Holographic Consequences of a No Transmission Principle
HHolographic Consequences of aNo Transmission Principle
Netta Engelhardt and Gary T. HorowitzDepartment of Physics, University of CaliforniaSanta Barbara, CA 93106, USA
Abstract
Two quantum field theories whose Hilbert spaces do not overlap cannot trans-mit a signal to one another. From this simple principle, we deduce some highlynontrivial consequences for holographic quantum gravity. These include: (1)certain cosmological bounces are forbidden, (2) generic singularities inside blackholes cannot be resolved, and (3) traversable wormholes do not exist. At theclassical level, this principle rules out certain types of naked singularities andsuggests that new singularity theorems should exist. [email protected], [email protected] a r X i v : . [ h e p - t h ] O c t ontents A true understanding of singularities, both in cosmology and in the black hole interior,requires a nonperturbative formulation of a quantum theory of gravity. Perhaps thebest-understood approach to quantum gravity is gauge/gravity duality, which relatesquantum string theory on asymptotically locally Anti-de Sitter (AdS) backgrounds –the bulk – to quantum field theories living on the conformal boundary of the bulk [1–3].At its strongest form, the duality provides a definition of nonperturbative quantumgravity in terms of a field theory. A powerful feature of this duality is that statementsthat are hard to establish on one side of the duality are often much easier to proveor derive on the dual counterpart. Unitarity of the evolution is an early example ofa simple statement about field theory with a highly nontrivial consequence for blackhole evaporation in the gravitational dual. This is the direction that we pursue in thispaper: deducing significant implications for quantum gravity from a simple field theoryprinciple.The principle we employ is intuitive: a signal cannot be transmitted between twoquantum field theories defined on separate Hilbert spaces. We call this the No Trans-mission Principle . In general, two quantum field theories have separate Hilbert spaceswhen they are defined on separate spacetimes. When the field theories in questionare conformally invariant, there is a subtlety: two spacetimes which a priori appeardisconnected can possibly be conformally mapped to one connected spacetime. Two Note that we do not restrict to conformal field theories, as gauge/gravity duality can also applyto theories without conformal invariance, provided that these theories have a UV fixed point. d -dimensional Minkowski space-time, for example, can be mapped into disjoint finite subsets of the same CFT on theEinstein static universe. The two original copies of the CFT are then defined on thesame Hilbert space, and signals can then propagate from one to the other through theCFT on the Einstein static universe. In the much more common situation in whichtwo CFTs cannot be conformally embedded in a larger CFT in this sense, their Hilbertspaces cannot overlap, and therefore no signals may be exchanged between the two (seeFig. 1 for an illustration).The application to holography is immediate. If two field theories with holographicduals do not share a Hilbert space, then no signal can be transmitted between theirbulk duals. This is reminiscent of the Gao–Wald theorem [4]: signals between twoboundary points travel faster on the boundary than in the bulk; here we say thatsignals can only propagate through the bulk if it can do so on the boundary. Thisresults in nontrivial constraints on the structure of bulk spacetimes with cosmologicalsingularities or black holes. We note that while certain black hole spacetimes mightappear to present a violation of this line of reasoning (e.g. Reissner-Nordstrom-AdS)by allowing signals to propagate from one asymptotic boundary to another through theinner horizon, the instability of the inner horizon guarantees that no actual violationsoccur. This is discussed at length in Sec 4.The power of the No Transmission Principle in holography is in its broad applicabil-ity to quantum string theory in any regime . The statement of the principle omits anymention of N or the ’t Hooft coupling, so any constraints obtained from this principleare valid when the bulk theory is anything from classical Einstein gravity to full quan-tum string theory. At the classical level, it implies new results about asymptoticallyAdS solutions of general relativity, which may presumably be proved using geometryalone. At the quantum level, it yields new insights into quantum gravity, which cannotbe obtained directly from our current understanding of the theory.We organize our investigation into two different categories based on the extent ofevolution of the dual field theory. In particular, we separately consider field theoriesin which evolution stops at some finite time – “singular” field theories – and ones inwhich evolution continues indefinitely, which we term “nonsingular” (in a conformally–invariant sense to be defined in Sec. 2). Application of the No Transmission Principleto two singular field theories yields constraints on the behavior and resolution of cos-mological singularities; the implications for two nonsingular field theories place limitson black hole singularity resolution and wormholes. Below we provide a heuristic dis-cussion of the principle’s implications separately for singular and nonsingular CFTs.2 ingular Field Theories: The No Transmission Principle yields a simple result whenapplied to singular field theories: if the boundary constituents are two singular fieldtheories, then their corresponding Hilbert spaces have no mutual overlap. In this case,the field theory singularity must extend through the bulk, cutting off bulk evolution atfinite time; the alternative would allow signals to propagate between the field theoriesthrough the bulk even though such propagation is forbidden due to the No TransmissionPrinciple.We may now investigate consequences of terminated bulk evolution in variousregimes of quantum string theory. First, at the level of classical General Relativity(large N and large coupling), we obtain nontrivial statements about the structure ofsingular solutions to general relativity: a cosmological singularity on the boundaryresults in a bulk singularity which extends to infinity and cannot end in the interior,leaving a smooth region behind. This agrees with intuition, as a spacetime which has asingularity with a “hole” would constitute a gross violation of cosmic censorship as wellas global hyperbolicity. Such a singularity would arise from smooth initial data in thepast and an observer traveling through the hole would be able to see the singularity onthe other side. There are various cosmological singularity theorems in general relativitywhich have a similar conclusion, but those assume the strong energy condition, whichis violated by reasonable and more pertinently extant matter such as massive scalarfields.At the level of classical string theory (infinite N and finite coupling), we find con-straints on stringy singularities. Certain types of cosmological singularities can besmoothed out by closed string tachyon condensation [5]. In the context of holography,the No Transmission Principle shows that a tachyon condensate cannot mediate signaltransmission from a collapsing spacetime to an expanding one.The application of this argument to full quantum gravity is of greatest interest.Gauge/gravity duality at finite N and finite coupling still has a fixed (classical) metricon the conformal boundary, but the metric in the interior is subject to arbitrarily largequantum and stringy fluctuations. The No Transmission Principle certainly appliesto field theories at finite N and coupling, leading to the conclusion that holographicquantum gravity forbids a large class of bounces through cosmological singularities .Cosmological bounces have been extensively investigated in prior literature, rangingfrom classical solutions which violate the strong energy condition [6, 7], to bounces innonholographic theories of quantum gravity, and even bouncing cosmologies in holog- We are only able to rule out holographic bounces for which the dual field theory is genuinelysingular. We are aware of only one example of a cosmological singularity in holography for which thedual field theory may be nonsingular in the sense of Sec. 3. See Sec. 3.4 for discussion.
