Energy Conditions and Their Cosmological Implications

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Provided by CERN Document Server ENERGY CONDITIONS AND THEIR COSMOLOGICAL IMPLICATIONS

MATT VISSER and CARLOS BARCELO´ Physics Department, Washington University, Saint Louis, Missouri 63130-4899, USA E-mail: [email protected], [email protected] Plenary talk delivered at Cosmo99, Trieste, Sept/Oct 1999. gr-qc/0001099; 28 January 2000; LATEX-ed February 19, 2000.

The energy conditions of general relativity permit one to deduce very powerful and general theorems about the behaviour of strong gravitational ﬁelds and cosmological geometries. However, the energy conditions these theorems are based on are beginning to look a lot less secure than they once seemed: (1) there are subtle quantum eﬀects that violate all of the energy conditions, and more tellingly (2), there are also relatively benign looking classical systems that violate all the energy conditions. This opens up a Pandora’s box of rather disquieting possibilities — everything from negative asymptotic mass, to traversable wormholes, to warp drives, up to and including time machines.

1 Introduction such as the singularity theorems (guaranteeing, under certain circumstances, gravitational collapse and/or the Einstein gravity (general relativity) is a tremendously existence of a big bang singularity),1,2 the positive en- complex theory even if you restrict attention to the ergy theorem, the non-existence of traversable worm- purely classical regime. The field equations are holes (topological censorship), and limits on the extent to which light cones can “tip over” in strong gravita- µν 8πGNewton µν G = 4 T . (1) tional fields (superluminal censorship). All these theo- c rems require some form of energy condition, some notion The left-hand-side, the Einstein tensor Gµν , is compli- of positivity of the stress-energy tensor as an input hy- cated enough by itself, but is at least a universal function pothesis, and the variety of energy conditions in use in of the spacetime geometry. In contrast the right-hand- the relativity community is driven largely by the tech- side, the stress-energy tensor T µν , is not universal but nical requirements of how much you have to assume to instead depends on the particular type of matter and in- easily prove certain theorems. teractions you choose to insert in your model. Faced with Over the years, opinions have changed as to how this situation, you must either resign oneself to perform- fundamental some of the specific energy conditions are. ing an immense catalog of special-case calculations, one One particular energy condition has now been completely special case for each conceivable matter Lagrangian you abandoned, and there is general agreement that another can write down, or try to decide on some generic features is on the verge of being relegated to the dustbin. There that “all reasonable” stress-energy tensors should satisfy, are however, some more general issues that make one and then try to use these generic features to develop gen- worry about the whole programme. Specifically: eral theorems concerning the strong-field behaviour of (1) Over the last decade or so it has become increas- gravitational fields. ingly obvious that there are subtle quantum effects that One key generic feature that most matter we run are capable of violating all the energy conditions, even across experimentally seems to share is that energy den- the weakest of the standard energy conditions. Now be- sities (almost) always seem to be positive. The so-called cause these are quantum effects, they are by definition “energy conditions” of general relativity are a variety of small (proportional toh ¯) so the general consensus for different ways of making this notion of locally positive en- many years was to not worry too much.3 ergy density more precise. The (pointwise) energy condi- (2) More recently,4,5 it has become clear that there tions take the form of assertions that various linear com- are quite reasonable looking classical systems, field the- binations of the components of the stress-energy tensor ories that are compatible with all known experimental (at any specified point in spacetime) should be positive, data, and that are in some sense very natural from a or at least non-negative. The so-called “averaged energy quantum field theory point of view, which violate all the conditions” are somewhat weaker, they permit localized energy conditions. Because these are now classical viola- violations of the energy conditions, as long as “on aver- tions of the energy conditions they can be made arbitrar- age” the energy conditions hold when integrated along ily large, and seem to lead to rather weird physics. (For null or timelike geodesics.1,2,3 instance, it is possible to demonstrate that Lorentzian- The refinement of the energy conditions paralleled signature traversable wormholes arise as classical solu- the development of powerful mathematical theorems, tions of the field equations.)5

