semiclassical geons at particle accelerators -...

arX

iv1

306

6537

v2 [

hep-

th]

31

Jan

2014

Prepared for submission to JCAP

Semiclassical geons at particle

accelerators

Gonzalo J Olmoa

D Rubiera-Garciab

aDepartamento de Fiacutesica Teoacuterica and IFIC Universidad de Valencia-CSIC Facultad deFiacutesica C Dr Moliner 50 Burjassot-46100 Valencia SpainbDepartamento de Fiacutesica Universidade Federal da Paraiacuteba 58051-900 Joatildeo Pessoa ParaiacutebaBrazil

E-mail gonzaloolmocsices drubierafisicaufpbbr

Abstract We point out that in certain four-dimensional extensions of general relativity con-structed within the Palatini formalism stable self-gravitating objects with a discrete mass andcharge spectrum may exist The incorporation of nonlinearities in the electromagnetic fieldmay effectively reduce their mass spectrum by many orders of magnitude As a consequencethese objects could be within (or near) the reach of current particle accelerators We providean exactly solvable model to support this idea

Keywords Modified gravity Palatini approach nonsingular spacetimes semiclassical geonsBorn-Infeld wormholes

ArXiv ePrint 13066537 [hep-th]

Contents

1 Introduction 1

2 Basics of Palatini gravity 4

3 The matter sector 6

4 Palatini gravity with matter 7

5 Solving the equations 8

51 The metric ansatz 852 General solution 9

6 The Born-Infeld model 10

7 Quadratic gravity coupled to Born-Infeld 11

8 Exactly solvable model 12

81 Metric components 13

82 Curvature invariants 1483 Wormhole structure and electric flux 1584 Horizons and mass spectrum when δ1 = δc 1785 Geons as point-like particles 19

9 Summary and conclusions 20

1 Introduction

Black holes are a genuine prediction of general relativity (GR) that set the classical limits ofvalidity of Einsteinrsquos theory itself in the high-curvature regime defined by their singularityThey also pose severe challenges to a consistent description of physical phenomena in whichgravitation and quantum physics must be combined to give a reliable picture Black holes arethus regarded as a means to explore new physics where gravitation approaches the quantumregime and to understand the fate of quantum information in the presence of event horizonsand singularities The existence of extra dimensions required by the stringM theory ap-proach would also affect the dynamics of black hole formationevaporation and the physics ofelementary particles at very high energies For these reasons the physics of black holes andin particular of microscopic black holes is nowadays a very active field of research from boththeoretical and experimental perspectives In fact it has been suggested that microscopicblack holes could be created in particle accelerators [1ndash3] and numerous studies have beencarried out to determine the feasibility of their production [4ndash8] and to characterize theirobservational signatures [9 10] In this sense the general view is that extra dimensions arerequired to reduce the effective Planck mass to the TeV energy scale [11ndash14] (see also the re-view [15]) which would make them accessible through current or future particle acceleratorsOnce produced the semiclassical theory predicts that angular momentum and charge shouldbe radiated away very rapidly yielding a Schwarzschild black hole as the late-time phase of

ndash 1 ndash

the process Since black holes are quantum mechanically unstable under Hawking decay theSchwarzschild phase could be observed through the detection of a burst of Hawking quanta

The above picture is expected to be robust for sufficiently massive black holes for whichthe backreaction on the geometry is weak and can be neglected In this approach onequantizes the matter fields on top of the classical geometry of GR and neglects the change inthe geometry produced by the quantum evaporation process It should be noted in this sensethat the renormalizability of the matter fields in a curved background requires a completionof the gravity Lagrangian that involves quadratic corrections in the curvature tensor [16 17]Moreover these corrections also arise in several approaches to quantum gravity such as thosebased on string theory [18 19] On consistency grounds the semiclassical description shouldthus take into account these corrections even in the case in which the matter backreaction isneglected However due to the fact that quadratic curvature corrections lead to higher-orderderivative equations which makes it difficult to find exact solutions and generates instabilitiesand causality violations the standard GR description is generally preferred

In this sense we have recently investigated [20ndash22] how the structure of black holes withelectric charge gets modified when a quadratic extension of GR is formulated agrave la Palatini iewithout imposing any a priori relation between the metric and affine structures of space-time[23] This approach avoids important shortcomings of the usual formulation of the quadratictheory in which the connection is defined a priori as the Christoffel symbols of the metricRelaxing the Levi-Civita condition on the connection one generically finds the existence ofinvariant volumes associated to the connection which define a metric structure algebraicallyrelated with that defined by the metric gmicroν [24] As a result the Palatini version yields second-order equations for gmicroν thus avoiding ghosts and other instabilities In fact in vacuum thefield equations exactly boil down to those of GR which guarantees that Minkowski is a stablevacuum solution and that there are no new propagating degrees of freedom beyond the typicalspin-2 massless gravitons The dynamics differs from that of GR in nonlinearities induced bythe matter on the right-hand side of the equations These nonlinear terms arise due to thenontrivial role played by the matter in the determination of the connection [24]

In our analysis [25] we found that spherically symmetric electro-vacuum solutions canbe naturally interpreted as geons ie as consistent solutions of the gravitational-electromagneticsystem of equations without sources This is possible thanks to the nontrivial topology ofthe resulting space-time which through the formation of a wormhole allows to define electriccharges without requiring the explicit existence of sources of the electric field In this sce-nario massive black holes are almost identical in their macroscopic properties to those foundin GR However new relevant structures arise in the lowest band of the mass and chargespectrum (microscopic regime) In particular below a certain critical charge qc = eNc withNc =

radic

2αem asymp 1655 where αem is the fine structure constant and e the electron chargeone finds a set of solutions with no event horizon and with smooth curvature invariants ev-erywhere Moreover the mass of these solutions can be exactly identified with the energystored in the electric field and their action (evaluated on the solutions) coincides with thatof a massive point-like particle at rest The topological character of their charge thereforemakes these solutions stable against arbitrary perturbations of the metric as long as thetopology does not change On the other hand the absence of an event horizon makes theseconfigurations stable against Hawking decay (regular solutions with an event horizon alsoexist though they are unstable) These properties together with the fact that these solutionslie in the lowest band of the charge and mass spectrum of the theory suggest that they canbe naturally identified as black hole remnants The existence of such solutions in a minimal

ndash 2 ndash

extension of GR demands further research to better understand their stability and chancesof being experimentally accessible This is the main motivation for this work

In this paper we study the effects that modifications on the matter sector of the theorystudied in [25] could have for the qualitative and quantitative stability of its solutions In[25] we considered the quadratic Palatini theory

Sg =c3

16πG

int

d4xradicminusg[

R+ l2P (aR2 + bRmicroνR

microν)]

(11)

where l2P = ~Gc3 coupled to a spherically symmetric sourceless electric field The quadraticcurvature terms are regarded as quantum gravitational corrections1 that vanish when ~ rarr 0In that limit the theory recovers the usual Einstein-Maxwell equations which have theReissner-Nordstroumlm black hole as a solution For finite ~ new solutions arise with the prop-erties summarized above Though the Reissner-Nordstroumlm solution is generally accepted asa valid solution of the classical Einstein-Maxwell system the fact is that in the innermostregions of such black holes the amplitude of the electric field grows without bound abovethe threshold of quantum pair production [29] which should have an impact on the effectivedescription of the electric field These effects were neglected in [25] Here we want to explorethe consequences for the existence and properties of the Palatini-Maxwell geons of [25] whennonlinearities in the description of the electromagnetic field are incorporated

It is well known that under certain conditions [30] the effects of the quantum vacuumcan be taken into account within the effective Lagrangians approach In this approach oncethe heavy degrees of freedom are integrated out in the path integral of the original actionof quantum electrodynamics a perturbative expansion leads to series of effective (classical)Lagrangians correcting the Maxwell one which take the form of powers in the field invariantsof the electromagnetic field [31] This is the case of the Euler-Heisenberg Lagrangian [32 33]These effective Lagrangians account at a purely classical level not only for the dynamics ofthe low-energy fields but also for the quantum interactions with the removed heavy-modesector Historically the nonlinear modifications on the dynamics of the electromagnetic fielddate back to the introduction of the Born-Infeld Lagrangian [34] aimed at the removal of thedivergence of the electronrsquos self-energy in classical electrodynamics It has been shown thatthis Lagrangian also arises in the low-energy regime of string and D-Brane physics [35ndash39] andit is often considered in the context of black hole physics in GR [40ndash42] and in modified theoriesof gravity [43ndash45] For concreteness and analytical simplicity in this work we will considerthe gravity theory (11) coupled to the Born-Infeld nonlinear theory of electrodynamics as away to test the robustness of our previous results under quantum-motivated modifications ofthe matter sector

The Born-Infeld theory recovers the linear Maxwell theory when a certain parameter β2

is taken to infinity (see section 6 below for details) For any finite value of β2 the nonlineari-ties of the matter sector enter in the construction of the geometry As a result here we willsee that the qualitative features of the solutions found in [25] persist for arbitrary values ofβ2 However from a quantitative point of view we find that the mass of the remnants can bemany orders of magnitude smaller than that found in the Maxwell case while the maximumcharge allowed to have a remnant can be much larger These results therefore put forwardthat the Maxwell case is the least favorable scenario from an experimental perspective and