Nonsingular Field Theories:
The No Transmission Principle may be similarly ap-plied to nonsingular field theories to yield fruitful results constraining the black holeinterior and forbidding traversable wormholes that connect two asymptotically AdS re-gions of spacetime. The line of reasoning is similar to the cosmological case: two fieldtheories on separate spacetimes which have maximal evolution – i.e. cannot be evolvedany further in any conformal frame – must have non-overlapping Hilbert spaces. Theytherefore cannot transmit signals to one another.The above argument immediately shows that quantum string theory forbids theresolution of a Schwarzschild black hole future singularity into a white hole singular-ity of another asymptotic region. That would allow signals to propagate from oneasymptotically AdS bulk to another, inducing forbidden transmission between nonsin-gular field theories. More generally, a charged or rotating AdS black hole has an innerhorizon which is known to be classically unstable [15, 16]: generic perturbations willturn this horizon into a weak null singularity [17, 18] (see Sec. 4 for more detail). TheNo Transmission Principle implies that there is no evolution past this singularity intoanother asymptotic region in holographic quantum gravity. Finally, the same line ofreasoning shows that traversable wormholes between two asymptotically AdS bulks areforbidden by quantum string theory, as such a setup would allow signals to propagatefrom one asymptotically AdS region to another.In the next section we introduce the No Transmission Principle in greater detail.Singular and nonsingular field theories are defined in Sec. 3, where the No TransmissionPrinciple is used to derive constraints on cosmological bounces. Sec. 4 contains appli-cations to nonsingular field theories, and Sec. 5 contains some concluding remarks.This includes a brief discussion of applications of the No Transmission Principle tosuperselection sectors in holography [19]. 4
No Transmission Principle
When can signals be exchanged between two quantum field theories? Excitations in onefield theory can only make sense in the other if the operator algebra of each field theoryacts on a common Hilbert space . More precisely, consider two field theories on fixedclassical, non-overlapping geometries of the same dimensionality with Hilbert spaces H and H , respectively. We will say that the two field theories are independent if thejoint Hilbert space of the system can be written as a tensor product: H = H ⊗ H ,and when acting on H , every operator of the first theory takes the form Φ ⊗ I , andevery operator of the second theory takes the form I ⊗ Φ , where I i is the identityoperator on H i . Otherwise, the two field theories are dependent . No Transmission Principle (NTP):
Two independent quantum field theories can-not transmit signals to one another.
