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PHYSICAL REVIEW D 67, 024002 ~2003!
Geometric dilaton gravity and smooth charged wormholes
Wolfgang Graf*Institut fur Theoretische Physik der Universita¨t Wien, Boltzmanngasse 5, A-1090 Wien, Austria
~Received 15 September 2002; published 8 January 2003!
A particular type of coupling of the dilaton field to the metric is shown to admit a simple geometricinterpretation in terms of a volume element density independent from the metric. For dimensionn54 twofamilies of either magnetically or electrically charged static spherically symmetric solutions to the correspond-ing Maxwell-Einstein-dilaton field equations are derived. Whereas the metrics of the ‘‘magnetic’’ spacetimesare smooth, asymptotically flat, and have the topology of a wormhole, the ‘‘electric’’ metrics behave similarlyas the singular and geodesically incomplete classical Reissner-Nordstro¨m metrics. At the price of losing thesimple geometric interpretation, a closely related ‘‘alternative’’ dilaton coupling can nevertheless be defined,admitting as solutions smooth ‘‘electric’’ metrics.
DOI: 10.1103/PhysRevD.67.024002 PACS number~s!: 04.40.Nr, 04.20.Cv, 04.20.Jb, 14.80.Hv
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I. INTRODUCTION
Einstein’s classical theory of gravity, based on a methas so far passed all experimental tests concerning thetion of bodies or the deflection of light, with ever increasiprecision~cf. Will @1#!. But there are some disturbing limtations of a more formal nature, such as generic singulariwhen not allowing ‘‘exotic matter.’’ This is most evident ithe gravitational collapse of stars and in the early phasethe Universe, notably at the ‘‘big bang singularity.’’ In fact,seems that we have come to live with this unsatisfactsituation as a necessary consequence of a classical detion. These problems can be attributed either to Einstetheory itself or to the inadequacy of the matter model usor to both. In the first case, singularities seem to be a genfeature ~somewhat attenuated by the expectation that twill in general be hidden behind an event horizon!, unlesssome other ‘‘pathologies’’ are accepted, such as exotic maor closed timelike lines. In the other case, for example,apparent accelerating cosmic expansion has led to a gealization involving a ‘‘variable cosmological constant’’L~‘‘quintessence,’’ ‘‘k essence’’!. Previously, a generalizatiobased on a ‘‘variable gravitational constant’’G has been con-sidered, well known as Brans-Dicke theory. Also, in orderinflation to work, some specific modifications must be maIn all these generalizations, an additional real scalar fiserving the particular purpose, plays a fundamental role.cept in exceptional cases, the singularity problem remainis, however, generally expected that a future reconciliationgravity and quantum theory will lead to a unique theowithout the above mentioned problems. Attempts in thisrection can be seen in the currently extensively stud‘‘string-inspired cosmologies,’’ which are based on some lenergy limit of string theory and an appropriate reductiondimensionn54. There a scalar field multiplying~as an ex-ponential! the Ricci scalar and/or the metric plays a promnent role, which is interpreted physically as the dilaton fieA big advantage of these approaches is the fact that e
*Electronic address: [email protected]
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tions closely related to Einstein’s are a necessary conquence.
Our aim is to show that already on a classical level,properly including the dilaton scalar, some of the singularproblems can be avoided. Firmly based on standard diffetial geometry and only loosely inspired by the dilaton scaof string theory, we formulate a theory of dilaton gravity. Acriteria for the soundness of our approach, both the formution of a minimal coupling scheme and the existence of ntrivial geodesically complete and asymptotically flat sotions of the corresponding field equations are taken.
We will proceed as follows. In Sec. II a particular form othe coupling of a scalar fieldf ~called ‘‘dilaton’’! to themetric tensorgik of a spacetime is proposed, which admitsstraightforward and unique geometric interpretation in terof an independentvolume element density. A dilatonic cou-pling scheme is formulated in order to accomodate additionongeometric fields. In Sec. III, the field equations corsponding to a Maxwell-Einstein-dilaton Lagrangian are drived, where also their ‘‘Einstein form’’ is given. In Sec. IVfamily of magnetically charged static spherically symmetsolutions is derived, closely related to the well-know‘‘string-inspired’’ charged black hole solutions of GarfinkleHorowitz, and Strominger@2#. These metrics are shown to bgeodesically complete~in fact, smooth! and asymptoticallyflat, each of them containing awormhole, with no exoticmatter being involved. The corresponding family of electcally charged solutions consist of singular metrics whhave either an event horizon or exhibit a naked singularAn ‘‘alternative’’ nongeometric coupling is shown to behowever, possible, admitting a family of smooth electricacharged solutions. In Sec. V, for the proposed geometric cpling the equivalence principle is shown to be fulfilled, in thsense that uncharged point particles still move on geodeIn order to derive expressions for the mass and some osignificant parameters, the parametrized post-Newtonianproximation is invoked. In terms of the basic parametersm,b, and g, it is shown that the derived solutions must bconsidered as viable with respect to present-day astronomempirical data. For the realm of elementary particles, itshown that for the smooth magnetic and electric wormhsolutions, significant effects could be expected at distan
©2003 The American Physical Society02-1
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WOLFGANG GRAF PHYSICAL REVIEW D67, 024002 ~2003!