1Other types of quantum corrections in the gravitational sector in scenarios inspired by loop quantumgravity have been considered in [26ndash28]

ndash 3 ndash

that quantum corrections in the matter sector can substantially improve the chances of exper-imentally detecting the kind of black hole remnants found in [25] In fact for specific valuesof the parameter β2 we find that the mass spectrum of the stable solutions can be loweredfrom the Planck scale sim 1019 GeV down to the TeV scale This means that ongoing exper-iments in particle accelerators such as the LHC could be used to explore new gravitationalphenomena directly related with the Planck scale within a purely four-dimensional scenario

The paper is organized as follows in Sec 2 we define our theory and provide the generalmetric and connection field equations We introduce the matter sector of our theory underthe form of a nonlinear electromagnetic field in Sec 3 and specify the Palatini field equationsfor this matter in Sec 4 These equations are solved in Sec 5 for a spherically symmetricmetric In Sec 6 we introduce the Born-Infeld theory In Sec 7 we formulate the metriccomponents for the quadratic Palatini theory and study an exactly solvable case in Sec 8We conclude in Sec 9 with some final remarks

2 Basics of Palatini gravity

Let us consider a general family of Palatini theories defined as

S[gΓ ψm] =1

2κ2

int

d4xradicminusgf(RQ) + Sm[g ψm] (21)

where f(RQ) represents the gravity Lagrangian κ2 is a constant with suitable dimensions(in GR κ2 equiv 8πG) Sm[g ψm] represents the matter action with ψm representing the matterfields gαβ is the space-time metric R = gmicroνRmicroν Q = gmicroαgνβRmicroνRαβ Rmicroν = Rρmicroρν and

Rαβmicroν = partmicroΓανβ minus partνΓ

αmicroβ + ΓαmicroλΓ

λνβ minus ΓανλΓ

λmicroβ (22)

In the above action (21) the independent connection Γλmicroν is for simplicity not directly coupledto the matter sector Sm In the electromagnetic case to be discussed in this paper this pointis irrelevant if the connection is symmetric but in more general cases the coupling must bespecified We also assume vanishing torsion ie Γλ[microν] = 0 though impose this condition a

posteriori once the field equations have been obtained (see [24] for details) This implies thatthe Ricci tensor is symmetric ie R[microν] = 0 Thus in what follows symmetry in the indicesof Rmicroν will be implicitly understood It must be stressed that the Palatini formulation of(21) is inequivalent to the metric approach which translates into a different structure of thefield equations [46 47] and consequently on the mathematical and physical features of thetheory

Under the above assumptions independent variations of the action (21) with respect tometric and connection yield

fRRmicroν minusf

2gmicroν + 2fQRmicroαR

αν = κ2Tmicroν (23)

nablaβ

[radicminusg (fRgmicroν + 2fQR

microν)]

= 0 (24)

where we have used the short-hand notation fR equiv dfdR and fQ equiv df

dQ As a first step to solve

Eq(23) we introduce the matrix P whose components are Pmicroν equiv Rmicroαg

αν which allows usto express (23) as

fRPmicroν minus f

2δmicroν + 2fQPmicro

αPαν = κ2Tmicro

ν (25)

ndash 4 ndash

In matrix notation this equation reads

2fQP2 + fRP minus f

2I = κ2T (26)

where T is the matrix representation of Tmicroν Note also that in this notation we have R = [P ]micro

micro

and Q = [P 2]micromicro Therefore (26) can be seen as a nonlinear algebraic equation for the object

P as a function of the energy-momentum tensor ie P = P (T ) We will assume that a validsolution to that equation can be found Expressing (24) as

nablaβ

[radicminusggmicroα (fRδνα + 2fQPα

ν)]

= 0 (27)

we see that the term in brackets only depends on the metric gmicroν and T which implies thatthe connection Γλmicroν appears linearly in this equation and can be solved by algebraic meansDefining now the object

Σαν = (fRδ

να + 2fQPα

ν) (28)

we can rewrite the connection equation in the form

nablaβ[radicminushhmicroν ] = 0 (29)

where radichhmicroν =

radicminusggmicroαΣαν (210)

Eq(29) implies that Γαmicroν is the Levi-Civita connection of hmicroν From the relation (210) onefinds that hmicroν and gmicroν are related by

hmicroν =gmicroαΣα

ν

radic

det Σ hmicroν =

(radic

det Σ)

Σmicroαgαν (211)

This puts forward that Σmicroν defines the relative deformation existing between the physical

metric gmicroν and the auxiliary metric hmicroν

Using the definition of Σmicroν and the relations (211) it is easy to see that (23) [or

alternatively (25)] can be written as PmicroαΣα

ν = Rmicroαhανradic

det Σ = f2 δνmicro + Tmicro

ν which allowsto express the metric field equations using hmicroν as follows

Rmicroν(h) =

1radic

det Σ

(

f

2δmicroν + κ2Tmicro

ν

)

(212)

Note that since RQ and Σ are functions of T then the right-hand side of (212) is completelyspecified by the matter whereas the left-hand side simply represents the Ricci tensor of themetric hmicroν The equations satisfied by hmicroν therefore are formally very similar to those foundin GR We emphasize that since hmicroν satisfies second-order equations the algebraic relationexisting between hmicroν and gmicroν gurantees that gmicroν also satisfies second-order equations Notealso that in vacuum (212) boils down to GR plus an effective cosmological constant (see [24]for details) As in previous works [20 25] here we shall take the strategy of solving the fieldequations in terms of hmicroν and then use Σ in (211) to obtain the physical metric gmicroν

ndash 5 ndash

3 The matter sector

For the sake of generality we define the matter sector of our theory by means of the action

Sm =1

8π

int

d4xradicminusgϕ(XY ) (31)

where ϕ(XY ) represents a (so far unspecified) Lagrangian of the two field invariants X =minus1

2FmicroνFmicroν and Y = minus1

2FmicroνlowastFmicroν that can be constructed with the field strength tensor

Fmicroν = partmicroAν minus partνAmicro of the vector potential Amicro and its dual lowastFmicroν = 12ǫmicroναβFαβ By now this

function is only constrained by parity invariance (ϕ(XY ) = ϕ(XminusY )) Let us stress that(31) is the natural generalization of the Maxwell action

Sm = minus 1

16π

int

d4xradicminusgFαβFαβ (32)

to the general charged case When ϕ(XY ) 6= X one says that (31) defines a nonlinear elec-trodynamics (NED) theory The energy-momentum tensor derived from (31) and appearingin Eq(212) is written as

Tmicroν = minus 1

4π

[

ϕXFmicroαFα

ν + ϕY FmicroαlowastFαν minus

δνmicro4ϕ(XY )

]

(33)

On the other hand the field equations for this NED matter are written as

nablamicro

(radicminusg(ϕXFmicroν + ϕY lowastF microν

)

) = 0 (34)

In what follows we shall only deal with purely electrostatic configurations for which the onlynonvanishing component of the field strength tensor is F tr(r) This ansatz implies that theY -invariant will play no role in the dynamics of the theory and can be safely neglected fromnow on In such a case assuming a line element of the form ds2 = gttdt

2 + grrdr2 + r2dΩ2

the field equations lead to

F tr =q

r2ϕXradicminusgttgrr

(35)

where q is an integration constant interpreted as the electric charge associated to a givensolution Writing X = minusgttgrr(F tr)2 it follows that for any spherically symmetric metric

ϕ2XX =

q2

r4 (36)

On the other hand the components of the energy-momentum tensor (33) for these electro-static solutions read in matrix form

Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)

where I and 0 are the identity and zero 2times 2 matrices respectively

In order to write the field equations associated to the matter source (31) we first needto find the explicit form of P for it which will allow us to construct Σ and compute itsdeterminant To do this we write (26) as

ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ

+ κ2 (ϕminus 2XϕX)

)

(310)

and we have defined κ2 = κ24π There are 16 square roots that satisfy this equation namely

radic

2fQ

(

P +fR4fQ

I

)

=

s1λminus 0 0 00 s2λminus 0 00 0 s3λ+ 00 0 0 s4λ+

(311)

where si = plusmn1 Agreement with GR in the low curvature regime (where fR rarr 1 and fQ rarr 0)requires si = +1 For this reason we simplify the notation and take

radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)

From this it follows that the matrix Σ is given by

Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)

and λplusmn are given in Eqs(39) and (310)

4 Palatini gravity with matter

The field equations in matrix form for the energy-momentum tensor (37) are written as

Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)

In order to solve them we must compute explicitly the objects λ2plusmn and σplusmn besides R and QTo do this we consider the quadratic Lagrangian

f(RQ) = R+ l2P (aR2 + bQ) (42)

and trace Eq(23) with gmicroν which yields

R = minusκ2T (43)

where T is the trace of the energy-momentum tensor of the matter In the Maxwell casethe Rminusdependent part of the gravity Lagrangian does not play a very relevant role in the

ndash 7 ndash

dynamics as a consequence of the tracelessness of the energy-momentum tensor entering inEq(43) However for NED the trace reads

T = 2T tt + 2T θθ =1

2π[ϕminusXϕX ] (44)

which is nonvanishing for any nonlinear function ϕ(XY = 0) Therefore NED make possibleto excite new dynamics associated to the Rminusdependent part of the Lagrangian The explicitexpression for Q can be obtained for the theory (42) by taking the trace in (312) and solvingfor Q which leads to the result