At first sight, the simplicity of this statement may belie its utility. The power ofthe principle stems from its applicability to gauge/gravity duality without reference tothe coupling or the number of degrees of freedom of the field theory. In the holographiccontext, it states that if two holographic field theories cannot influence one anotheron the boundary, then they cannot do so through the bulk, be it classical, stringy, orquantum.In the context of holography, we will often be interested in quantum field theoriesthat are conformally invariant, so we will give a few examples of dependent and inde-pendent CFTs. The simplest example can be constructed from two copies of N = 4super Yang-Mills (SYM) in the ground state on (3+1)-dimensional Minkowski space.As is well known, each copy is holographically dual to a Poincar´e patch of pure AdS .Explicitly, the Minkowski metric can be written as a conformal rescaling of part of theEinstein static universe: ds flat = (cid:20) sec (cid:18) T + R (cid:19) sec (cid:18) T − R (cid:19)(cid:21) (cid:0) − dT + dR + sin Rd Ω (cid:1) . (1)where − π < T ± R < π and
R >
0. Thus each copy of N = 4 SYM on Minkowskispace is a subset of N = 4 SYM on the Einstein static universe. The two copies aredependent if they share a Hilbert space, which can only happen if the above conformaltransformation maps them onto the same N = 4 SYM on the Einstein static universe. We thank D. Jafferis for suggesting this formulation. We will not consider backreaction, and for our purposes these spacetimes need not be dynamical. a) (b) Figure 1: Two copies of a CFT can be mapped to (a) disjoint subsets of the same CFTin the Einstein static universe if they are dependent, or (b) two separate Einstein staticuniverses if they are independent.They are independent if it maps them to two separate N = 4 SYM on two distinctcopies of the Einstein static universe. In the former case, the two copies of N = 4SYM on Minkowski space are holographically dual to two Poincar´e patches of the sameglobal AdS spacetime (see Fig. 1(a)). These CFTs can communicate both through theboundary and through the bulk. In the latter case (see Fig. 1(b)), they are each dual toa Poincar´e patch of a different global AdS spacetime: the CFTs cannot communicatein any way, either through the bulk or the boundary, in agreement with the NTP.This construction is not limited to Minkowski space: de Sitter (dS), for example, isalso conformally related to a finite patch of the Einstein static universe, so two CFTsliving on two copies of dS, or one copy on dS and the other on Minkowski space, canall in principle be dependent – that is, as long as the two copies of the CFT can beconsistently mapped to the same Einstein static universe.It is worth noting that any two nonconformal quantum field theories that liveon separate, maximally extended spacetimes, must be independent. Conversely, as acaution we note that simply taking any two finite subsets of a CFT and conformallyrescaling them into two complete universes does not always yield dependent CFTs.6 =r t=0CFT CFT Figure 2: By excising a constant time ( t = 0, r > r ) surface from global AdS, we mayartificially obtain a setup which a priori seems to have two independent CFTs, andyet bulk signals from CFT to CFT are allowed. This is not in violation of the NoTransmission Principle, since the two CFTs are simply subsets of the CFT that liveson the complete Einstein static universe.The problem is the following: consider a Cauchy surface in the original CFT and twoopen subsets S and S . Any pure state defined in one of the subsets will be singularin the larger CFT, since there are no correlations across the boundary of S i [20].For example, the quantum stress tensor will diverge on this boundary. To avoid thisproblem, dependent CFTs must each contain an entire Cauchy surface of the largerCFT (or all but a set of measure zero, as in the case of Minkowski spacetime).As a final, somewhat pathological example, consider a holographic field theory onthe Einstein static universe dual to global AdS, and excise the region r > r from theconstant time slice t = 0 in the bulk geometry. The result naturally picks out twodual field theories on two separate universes: the field theory that lives on t <
0, andthe field theory that lives on t > t < t > t = 0 slice not excised.While these examples show that there exist dependent field theories, the class ofpairs of field theories that are independent is by far larger. We now turn to derivingconstraints on the bulk spacetime from the No Transmission Principle.7 The No Transmission Principle For Singular FieldTheories
A key question in the study of cosmological singularities is whether quantum gravitymediates a bounce from a big crunch. Is a cosmological singularity a true end tospacetime or is there another semiclassical region of spacetime before a big bang orafter a big crunch? We will show in this section that a large class of holographicbounces are forbidden. For completeness, we will also discuss one case of a quantumbounce which is not yet ruled out. There is evidence that it is, but further work isneeded to settle the question.The crux of the argument is that the NTP implies that the bulk evolves forward intime only if the dual field theory evolves forward in time. If the field theory evolutionends, bulk evolution must stop; it is this principle which generally forbids bounces. Aquantum field theory evolution will end only if it becomes singular in a suitable sense.For quantum field theories which are not conformally invariant, the definition is simple:a field theory is singular if its evolution stops at some finite time. This can be a resultof a curvature singularity in the underlying spacetime or a sickness in the field theory(e.g. a Hamiltonian which is unbounded from below). We will review several waysin which this can happen, but first we must address the following issue: since manyholographic setups involve CFTs, any finite time evolution can be rescaled to infinitetime via a conformal mapping. We will therefore need a conformally-invariant way ofdefining arrested evolution.We consider CFTs on maximally conformally extended spacetimes, i.e., spacetimeswhich cannot be conformally embedded as a proper subset of a larger spacetime. Weassume that the topology is X × R , where X is a compact ( d − and that the metric is globally hyperbolic: there exists a Cauchy surface C withunique classical evolution from initial data on C to anywhere in the spacetime. In fact,the spacetime can be foliated by a one parameter family of Cauchy surfaces {C t } eachwith topology X [21]. Given a conformal metric on X × R , we pick a conformal framein which the volume of C t is bounded from above and below by nonzero constants for all t ; such frames exist for any conformal metric (this excludes a conformal frame in whichthe CFT lives in a spacetime like de Sitter). We will call this a standard conformalframe . The notion that evolution ends in finite time in a standard conformal frame is We define a bounce to be one in which signals can propagate from the past semiclassical spacetimeto the future. If this is not possible, then operationally the spacetimes are not connected. Spacetimes like Minkowski space that can be conformally mapped to a compact universe areincluded.
Definition:
If the evolution of the CFT extends to infinite future proper time (in astandard conformal frame), then the CFT is future-complete. If in this frame, evolu-tion cannot be extended to infinite proper time, the CFT is future-singular. Similarly, ifthere is past-infinite evolution of the CFT, the CFT is past-complete, and past-singularotherwise.