1 Faced with this situation, you will either have to ativity community. The weakest of these is the NEC, learn to live with some rather peculiar physics, or you and it is in many cases also the easiest to work with and will need to make a radical reassessment of the place of analyze. The standard wisdom for many years was that the energy conditions in general relativity and cosmology. all reasonable forms of matter should at least satisfy the NEC. After it became clear that the NEC (and even the 2 Energy conditions ANEC) was violated by quantum effects two main lines of retrenchment developed: 3 To set some basic nomenclature, the pointwise energy (1) Many researchers simply decided to ignore quan- conditions of general relativity are:1,2,3 tum mechanics, relying on the classical NEC to prevent grossly weird physics in the classical regime, and hop- Trace energy condition (TEC), now abandoned. ing that the long sought for quantum theory of gravity Strong energy condition (SEC), almost abandoned. would eventually deal with the quantum problems. This Null energy condition (NEC). is not a fully satisfactory response in that NEC viola- Weak energy condition (WEC). tions already show up in semiclassical quantum gravity Dominant energy condition (DEC). (where you quantize the matter fields and keep gravity classical), and show up at first order inh ¯. Since semi- All of these energy conditions can be modified by aver- classical quantum gravity is certainly a good approxi- aging along null or timelike geodesics. mation in our immediate neighborhood, it is somewhat The trace energy condition is the assertion that the disturbing to see widespread (albeit small) violations of trace of the stress-energy tensor should always be nega- the energy conditions in the here and now. Many experi- tive (or positive depending on metric conventions), and mental physicists and observational astrophysicists react was popular for a while during the 1960’s. However, once quite heatedly when the theoreticians tell them that ac- it was realized that stiff equations of state, such as those cording to our best calculations there should be plenty of appropriate for neutron stars, violate the TEC this en- “negative energy” (energy densities less than that of the ergy condition fell into disfavour.6 It has now been com- flat-space Minkowski vacuum) out there in the real uni- pletely abandoned and is no longer cited in the literature verse. However, to avoid the conclusion that quantum — we mention it here as a concrete example of an energy effects can and do lead to locally negative energy den- condition being outright abandoned. sities, and even violations of the ANEC, requires truly The strong energy condition is currently the subject radical surgery to modern physics, and in particular we of much discussion, sometimes heated. (1) The most would have to throw away almost all of quantum field naive scalar field theory you can write down, the min- theory. imally coupled scalar field, violates the SEC,1 and indeed (2) A more nuanced response is based on the Ford– curvature-coupled scalar field theories also violate the Roman Quantum Inequalities.10 These inequalities, which SEC; there are fermionic quantum field theories where are currently still being developed and extended, and interactions engender SEC violations,7 and specific mod- whose implications are still a topic of considerable activ- els of point-like particles with two-body interactions that ity, are based on the facts that while quantum-induced violate the SEC.8 (2) If you believe in cosmological infla- violations of the energy conditions are widespread they tion, the SEC must be violated during the inflationary are also small, and on the observation that a negative epoch, and the need for this SEC violation is why in- energy in one place and time always seems to be compen- flationary models are typically driven by scalar inflaton sated for (indeed, over-compensated for) by positive en- fields. (3) If you believe the recent observational data ergy elsewhere in spacetime. This is the so-called Quan- regarding the accelerating universe, then the SEC is vio- tum Interest Conjecture.10 While the positive pay-back lated on cosmological scales right now! 9 (4) Even if you is not enough to prevent violation of the ANEC (based are somewhat more conservative, and regard the alleged on averaging the NEC along a null geodesic) the hope is present-day acceleration of the cosmological expansion that it will be possible to prove some type of space-time as “unproven”, the tension between the age of the oldest averaged energy condition from first principles, and that stars and the measured present-day Hubble parameter such a space-time averaged energy condition might be makes it very difficult to avoid the conclusion that the sufficient to enable us to recover the singularity/positive- SEC must have been violated in the cosmologically recent mass/censorship theorems under weaker hypotheses than past, sometime between redshift 10 and the present.9 Un- currently employed. der the circumstances it would be rather quixotic to take A fundamental problem for this type of approach the SEC too seriously as a fundamental guide. is the more recent realization that there are also seri- In contrast, the null, weak, and dominant energy ous classical violations of the energy conditions.4,5 These conditions are still extensively used in the general rel- classical violations can easily be made arbitrarily large,