Q =T 2

4+

(T tt minus T θθ )2

(1minus (2a+ b)χT )2 (45)

where we have defined Q = Qκ4 and introduced the parameter χ = κ2l2P Next we can writeλplusmn in Eqs(39) and (310) as

λ+ =1

2radic2blP

(

1minus 4χ(aT θθ + (b+ a)T tt )minus (b+ 2a)2χ(T tt + T θθ )

2

1minus 2χ(b+ 2a)(T tt + T θθ )

)

(46)

λminus = λ+(Ttt harr T θθ ) (47)

and therefore the factors σplusmn appearing in (313) are obtained as

σ+ = 1 + 2χ

[

minus[aT + bT tt ] +(b+2a)(b+4a)

4 χT 2

1minus (b+ 2a)χT

]

(48)

σminus = σ+(Ttt harr T θθ ) (49)

When b = 0 we obtain σplusmn = 1 minus 2aχT which corresponds to the theory f(R) = R + al2PR2

coupled to NED charged matter a case studied in [48] When a = 1 b 6= 0 and vanishingtrace (T = 0 corresponding to Maxwell theory) we obtain σ+ = 1 + χT tt and since T tt =

minusT θθ = minus q2

8πr4 defining a variable z4 = 4π

κ2l2P β2 r

4 we achieve the result σ+ = 1 + 1z4 and

Q = κ4q4

r4 in agreement with the result obtained in [20] for the Maxwell case

5 Solving the equations

51 The metric ansatz

We shall first formally solve the field equations (212) which are valid for any f(RQ) gravitytheory and any ϕ(XY = 0) matter Lagrangian densities Later we will consider the particularcase of Lagrangian (42) For this purpose it is convenient to introduce two different lineelements in Schwarzschild-like coordinates one associated to the physical metric gmicroν

ds2 = gttdt2 + grrdr

2 + r2dΩ2 (51)

and another associated to the auxiliary metric hmicroν

ds2 = httdt2 + hrrdr

2 + r2dΩ2 (52)

with dΩ2 = dθ2 + sin2(θ)dφ2 The relation between these two line elements is nontrivial dueto the relation gmicroν = Σαmicrohαν

radicdetΣ To be precise they are related as

ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)

From (53) it is clear that gtt = httσ+ grr = hrrσ+ and that r2 and r2 are related byr2 = r2σminus We are particularly interested in those cases in which the function σminus may vanishat some r = rc gt 0 When this happens we find a center in the geometry defined by hmicroν (withr = 0) and a two-sphere of area A = 4πr2c in the physical geometry As we will see in furtherdetail latter the physical geometry has a minimum at r = rc and therefore that surfacecan be identified as the throat of a wormhole This can be easily seen by introducing a newcoordinate dx2 = dr2σminus (with x isin]minusinfininfin[) The vanishing of σminus at rc is simply interpreted

as the point where the radial function r2(x) reaches a minimum (drdx = σ12minus rarr 0) [see

Fig2 below]

52 General solution

Since the field equations take a simpler form in terms of the metric hmicroν see (41) we will usethe line element (52) to solve for the metric For this purpose it is convenient to use r asthe radial coordinate which brings (52) into


2 + r2dΩ2 (54)

where hrrdr2 = hrrdr

2 Considering the redefinitions htt = minusA(r)e2ψ(r) hrr = 1A(r) thecomponents of the tensor Rmicro

ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1

r2[1minusA(1 + rψr)minus rAr] (57)

Now taking into account the symmetry Ttt = Tr

r of the NED energy-momentum tensor thatholds for electrostatic spherically symmetric solutions it is easily seen that the substractionRt

t minusRrr in the field equations (55) and (56) leads to ψ = constant and can be set to zero

through a redefinition of the time coordinate like in GR This leaves a single equation to besolved namely

1

r2(1minusA(r)minus rAr) =

1

2σ+σminus

(

f +κ2

4πϕ

)

(58)

Taking the usual ansatz

A(r) = 1minus 2M(r)

r (59)

and inserting it into (58) we are led to

Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash

Using the relation between coordinates r2 = r2σminus we have drdr = σ

12minus

(

1 +rσ

minusr

2σminus

)

and with

this we collect the final result

Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(

1 +rσminusr2σminus

)

(511)

Let us recall that in the above derivation only two assumptions have been made namelyspherical symmetry of the space-time and symmetry of the energy-momentum tensor of thematter Tt

t = Trr and thus are valid for any f(RQ) theory coupled to arbitrary charged

NED matter ϕ(XY = 0) Moreover Eq(511) completely determines the associated spheri-cally symmetric static solutions through a single function M(r) once the gravity and matterLagrangians are given

Upon integration of the mass function (511) one finds an expression of the formM(r) = M0 + ∆M where M0 is an integration constant identified as the Schwarzschildmass M0 equiv rS2 and ∆M represents the electromagnetic contribution In order to deal withdimensionless magnitudes we find it useful to introduce a function G(r a b ) as follows

M(r)

M0= 1 + δ1G(r a b ) (512)

where δ1 is a constant parameter and G(r a b ) encapsulates all the information on thegeometry

6 The Born-Infeld model

The Born-Infeld (BI) Lagrangian [34] is given in its full form by

ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)

where β is the BI parameter such that (61) recovers the Maxwell Lagrangian in the β rarr infinlimit The BI Lagrangian arises in the low-energy regime of string and D-brane physics whereβ is related to the inverse string tension αprime as β = 2παprime [35ndash39] When coupled to gravitythe black holes associated to this Lagrangian have been studied in depth [40ndash45]

In order to deal with dimensionless variables it is useful to define two length scalesr2q equiv κ2q24π and l2β = (κ2β24π)minus1 in terms of which we can define the dimensionless

variable z4 = r4

r2q l2β Using this the electromagnetic field equation (35) provides the expression

X

β2=

1

1 + z4 (62)

which allows to express the components of the energy-momentum tensor associated to theLagrangian (61) as

κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(

1minus z2radicz4 + 1

)

(63)

ndash 10 ndash

The trace (44) can thus be written as

κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)

which is nonvanishing as expected Note in passing that in flat space the energy densityassociated to a point-like charged field F tr(r) in this theory is obtained as

ε(q) = 4π

int infin

0drr2T tt (r q) = β12q32

int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32

3Γ(34)2 asymp 123605 which yields a finite total energy

7 Quadratic gravity coupled to Born-Infeld

In this section we analyze the gravity theory (42) coupled to Born-Infeld NED introducedin Sec 6 This will allow us to present the basic equations of the problem and some usefultransformations that will considerably simplify the algebraic expressions involved Particularcases of interest will be considered in full detail later

From the definitions introduced in Sec 3 the components of the metric gmicroν take thegeneral form

gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(

1 +zσminusz2σminus(z)

)2

(71)

A(z) = 1minus 1 + δ1G(z)

δ2zσminus(z)12 (72)

where we have defined

δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)

and the function G(z) satisfies

Gz =z2σ

12minus

σ+

(

1 +zσminusz2σminus

)

(

l2βf + ϕ)

(74)

In this expression we have denoted ϕ = ϕβ2

In terms of the variable z the expressions for σplusmn and Gz for the theory (42) for arbitraryvalues of the parameters a and b are very cumbersome which complicates the identificationof models that may contain geonic wormholes of the kind found in [25] For this reason wefind it very useful to perform a change of variable and introduce some redefinitions of theparameters of the theory In particular we consider the transformations

z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash

where λ equiv l2P l2β With these definitions we find that Gǫ = Gzdzdǫ reads as

Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)

whereas σplusmn become

σ+ =ǫ(2 + ǫ)(16γ1 + ǫ(2 + ǫ+ 8γ1 minus 4γ2)) + 32γ1γ2

ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and

σminus =ǫ(2 + ǫ)(8γ2 + ǫ(2 + ǫminus 8γ1 + 4γ2)) + 32γ1γ2

ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)

Note that when ǫ ≫ 1 we have σplusmn rarr 1 and Gǫdǫdz asymp 2radicǫ which nicely recovers the GR

limit when (75) is used

8 Exactly solvable model

To proceed further it is convenient to specify particular values for the parameters a and b of

the gravity Lagrangian (42) For simplicity we consider the case f(RQ) = Rminus l2P2 R

2+ l2PQwhich provides useful simplifications that allow to make significant progress using analyticalmethods This model corresponds to a = minusb2 and b = 1 (or equivalently γ1 = 0 andγ2 = minusλ where λ gt 0) Any other choice of b gt 0 would be physically equivalent to thischoice up to a rescaling of the parameter λ It should be noted that the choice b gt 0 wascrucial in [49 50] to obtain bouncing cosmological models without big bang singularity Thismotivates our choice2 For our choice of parameters the general expressions given above boildown to

Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)

σminus = 1minus ǫcǫ (83)

where ǫc = 4λ represents the point where σminus vanishes Similarly as in the Maxwell case [25]the existence of a zero in σminus implies that the region ǫ lt ǫc is not physically accessible3 ie the

geometry is only defined for ǫ ge ǫc In terms of z the point ǫc corresponds to zc =radic2λ

(1+4λ)14

2We note that exact solutions with b lt 0 can also be found though they lead to physically uninterestingsituations