Three different types of singular CFTs have been discussed in the literature: (1)the CFT could have a potential which is unbounded from below so observables run offto infinity in finite time, (2) the CFT could live on a spacetime with a cosmologicalsingularity in the standard conformal frame (see Sec. 3.1 for a precise definition), (3)the CFT could be coupled to a time dependent source for a relevant deformation whichbecomes singular in a finite time. We now briefly review examples that illustrate thesethree possibilities. In each case the dual bulk geometry has a cosmological singularity.An example of the first case is given in [22, 23] where a stable (2 + 1)-dimensionalCFT is deformed by a marginal triple trace term: S = S + k (cid:82) O where O is adimension one operator. This adds a potential which is unbounded from below. Asimilar example is studied in [13], where a (3 + 1)-dimensional CFT is deformed by amarginal double trace term: S = S + k (cid:82) ˜ O where ˜ O is a dimension two operator.Since k >
0, this again corresponds to adding a potential which is unbounded frombelow. In both cases, the bulk dual consists of gravity coupled to a scalar field φ with potential V ( φ ) coming from a consistent truncation of supergravity. It was shownthat there are solutions in which smooth asymptotically AdS initial data evolves intoa cosmological singularity in the bulk.Examples of the second possibility, that the field theory spacetime is itself singular,were discussed in [24–26] where the CFT was defined on the Kasner universe: ds = − dt + d − (cid:88) i =1 t p i dx i , (2)where (cid:80) p i = (cid:80) p i = 1 and the x i ’s are periodically identified (to make space com-pact). This metric describes an anisotropic, homogeneous cosmology with a curvaturesingularity at t = 0. This metric is not in a standard conformal frame (the volume ofspatial slices vanishes at t = 0), but is conformally related to a standard conformalframe by the conformal factor t − / ( d − . There is a simple bulk dual to this setup, with9he metric: ds = 1 z (cid:32) dz − dt + d − (cid:88) i =1 t p i dx i (cid:33) , (3)which solves the vacuum Einstein equations with a negative cosmological constant.This bulk metric has a cosmological singularity at t = 0 for all z , as well as a singularityat the Poincare horizon, z = ∞ . Should the additional singularity be of concern, thereis a related solution, the Kasner-AdS Soliton, which is also dual to a field theory onEq. (2) (with one of the p i = 0) [27] ds = 1 z (cid:34)(cid:0) − z d − (cid:1) dθ − dt + d − (cid:88) i =1 t p i dx i + dz − z d − (cid:35) . (4)With a suitable period for θ , space now smoothly caps off at z = 1.Finally, we give an example of singular field theories in which the singularity stemsfrom a singular time-dependent source [28–30]. A relevant perturbation of a CFT onde Sitter (with a constant coefficient of appropriate sign) is dual to a bulk with mattercoupled to gravity . Within the future lightcone of a point in the bulk, the solution isdescribed by an open FRW universe with a big crunch extending all the way through thebulk. Similarly, within the past lightcone, there is an expanding open FRW universewith an initial big bang extending all through the bulk. For small coefficients, thedeformed CFT on de Sitter space is perfectly well-defined, but if one now conformallymaps de Sitter to a standard conformal frame like the Einstein static universe, thecoefficient of the relevant operator gets multiplied by a power of the conformal factorand diverges in finite time both in the past and the future: the CFT is being drivenby a source that becomes singular [30].We will establish a correspondence between singular CFTs and singular bulks inthe sections below. We first consider bounces in a bulk described by classical generalrelativity, then in a special case of classical string theory, and finally in full quantumgravity. We argue below that a singular holographic field theory is dual to an asymptoticallyAdS spacetime with a cosmological singularity – a singularity which extends out toinfinity and cuts off all further evolution. More precisely, let Σ be a complete spacelike For some deformations, this is dual to gravity coupled to the same scalar fields used in the firstexample above, and the bulk solutions are in fact identical to the ones found in [13, 22, 23]. g AB = Ω g AB be the rescaled bulk metric with timelikeboundary at infinity attached. Definition:
The bulk spacetime has a future cosmological singularity (big crunch) ifit is maximally extended and the length of all future directed timelike curves from Σ isbounded when computed with the rescaled metric ˜ g AB . A big bang singularity is defined similarly with “future” replaced by “past”.One can characterize the region of the bulk described by the dual field theory asfollows: given a conformal boundary B to an asymptotically AdS spacetime, the dualfield theory on B describes the region of the bulk spanned by all spacelike surfaceswhich end on B .In the particular case where a holographic field theory is singular due to a cosmolog-ical singularity on the boundary, we obtain a purely geometric result which relates theconformal geometry of the boundary to the geometry of the bulk. To state it formally,we first define what we mean by a boundary cosmological singularity: Definition:
The conformal boundary B of an asymptotically AdS spacetime has a fu-ture cosmological singularity (a big crunch) if in any standard conformal frame, throughevery point of any Cauchy surface C t , there exists a future incomplete timelike geodesic. Once again, a big bang singularity is defined similarly with “future” replaced by “past”.Note that we cannot require that the length of all timelike curves (or even all timelikegeodesics) be bounded in this case, since we can rescale the metric by a function thatdiverges locally near the singularity. We now state a purely geometric implication ofthe NTP:
Conclusion:
Consider any supergravity theory arising in the low energy limit of stringtheory. If the conformal boundary metric of an asymptotically AdS solution has a cos-mological singularity, then the bulk solution must also have a cosmological singularity.