2 and appear to be unconstrained by any form of energy without something like an axion to solve the so-called inequality. The simplest source of classical energy con- “strong CP problem”. Still, the axion has not yet been dition violations is from scalar fields, so we shall first directly observed experimentally. present some background on the usefulness and need for A fourth candidate scalar field of phenomenological scalar field theories in modern physics. interest specifically within the astrophysics/cosmology community is the so-called “inflaton”. This scalar field is 3 Scalar Fields: Background used as a mechanism for driving the anomalously fast expansion of the universe during the inflationary era. While Scalar fields play a somewhat ambiguous role in modern observationally it is a relatively secure bet that some- theoretical physics: on the one hand they provide great thing like cosmological inflation (in the sense of anoma- toy models, and are from a theoretician’s perspective al- lously fast cosmological expansion) actually took place, most inevitable components of any reasonable model of and while scalar fields of some type are the only known empirical reality; on the other hand the direct experi- reasonable way of driving inflation, we must again admit mental/observational evidence is spotty. that direct observational verification of the existence of The only scalar fields for which we have really direct the inflaton field (and its variants, such as quintessence) “hands-on” experimental evidence are the scalar mesons is far from being accomplished. (pions π; kaons K; and their “charmed”, “truth” and A fifth candidate scalar field of phenomenological in- “beauty” relatives, plus a whole slew of resonances such terest specifically within the general relativity commu- 0 11 as the η, f0, η , a0, ...). Not one of these particles nity is the so-called “Brans–Dicke scalar”. This is per- are fundamental, they are all quark-antiquark bound haps the simplest extension to Einstein gravity that is states, and while the description in terms of scalar fields not ruled out by experiment. (It is certainly greatly is useful when these systems are probed at low mo- constrained by observation and experiment, and there menta (as measured in their rest frame) we should cer- is no positive experimental data guaranteeing its exis- tainly not continue to use the scalar field description tence, but it is not ruled out.) The relativity community once the system is probed with momenta greater than views the Brans–Dicke scalar mainly as an excellent test- ¯h/(bound state radius). In terms of the scalar field itself, ing ground for alternative ideas and as a useful way of this means you should not trust the scalar field descrip- parameterizing possible deviations from Einstein gravity. tion if gradients become large, if (And experimentally and observationally, Einstein gravity still wins.) ||φ|| ||∇φ|| > . (2) Finally, the membrane-inspired field theories (low- bound state radius energy limits of what used to be called string theory) are Similarly you should not trust the scalar field description literally infested with scalar fields. In membrane theo- if the energy density in the scalar field exceeds the criti- ries it is impossible to avoid scalar fields, with the most cal density for the quark-hadron phase transition. Thus ubiquitous being the so-called “dilaton”. However, the scalar mesons are a mixed bag: they definitely exist, and dilaton field is far from unique, in general there is a large we know quite a bit about their properties, but there class of so-called “moduli” fields, which are scalar fields are stringent limitations on how far we should trust the corresponding to the directions in which the background scalar field description. spacetime geometry is particularly “soft” and easily de- The next candidate scalar field that is closest to ex- formed. So if membrane theory really is the fundamental perimental verification is the Higgs particle responsible theory of quantum gravity, then the existence of funda- for electroweak symmetry breaking. While in the stan- mental scalar fields is automatic, with the field theory dard model the Higgs is fundamental, and while almost description of these fundamental scalars being valid at everyone is firmly convinced that some Higgs-like scalar least up to the Planck scale, and possibly higher. field exits, there is a possibility that the physical Higgs (For good measure, by making a conformal transfor- (like the scalar mesons) might itself be a bound state mation of the spacetime geometry it is typically possible of some deeper level of elementary particles (e.g., techni- to put membrane-inspired scalar fields into a framework color and its variants). Despite the tremendous successes which closely parallels that of the generalized Brans– of the standard model of particle physics we do not (cur- Dicke fields. Thus there is a potential for much cross- rently) have direct proof of the existence of a fundamental pollination between Brans–Dicke inspired variants of gen- Higgs scalar field. eral relativity and membrane-inspired field theories.) A third candidate scalar field of great phenomeno- So overall, we have excellent theoretical reasons to logical interest is the axion: it is extremely difficult to expect that scalar field theories are an integral part of see how one could make strong interaction physics com- reality, but the direct experimental/observational verifi- patible with the observed lack of strong CP violation, cation of the existence of fundamental fields is still an