3We note that in a spherically symmetric space-time the area of the two spheres is given by A = 4πr2(t x)where (t x) are generic coordinates of the non-spherical sector In static situations we have r2(t x) rarr r2(x)and one might want to use the function r(x) as a coordinate of the non-spherical sector (canonical coordinates[51]) This is possible as long as drdx 6= 0 If in some interval drdx = 0 at some point then the functionr cannot be used as a coordinate on the whole interval We will see later that in our problem the functionr(x) reaches a minimum (drdx = 0 and d2rdx2 gt 0) and cannot cover the region r lt rc (see Fig2) Thisexplains why ǫ cannot be extended below ǫc The existence of a minimum area signals the existence of aminimum volume in the theory an aspect related with the properties of the connection [24]

ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ

Figure 1 Evolution of the function G(z) for different values of λ The GR limit corresponds to

λ = 0 For a given λ the minimum occurs at zc =radic

2λ(1+4λ)14

and corresponds to G(zc) = minus1δc [see

(87)] Note that in GR the geometry extends down to z = 0 whereas for λ gt 0 we find that z cannotbe smaller than zc which manifests the existence of a finite structure (or core) replacing the centralpoint-like singularity

Recall that the GR limit is recovered when ǫ ≫ ǫc For completeness we note that for thismodel Gz(λ) takes the form

Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic

z2 minus 2λW [z] (84)

where W [z] equivradic1 + z4 minus z2 From this expression one readily verifies that the GR limit is

correctly recovered in the limit λrarr 0

Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)

The integration of (81) is immediate and yields

G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)

where 2F1 represents a hypergeometric function and δc is an integration constant In orderto recover the correct behavior at infinity G(z) asymp minus1z one finds that

δc =(1 + 4λ)14

2nBI (87)

where nBI was defined in (65)

81 Metric components

When γ1 = 0 one finds that grr = minus1(gttσminus) It is thus useful to define a new radial coordi-nate such that grrdr

2 = minusdx2gtt which brings the line element into its usual Schwarzschild-like form

ds2 = gttdt2 minus 1

gttdx2 + r2(x)dΩ2 (88)

ndash 13 ndash

All the information about the geometry is thus contained in the functions gtt = minusAσ+and r2(x) In our case the relation between x and r can be explicitly written in terms ofa hypergeometric function whose asymptotic behaviors are x asymp r for large values of r andxrc asymp x0(λ) + αλ

radicz minus zc + where x0(λ) asymp minus1694λ(1 + 4λ)minus34 and αλ gt 0 as z rarr zc

From the previous results one can obtain series expansions that allow to study theasymptotic behavior of the geometry when z rarr infin and also in the vicinity of the corenamely z rarr zc In the far limit we find

gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)

This expression exactly recovers the result of GR coupled to the BI theory In the region nearthe core the metric takes the form

gtt asymp(1 + 2λ)32 (δc minus δ1)

234λ14(1 + 4λ)118δ2δcradicz minus zc

+(1 + 2λ)

(

2δ1 minusradic1 + 4λδ2

)

(1 + 4λ)32δ2+O(

radicz minus zc) (810)

From this expression we see that in general the metric diverges at zc as sim 1radicz minus zc

However for the choice δ1 = δc one finds a finite result everywhere

gtt asymp(1 + 2λ)

(

minusradic1 + 4λδ2 + 2δc

)

(1 + 4λ)32δ2+O(z minus zc) (811)

Note that the choice δ1 = δc allows to turn the metric near z = zc into Minkowskian form byjust rescaling by a constant factor the t and x coordinates To better understand the detailsof the geometry in this region we proceed next to evaluate the curvature invariants of themetric

82 Curvature invariants

To characterize the geometry in a coordinate-independent manner we consider the scalarsR(g) equiv gmicroνRmicroν(g) Q(g) equiv Rmicroν(g)R

microν(g) and K(g) equiv Rαβγδ(g)Rαβγδ(g) constructed out of

the physical metric gmicroν Expansion of these scalars in the far region (z ≫ zc) leads to

R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)

where KGR represents the GR value of the Kretschmann scalar It is worth noting that inQ(g) and in K(g) the Planck scale corrections decay at a slower rate than those due to theBorn-Infeld contribution whereas in R(g) the Planckian contribution decays much faster

ndash 14 ndash

On the other hand in the region z asymp zc we find that in general the expansions of theseinvariants around z = zc can be written as

R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)

Q = (δ1 minus δc)2

[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)

K = (δ1 minus δc)2

[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)

where αi ζi ξi microi νi and Ci are constants that depend on (λ δ1 δ2 δc) We thus see thatthese three curvature invariants are divergent at z = zc with the leading terms growing assim 1(z minus zc)

3 in the worst case This behavior is similar to that found in [25] for the case ofMaxwell electrodynamics and contrasts with the results of GR where the divergence growsas 1z8 in the case of Maxwell and as 1z4 in the case of Born-Infeld Of particular interestis the case δ1 = δc for which the expansions of the curvature invariants around z = zc read

R asymp4(

16λ2 + 11λ+ 3)

δc minus 3δ2radic4λ+ 1

(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(

δ2radic4λ+ 1minus 2δc

)2

λ2δ22+

16(

(1minus 4λ)δc + 3δ2λradic4λ+ 1

)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4

This confirms that like in the Maxwell case [25] there exist config-

urations for which the geometry at z = zc is completely regular

83 Wormhole structure and electric flux

The existence of solutions with a smooth geometry at z = zc indicates that one can still usethe field equations of the theory to investigate the physical processes going on in that regionIn particular one may be interested in the nature of the sources that generate the electricfield In this sense one should note that we are dealing with sourceless electromagnetic fieldequations This fact together with the constraints imposed by the field equations that forcez to be greater or equal than zc suggests that the spherically symmetric electric field couldhave a topological origin rather than being created by a distribution of charges an idea datingback to Misner and Wheeler [52] and that our model implements in a natural way In factthe absence of singularities at z = zc drive us to consider an extension of the geometry allowed

ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4

Figure 2 Representation of the radial function z in terms of x The minimum of the curve represents

the wormhole throat where dzdx = σ12minus

= 0 and d2zdx2 gt 0 Note that Gx is smooth and finiteeverywhere

by the definition of the coordinate x introduced in (88) such that x can be extended to thewhole real axis while z remains bounded to the region z ge zc This can be done by consideringthe relation dx2 = dz2σminus and noting that two signs are possible in the resulting definition

of x(r) namely dx = plusmndzσ12minus In our analysis of the curvature invariants and the metric

above we assumed dx = +dzσ12minus The existence of nonsingular solutions motivates the

consideration of dx = minusdzσ12minus as a physically meaningful branch of the theory This leadsto (see Fig2)

x(z) =

F (zλ) if x ge xc2xc minus F (zλ) if x le xc

(821)

where F (zcλ) = xc =radic

π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)

Therefore to cover the whole geometry one needs two charts if the set (t r) is used ascoordinates (one for the interval x ge xc and another for x le xc) or a single chart if (t x) isused instead

The bounce of the radial function r2(x) (see Fig2) puts forward that our spacetime has agenuine wormhole geometry (see [53] for a review of wormhole solutions in the literature) Asa result the spherically symmetric electric field that local observers measure is not generatedby a charge distribution but by a sourceless electric flux trapped in the topology [52] In thissense an observer in the x gt xc region finds that the electric flux across any closed 2-surfacecontaining the wormhole throat in its interior is given by Φ equiv

int

S ϕX lowast F = 4πq where lowastFrepresents the 2-form dual to Faradayrsquos tensor If the integration is performed in the x lt xcregion and the orientation is assumed such that the normal points in the direction of growthof the area of the 2-spheres then the result is Φ = minus4πq because the orientation differs insign with that chosen on the other side of the throat This shows that no real sources generatethe field which is fully consistent with the sourceless gravitational-electromagnetic equationsof our theory

ndash 16 ndash

We note that the electric flux per surface unit flowing through the wormhole throat atz = zc takes the form

Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)

This quantity represents the density of lines of force at zc For any fixed λ it turns out to bea universal constant that only depends on ~ c and G When λrarr infin which is equivalent totaking the limit in which the Born-Infeld theory recovers Maxwellrsquos electrodynamics (lβ rarr 0)we reproduce the result obtained in [25] If one considers the limit λrarr 0 which correspondsto the coupling of the Born-Infeld theory to GR (lP rarr 0) one finds that the density oflines of force tends to zero This situation can be seen as the limit in which the wormholedisappears (zc rarr 0) which is consistent with studies showing the impossibility of generatingwormholes supported by nonlinear theories of electrodynamics in GR [54 55] As a resultin the case of GR (coupled to Maxwell or to BI) one needs to find a source for the existingelectric fields This leads to a well-known problem plagued by inconsistencies related to theimpossibility of having a point-like particle at rest at r = 0 whose energy and charge matchthose appearing in the geometry and at the same time being a solution of the Einstein fieldequations everywhere The wormhole structure that arises in our theory therefore naturallyavoids the problem of the sources that one finds in GR [19]

The fact that for any λ gt 0 the density of lines of force at the wormhole throat is aconstant independent of the particular amounts of charge and mass strongly supports theview that the wormhole structure exists even in those cases in which δ1 6= δc To the light ofthis result the question of the meaning and implications of curvature divergences should bereconsidered since their presence seems to pose no obstacle to the existence of a well-definedelectric flux through the z = zc surface