of roughly the order of the classical electron radius. Forwormhole solutions it is shown that their mass parameterzero as a lower bound. Also an explanation of the ‘‘repsive’’ character of the dilaton involved in the wormhole slutions is given. The concluding Sec. VI formulates the mconclusions and points to some important open question
II. GEOMETRIC DILATON COUPLING
A. Volume geometry: Hodge duality
As is well-known~cf., e.g., Abraham, Marsden, and Rat@3# for the closely related concept of avolume manifold!, avolume element density~VED! is geometrically a nondegenerate smooth n-form density volume, that is, undeorientation-preserving coordinate transformations it behaas a conventionaln form, whereas under orientation revesion it gets an extra factor21. Such a VED already allowsone to invariantly express the divergence divv of a vectorfield v as~div v! volªd(v • vol), where the dot denotes thcontraction of a differential form by a vector field. This denition is well known in Hamiltonian mechanics, whereplays a major role. Also the Gauss integral theorem canready be formulated. Note that no metric has been involso far.
It makes sense to speak of a ‘‘positive’’ VED, anany two such VEDs differ only by a positive functionl.Assuming that we also have a nondegenerate metricgik ,we can therefore always write for a general VED, v5 ugu1/2ef dx1`•••`dxn, where we have convenientlset l5ef, with some scalar functionf.1 This functionalform ensures positivity ofl, when f is continuous. Ofcourse we could also have chosen any other smooth mtone positive function off, but this would not introduceanything new, as effectively only the ‘‘dilaton factor’’ef
matters. The scalar fieldf thus represents anessentiallyunique new geometrical degree of freedom.
Its occurrence in the form of the factoref strongly re-minds one of the dilaton factor appearing in the reducLagrangians for the low energy limit of string theo~LELST!.2
For a volume manifold with a nondegenerate metric,notion of theHodge dualof a differential form has to besligthly generalized. Recall that the dual!F of a plainp-form F is the result of the following construction, givewith respect to some coordinate basis:
Fi 1••• i p→F j 1••• j p
ªgi 1 j j•••gi pj pFi 1••• i p
→!F j p11••• j n
ªvolj 1••• j pj p11••• j nF j 1••• j p
[ugu1/2ef « j 1••• j pj p11••• j nF j 1••• j p, ~1!
where« denotes the permutation symbol. The plainp-form F
1As is common practice in the physics literature, we will denothe corresponding coefficient of then form dx1`•••`dxn as ‘‘sca-lar density,’’ e.g., the LagrangianL.
2Although there the equivalent factore22f seems to be morenatural.
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is thus mapped to a (n2p)-form density!F. As this map isone-to-one, it can be inverted to map ap-form H density to aplain (n2p)-form !21H. Unfortunately, the nomenclaturof the two different types of differential forms is far fromstandard3 and so we will adhere to de Rham@4#, denoting theforms we called ‘‘plain’’ with even typeand the ‘‘form den-sities’’ with forms of odd type.
Consequently, we now define the generalized Hodge dof any formF ~even or odd! as
* F5H !F, F even,
!21F, F odd.~2!
This definition makes the duality operator trivially idemptent, *2F5F.4
Alternatively, we could also define the Hodge dualityF→* F as theunique isomorphismfrom the vector space oeven~odd! p forms to the ‘‘dual’’ vector space of odd~even!(n2p) forms, whose restriction to even forms gives
F`* G5~F,G! vol, F,G even. ~3!
Here the round bracket denotes the scalar product of fobased on the Riemann metric. As a consequence, we havodd forms the corresponding relation
* F`G5~* F,* G! vol, F,G odd. ~4!
Based on the Hodge duality for differential forms, the opetors for thedivergenced and theLaplacianD for differentialforms can now be defined asdª* d* and Dªdd1dd,whered denotes the operator ofexterior derivative, which isvalid for forms of any even/odd type.5
Why this insistence on differential forms? The main reson is that the Lagrangian scalar density is geometricmore properly understood as an odd form of maximal degn, the energy momentum tensor being a covector-valued(n21) form. Of course, the electromagnetic Maxwell fielto be extensively used later together with its Hodge dualto be understood as an even 2 form. Also we need the digence of a vector field and of a two form, as well as tLaplacian of a scalar field.
B. General dilaton coupling
Be it from a five-dimensional Klein-Kaluza~KK ! reduc-tion, or from a LELST compactified ton54, theLagrangians have the generic form6
3Also the following designations are common: pseudoformtwisted forms, Weyl tensors, and oriented tensors.
