general a long rangeinteraction due to an effective mass density ρ + Pi Pi. It gives, however, a zero effect for gauge cosmicstrings and global monopoles.

For the contributions of χ and ψµν terms we have that:(i) If the stress energy momentum tensor vanishes (or falls offsufficiently rapidly) outside a localizedsource then at distances d from the source, these contributions are of short range ∝exp(−md)/dpwhere m stands for the mass m0 (or m1) and p is a parameter depending on the symmetry of thespacetime and is equal to 0, 12 and 1 for plain domain walls, strings and monopoles respectively. (ii) If the sources are not localized then for the χ, ψµν contributions there are two characteristicregimes:(a) At distances d ≫max(m−10 , m−11 ) the mass terms dominate over the derivative terms in the twolast equations of (56).

Thus, asymptotically at large distances we have long range contributionsΦ(χ)N≈−κ(ρ −Pi Pi)/(6m20) = κT/(6m20), as we found in this paper, and Φ(ψµν)N≈κ(2ρ +Pi Pi)/(3m21). (b) At distances d ≪min(m−10 , m−11 ) the derivative terms dominate over the mass terms.

Theinteresting thing to note here is that at such distances the total ‘Newtonian’ potential in Eq. (55)satisfies ∇2ΦN ≈0.

This has implications for the differentiability of the spacetime metric andimplies drastic changes in the singularity structure of gravity at short distances. For example,the gravitational potential of a point massive particle is finite at the origin in contrast to the 1/rCoulomb behavior in the Newtonian theory.These considerations are in agreement with the results of the previous sections and the results of[11] for gauge cosmic strings.They may be particularly relevant to the study of the evolution oftopological defects in the very early universe: (a) for structure formation scenarios based on globaldefects where the long range modifications of the quadratic theories may play an important role; (b)for collisions of cosmic strings where the drastic short range modifications may change significantly thepredictions of these simulations for the evolution parameters of a string network.

Thus it is interestingto study further topological defects and collisions of cosmic strings in quadratic gravitational theoriesand implement appropriate modifications in future numerical simulations. The outcome of such aninvestigation confronted with observation, may, among other things, allow one to put constraints onthe m0, m1 parameters of quadratic gravitational theories.ACKNOWLEDGMENTSThis work was supported by the European Community DG XII.

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