84 Horizons and mass spectrum when δ1 = δc

From the analysis in sections 81 and 82 one can easily verify that regardless of the valueof δ1 a few zc units away from the center the curvature invariants and the location of theexternal event horizon rapidly tend to those predicted by GR However for solutions withδ1 = δc one finds that the event horizon may disappear in certain cases which brings abouta new kind of gravitating object whose internal and external properties differ from thosetypically found in GR In fact one can verify numerically that for such configurations theexistence of the event horizon crucially depends on the sign of the leading term of gtt in (811)If this term is positive then there exists an external horizon but if it is negative then thehorizon is absent This implies that the horizon disappears if the condition

2δc lt δ2radic1 + 4λ (823)

is satisfied This condition can be put in a more intuitive form if one expresses the chargeas q = Nqe with Nq representing the number of elementary charges e which leads to rq =2lPNqN

Mq where NM

q equivradic

2αem asymp 1655 and αem is the fine structure constant Taking

now into account the definitions given in (73) one finds that δ1δ2 = λ12NqNMq which

allows to rewrite (823) as

Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash

Therefore if the number of charges of the object is smaller than NBIq = NM

q

radic

1 + 14λ the

object has no external event horizon and appears to an external observer in much the sameway as a massive charged elementary particle

The mass spectrum of these objects can be obtained directly from the regularity condi-tion δ1 = δc which establishes a constraint between the amount of charge and mass of thosesolutions Using again the notation q = Nqe we find

MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)

where nBI asymp 123605 was defined in (65) This mass spectrum can also be written as

MBI =

(

4λ

1 + 4λ

)14

MM (826)

where MM = (NqNMq )32nBImP is the mass spectrum of the Maxwell case found in [25]

Note that the limit λ rarr infin nicely recovers the result of the Maxwell case (recall that λ equivl2P l

2β) In general we see that for any finite λ the mass corresponding to a given charge

q is smaller in the BI than in the Maxwell case This manifests that the quantum effectsresponsible for the nonlinearities of the matter field have an impact on its energy density andresult in a spectrum of lighter particles

To illustrate this last point let us estimate some typical scale for the parameter β Thiscan be done from the effective Lagrangians scheme of quantum electrodynamics [31] whichis a reliable approximation as far as the maximum allowed electric field does not exceed therange sim 1016 minus 1018 and that allows to employ (nonlinear) classical Lagrangians as suitablephenomenological descriptions of the electromagnetic field encoding quantum vacuum effectsNoting that β is identified as the maximum value of the electric field in BI theory attainedat the center [see Eq(62)] this sets a maximum scale for β in the range above (note alsothat the expansion of BI Lagrangian for small β2 coincides for Y = 0 and to lowest order inthe perturbative expansion [31] with the Euler-Heisenberg effective Lagrangian of quantumelectrodynamics [32 33] modulo a constant) Since β2 has dimensions of q2cr4 where c isthe speed of light we can see the value of β2 as the intensity of electric field (squared) thatan electron generates at a distance re such that r4e = e2cβ2 = αem~β

2 From the definitionof λ equiv l2P l

2β and l2β = 4π(κ2β2) = c3(2Gβ2) we find that l2β = c3r4e(2Gαem~) = r4e2l

2P

With this result we can write λ = 2l4P r4e For β = 1018 we get re asymp 10minus18 m which leads to

λ sim (10minus17)4 The smallness of this parameter implies that MBI asymp (4λ)14MM asymp 10minus17MM ie the mass has been lowered by 17 orders of magnitude This brings the mass spectrumof our solutions from the Planck scale 1019 GeV down to the reach of current particleaccelerators sim 102 GeV which shows that the Planck scale phenomenology of Palatini gravitycan be tested and constrained with currently available experiments

We note that the matter model used in our discussion is far from being completelysatisfactory and that a number of corrections might be important in the range of scalesconsidered In fact more accurate descriptions of the quantum vacuum effects would requiretaking into account strong field and nonperturbative effects aspects that are still underintense scrutiny (see for instance [56ndash58]) In this sense the BI toy model discussed heremust be seen as an approximation which by no means incorporates all the effects expectedto be relevant in this problem Nonetheless it puts forward the important phenomenological

ndash 18 ndash

effects that non-linearities in the matter sector could have if at relatively low energies thequantum gravitational degrees of freedom required a metric-affine structure to accuratelydescribe the space-time dynamics

85 Geons as point-like particles

The regular solutions that we are considering represent an explicit realization of the concept ofgeon introduced by Wheeler in [59] These objects were defined as self-gravitating nonsingularsolutions of the sourceless gravitational-electromagnetic field equations In our case howeverthe regularity condition seems not to be so important to obtain consistent solutions to thegravitational-electromagnetic system of equations since as we have shown the existence ofcurvature divergences at the wormhole throat has no effect on the properties of the electric fieldthat flows through it Nonetheless the existence of completely regular horizonless solutionsthat could be interpreted as elementary particles with mass and charge suggests that suchsolutions could play a more fundamental role in the theory Further evidence in this directioncan be obtained by evaluating the full action on the solutions that we have found Thiscomputation will give us information about the total electric and gravitational energy storedin the space-timeWith elementary manipulations we find that

S =1

2κ2

int

d4xradicminusg(Rminus l2P

2R2 + l2PQ) +

1

8π

int

d4xradicminusgϕ = minus

int

d4xradicminusgT tt σ+ (827)

where we have replaced R and Q by their respective expressions (43) and (45) and havealso used that ϕ8π = T θθ Carrying out explicitly the integration over the whole spacetimein terms of the variable dx2 = dz2σminus leads to

S = minus(rqlβ)32

int

dt

int +infin

minusinfindxz2dΩT tt σ+ = minus8π(rqlβ)

32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt

where I = minusκ2l2βint +infinzc

dzz2T tt σ+σ12minus only depends on λ In the second step in (828) we

have introduced a factor 2 in order to take into account the two sides of the wormhole andin the last step we have used the definitions of Eqs(73) and (63) Note that M representsthe integration constant that we identified with the Schwarzschild mass of the unchargedsolution Making use of the variable ǫ and applying the theorem of residues we find I =nBI(1 + 4λ)14 = 1(2δc) [see Eq(87)] and thus we collect the final result

S = 2Mc2δ1δc

int

dt (829)

This implies that when the regularity condition δ1 = δc holds the resulting action reads

S = 2MBIc2int

dt (830)

with MBI being the mass defined in (825) Remarkably this is just the action of a point-likemassive particle at rest Taking into account that local observers can only be on one of thesides of the wormhole it follows that one factor MBI comes from integrating on one side of

ndash 19 ndash

the space-time and the other comes from integrating on the other side In this sense thespatial integral

E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)

can be seen as the addition of the electromagnetic energy plus the gravitational binding energygenerated by the electric field Since in the regular cases δ1 = δc the total energy producedby the electric field is equal to the Schwarzschild mass of the object M = MBI it followsthat these configurations are in fact geonic-like gravitational solitons This result furthersupports our view that these objects with or without an event horizon have particle-likeproperties Additional evidence relevant for this discussion comes from the fact that thehypersurface x = xc where the wormhole throat is located changes from space-like to time-like when the horizon disappears This means that the wormhole throat in the horizonlesssolutions follows a time-like trajectory like any massive physical particle Therefore to anexternal observer with low resolution power such objects would appear as massive chargedpoint-like particles

It should be stressed that in the context of GR the identification between the totalenergy and the Schwarzschild mass is also obtained in some models where an exotic dust anda Maxwell field are considered [see eg [60])] Similarly as in our case the factor 2M comesfrom the integration of the radial coordinate along its whole range of definition and can beinterpreted as corresponding to a pair of particles located on each side of the wormhole

9 Summary and conclusions

In this paper we have considered the coupling of a nonlinear theory of electrodynamics toa quadratic extension of GR formulated agrave la Palatini Unlike in the standard formulationof quadratic gravity in which fourth-order equations govern the behavior of the metric inthe Palatini formulation we find second-order equations for the metric This follows from thefact that the connection can be solved by algebraic means as the Levi-Civita connection of anauxiliary metric hmicroν [24] Since this metric satisfies second-order equations and is algebraicallyrelated with gmicroν the theory turns out to have the same number of propagating degrees offreedom as GR Moreover in vacuum the field equations of our theory exactly become thoseof GR which is a manifestation of the observed universality of Einsteinrsquos equations in Palatinitheories [61 62] The simplicity of the field equations governing hmicroν has allowed us to findexact formal solutions for arbitrary gravity Lagrangian f(RQ) and arbitrary electrodynamicstheory ϕ(XY = 0) in the electrostatic case

The introduction of a nonlinear electrodynamics theory is motivated by the need toconsider quantum corrections in the matter sector at energies above the pair-productionthreshold We were particularly interested in determining if the geon-like solutions found in apurely Maxwell context in the quadratic Palatini theory would persist under quantum-inducedmodifications of the matter sector We have found that this is indeed the case Geon-likesolutions consisting on a wormhole supported by a Born-Infeld electrostatic field exist in afamily of gravity models whose parameters also yield nonsingular bouncing cosmologies (b gt0) [49] For simplicity we have considered a specific choice of parameters (a = minusb2 b = 1)which yields exact analytical solutions and allows to obtain definite predictions about thecharge and mass spectrum of the theory