4Note that when distinguishing even and odd type forms,!2 doesnot make sense.
5The more precise definition ofd by de Rham introduces an extrfactor21, depending on the dimensionality of the manifold and tsignature of the metric.
6We use the conventions of Misner, Thorne, and Wheeler@7#throughout, the squares denoting the conventional metric scproduct.
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GEOMETRIC DILATON GRAVITY AND SMOOTH . . . PHYSICAL REVIEW D67, 024002 ~2003!
L5ugu1/2eaf @R2b~¹f!22egfF2#. ~5!
The constant parametersa, b, andg depend on the particular higher-dimensional base theory and the chosen reducOf course, from a LELST many more scalar fields and asymmetric tensor fields of different degrees~‘‘moduli’’ ! willappear, but we keep only the dilaton scalarf and a rank-2antisymmetric tensor fieldF, later to be interpreted as thMaxwell field two form.7 R denotes the omnipresent Riemann curvature scalar of general relativity. We will referthis class of Lagrangians as Maxwell-Einstein-dilat~MED! Lagrangians.
Often, more general Lagrangians are studied, containfree functions~cf. Esposito-Fare´se and Polarski@5#, and ref-erences therein!, but our choice is already general enoughcover the most important applications as special casesparticular, the class of Bergmann-Wagoner Lagrangianstensively studied for about 30 years by Bronnikovet al. ~cf.@6#! should be mentioned, having the general formL5ugu1/2@ f (f)R1h(f)(¹f)22F2# ~in particular, with h51 and f 512jf2, wherej is a constant parameter!. Al-though diverse charged wormhole solutions have beentained, they all violate some of the energy conditions. Nthat the class of MED Lagrangians considered in our wessentially differs from Bronnikov’s class, which completeexcludes ‘‘string-inspired’’ Lagrangians.
As convenient for dimensionally reduced Lagrangiathe relativistic gravitational constantk ~as well as any factor1/2) is assumed to be absorbed intoF2. For example, the KKreduction leads toa51, b50, g52, whereas a typicaLELST reduction hasa522, b524, g50. With a scalardefined byFªef, we can also deduce the scalar-tensBrans-Dicke Lagrangian from this form, with parametersa51, b5v5” 3/2, g521. Similarly, for b53 we get a con-formal scalar coupling.
However, all these MED Lagrangians can be transforma moduloa trivial divergence by a Weyl conformal transfomationgik8 5gik eaf into a so-calledEinstein frame, charac-terized bya850. For dimensionn54, this results inb85b13/2a2,g85g1a. In such a frame, formally the conventional Einstein Lagrangian is obtained, with a masslKlein-Gordon~KG! field f and a Maxwell fieldF gravita-tionally coupled with an effective coupling ‘‘constante(a1g)f. The scalar fieldf is ghost-free, i.e.,nonexotic, ifand only if b8>0.
More generally, for any dimensionn, a dilaton-based conformal transformation of the metric,gik8 5gik elf (l a con-stant parameter!, leads to the following transformation behavior of the parameters of the MED Lagrangian density
a→a85a2l,
b→b85b1~n21!~n22!/4 ~a22a82!, ~6!
g→g85g1l.
7Sometimes ‘‘potential’’ termsV(f) also appear or are introduce‘‘by hand.’’
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Although the causal structure is not altered as long asf iscontinuous,8 some basic metric-based relations~e.g., lengthand ‘‘straightness’’! are not. For the physical interpretatiotherefore some conformal frame has to be taken as thedamental one, characterized by a particular form of thegrangian density. This will depend on the particular couplito other fields, in particular, to point masses. However‘‘string-inspired’’ dilaton theories most authors agree thatEinstein frame should be taken, primarily justified by tavailability of the familiar interpretatory apparatus of thclassical Einstein gravity.9
C. Geometric dilatonic coupling and minimal coupling scheme
Let us now introduce a particularly simple coupling, chaacterized by the parametersa51, b50, g50. Evidently,the dilaton enters the corresponding MED Lagrangian inmode which can be interpreted geometrically in terms ogeneral metric-based volume element density, as previodescribed. The Lagrangian simply becomesL5ugu1/2ef (R2F2). Let us call such a coupling ageometric dilaton cou-pling ~GDC! and the corresponding conformal frame aGframe (G for ‘‘geometric’’!.
The particularly simple form of the GDC suggests tfollowing minimal geometric dilatonic coupling schem~MGDCS!: assuming we have already a Lagrangian denwithout dilaton and satisfying the prerequisites of genecovariance,L05ugu1/2L, we get a GDC by just correctingany occurence of the Riemann volume element densityugu1/2
by the dilaton factoref: LªL 0 ef5ugu1/2ef L, even whenit explicitly occurs insideL.