It is easy to see that this follows immediately from the NTP. If the conformal boundaryhas a cosmological singularity in the future (as defined above), evolution of the holo-graphic field theory dual must end in finite time. This is the second type of singularfield theory discussed in Sec. 3. If the bulk dual does not have a big crunch singularity,then evolution in the bulk can continue into another region of spacetime with its own11 FT CFT (a) CFT CFT (b) Figure 3: (a) In a bulk in which a singularity that extends to the boundary has ahole, signals can travel from the bulk region dual to CFT to the bulk region dual toCFT . This is forbidden by the No Transmission Principle, as CFT and CFT areindependent. (b) The NTP implies that any signal in the bulk region dual to CFT must terminate before it reaches the bulk region dual to CFT : the singularity has nohole.asymptotic boundary metric. This would require that the bulk be dual to two field the-ories: the original future-singular field theory, and another, past-singular field theorywhich can receive causal signals via the bulk from the future-singular field theory. SeeFig. 3(a). The two field theories are manifestly independent, so any communicationbetween them violates the NTP. This contradiction is avoided only if the bulk has acosmological singularity, as in Fig 3(b), establishing the above result.The above implication is a novel singularity “theorem”: a boundary singularityguarantees a bulk singularity. Since it is purely a geometric statement, it is reasonableto expect that it can be shown just using techniques from classical general relativity.Ideally, the restriction to supergravity theories would be replaced by a more generalcondition like the Null Energy Condition ( T ab (cid:96) a (cid:96) b ≥ (cid:96) a ). Since the resultshows that the singularity cannot end in a naked singularity in the interior as illustratedin Fig. 3, it also incorporates a (very limited) form of cosmic censorship.The above conclusion is not obvious: there are bulk spacetimes which are nonsin-gular, but still have a conformal boundary with a cosmological singularity. As argued12bove, the NTP shows that these are not solutions to string theory. As an example,consider the following spacetime: ds = 1 z (cid:34) − dt + d − (cid:88) i =1 ( t + z ) p i dx i + dz (cid:35) (5)where (cid:80) p i = 0. This form of the metric is intended to apply only in the asymptotic(small z ) region. (For large z , it can be modified in any nonsingular manner.) Thebulk spacetime given by (5) is suggestive of a regulated version of the AdS Kasnermetric in which the cosmological singularity is replaced by a bounce. The metric onthe boundary z = 0 is still singular and is simply Kasner (2) in a standard conformalframe. The NTP, as argued above, shows that this metric is forbidden as a solution togeneral relativity with matter coming from string theory. Since this spacetime violatesthe Null Energy Condition, the results of the NTP agree with conventional wisdom,that any reasonable classical geometry arising from string theory should obey the NullEnergy Condition.We have focused above on the case in which the field theory has a cosmologicalsingularity, but the conclusion holds whenever the field theory is singular, irrespectiveof the type of singularity: a portion of a bulk spacetime which is dual to a future-singular field theory can have no overlap with a portion of the bulk which is dual to apast-singular field theory. In this section, we explore one implication of the No Transmission Principle for holog-raphy with infinite N and finite coupling, dual to a classical, but stringy bulk. Canclassical stringy effects mediate evolution through the singularity? Since the argumentin the previous section made no use of N or the ’t Hooft coupling, we conclude thatthe same line of reasoning shows that the answer is no.We illustrate this with an example of one classical stringy effect that can be ex-plicitly ruled out as mediating evolution through a singularity: tachyon condensation.When one direction of space is compactified to a circle, string theory has states corre-sponding to winding modes around the circle. These winding modes become tachyonicwhenever the size of the circle is smaller than the string scale and fermions are anti-periodic around this circle. A space that has a circle at infinity which is larger than thestring scale and slowly shrinks to become smaller than the string scale in the interiorhas a tachyon instability; this instability results in the circle pinching off [31].13n the context of cosmological singularities, the tachyon instability can manifest asa result of the circle shrinking in time. An explicit instance of this was worked outin [5] for the Milne spacetime [32]: ds = − dτ + τ dχ . (6)These coordinates cover the interior of the light cone of two dimensional Minkowskispace. Compactifying χ with period χ forms a Lorentzian cone, as illustrated inFig. 4(a). Since the length of the circle is L ( τ ) = | τ | χ , its rate of change is ˙ L = ± χ .It was shown in [5] that if χ is sufficiently small, the string spectrum can be reliablycomputed and a winding string mode becomes tachyonic when L reaches the stringscale . When the tachyon condenses, spacetime ends. All string modes (includingthe graviton) become massive and there is no low energy spacetime description of thephysics [5]. This is a classical string theory effect which does not involve quantumgravity.The above metric with −∞ < τ < + ∞ describes two cones attached at a singulartip (Fig. 4(a)). Although the tip is only a conical singularity (since the curvaturevanishes for all nonzero τ ), in classical string theory it is unstable to developing largecurvature: the slightest perturbation with momentum around the circle will becomestrongly blue-shifted near τ = 0 and cause the curvature to become large. The NTPshows that, at least in the context of holography, the tachyon instability (which canset in before the curvature becomes large) will not allow signals to propagate from thepast cone to the future cone through the tachyon condensate.More explicitly, consider AdS in Poincare coordinates and write the Minkowskimetric on each radial slice in Milne coordinates: ds = 1 z (cid:2) − dτ + τ dχ + dz (cid:3) (7)where χ is compactified with period χ . The bulk spacetime has a cosmological singu-larity at τ = 0: this is a simple example of an AdS cosmology. The dual field theoryconsists of two independent field theories on two Lorentzian cones which are attachedat their tip. In a standard conformal frame, the past cone by itself is conformal toan