In this respect we have found that the mass spectrum of the regular geons the oneswithout curvature divergences at the wormhole throat can be several orders of magnitude

ndash 20 ndash

smaller than those found in Maxwellrsquos theory For values of the parameters reaching the limitof validity of the approximations where the use of classical nonlinear Lagrangians such as theBorn-Infeld model is justified we have shown that the mass spectrum can be lowered fromthe Planck scale down to the GeV scale There is a simple physical reason for this effectFor small values of λ the density of lines of force at the wormhole throat [see Eq (822)]decreases Since as we have shown the mass of these objects is due to the energy stored inthe electrostatic field lowering the density of lines of force implies lowering their mass Inthe GR limit λrarr 0 the wormhole closes and these objects disappear from the spectrum ofsolutions of the theory

In our view the particular combination of theories considered here must be regarded asa toy model that could be improved in different ways Nonetheless it allows to put forwardthat new avenues to the generation of stable massive particles from a quantum gravitationalperspective are possible and that Planck scale physics can be brought into experimental reachwithin a purely four-dimensional scenario The potential implications that the existenceof stable massive particles such as the horizonless geons found here could have for themissing matter problem in astrophysics and cosmology [63 64] demand further research inthis direction In particular improvements in the treatment of the quantum correctionsaffecting the matter sector and a rigorous derivation of the semiclassical corrections expectedin Palatini backgrounds should be considered in detail in the future

To conclude we underline that the results of this paper indicate that the existence ofgeons is not directly tied to the particular matter source considered because such solutionsdo not arise in GR [54 55] but rather to the Planck-corrected Palatini model employed Thematter is needed to excite the new gravitational dynamics (recall that in vacuum the theoryboils down to GR) but the existence of wormholes to support the geons must be attributed tothe quantum-corrected gravity Lagrangian and the way Palatini dynamics affects the structureof space-time This suggests that similar phenomenology could be expected when non-abeliangauge fields are added to the matter sector and also in other approaches based on the premisethat metric and affine structures are a priori independent [65]

Acknowledgments

G J O is finantially supported by the Spanish grant FIS2011-29813-C02-02 and the JAE-docprogram of the Spanish Research Council (CSIC) D R -G is supported by CNPq (Brazilianagency) through grant number 5610692010-7 and acknowledges the hospitality and partialsupport of the theoretical physics group at the University of Valencia where this work wasinitiated We are indebted to E Guendelman and M Vasihoun for useful discussions andcomments and to F S N Lobo for his critical reading of the manuscript

References

[1] S Dimopoulos and G Landsberg Black holes at the LHC Phys Rev Lett 87 (2001) 161602[arXivhep-ph0106295]

[2] S B Giddings and S D Thomas High-energy colliders as black hole factories the end ofshort distance physics Phys Rev D 65 (2002) 056010 [arXivhep-ph0106219]

[3] D M Eardley and S B Giddings Classical black hole production in high-energy collisionsPhys Rev D 66 (2002) 044011 [arXivgr-qc0201034]

ndash 21 ndash

[4] L Rezzolla and K Takami Black-hole production from ultrarelativistic collisions Class QuantGrav 30 (2013) 012001 [arXiv12096138 [gr-qc]]

[5] W E East and F Pretorius Ultrarelativistic black hole formation Phys Rev Lett 110 (2013)101101 [arXiv12100443 [gr-qc]]

[6] M Cavaglia Black hole and brane production in TeV gravity A Review Int J Mod Phys A18 (2003) 1843 [arXivhep-ph0210296]

[7] H Yoshino and Y Nambu Black hole formation in the grazing collision of high-energyparticles Phys Rev D 67 (2003) 024009 [arXivgr-qc0209003]

[8] B Koch M Bleicher and S Hossenfelder Black hole remnants at the LHC JHEP 0510 (2005)053 [arXivhep-ph0507138]

[9] V Khachatryan et al [CMS Collaboration] Search for Microscopic Black Hole Signatures atthe Large Hadron Collider Phys Lett B 697 (2011) 434 [arXiv10123375 [hep-ex]]

[10] M Cavaglia R Godang L M Cremaldi and D J Summers Signatures of black holes at theLHC JHEP 0706 (2007) 055 [arXiv07070317 [hep-ph]]

[11] X Calmet S D H Hsu and D Reeb Quantum gravity at a TeV and the renormalization ofNewtonrsquos constant Phys Rev D 77 (2008) 125015 [arXiv08031836 [hep-th]]

[12] I Antoniadis N Arkani-Hamed S Dimopoulos and G R Dvali New dimensions at amillimeter to a Fermi and superstrings at a TeV Phys Lett B 436 (1998) 257[arXivhep-ph9804398]

[13] N Arkani-Hamed S Dimopoulos and G Dvali The Hierarchy problem and new dimensions ata millimeter Phys Lett B 429 (1998) 263 [arXivhep-ph9803315]

[14] L Randall and R Sundrum A Large mass hierarchy from a small extra dimension Phys RevLett 83 (1999) 3370 [arXivhep-ph9905221]


[16] L Parker and D J Toms Quantum field theory in curved spacetime quantized fields andgravity (Cambridge University Press 2009)

[17] N D Birrell and P C W Davies Quantum fields in curved space (Cambridge UniversityPress 1982)

[18] M Green J Schwarz and E Witten Superstring Theory (Cambridge University PressCambridge 1987)

[19] T Ortin Gravity and strings (Cambridge Monographs on Mathematical Physics CambridgeUniversity Press 2004)

[20] G J Olmo and D Rubiera-Garcia Reissner-Nordstroumlm black holes in extended Palatinitheories Phys Rev D 86 (2012) 044014 [arXiv12076004 [gr-qc]]

[21] G J Olmo and D Rubiera-Garcia Nonsingular black holes in quadratic Palatini gravity EurPhys J C 72 (2012) 2098 [arXiv11120475 [gr-qc]]

[22] G J Olmo and D Rubiera-Garcia Nonsingular charged black holes agrave la Palatini Int J ModPhys D 21 (2012) 1250067 [arXiv12074303 [gr-qc]]

[23] G J Olmo Palatini Approach to Modified Gravity f(R) Theories and Beyond Int J ModPhys D 20 (2011) 413 [arXiv11013864 [gr-qc]]

[24] G J Olmo and D Rubiera-Garcia Importance of torsion and invariant volumes in Palatinitheories of gravity [arXiv13064210 [hep-th]]

[25] F N Lobo G J Olmo and D Rubiera-Garcia Semiclassical geons as solitonic black holeremnants in press [arXiv13062504 [hep-th]]

ndash 22 ndash

[26] L Modesto Loop quantum black hole Class Quant Grav 23 (2006) 5587[arXivgr-qc0509078]

[27] L Modesto and I Premont-Schwarz Self-dual Black Holes in LQG Theory andPhenomenology Phys Rev D 80 (2009) 064041 [arXiv09053170 [hep-th]]

[28] L Modesto Super-renormalizable Quantum Gravity Phys Rev D 86 (2012) 044005[arXiv11072403 [hep-th]]

[29] J Schwinger On gauge invariance and vacuum polarization Phys Rev 82 (1951) 664

[30] G V Dunne Heisenberg-Euler effective Lagrangians Basics and extensions (Shifman M(ed) et al From fields to strings vol 1 445-522) [arXivhep-th0406216]

[31] A Dobado A Goacutemez-Nicola A L Maroto and J R Pelaacuteez Effective Lagrangians for theStandard Model (Springer-Verlag Berlin Heidelberg 1997)

[32] W Heisenberg and H Euler Consequences of Diracrsquos theory of positrons Z Phys 120 (1936)714 [arXivphysics0605038]

[33] Z Bialynicka-Birula and I Bialynicki-Birula Nonlinear effects in Quantum ElectrodynamicsPhoton propagation and photon splitting in an external field Phys Rev D 2 (1970) 2341

[34] M Born and L Infeld Foundations of the new field theory Proc R Soc London A 144 (1934)425

[35] E S Fradkin and A A Tseytlin Nonlinear Electrodynamics from Quantized Strings PhysLett B 163 (1985) 123

[36] R G Leigh Dirac-Born-Infeld Action from Dirichlet Sigma Model Mod Phys Lett A 4

(1989) 2767

[37] E Witten Bound states of strings and p-branes Nucl Phys B 460 (1996) 335[arXivhep-th9510135]

[38] A A Tseytlin On nonAbelian generalization of Born-Infeld action in string theory NuclPhys B 501 (1997) 41 [arXivhep-th9701125]

[39] M Perry and J H Schwarz Interacting chiral gauge fields in six-dimensions and Born-Infeldtheory Nucl Phys B 489 (1997) 47 [arXivhep-th9611065]

[40] I H Salazar A Garcia and J Plebanski Duality Rotations And Type D Solutions To EinsteinEquations With Nonlinear Electromagnetic Sources J Math Phys 28 (1987) 2171

[41] M Demianski Static Electromagnetic Geon Found Phys 16 (1986) 187

[42] N Breton Born-Infeld black hole in the isolated horizon framework Phys Rev D 67 (2003)124004 [hep-th0301254]

[43] D L Wiltshire Black Holes In String Generated Gravity Models Phys Rev D 38 (1988) 2445