Taking the standard Maxwell-Einstein Lagrangian asprototypical example, we get as a result of the MGDCSGDC Lagrangian:
MGCDS: L5ugu1/2~R21/2F2!°L5ugu1/2ef ~R21/2F2!. ~7!
Similarly, a massive point particle could also be incorprated, to giveL5ugu1/2ef R2m( x2)1/2dT , where the Diracdelta distribution is supported by the world-lineT of theparticle, given byT:t°xi(t). Note that the mass term doenot acquire a dilaton factor, as it does not contain the facudetgu1/2.
III. GDC FIELD EQUATIONS
The field equations derived from the geometricacoupled Maxwell-Einstein-dilaton Lagrangian L5ugu1/2ef (R21/2 F2) are, up to a factorugu1/2ef,
Gik2s2 Q ik8 2s Q ik9 5Mik , ~8!
R51/2F2, ~9!
05div F, ~10!
8Which is, however, not in general the case.9Compare, e.g., Gasperini and Veneziano@8#.
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where
MikªFir Fks grs21/4F2 gik , ~11!
Q ik9 ª¹i¹kf2¹2f gik , ~12!
Q ik8 ª¹if¹kf2~¹f!2 gik , ~13!
and
sª~n22!/2. ~14!
Here divF denotes the dilaton-generalized divergence ofform, which in a coordinate base can also be written as
~div F ! iªugu21/2e2f ] j~ ugu1/2ef Fi j 
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GEOMETRIC DILATON GRAVITY AND SMOOTH . . . PHYSICAL REVIEW D67, 024002 ~2003!
The parametersq,r 1 ,r 2 are, however, restricted by14
q252 r 1r 2 /~11a2!. ~31!
Mapping thea251/3 GHS solution back to theG frame, wethen get the following ‘‘magnetic’’ family of solutions:
ds252X1dt21~X1X2!21dr21r 2dV2, ~32!
ef5X21/2, F5Fmªq sinu du`dw, ~33!
where
q2ª3/2r 1r 2 . ~34!
However, in these coordinates the metric isnot regular atr5r 2 , and the dilaton scalar does not even exist forr,r 2 . Introducing a new coordinater ~solution of dr/dr5« X2
21/2) by means of
r~r !5«r 2 „j1/2~j21!1/21 ln~j1/21~j21!1/2!…, jªr /r 2 ,~35!
removes these drawbacks. The parameter«561 character-izes the twor branches joined atr50 and mapping to thesingle r range r>r 2.0. The inverse functionr (r) issmooth inr. In these new cordinates~which now properlyinclude the locusr 5r 2 , respectively,r50), the resultingmetric
ds252X1dt21X121dr21r 2~r!dV2 ~36!
is nondegenerate15 and can be shown to besmooth in aneighborhood ofr50 which does not includer5r(r 1).16
Also, ef becomes smooth there, and the expression forMaxwell field Fm remains unchanged~and smooth!.
As the metric is symmetric underr→2r, and there isnow always a two sphere with minimal areaA54pr 2
2 @cor-responding to a ‘‘radius’’r (0)5r 2], the geometric interpre-tation is that of awormholewith throat atr50. This notionof wormhole, based on alocal reflection symmetry, is, how-ever, different and more general than the usual one, whonly allows wormholes withtimelike throats ~e.g., Visser@12#, compare also Hayward@13#!.
If r 1>r 2 the locusr 5r 1 can be shown to be aregularevent horizon~even in the ‘‘degenerate’’ caser 15r 2), andthe metric can be smoothly extended through it by standprocedures~e.g., by using Eddington-Finkelstein coordnates!. For r 1,r 2 there is no black hole and the complemetric can be expressed in the single coordinate chart gabove. The wormhole topology~connecting universes withtheir own asymptotic regions! is common to all metrics ofthis family and can in this sense be considered asgenericamong the class of SSS solutions.
14Despite the suggestive notation, it isnot required that r 1
>r 2 .15Except for the usual easily removable degeneracy at thez axis.16Note the close similarity of the form of this metric wit
Schwarzschild’s, to which it reduces forr 250.
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The Carter-Penrose~CP! diagrams corresponding to thesextensions fall into the three distinct types I:r 2.r 1>0, II:r 25r 1.0, III: r 1.r 2.0, and are shown in Fig. 1~wherewe also used the more physical characterization by meanchargeq versus massm, to be justified later!. Depending onthe type of the solution, the throat of the wormhole istime-like for type I, null for type II ~coinciding with the eventhorizon!, andspacelikefor type III.
B. Gauge potential for the magnetic GDC solution
Although the question of an appropriate gauge potentiamost often ignored~being trivial in the ‘‘electric’’ case!, wewill now exhibit a smooth potential in the sense ofU(1)-gauge theory. The existence of such a potential mathe smooth SSS solution complete. Consider theu(1)-valued~i.e., purely imaginary!