3 open question. Nevertheless, we think it fair to say that There are a few hidden subtleties here: First because there are excellent reasons for taking scalar fields seri- of the curvature coupling term ξRφ2 the stress-energy ously, and excellent reasons for thinking that the gravi- tensor for the scalar field contains a term proportional to tational properties of scalar fields are of interest cosmo- the Einstein tensor logically, astrophysically, and for providing fundamental φ ∇ ∇ − 1 ∇ 2 − probes of general relativity. Tab = aφ bφ gab( φ) gabV (φ) (11) 2 2 c + ξ Gab φ − 2∇a(φ ∇bφ)+2gab∇ (φ∇cφ) . 4 Scalar Fields and Gravity Second, the way we have defined T matter it is to be cal- In setting up the formalism for a scalar field coupled to culated from Sbulk by simply ignoring the factor f(φ). gravity we first need to specify a few conventions: the Because the RHS contains a term proportions to the − metric signature will be taken to be ( , +, +, +) and Einstein tensor it is best to rearrange the gravity EOM we adopt Landau–Lifshitz spacelike conventions (LLSC) 12 by isolating all such terms on the LHS, in which case the This is equivalent to MTW conventions with latin gravity EOM is equivalent to indices for the tensors, and is equivalent to Flanagan– 4 Wald. In the MTW classification this corresponds to φ κ matter κGab =[Teff ]ab + f(φ) T . (12) (+g,+Riemann, +Einstein). We take the total action κ − ξφ2 ab to be: Here the “effective” stress-energy for the scalar field is S = S + S + Sbulk. (3) g φ ( The gravity action is standard φ κ ∇ ∇ − 1 ∇ 2 − Z [Teff]ab = 2 aφ bφ gab( φ) gabV (φ) √ κ − ξφ 2 4 1 ) Sg = d x −g κR, (4) 2 c − ξ [2∇a(φ ∇bφ) − 2gab∇ (φ∇cφ)] . (13) with the ordinary Newton constant being defined by

c4 Note that this effective stress-energy can change sign for κ = . (5) 8πGNewton certain values of the scalar field, which is our first hint of peculiar physics. More importantly, observe that it We shall permit the scalar field to exhibit an arbitrary is the energy conditions defined in terms of this effec- curvature coupling ξ tive stress-energy tensor that are the physically inter- Z √ esting ones: energy conditions imposed upon this ef- d 1 2 2 Sφ = d x −g − (∇φ) + ξRφ − V (φ) . (6) fective stress-energy imply constraints on the Einstein 2 tensor, placing constraints on the curvature, which is Finally the action for ordinary bulk matter is taken as what is really needed for deriving singularity/positive- Z mass/censorship theorems. √ d In addition, the “effective” gravitational coupling of Sbulk = d x −gf(φ) Lmatter. (7) “normal matter” to the gravitational field is

2 Here we assume f(φ) is algebraic, and that Lmatter does κ − ξφ κeff = . (14) not involve φ. We also assume for technical reasons f(φ) that Lmatter does not contain any terms involving second derivatives of the metric. (For example, let Lmatter In terms of Newton’s constant be the Lagrangian of the ordinary standard model of par- eff κ G (φ)=GNewton f(φ) . (15) ticle physics.) Under these circumstances the equations Newton κ − ξφ2 of motion (EOM) for gravity can be written It is particularly convenient to pick f(φ)=(κ − ξφ2)/κ. φ matter κGab = Tab + f(φ) Tab . (8) With this simplifying choice bulk matter couples to gravity in the ordinary way, while the only gravitational pe- The EOM of the φ field are culiarities are now concentrated in the gravity-φ sector. 2 0 0 In the remainder of the technical discussion we shall, (∇ − ξR)φ − V (φ)+f (φ) Lmatter =0. (9) for simplicity, treat the normal matter in the test-field The EOM for the bulk matter fields can be phrased as limit. That is, whatever normal matter is present is as- sumed to be sufficiently diffuse to not appreciably affect ab ∇b f(φ) Tmatter =0. (10) the spacetime geometry, while test particles of normal