infinite cylinder: ds = − dt + dχ with t = ln τ ; the field theory on the pastcone itself is nonsingular. In the bulk, tachyon condensation replaces the singularityat τ = 0 with a tachyon condensate when | τ | L/z < (cid:96) s , see Fig 4(b). If signals in thebulk could propagate through the tachyon condensate, it would violate the NTP. Note We are ignoring the extra spatial dimensions which will play no role in this discussion. FT CFT χτ =0 (a) ? CFT CFT τ =0 (b) Figure 4: (a) Two independent CFTs on the past and future Lorentzian cone, witha conical singularity in between them. (b) The bulk dual with a tachyon condensatemodifying the near-singularity region. It may a priori seem that the tachyon could allowsignals to propagate through, but the No Transmission Principle forbids the signal inthe shaded region from emerging in the bulk region dual to CFT .that although the tachyon condensate extends out to infinity, it is confined to a region | τ | < z(cid:96) s /L . At the conformal boundary, then, it is concentrated at the tip and doesnot affect the geometry of the cone, or the infinite static cylinder. A holographic field theory at finite N and finite coupling is dual to a bulk spacetimewith all quantum string theory effects included. If the dual field theory is singular,the No Transmission Principle implies that signals cannot be transmitted throughcosmological singularities – even in a quantum stringy bulk. In other words, suchbounces are forbidden. A potential objection to this argument is that holographymight fail in this context. After all, gravitational excitations are no longer confinedinside a finite box. Since quantum gravity effects are strong near the singularity, andthe singularity extends all the way out to infinity, perhaps decoupling fails and theboundary metric will be subject to quantum fluctuations , rather than being a fixed, We thank S. Hartnoll for raising this point. not become large. To see this, recall that on any scale smaller than the boundary curvature,the boundary metric looks flat, so the bulk metric resembles standard AdS in thatasymptotic region.More explicitly, for any nonsingular d -dimensional boundary metric g µν , introduceFefferman-Graham coordinates in which the bulk metric takes the asymptotic form ds = 1 z [ dz + g µν dx µ dx ν ] . (8)The scalar curvature of this metric is R = − d ( d + 1) + z R where R is the scalarcurvature of g µν . So as long as R remains finite, it yields only a subleading contributionto the asymptotic bulk curvature. (A similar result holds for the full Riemann tensor.)This implies that if the bulk metric has large finite curvature that continues into theasymptotic region (as expected for any quantum backreaction near a cosmologicalsingularity), the boundary metric must be singular.Quantum gravity effects are expected to be significant only within a bounded dis-tance to the singularity. Suppose that the singularity is at t = 0 and the quantumeffects are significant within a proper time τ of the singularity. Applying a conformalrescaling which produces a finite boundary metric, we find that all effects within aproper time τ of the asymptotic singularity are shrunk to zero proper time on theboundary. In other words, they are compressed to the t = 0 singularity in the bound-ary metric, and the boundary metric at any nonzero time does not fluctuate. This canalso be seen by conformally rescaling any singular boundary metric to a (de Sitter-like)frame in which the singularity is pushed off to infinity. In this frame, the curvatureon the boundary remains bounded, and the standard rules of holography should apply.Quantum gravity effects in the bulk should not affect this boundary metric. A rescalingback to the singular boundary metric shows that the boundary metric away from thesingularity remains fixed.Thus, the basic rules of holography still apply. Our conclusion from the NTP holds:16 singular field theory cannot be holographically dual to a cosmological bounce in fullquantum string theory. We are aware of only one construction in which the bulk spacetime has a cosmologicalsingularity, but it is not immediately obvious that the boundary field theory is singular.Consider N = 4 super Yang-Mills theory on four-dimensional Minkowski space witha time dependent coupling g Y M ( t ) that vanishes at t = 0 [33]; this is an exampleof a marginal coupling which is taken to extreme values. The bulk dual consists ofgravity coupled to a massless scalar field (the dilaton). Since g Y M determines theasymptotic value of the dilaton via e φ = g Y M , the vanishing of the coupling requiresthat φ ( t ) → −∞ as t →
0. The backreaction of this diverging scalar field causes acosmological singularity in the bulk. For example, one of the bulk solutions that hasbeen found is ds = 1 z [ dz + | t | η µν dx µ dx ν ] , e φ ( t ) = g s | t | √ (9)which has a cosmological singularity at t = 0. In the field theory, it is not obviousthat evolution must stop. Indeed the vanishing of the coupling might suggest thatthe theory just becomes free near t = 0. It was shown in [33] that this is not thecase. The fields become large near t = 0 and the interactions never become negligible.Furthermore, they present strong evidence that evolution must end. The state appearsto become singular with diverging energy and a wildly oscillating phase as t →
0. Thisis not yet a proof that evolution must end since effects of renormalization were notfully taken into account. If it could be shown that (at finite N and coupling) thereis some choice of g Y M ( t ) such that evolution is well defined through a time when thecoupling vanishes, then evolution in the bulk must continue as well. This would be thefirst example of a holographic bounce. Note that this is not in contradiction with theabove discussion, which assumed that the dual field theory was singular.If g Y M ( t ) becomes very small but stays bounded away from zero, then the fieldtheory is clearly nonsingular and evolution can continue for all time. This, however,tells us nothing about cosmological bounces: the bulk solution now contains a largeblack hole [33], and the singularity does not enter the asymptotic region. The factthat the field theory evolution continues for all time simply reflects future-infiniteevolution outside of the black hole. This is very similar to our discussion in the previous Note that the boundary metric in Fefferman-Graham coordinates is singular, consistent with thegeneral result discussed above. But since the singularity is entirely in the conformal factor, one canview the super Yang-Mills theory as living in Minkowski spacetime.