A6ª2 in/2 ~cosq71!dw, ~37!
where the upper sign refers to the upper hemisphereq5” pand the lower sign to the lower hemisphereq5” 0. Evidently,FªdA5 in/2 sinq dq`dw. The transition function for thepotential in the overlap of the two hemispheres is givenSªeinw: A15A21S21dS. For consistencynPN musthold ~cf. Gockeler and Schu¨cker @14# for more details!. Re-verting to the corresponding real field,iFªF, this amountsto q5n/2. Taking properly into account the terms appeariin the ‘‘gauge derivative’’ for an electrically charged particin the field of a magnetic monopole,¹5]1 ie/\ A, we ob-tain Dirac’s quantization condition: pq/\5n/2. For n51and p5e the minimal magnetic charge isg51/2e/\'68.5e, giving the factor (g/e)2'4.73103 needed inSec. V.
C. Singular GDC solutions of electric type
As already shown by Garfinkle, Horowitz, anStrominger @2#, from a magnetically charged solutio(g,f,F) of their E frame field equations, an electricall
FIG. 1. Carter-Penrose diagrams for the extended wormhmetrics. Thick lines: null infinity, thin lines: event horizon, dashlines: wormhole throat, circles:i 6,i 0.
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charged one can formally be obtained by taking (g,2f,* F), where *F is the ~generalized! Hodge dual ofF.17
But then, the transformation back to theG frame inevitablyleads to a metric immanently degenerate atr 5r 2
ds25X2~2X1dt21X121dr21r 2~r!dV2!, ~38!
which is the image of the degenerate conformal mappwith factor X2 of the smooth ‘‘magnetic’’ metric considerebefore.18 Dilaton factor and Maxwell field are given by
ef5X221/2, F5Feªp/r 2 X2
1/2dr`dt, ~39!
wherep253/2r 1r 2 . A gauge potential forF is
Aeª«p/r 2 X2 dt. ~40!
Note that no charge quantization is involved and thatF van-ishes at the singularity of the metric,r50 (r 5r 2), in fact,both F and its potential are smooth there.
If r 1.r 2 and r 1.0, the locusr 5r 1 is a regular eventhorizon, hiding the spacelike singularity. For the ‘‘degenate’’ caser 15r 2 , there is still a horizon, but the singularitbecomes timelike. Forr 1,r 2 , the singularity is timelikeand even naked. The corresponding CP diagrams agreethose of the Reissner-Nordstro¨m family of solutions, exceptfor the case r 1.r 2 , where the diagram agrees witSchwarzschild’s, which has a spacelike singularity.
D. Alternative dilaton coupling
A closer look at the general GHS solution reveals tonly the choicea251/3 allows one to remove the offendincommonX2 factor from the SSS metric by an appropriaconformal transformation. This again is given by the safactor e22aw, as from theG frame to theE frame, resultingin the ‘‘alternative’’ dilaton coupling~ADC! Lagrangian
L5ugu1/2~e2f R21/2ef F2!. ~41!
Evidently, it cannotdirectly be interpreted in terms of a voume manifold and corresponding coupling scheme. The fiequations corresponding to this alternative Lagrangianthen
Gik2Q ik8 1Q ik9 5e2f Mik , ~42!
R521/2e2f F2, ~43!
05div ~e2f F !, ~44!
with the corresponding dilaton equation derived from the
hf521/6e2f F2. ~45!
Here the divergence and the Laplacian are defined basedvolume element densityugu1/2e2f. Except for the manifes
17A manifestation of the ‘‘weak/strong coupling duality’’ of strintheory.
18Note thatX2 is non-negative, considered as a function ofr.
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appearance of dilaton factors the essential change is areversal in the dilaton equation.
This shows that smooth electric wormhole solutionspossible when sacrificing the geometrical interpretatiTheir metrics agree with those of the smooth GDC worholes. The coupling is again ghost-free and the materialstill obeys the energy conditions. Among the class of MELagrangians considererd, it is the only Lagrangian wsmooth SSS solutions.
An ‘‘alternative’’ coupling scheme, involving arbitrarynongeometrical fields, could, however, be formulated tentively as follows: ~i! apply the standard GDC scheme,~ii !denote the map from theG frame to theE frame byE, and~iii ! define the ‘‘alternate’’A frame ~and corresponding Lagrangian! as the image of theG frame under the mapE 2.
When including point masses, such a coupling schewould lead to nongeodesic behavior for their trajectories~seeSec. V!. This could be considered as a drawback. Butmain objection against this coupling is of course that it hbeen deliberately constructed so as to possess smoothtrically charged SSS solutions, and also its lack of any dirgeometric interpretation.
E. Comparison with other SSS solutions
As already noted by Garfinkle, Horowitz, and Stromingall the nontrivialE-frame metrics of the GHS family of solutions are either geodesically incomplete and/or singuwith the exception of the ‘‘cornucopion’’ metric, which is aextreme solution fora51 interpreted in the string frame.