4 matter can still be used as convenient probes of the space- Thus any static field with a positive potential violates the time geometry. Making this test-matter approximation SEC. (For example, a slowly changing scalar with a mass thus simplifies life to the extent that we are looking at a or quartic self-interactions, this is the simplest toy model gravity EOM for cosmological inflation and/or quintessence; also any φ κGab =[Teff]ab, (16) positive cosmological constant violates the SEC.) coupled to the scalar EOM Adding a non-minimal coupling (ξ =6 0) is not help- ful, though it does make the algebra messier. (In particu- (∇2 − ξR)φ − V 0(φ)=0. (17) lar, adding non-minimal coupling will not by itself switch off cosmological inflation.) The key point regarding the This system is now sufficiently simple that a number of SEC is that a positive potential energy V (φ) > 0 tends exact analytic solutions are known.5,13,14,15 We shall not to violate the SEC, and you can always think of getting re-derive any of these exact analytic solutions but instead SEC violations by going to a region of field space with will quote them as examples when we want to illustrate high potential energy. 5 some aspect of the generic situation. Though the SEC violations we are talking about here are most commonly used in building models of cosmolog- 5 Scalar Fields and energy conditions ical inflation, perhaps the most serious long term issue is that the singularity theorem we use to prove the existence 5.1 SEC of a big bang singularity in FLRW cosmologies depends Technically the SEC is defined (in 4 dimensions) by using crucially on the SEC. Not only do SEC violations per- the trace-reversed stress energy tensor mit cosmological inflation but they also open the door to replacing the big bang singularity with a “bounce”, 1 or “Tolman wormhole”.7,8,16 In fact it is now known that T ≡ T − g tr(T ). (18) ab ab 2 ab generically the SEC is the only energy condition you need to violate to get a bounce, and that it is still possible to The SEC is then the assertion that for any timelike vector satisfy all the other energy conditions at and near the bounce.16 (In counterpoint, if one replaces the SEC by a b ≥ T ab v v 0. (19) strong enough inhomogeneity assumptions then it is possible to prove that certain classes of chaotic inflationary The reason for wanting this condition is that, via the models must nevertheless possess a big-bang singularity.) Einstein equations, the SEC would imply the Ricci convergence condition Note that scalar fields do not guarantee the preven- tion of a big bang singularity, they merely raise this pos- a Rab v ≥ 0. (20) sibility — and this issue is interesting enough to warrant further investigation. Another interesting point is that This convergence condition on the Ricci curvature tensor the reason that most inflationary models are based on is then used to prove that nearby timelike geodesics are minimally coupled scalars (ξ = 0) is purely a histori- always focussed towards each other, and this focussing cal one, there simply was no need to go to non-minimal lemma is then a first critical step in proving singularity coupling to get the SEC violations that are needed for theorems and the like. Unfortunately if you actually cal- cosmological inflation, and it is only once you get down culate this quantity for the effective stress-energy of the to rather specific model building that non-minimal cou- scalar field you find pling becomes interesting. ( κ [T φ ] va vb = (va∇ φ)2 − V (φ) (21) eff ab κ − ξφ2 a ) 5.2 NEC a b c − ξ 2v v ∇a(φ∇bφ) −∇ (φ∇cφ) . The NEC has two great advantages over the SEC: it is the simplest energy condition to deal with algebraically, It’s very easy to make this quantity negative. There is 1 and because it is the weakest pointwise energy condition a specific example in Hawking–Ellis, page 95; flat space 1 2 2 it leads to the strongest theorems. The NEC is the as- with ξ =0andV (φ)= m φ , but the phenomenon is 2 sertion that for any null vector ka we should have much more general. For minimal coupling (ξ =0)wesee

φ a b a∇ 2 − a b ≥ [Teff]ab v v =(v aφ) V (φ). (22) Tab k k 0. (23) 5 Unfortunately when we actually calculate this quantity Integrate by parts, discarding the boundary terms (that for the scalar field we find is, assume sufficiently smooth asymptotic behaviour) ( ( ) I 2 2 2 2 φ a b κ a 2 κ dφ 4ξ φ (dφ/dλ) [Teff]ab k k = (k ∇aφ) I κ − ξφ2 = − 2 + − 2 dλ. (30) ) κ ξφ dλ κ ξφ a b −ξ 2k k ∇a(φ∇bφ) . (24) Now assemble the pieces:

I 2 κ[κ − ξ(1 − 4ξ)φ2] dφ For minimal coupling I = dλ. (31) (κ − ξφ2)2 dλ [T φ ] ka kb =(ka∇ φ)2 ≥ 0. (25) eff ab a The integrand is not positive definite, and ANEC can be The accident that the NEC is satisfied for minimal cou- violated, provided there is a region along the geodesic pling led to a situation where researchers just did not where 2 look under enough rocks to see where the problems lay. ξ(1 − 4ξ) φ >κ. (32) For non-minimal coupling we start, as a convenience, This can only happen for ξ ∈ (0, 1/4), so there is some- by extending k to be a geodesic vector field around the thing very special about this range of curvature cou- point of interest, so that ka∇ kb = 0. Then using the a plings. In particular 1/6 ∈ (0, 1/4), so conformal cou- affine parameter λ we have ka∇ = d/dλ so that a pling in d = 4 lies in this range. This is important be- ( ) 2 cause there are technical issues in quantum field theory κ dφ d2[φ2] φ a b − that seem to almost automatically imply that real phys- [Teff ]ab k k = − 2 ξ 2 . κ ξφ dλ dλ ical scalars should be conformally coupled. (Conformal (26) coupling ξ =1/6 is typically a infrared fixed point of the Pick any local extremum of φ along the null geodesic, renormalization group flow, furthermore setting ξ =1/6 then and then going to flat spacetime automatically repro- duces the so-called “new-improved stress-energy tensor”, κ d2[φ2] φ a b − a stress-energy tensor that is much better behaved in [Teff]ab k k = − 2 ξ 2 . (27) κ ξφ dλ quantum field theory than the unimproved minimally- coupled stress-energy tensor.)5 It is easy to make this negative (for any ξ =0).If6 ξ<0 Also note that ANEC violations require consider any local maximump of the field φ,theNECis violated. If ξ>0andφ< κ/ξ then anyp local minimum 2 κ κ φ > > . (33) does the job, while if ξ>0andφ> κ/ξ one again ξ(1 − 4ξ) ξ needs a local maximum of φ to violate the NEC. Now classical violations of the NEC are much more This implies that the prefactor in the effective stress- disturbing than classical violations if the SEC. In par- energy tensor for scalars must go through a zero and ticular, traversable Lorentzian-signature wormholes are become negative in order to get ANEC violations. Thus known to be associated with violations of the NEC (and ANEC violations are considerably more constrained than ANEC),17 as are warp-drives,18,19 time machines,3 and NEC violations, and the ANEC violating regions are al- similar exotica — this should start to make you feel just ways associated with regions where the effective stress- a little nervous. energy has “reversed sign”. Furthermore in the ANEC violating region 5.3 ANEC κ φ2 > > 16κ, (34) The ANEC is technically much more interesting. Pick ξ(1 − 4ξ) some complete null geodesic γ and consider the integral I implying that the scalar field must take on enormous (super–Planckian) values in order to provide ANEC vi- I φ a b = [Teff]ab k k dλ. (28) olations. Now super–Planckian values for scalar fields are not that unusual, they are part and parcel of many Then (though not all) inflationary models, and the standard ( ) I 2 lore in cosmology is to not worry about a super-Planckian I κ dφ − d dφ value for the scalar field unless the energy density is also = 2 2ξ φ dλ. (29) κ − ξφ dλ dλ dλ super–Planckian.