In this section we discuss consequences of the No Transmission Principle for nonsingularholographic field theories. We find the exclusion of the following two a priori possiblescenarios: (i) evolution through black hole singularities to a new asymptotically AdSregion of spacetime, (ii) existence of traversable wormholes connecting two asymptoticregions. Here we will assume that the CFT and its superselection sectors containcomplete information about the black hole interior. We discuss the alternative inSec. 5.
No Evolution through Black Hole Singularities : Could a signal entering an AdSblack hole pass through the black hole and reemerge in another asymptotically AdSspacetime? In the simplest case of a Schwarzschild black hole in global AdS, it is apriori possible that quantum gravity could mediate evolution through the singularityinto another asymptotic region (see Fig. 5). This kind of evolution, however wouldresult in signals propagating from the field theory dual to the black hole to the fieldtheory dual to the white hole. Since the two field theories each live on a completeEinstein static universe, they are independent, and this is immediately forbidden bythe NTP. We find that in any regime of string theory, the black hole singularity inSchwarzschild-AdS does not admit evolution to another asymptotic region.Generic black holes, however, have some nonzero rotation (and possibly nonzerocharge). The resulting causal structure appears at first to facilitate the transmission ofsignals between asymptotic regions even at the classical level. The Penrose diagram fora charged and rotating AdS black hole contains multiple asymptotic regions (see Fig. 6).Each asymptotic region is dual to a CFT on a complete Einstein static universe, andeach of the dual CFTs is clearly independent of all of the others. It may prima facieappear that this system presents a violation of the NTP: signals can travel from oneasymptotic region to another through the bulk, as illustrated in Fig. 6. These signalsresult in a transmission of information which cannot be explained by entanglementand is of course forbidden by the NTP. This puzzle is resolved by the instability of theinner horizon of such black holes: any excitation that enters the black hole will cause18 FT CFT Figure 5: Resolution of the Schwarzschild black hole singularity allowing evolution toanother asymptotic region would violate the No Transmission Principle.the inner horizon to develop a curvature singularity [15–18]. The reader may raise anobjection at this point: normally, a field theory excitation with energy less than N p for some suitable power p does not result in any backreaction on the bulk metric in thelarge N limit. This argument, however, assumes bulk stability. In the presence of aninstability, the curvature diverges for any excitation at finite N , so the natural large N limit contains a singularity for any excitation .The nature of the so-called weak null singularity that forms on the inner horizon hasraised speculation regarding possible evolution beyond it. While the Christoffel symbolsat the singularity are not square integrable, the metric at the singularity remainscontinuous. Since the total tidal distortion remains finite across the singularity, therehas been some debate on the possibility that signals could pass through the singularityeven in the purely classical regime. We claim that this cannot happen: such a scenariowould result in the exchange of signals between independent field theories, and istherefore ruled out by the NTP. Most significantly, evolution through the weak nullsingularity to another asymptotic region is ruled out by the NTP even in full quantumgravity.Finally, we comment on the asymptotic regions of an eternal AdS black hole whichare not causally connected, where the maximally extended spacetime is not a purestate of a single field theory. The above argument is unaffected by these additionalasymptotic regions: given any entangled state, an excitation can always be addedto one field theory and allowed to propagate. Alternatively, one can combine the We thank Don Marolf for a discussion on this point. nn e r ho r i z on i nn e r ho r i z onou t e r ho r i z on ou t e r ho r i z on CFT CFT Figure 6: A conformal diagram of a charged black hole, with the possible path ofa signal between independent CFTs. Any such signal, however, collapses the innerhorizon into a null singularity. The No Transmission Principle implies that there is noevolution past this null singularity.two field theories dual to the asymptotic regions on either side of the Einstein-Rosenbridge into one Hilbert space, and apply the NTP to these larger Hilbert spaces. Thisminor complication may at any rate be avoided via the formation of black holes fromthe collapse of a charged spherical shell. Such systems initially have only a singleasymptotic region, and can be dual to a field theory in a pure state. If the local chargedensity of the shell is greater than its local mass density, then the shell will reach aminimum radius inside the horizon and bounce out into another asymptotic region(provided it is given sufficient kinetic energy at collapse) [34]. With AdS boundaryconditions, the shell will continue to oscillate and execute an infinite number of bounces.The spacetime outside the shell is exactly Reissner-Nordstrom-AdS, and the innerhorizon is again unstable. So this provides a pure state, or one sided, version of theargument that there is no evolution through inner horizon singularities.Of course small black holes can evaporate in quantum gravity, and the fact that theevolution must be unitary already provided an indication that no information couldpass through the black hole into another universe.20 FT CFT Figure 7: A traversable wormhole obtained by identifying a spatial surface in two sepa-rate AdS universes. The orange signal (single-arrow), initially dual to some excitationin CFT , propagates into CFT ; the purple signal (double-arrow), initially dual to anexcitation in CFT , propagates from CFT to CFT . This