To my knowledge all other static spherically symmetsolutions directly or indirectly related to Maxwell-Einsteindilaton gravity violate some of the energy conditions, amust be considered as classically ‘‘unphysical.’’19 Thereforewe will not dicuss them here.
Unfortunately, general enough existence or no-go threms do not yet exist, save for particular couplings andcorresponding conformal frames. For example, forclosely related vacuum Brans-Dicke theory, it has beshown by Nandi, Bhattacharjee, Alam, and Evans@15# thatwhile in the Jordan frame there do exist wormhole-solutiofor the ~unphysical! range23/2,v,24/3 of the BD pa-rameter, which are, however, plagued by ‘‘badly diseasenaked singularities, in theE frame there do not exist sucsolutions at all, unless energy-violating regions are delibately introduced.
However, we must admit that while our GDC/ADCwormhole solutions are smooth as regards metric, voluelement density~i.e., dilaton factoref, respectivelye2f),and gauge potentials, they are not, when instead considethe dilaton scalarf itself, which diverges to2` ~respec-tively 1`) at the throat of the wormhole. Although thiposes no problem for the smoothness of the correspon
19In fact, Bronnikov’s wormhole solutions turn out to be highunstable~cf. @6#!. This is also the case for the recently foun‘‘ghostly’’ massless wormhole solution of Armenda´riz-Picon @16#,as discussed by Shinkai and Hayward@17#.
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TABLE I. PPN parameters for charged static spherically symmetric metrics.
2m b g g asymptotics
GDC/magneticADC/electric r 1 111/6e2 111/6e2 O(e2)GDC/electric r 11r 2 111/3e2 (11x)21 n. a.GHS/E frame r 1 111/4e2 1 O(e0)GHS/S frame r 12r 2 1 (11x)/(12x) O(e1)Reissner-Nordstro¨m r 11r 2 121/2e2 1 O(e0)
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dilaton factors~which just smoothly vanish there!, the geo-metric interpretation in terms of a corresponding nondegerate volume element density cannot anymore be strimaintained, as it is not manifestly positive. This must in fabe considered a flaw, albeit not a serious one, as the phyinterpretation does not suffer.
V. INTERPRETATION
A. GDCÕADC and Einstein’s equivalence principle
Let us emphasize that the GDC theory presented herstill a metric theory, in the sense that massive point sourmove on timelikegeodesicsof the metric, when expressed ia G frame. This can be most easily seen by noting thatvolume integral over the GDC Lagrangian for a point pticle effectively splits into the volume integral over the ‘‘gemetric’’ part of the Lagrangian plus a conventional line intgral m*( x2)1/2dt, which by variation of the arc length theleads to the standard geodesic equation.20
As the alternative coupling scheme seems to be tooficial to be really believed, we introduce the point particterm ‘‘by hand’’ into theA frame Lagrangian. Therefore Einstein’s equivalence principle~EEP! is satisfied both for theGDC and~trivially ! ADC formulations.
However, for both the standard GHS and the generaliGHS solution the validity of the EEP is undecided, asequivalent to a coupling scheme to external sources hasbeen formulated. In the following we will therefore simpassumethe EEP to hold also for the corresponding strininspired theory. This will allow us in the following to interpret the parametrized post-Newtonian~PPN! parameterm forall solutions considered as the mass of the gravitatiosource.
B. PPN viability and experimental verifiability
However, the compatibility with the EEP is not sufficiefor the viability of a generalized theory of gravity, as it donot refer to any particular solution. A framework to chejust this ~in particular, SSS solutions! is well known underthe name of parametrized post-Newtonian~PPN! approxima-tion ~cf. Will @18#!. We will give in tabular form only theresulting most important parameters massm, b, andg, andcompare them on the one hand with the corresponding
20This can also be justified with a ‘‘pure dust’’ model of mattebased on relativistic thermodynamics~paper in preparation!.
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rameters for thea251 GHS solutions~parametrized by(M ,Q,f0),21 where Mªr 1/2, q2
ªr 1r 2 ,22 qªQe2f0),both in theE frame and in theS frame, and on the other hanwith the PPN parameters of the Reissner-Nordstro¨m metric,also given in the same form23
ds252l2dt21l22dr21r 2dV2,
where l2ª~12r 1 /r !~12r 2 /r !. ~46!
In this PPN approximation, the metric has the followin‘‘isotropic’’ form:
ds252@122m/r 12~m/r !2 b#dt21~112m/r g!
3~dr21r 2dV2!. ~47!
Setting eªq/m, xªr 2 /r 1 and assumingm5” 0, r 15” 0,we can collect the results in the Table I, where alsoasymptotic behavior ofg is given fore→`. As for the elec-tric GDC solutione is limited by e2<3/2, in this case anasymptotic expression does not make sense. The metricthe GDC and ADC wormholes agree, so they share a cmon table entry.