6 6 Conclusions fairs: either we can learn to live with wormholes, and other strange physics engendered by energy condition vi- Even with the caveats provided above, the fact that olations, or we need to patch up the theory. We cannot the ANEC can be violated by classical scalar fields is just say that some improved version of the energy con- significant and important — in particular the ANEC ditions will do the job for us, since we already have an is the weakest of the energy conditions in current use, explicit solution of the Einstein equations that contains and violating the ANEC short circuits all the standard traversable wormholes — we would have to do something singularity/positive-mass/censorship theorems. This ob- more drastic, like attack the notion of a scalar field, or servation piqued our interest and we decided to see just forbid conformal couplings [we would need to forbid the how weird the physics could get once you admit scalar entire range ξ ∈ (0, 1/4)], or forbid super-Planckian field 5 fields into your models. values — each one of these particular possibilities how- In particular, it is by now well-known that ever is in conflict with cherished notions of some segments traversable wormholes are associated with violations of of the particle physics/ membrane theory/ relativity/ as- the NEC and ANEC, so we became suspicious that there trophysics communities. Most physicists would be loathe might be an explicit class of exact traversable worm- to give up the notion of a scalar field, and conformal hole solutions to the coupled gravity-scalar field system.3 coupling is so natural that it is difficult to believe that In a recent paper5 we presented such a class of solu- banning it would be a viable option. Banishing super– tions — for algebraic simplicity we restricted attention Planckian field values is more plausible, but this runs to conformal coupling (ξ → 1/6) and set the potential afoul of at least some segments of the inflationary com- to zero V (φ) → 0. We found a three-parameter class munity. of exact solutions to the coupled Einstein–scalar field equations with the three parameters being the mass, the References scalar charge, and the value of the scalar field at infin- ity. Within this three-dimensional parameter space we 1. S.W. Hawking and G.F.R. Ellis, The large scale found a two-dimensional subspace that corresponds to structure of spacetime, (Cambridge University 5 exact traversable wormhole solutions. Press, England, 1973). The simplifications of conformal coupling and zero 2. R.M. Wald, General Relativity, (University of potential were made only for the sake of algebraic sim- Chicago Press, Chicago, 1984). plicity and we expect that there are more general classes 3. M. Visser, Lorentzian wormholes, (AIP Press, New of wormhole solutions waiting to be discovered. In ad- York, 1995). dition, deviations from spherical symmetry are also of 4. E.E. Flanagan and R.M. Wald, Phys. Rev. D 54, interest. It should be borne in mind that although 6233 (1996). the present analysis indicates that there must be super- 5. C. Barceló and M. Visser, Phys. Lett. B466, 127 Planckian scalar fields somewhere in the wormhole space- (1999). time, this does not necessarily mean that a traveller needs 6. Ya.B. Zeldovich, and I.D. Novikov, Stars and Rel- to traverse one of these super-Planckian regions to cross ativity (Relativistic Astrophysics, Vol. 1), (Univer- to the other side: If the wormhole is not spherically sym- sity of Chicago Press, Chicago, 1971), see esp. p. metric it is typically possible to minimize the spatial ex- 197. 20 tent of the regions of peculiar physics. Work on these 7. B. Rose, Class. Quantum Grav. 3, 975 (1986); 4, topics is continuing. 1019 (1987). Now traversable wormholes, while certainly exotic, 8. L. Parker and Y. Wang, Phys. Rev. D 42, 1877 are by themselves not enough to get the physics com- (1990). munity really upset: The big problem with traversable 9. M. Visser, Science 276, 88 (1997). wormholes is that if you manage to acquire even one 10. L.H. Ford and T.A. Roman, Phys. Rev. D 51, inter-universe traversable wormhole then it seems almost 4277 (1995); D 60, 104018 (1999); absurdly easy to build a time machine — and this does L.H. Ford, M.J. Pfenning, and T.A. Roman, Phys. get the physics community upset.3 There is a conjec- Rev. D 57, 4839 (1998). ture (Hawking’s Chronology Protection Conjecture)that 11. Particle Data Group, http://pdg.lbl.gov, Euro. quantum physics will save the universe by destabilizing Phys. J. C3, 1 (1998). the wormhole just as the time machine is about to form, 12. C.W. Misner, K.S. Thorne, and J.A. Wheeler, but it must certainly be emphasized that there is con- Gravitation, (Freeman, San Francisco, 1973). siderable uncertainty as to how serious these causality 13. I.Z. Fisher, Zh. Exp. Th. Fiz. 18, 636 (1948); problems are.3 gr-qc/9911008. There are two responses to the current state of af- 14. O. Bergmann and R. Leipnik, Phys. Rev. 107,

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