violates the No TransmissionPrinciple. No Traversable Wormholes in Quantum Gravity : A traversable wormhole maybe constructed from two copies of global AdS, each with a dual CFT on S d − × R :excise a unit ball on a time slice t = 0 from each spacetime and glue the bottom edgeof one to the top edge of the other (see Fig. 7). A signal with t < t > r = 1. Could this singularitybe smoothed out in quantum gravity? The answer is clearly no, since this wormholeviolates the No Transmission Principle.Now consider a permanent traversable wormhole: a static bulk spacetime with two(or more) asymptotically AdS regions each with a dual field theory . A bulk excitationnear the first boundary can propagate to a second asymptotic region through thewormhole. Such an excitation initially corresponds to exciting a field theory dual to oneasymptotic region; in this initial stage, the field theory on the second asymptotic regionis ignorant of this excitation. Once the excitation propagates through the wormhole,however, it is identical to an excitation created by the second field theory, and the The dual field theories will almost certainly be in some entangled state; however, signals cannotbe transmitted via entanglement. . We have deduced several nontrivial consequences of the simple statement that twoquantum field theories cannot transmit signals to one another unless they share a com-mon Hilbert space (and hence are both part of a larger field theory). It is rathersurprising that a number of deep properties of full quantum gravity follow immedi-ately from this simple statement. Of course our conclusions are restricted to the the-ory of quantum gravity with asymptotically AdS boundary conditions obtained fromgauge/gravity duality; other theories of quantum gravity might behave differently.In Sec. 4 we assumed that the dual field theory contains complete information aboutthe black hole interior. This is almost certainly true for eternal black holes with twoasymptotic regions on an initial static surface: it is generally agreed that a maximallyentangled state of the two theories corresponding to a thermofield double describesthe region inside as well as outside the horizon [36]. If the dual field theory does notdescribe the interior of a single-sided black hole, then we cannot rule out the possibilitythat quantum gravity permits signals to propagate through black hole or cosmologicalsingularities to a region inside another horizon.As is standard in holography, we have been assuming that if an asymptoticallyAdS spacetime has several asymptotic regions, there is a separate dual field theoryassociated with each one. In particular, there is no coupling between them. If suchcouplings did exist, the No Transmission Principle would clearly not apply. Could ourconclusions be avoided by simply extending holography to allow couplings between thedual theories? The answer is no. Most of our examples involve bulk spacetimes inwhich one asymptotic region is to the future of the other. In this case any couplingbetween the dual field theories would violate causality in the bulk, since they wouldallow signals to be sent into the past as well as the future. The one exception is thepossibility of a traversable wormhole; even this case, however, cannot be described byany simple coupling between the dual theories, such as the product of a single traceoperator in one theory with a single trace operator in the other. This type of double A classical solution describing a traversable wormhole has recently been constructed using mattersatisfying the null energy condition, but it involves closed timelike curves [35].
CF T and that of CF T .This would contain nontrivial physics which is not captured by the dual field theory,contradicting the statement that the two dual descriptions are equivalent.The ad hoc rule fairs no better with the singular field theories discussed in Sec. 3.These theories are driven by external sources which diverge in finite time, or by thebackground metric itself becoming singular. Since normal evolution ends in finite time,the limiting state is likely to be very badly behaved. Often the energy, and possiblyother expectation values, diverge. (Indeed, a typical state of finite energy is expectedto describe a black hole in the bulk.) It is unlikely that one can match such a singularend state with some initial state of another copy of the dual theory. We conclude by noting that our results support the idea of superselection sectorsin holography [19]. Given CFTs on two disjoint copies of Minkowski spacetime, localobservables cannot distinguish whether they share a Hilbert space or not; ancillaryinformation is necessary to determine whether the bulk geometry is one copy of AdS When the Hamiltonian is unbounded from below, one might try to construct a well definedoperator via a self-adjoint extension. While this is straightforward in quantum mechanics, previousattempts to define it in quantum field theory do not lead to bouncing cosmologies [13, 23], and it maybe impossible to do so.
23r two. This additional data is contained in non-local observables such as two-pointcorrelators or mutual information. In other words, the knowledge of whether the twoCFTs are dependent or independent is not accessible to local observables. This is rem-iniscient of the superselection sectors of [19], where it was shown that local observablesof two entangled CFTs cannot determine whether the two CFTs are dual to a wormholeor two disconnected universes.
Acknowlegements
It is a pleasure to thank W. Donnelly, D. Engelhardt, D. Jafferis, S. Fischetti, S.Hartnoll, J. Maldacena, D. Marolf, J. Polchinski, H. Reall, and A. Wall for discussions.This work began during the KITP program on Quantum Gravity Foundations: UVto IR, and GH thanks the KITP for its hospitality. This work was supported in partby NSF grant PHY-1504541. The work of NE was supported by the NSF GraduateResearch Fellowship under grant DE-1144085 and by funds from the University ofCalifornia.
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