Before proceeding, let us observe that theS-frame GHS‘‘cornucopion’’ solution, being degenerate (r 15r 2), hasvanishingPPN massm and so does not fit into the table (band g undefined!. This applies also to both the zero-mamagnetic GDC and electric ADC solutions, wherer 150.
For heavenly bodies, where most empirical data comfrom, we haveueu5uq/mu!1. Assumingr 1'm, we alsohave xªr 2 /r 1[2/3 (q/r 1)2'e2(m/r 1)2!1, so all thesemetrics areequally viable, havingb'g'1.
However, applied to charged elementary particles~whereup to now no PPN data seems to exist!, the correspondinggshow significant differences in their dependence on the ‘‘scific charge’’ ratio q/m. This is most pronounced for thsmooth SSS metric, whereg;e2. For the smooth electricmetric, taking the charge and mass of the electron, we gefactor e2'1040, which is essentially the smallest of Diracbig numbers.
21M is not necessarily to be identified with the PPN mass.22Note that we have to rescaleF with 1/A2 in order to have
common conventions.23Note that this form of the metric is only possible ifm2>q2, i.e.,
e2<1.
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This would lead for the electric ADC metric to significaeffects near the wormhole throatr'r 2'q2/m, which forthe electron is of the order of theclassical electron radiusr El'3310213 cm. For the magnetic GDC metric, this esmate does change less than one order of magnitude, wtaking q5e/2a ~corresponding the Dirac’s charge quantiztion! and assuming the mass of the proton,m'1836me ,giving a throat ‘‘radius’’ of about 8310213 cm. The energiescorresponding to these distances~a few MeV! would be wellwithin the reach of present-day experimental technology.
In comparison, for theS-frame GHS metric deviationswould appear only at distances~and corresponding energie!of the order ofr'e, which is even one order of magnitudsmaller than the Planck lengthr Pl'2310233 cm.
Of course, magnetic monopoles have not been obse~not even by their characteristic electromagnetic signatu!.However, for the possible dilatonic effects of electricacharged particles the situation looks more favorable. A reistic estimate of the effects involved would have to take inaccount a~not yet existing! quantum field theory adapted tcurved backgrounds with nontrivial topology.
C. Zero-mass solutions; boundedness of mass
From a formal geometric viewpoint, the classicReissner-Nordstro¨m family of metrics are equally meaningful for zero and even negative values of their mass pareters; only the character of their everpresent singularitier 50 change to the worse. From a physical standpoint,would like to be able to exclude such negative-mass stions. In the classical Einstein theory for isolated systeand assuming suitable energy conditions, this has bachieved only fairly recently~see references given in Wal@9#!.
The boundedness of the mass,m>0, is, however, guar-anteed by the families of charged SSS solutions descrabove, when insisting onsmoothness.
First, the particular solutions withr 150, r 2.0 havesmooth metrics withvanishing mass, m50. As the corre-sponding chargeq necessarily vanishes, they arevacuum so-lutions of GDC/ADC gravity.24
Now, let us assume negative massm, i.e., r 1,0. Thenwe must necessarily also haver 2,0 if the solution is to becharged. But then the transformationr °r(r ) to regular co-ordinates fails to produce a metric locally regular atr(0),regularity can also not be achieved by any other map.main the uncharged possible solutions withm,0. Thismeans necessarilyr 250, but this is exactly the negativemass Schwarzschild metric, which is well known to havenaked singularity.25 Therefore as claimed, for the family osmooth SSS solutions there must be thelower mass boundm>0, in order that the metric remains smooth. The non
24Closely related uncharged massless wormhole solutions hbeen investigated very recently by Armenda´riz-Picon @16#, althoughfor a ‘‘ghostly’’ KG/dilaton scalar.
25The positive-mass Schwarzschild metrics would also becluded by smoothness.
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smooth SSS metrics can thus form5” 0 be characterized bythe two physically meaningful parametersm, q, wherem.0, q5” 0, whereas the massless solutions (m50) are un-charged (q50) and are uniquely characterized by their ‘‘sclar charge’’r 2>0.
Incidentally, the conserved total energy residing in tMaxwell field can be straightforwardly calculated, givingE53m for both types I and II andE5q2/m,3m for type III,indicating that there is agravitational binding energyDgE<2m, saturated for type I and II. Note that there is constency in the sense that the vanishing of the massm corre-sponds to the vanishing both of the chargeq and the Max-well field F.