[44] M Aiello R Ferraro and G Giribet Exact solutions of Lovelock-Born-Infeld black holes PhysRev D 70 (2004) 104014 [arXivgr-qc0408078]

[45] M H Dehghani N Alinejadi and S H Hendi Topological Black Holes in Lovelock-Born-InfeldGravity Phys Rev D 77 (2008) 104025 [arXiv08022637 [hep-th]]

[46] Q Exirifard and M M Sheikh-Jabbari Lovelock gravity at the crossroads of Palatini andmetric formulations Phys Lett B 661 (2008) 158 [arXiv07051879 [hep-th]]

[47] M Borunda B Janssen and M Bastero-Gil Palatini versus metric formulation in highercurvature gravity JCAP 0811 (2008) 008 [arXiv08044440 [hep-th]]

[48] G J Olmo and D Rubiera-Garcia Palatini f(R) black holes in nonlinear electrodynamicsPhys Rev D 84 (2011) 124059 [arXiv11100850 [gr-qc]]

ndash 23 ndash

[49] C Barragan and G J Olmo Isotropic and Anisotropic Bouncing Cosmologies in PalatiniGravity Phys Rev D 82 (2010) 084015 [arXiv10054136 [gr-qc]]

[50] C Barragan G J Olmo and H Sanchis-Alepuz Bouncing Cosmologies in Palatini f(R)Gravity Phys Rev D 80 (2009) 024016 [arXiv09070318 [gr-qc]]

[51] H Stephani D Kramer M Maccallum C Hoenselaers and E Herlt Exact Solutions ofEinsteinrsquos Field Equations (Cambridge University Press Cambridge 2003)

[52] C W Misner and J A Wheeler Classical physics as geometry Gravitation electromagnetismunquantized charge and mass as properties of curved empty space Ann Phys 2 (1957) 525

[53] F S N Lobo Exotic solutions in General Relativity Traversable wormholes and rsquowarp driversquospacetimes [arXiv07104474 [gr-qc]]

[54] A V B Arellano and F S N Lobo Non-existence of static spherically symmetric andstationary axisymmetric traversable wormholes coupled to nonlinear electrodynamics ClassQuant Grav 23 (2006) 7229 [arXivgr-qc0604095]

[55] A V B Arellano and F S N Lobo Evolving wormhole geometries within nonlinearelectrodynamics Class Quant Grav 23 (2006) 5811 [arXivgr-qc0608003]

[56] F Bastianelli J M Davila and C Schubert Gravitational corrections to the Euler-HeisenbergLagrangian JHEP 0903 (2009) 086 [arXiv08124849 [hep-th]]

[57] J M Davila and C Schubert Effective action for Einstein-Maxwell theory at order RF4Class Quant Grav 27 (2010) 075007 [arXiv09122384 [gr-qc]]

[58] B Goncalves G de Berredo-Peixoto and I L Shapiro One-loop corrections to the photonpropagator in the curved-space QED Phys Rev D 80 (2009) 104013 [arXiv09063837 [hep-th]]

[59] J A Wheeler Geons Phys Rev 97 (1955) 511

[60] K A Bronnikov V N Melnikov G N Shikin and K P Staniukowicz ScalarElectromagnetic And Gravitational Fields Interaction Particle - Like Solutions Ann Phys(NY) 118 (1979) 84

[61] M Ferraris M Francaviglia and I Volovich The Universality of vacuum Einstein equationswith cosmological constant Class Quant Grav 11 (1994) 1505 [arXivgr-qc9303007]

[62] A Borowiec M Ferraris M Francaviglia and I Volovich Universality of Einstein equationsfor the Ricci squared Lagrangians Class Quant Grav 15 (1998) 43 [arXivgr-qc9611067]

[63] N S Kardashev I D Novikov and A A Shatskiy Astrophysics of Wormholes Int J ModPhys D 16 (2007) 909 [arXivastro-ph0610441]

[64] M Yu Khlopov Primordial Black Holes Res Astron Astrophys 10 (2010) 495[arXiv08010116 [arXivastro-ph]]

[65] A Ashtekar Lectures on Non-perturbative Canonical Gravity (World Scientific Singapore1991)

ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution







84 Horizons and mass spectrum when 1= c



Contents

1 Introduction 1

2 Basics of Palatini gravity 4

3 The matter sector 6

4 Palatini gravity with matter 7

5 Solving the equations 8

51 The metric ansatz 852 General solution 9

6 The Born-Infeld model 10

7 Quadratic gravity coupled to Born-Infeld 11

8 Exactly solvable model 12

81 Metric components 13

82 Curvature invariants 1483 Wormhole structure and electric flux 1584 Horizons and mass spectrum when δ1 = δc 1785 Geons as point-like particles 19

9 Summary and conclusions 20

1 Introduction

Black holes are a genuine prediction of general relativity (GR) that set the classical limits ofvalidity of Einsteinrsquos theory itself in the high-curvature regime defined by their singularityThey also pose severe challenges to a consistent description of physical phenomena in whichgravitation and quantum physics must be combined to give a reliable picture Black holes arethus regarded as a means to explore new physics where gravitation approaches the quantumregime and to understand the fate of quantum information in the presence of event horizonsand singularities The existence of extra dimensions required by the stringM theory ap-proach would also affect the dynamics of black hole formationevaporation and the physics ofelementary particles at very high energies For these reasons the physics of black holes andin particular of microscopic black holes is nowadays a very active field of research from boththeoretical and experimental perspectives In fact it has been suggested that microscopicblack holes could be created in particle accelerators [1ndash3] and numerous studies have beencarried out to determine the feasibility of their production [4ndash8] and to characterize theirobservational signatures [9 10] In this sense the general view is that extra dimensions arerequired to reduce the effective Planck mass to the TeV energy scale [11ndash14] (see also the re-view [15]) which would make them accessible through current or future particle acceleratorsOnce produced the semiclassical theory predicts that angular momentum and charge shouldbe radiated away very rapidly yielding a Schwarzschild black hole as the late-time phase of

ndash 1 ndash





radic


ndash 2 ndash



Sg =c3

16πG

int

d4xradicminusg[


microν)]

(11)






ndash 3 ndash





S[gΓ ψm] =1

2κ2

int






λmicroβ (22)




fRRmicroν minusf



nablaβ


microν)]

= 0 (24)





fRPmicroν minus f


αPαν = κ2Tmicro

ν (25)

ndash 4 ndash


2fQP2 + fRP minus f

2I = κ2T (26)


micro



nablaβ


ν)]

= 0 (27)


Σαν = (fRδ

να + 2fQPα

ν) (28)







ν

radic

det Σ hmicroν =

(radic

det Σ)









Rmicroν(h) =

1radic

det Σ

(

f


ν

)

(212)


ndash 5 ndash

3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution














radic


ndash 2 ndash



Sg =c3

16πG

int

d4xradicminusg[


microν)]

(11)






ndash 3 ndash





S[gΓ ψm] =1

2κ2

int






λmicroβ (22)




fRRmicroν minusf



nablaβ


microν)]

= 0 (24)





fRPmicroν minus f


αPαν = κ2Tmicro

ν (25)

ndash 4 ndash


2fQP2 + fRP minus f

2I = κ2T (26)


micro



nablaβ


ν)]

= 0 (27)


Σαν = (fRδ

να + 2fQPα

ν) (28)







ν

radic

det Σ hmicroν =

(radic

det Σ)









Rmicroν(h) =

1radic

det Σ

(

f


ν

)

(212)


ndash 5 ndash

3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution












Sg =c3

16πG

int

d4xradicminusg[


microν)]

(11)






ndash 3 ndash





S[gΓ ψm] =1

2κ2

int






λmicroβ (22)




fRRmicroν minusf



nablaβ


microν)]

= 0 (24)





fRPmicroν minus f


αPαν = κ2Tmicro

ν (25)

ndash 4 ndash


2fQP2 + fRP minus f

2I = κ2T (26)


micro



nablaβ


ν)]

= 0 (27)


Σαν = (fRδ

να + 2fQPα

ν) (28)







ν

radic

det Σ hmicroν =

(radic

det Σ)









Rmicroν(h) =

1radic

det Σ

(

f


ν

)

(212)


ndash 5 ndash

3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution














S[gΓ ψm] =1

2κ2

int






λmicroβ (22)




fRRmicroν minusf



nablaβ


microν)]

= 0 (24)





fRPmicroν minus f


αPαν = κ2Tmicro

ν (25)

ndash 4 ndash


2fQP2 + fRP minus f

2I = κ2T (26)


micro



nablaβ


ν)]

= 0 (27)


Σαν = (fRδ

να + 2fQPα

ν) (28)







ν

radic

det Σ hmicroν =

(radic

det Σ)









Rmicroν(h) =

1radic

det Σ

(

f


ν

)

(212)


ndash 5 ndash

3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











2fQP2 + fRP minus f

2I = κ2T (26)


micro



nablaβ


ν)]

= 0 (27)


Σαν = (fRδ

να + 2fQPα

ν) (28)







ν

radic

det Σ hmicroν =

(radic

det Σ)









Rmicroν(h) =

1radic

det Σ

(

f


ν

)

(212)


ndash 5 ndash

3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










3 The matter sector


Sm =1

8π

int







Sm = minus 1

16π

int



Tmicroν = minus 1

4π

[

ϕXFmicroαFα


δνmicro4ϕ(XY )