D. Energy conditions; repulsion
Due to fairly general theorems, in the context of classigravity ~e.g., Friedman, Schleich and Witt@19#, cf. also Vis-ser’s monography@12#!, wormhole metrics like the smoothSSS metrics would necessarily somewhere exhibit ‘‘exomatter,’’ in the form of regions with negative energy densof the material source.26 The Maxwell stress tensorMik evi-dently obeys automatically even the dominant energy contion ~DEC! and was used as the only material source insystem of field equations; moreover it was coupled in‘‘orthodox’’ way ~up to a dilaton factor in the case of thelectric ADC solution! on the right-hand side of the fieldequation. G2Q82Q95M , respectively G2Q81Q95e2fM . Therefore it can be said that our solutions donotcontain any exotic matter, their material sources obeyingDEC. This is also evident in theE frame, where the KGstress tensor~adding to the Maxwell stress tensor multiplieby the dilaton factor! obeys the DEC. In fact, the energtensor for our dilaton metrics does even obey the strongergy condition~SEC!, which plays a prominent role in thesingularity theorems of classical gravity.
So how does it come that there seems to be a ‘‘repulsforce’’ holding open the throat of the wormhole? This canseen by invoking the Raychaudhuri identity for geodesiFirst note that this identity cannot be applied directly in tG frame, as the field equations involve extra geometric terbesides the Einstein tensor. Mapping geodesics from thGframe or from theA frame to theE frame ~where Einstein’sequations formally hold! will result in additional nongeode
sic terms:uW 50→uW 51/2@(u•df) uW 1g21df#. Such terms;df are also known from Nordstro¨m’s scalar theoryof gravitation ~a precursor of Einstein’s metric theory!.27
These additional terms also prevent the Raychaudidentity from being applied. For the smooth SSS metr~and r .2m) they have a repulsive effect,g21df51/2r 2 /r 2 (X1 /X2)1/2nr (nrªX1
1/2]r), which for type I
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26The few known ‘‘nonexotic’’ wormhole solutions of the classicMaxwell-Einstein theory~e.g., Schein and Aichelburg@20#! in factbreak some of the standard assumptions, like having closed timelines.
27For a short history of scalar-tensor theories of gravitation,Brans@21#.
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and type III becomes unbounded at the former locus ofthroat, where the metric is now singular. Therefore whinterpreted in theE frame, test particles seem to be alwaeffectively repelledby the object ‘‘sitting’’ atr50. In con-trast, being driven bye2f, the singular electric SSS GDCsolution would always appear to attract the test particle.
VI. CONCLUSIONS
With respect to the criteria mentioned in the Introductiowe have been able to show that among the class of MaxwEinstein-dilaton Lagrangians there exist two essentiallyferent couplings allowing for well-behaved static sphericasymmetric solutions. However, only the GDC Lagrangiadmits a simple coupling scheme. Moreover, it has an imdiate geometric interpretation in terms of a volume manifo
When it comes to making a choice between alternaLagrangian-based theories of dilaton gravity, of course itpends on the weight one is willing to give to the existencewell-behaved~i.e., smooth! solutions and to the generality oMED Lagrangians. In the context of classical gravity, t‘‘regularizing’’ nature of the GDC/ADC dilaton could bewelcomed as a ‘‘new degree of freedom’’ to tame somethe inherent divergences. The main obstacle is, however,the ‘‘magnetic’’ nature of the geometric GDC solutionwhereas for the ‘‘electric’’ ADC solutions, it is their lack ogeometric interpretation and the nonuniqueness of a cosponding coupling scheme. There are no indicationsmore ‘‘sophisticated’’ Lagrangians could resolve this dlemma.
The stability of the smooth wormhole solutions has nbeen touched in our work, and constitutes the major oissue. However, in view of the fulfillment of all the energconditions, the prospects seem to be promising.
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It would be also be highly desirable to have generalsta-tionary spherically symmetric GDC/ADC solutions. Sombasic questions in this context: are there still smooth charwormhole solutions? Do nontrivial geometric vacuum sotions again exist? Does the ‘‘magnetic-electric dilemma’’ spersist? Can stability be proven to hold in a general sen
A satisfactory answer to these questions would of couchallenge the role of classical relativity and the correspoing ~nondilatonic! black holes.
Applications to the different setting of cosmology woube particularly interesting: there is the possibility that tdilaton could again act repulsively, thus contributing to tobserved accelerated cosmic expansion.
The diverse global spacetime models derivable frombasic extensions described here could also serve as ‘‘sboxes’’ to develop and test some other fundamental theoin situations where the two-dimensional formulations arelimited to be realistic.
ACKNOWLEDGMENT
I want to express my gratitude to Peter Aichelburg fuseful suggestions. After submitting this paper for publiction, I learned that a MED Lagrangian essentially equivalto ours was already postulated by Cadoni and Mignemi@22#,with the primary aim of having a 4D generalization of the 2Jackiw-Teitelboim theory.28 Although they emphasized th2D aspects, they also noted that the corresponding 4D fiequations admit nonsingular~but apparently geodesically incomplete! magnetically charged black hole solutions.
28For a recent review of 2D dilaton gravity, see Grumiller, Kummer, and Vassilevich@23#.
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