]

(33)


nablamicro


)

) = 0 (34)


2 + grrdr2 + r2dΩ2


F tr =q


(35)


ϕ2XX =

q2

r4 (36)


Tmicroν =

1

8π

(

(ϕminus 2XϕX)I 0

0 ϕI

)

(37)



ndash 6 ndash

2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










2fQ

(

P +fR4fQ

I

)2

=

(

λ2minusI 0

0 λ2+I

)

(38)

where

λ2+ =1

2

(

f +f2R4fQ

+ 2k2T θθ

)

=1

2

(

f +f2R4fQ

+ κ2ϕ

)

(39)

λ2minus =1

2

(

f +f2R4fQ

+ 2k2T tt

)

=1

2

(

f +f2R4fQ


)

(310)


radic

2fQ

(

P +fR4fQ

I

)

=


(311)


radic

2fQ

(

P +fR4fQ

I

)

=

(

λminusI 0

0 λ+I

)

(312)


Σ =fR2I +

radic

2fQ

(

λminusI 0

0 λ+I

)

=

(

σminusI 0

0 σ+I

)

(313)

where σplusmn =(

fR2 +

radic

2fQλplusmn

)




Rmicroν(h) =

1

2σ+σminus

(

(f + 2κ2T tt )I 0

0 (f + 2κ2T θθ )I

)

(41)


f(RQ) = R+ l2P (aR2 + bQ) (42)


R = minusκ2T (43)


ndash 7 ndash


T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











T = 2T tt + 2T θθ =1



Q =T 2

4+


(1minus (2a+ b)χT )2 (45)


λ+ =1

2radic2blP

(


2


)

(46)



σ+ = 1 + 2χ

[


4 χT 2

1minus (b+ 2a)χT

]

(48)






κ2l2P β2 r


Q = κ4q4






2 + r2dΩ2 (51)



2 + r2dΩ2 (52)



ndash 8 ndash

gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










gmicroν =

gtt 0 0 00 grr 0 00 0 r2 00 0 0 r2 sin2 θ

=

httσ+

0 0 0

0 hrrσ+

0 0

0 0 hθθσminus

0

0 0 0hφφσminus

(53)



Fig2 below]

52 General solution



2 + r2dΩ2 (54)



ν(h) become

Rtt = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr +2

r

)

]

(55)

Rrr = minus 1

2hrr

[ArrA

minus(

ArA

)2

+ 2ψrr +

(

ArA

+ 2ψr

)(

ArA

+ ψr

)

+2

r

ArA

]

(56)

Rθθ =

1






1


1

2σ+σminus

(

f +κ2

4πϕ

)

(58)


A(r) = 1minus 2M(r)

r (59)


Mr =r2

4σ+σminus

(

f +κ2

4πϕ

)

(510)

ndash 9 ndash


12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











12minus

(

1 +rσ

minusr

2σminus

)

and with


Mr =

(

f + κ2

4πϕ)

r2σ12minus

4σ+

(


)

(511)





M(r)

M0= 1 + δ1G(r a b ) (512)




ϕ(XY ) = 2β2

(

1minus

radic

1minus X

β2minus Y 2

4β4

)

(61)



variable z4 = r4


X

β2=

1

1 + z4 (62)


κ2T tt =1

l2β

(

1minusradicz4 + 1

z2

)

κ2T θθ =1

l2β

(


)

(63)

ndash 10 ndash


κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











κ2T = minus 2

l2β

(

z2 minusradic1 + z4

)2

z2radic1 + z4

(64)


ε(q) = 4π

int infin


int infin

0dt(radic

t4 + 1minus t2) = nBIβ12q32 (65)

where nBI =π32





gtt = minus A(z)

σ+(z) grr =

σminus(z)

σ+(z)A(z)

(


)2

(71)




δ1 =1

2rS

radic

r3qlβ

δ2 =

radic

rqlβ

rS (73)


Gz =z2σ

12minus

σ+

(


)

(

l2βf + ϕ)

(74)



z =ǫ12radic

2(1 + ǫ)14(75)

γ1 = (2a+ b)λ (76)

γ2 = (4a+ b)λ (77)

ndash 11 ndash


Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











Gǫ =

radicǫ(

ǫ(2 + ǫ)2 + 8γ1) (

(2 + ǫ)3 minus 8γ1(4 + 3ǫ))

2radic2(1 + ǫ)74(2 + ǫ)2(ǫ(2 + ǫ) + 8γ1)2

σ+

σ12minus

(78)



ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (79)

and


ǫ(2 + ǫ) (2ǫ+ ǫ2 + 8γ1) (710)







Gǫ =2 + ǫ+ ǫc

2radic2(1 + ǫ)74

radicǫminus ǫc

(81)

σ+ = 1 +ǫc

2 + ǫ(82)




(1+4λ)14



ndash 12 ndash

Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










Λ=0

Λ=10-2

Λ=10-1

Λ=1

Λ=101

05 10 15 20z

-20

-15

-10

-05GzD

GzD for various Λ



2λ(1+4λ)14




Gz =2zW [z]

(

z2 + (1 + 2λ)W [z])

radic1 + z4

radic




Gz asymp 2(radic

1 + z4 minus z2)

+O(λ) (85)


G(ǫ) = minus 1

δc+

radic

2(ǫminus ǫc)

3(1 + ǫ)34

[

1 + 2F1

[

1

23

43

2minus(ǫminus ǫc)

1 + ǫc

]

2

(

1 + ǫ

1 + ǫc

)

34]

(86)


δc =(1 + 4λ)14

2nBI (87)







ndash 13 ndash




gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution













gtt asymp minus1 +1

δ2zminus δ1δ2z2

+λ

z4+O

(

1

z5

)

(89)




+(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(




gtt asymp(1 + 2λ)

(


)

(1 + 4λ)32δ2+O(z minus zc) (811)






R(g) =r4q l

2β

2r8

(

1minus 28l2Pr2

+

)

(812)

Q(g) =r4qr8

(

1minus 16l2Pr2

+

)

(813)

K(g) = KGR +144rSr

2q l

2P

r9+ (814)


ndash 14 ndash


R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











R = (δ1 minus δc)

[

α1

(z minus zc)3

2

+α2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C1 + (815)


[

ζ1(z minus zc)3

+ζ2

(z minus zc)2+

ζ3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ξ1

(z minus zc)3

2

+ξ2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C2 + (816)


[

micro1(z minus zc)3

+micro2

(z minus zc)2+

micro3(z minus zc)1

]

+

+ (δ1 minus δc)

[

ν1

(z minus zc)3

2

+ν2

(z minus zc)1

2

+O[(z minus zc)1

2 ]

]

+ C3 + (817)



R asymp4(

16λ2 + 11λ+ 3)


(

8λ2 + 4λ+ 1)

3λ(2λ + 1)2δ2+O(z minus zc) (818)

Q asymp D(9(

160λ5 + 232λ4 + 160λ3 + 60λ2 + 12λ+ 1)

δ22

minus 12radic4λ+ 1

(

104λ4 + 128λ3 + 78λ2 + 23λ + 3)

δcδ2

+ 2(

688λ4 + 976λ3 + 652λ2 + 204λ+ 27)

δ2c)

(819)

+ O(z minus zc)

K asymp 4λ+ 1

λ2+

2(


)2

λ2δ22+

16(


)2

9(2λ + 1)4δ22+ O(z minus zc) (820)

where D = 19δ2

2λ2(2λ+1)4





ndash 15 ndash

z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










z=zxD

x0

dG

dx

-3 -2 -1 1 2 3 4x

1

2

3

4








x(z) =


(821)


π8

Γ[minus 1

4 ]Γ[ 14 ]

z2c(1+4λ)14

and

F (zλ) = xc +

radicǫminus ǫc

2radic2(1 + ǫ)14 (1 + ǫc)

(

2 +2 F1

[

1

41

23

2minus ǫminus ǫc

1 + ǫc

]

ǫc

(

1 + ǫ

1 + ǫc

)

14)



int


ndash 16 ndash


Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











Φ

4πr2cz2c

=

(

4λ

1 + 4λ

)12radic

c7

2(~G)2 (822)







Mq where NM

q equivradic




Nq lt NMq

radic

1 +1

4λ (824)

ndash 17 ndash


q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











q

radic

1 + 14λ the



MBI =

(

1 +1

4λ

)12( Nq

NBIq

)32

nBImP (825)


MBI =

(

4λ

1 + 4λ

)14

MM (826)








2P




ndash 18 ndash




S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution













S =1

2κ2

int


2R2 + l2PQ) +

1

8π

int


int



S = minus(rqlβ)32

int

dt

int +infin


32

int

dt

int +infin

zc

dzz2

σ12minus

T tt σ+ (828)

= 4Mc2δ1I

int

dt




S = 2Mc2δ1δc

int

dt (829)


S = 2MBIc2int

dt (830)


ndash 19 ndash


E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution











E =

int

dxz2dΩ

[

1

2κ2f(RQ) +

ϕ

8π

]

(831)







ndash 20 ndash




Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution













Acknowledgments


References




ndash 21 ndash























ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution
































ndash 22 ndash












(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution





















(1989) 2767













ndash 23 ndash


















ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution



























ndash 24 ndash

1 Introduction


3 The matter sector




52 General solution










semiclassical geons at particle accelerators -...

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