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JHEP11(2019)037 renders Springer m propagating . This setup 1 October 27, 2019 m November 7, 2019 S : September 24, 2019 : × : Accepted Published Received Published for SISSA by https://doi.org/10.1007/JHEP11(2019)037 quotient of (rotating) BTZ 2 Z seanmcbride@.ucsb.edu , . 3 1908.03998 The Authors. c Black Holes, Gauge- correspondence

A new class of traversable was recently constructed which relies , [email protected] Santa Barbara, CA 93106, U.S.A. E-mail: Department of Physics, University of California, Open Access Article funded by SCOAP may arise for higher-spin fields. Keywords: ArXiv ePrint: the limit whereformulation the of background the becomes method ofenergy extremal. images of for A fermions fields in keytypically curved to causing technical . have the step We important sign find is kinematic ofsign the the differences of the stress- the integrated from null that delay/advance) stress-energy to of (and vary scalar thus around in fields, the many throat cases of the the . Similar effects Here we beginon with a a classical four-dimensional backgroundallows Weyl defined one fermion to by compute field a theconditions of fermion around stress-energy any a mass exactly. non-contractiblethe For , appropriate associated perturbative boundary wormhole back-reaction traversable at and any suggests it can become eternally traversable at Abstract: only on local bulk dynamics rather than an explicit coupling between distinct boundaries. Donald Marolf and Sean McBride Simple perturbatively traversable wormholes fromfermions bulk JHEP11(2019)037 ] ]. 5 25 13 23 – 8 , 1 7 11 Adding quantum corrections allows 22 1 10 12 8 – 1 – 3 25 isometry 3 J d 3 6 -Wormholes 17 ]. 2 26 16 i d Z ] as a potential tool for producing superluminal travel. In classi- 10 kk 6 , 9 T h 1 20 in KKZBO i kk T h Traversable wormholes can be constructed if one drops the requirement of global hyperbolicity, e.g. by B.1 Solution inB.2 Minkowski space Solution inB.3 AdS 3.1 Spinor extensions3.2 of the Computing 3.3 General features 3.4 Numerical results 2.1 Review of2.2 AdS Explicit calculation 2.3 Subtleties of2.4 spinor representations The KKZBO spacetime and back-reaction 1 fluctuations that violate theorems null energy continue condition, to though apply theis in topological non-negative contexts so censorship where that the- the the integrated averaged null null energy energy along condition causal (ANEC)introducing holds. NUT charge [ cal , wormholesthe are null nontraversable energy due condition to (NEC),With which constraints the implies on the assumption causality topological ofstart from censorship and global theorems end [ hyperbolicity, at theseflat the and theorems boundary asymptotically require anti-de to Sitter causal be (AdS) contexts. curves deformable to that the boundary in both asymptotically 1 Introduction Wormholes have long beenand a in source science fiction of [ fascination both in the scientific literature [ 4 Conclusion A Geodesic in Kruskal-like coordinates B Spinor propagator in AdS 3 and stress-energy on the KKZBO spacetime Contents 1 Introduction 2 Preliminaries and review JHEP11(2019)037 ] ) ]) 22 Jx 24 ( . For φ 2 , where Z ] and for 2 Z 17 ) to – x ˜ ( ] was able to M/ 15 φ 23 ] (see also [ ]. For example, = 23 11 M maps J ] conjectures. ]. 21 8 isometry , 7 2 , though it also includes what one Z Z and dual SYK models [ 2 = ] were of the form ]. 1 ] in using the term wormhole to refer to 23 13 π 23 . For linear quantum fields, the sign of the 2 boundaries. This nonlocal coupling induces Z approximation was used to construct a time- 3 – 2 – = 2 ] and GR = QM [ 1 π 20 , examples of such torsion wormholes, [ 19 satisfies periodic or anti-periodic boundary conditions 3 considered by [ φ ]. Furthermore, since such traversable wormholes provide M 18 . Here we follow [ ] by introducing a time-dependent coupling between the two 14 M ], a nearly-AdS 22 contains black holes with Killing horizons and well-defined Hartle- ˜ M ]. Traversable wormholes are thus allowed, so long as it takes longer to wormholes in [ 3 12 ), and thus whether Jx ( φ − In certain asymptotically AdS The class of backgrounds Nonlocal boundary couplings of the sort used in GJW were expected to model more Consistent with such expectations, traversable wormholes were constructed by Gao, More generally, however, one might expect the ANEC to hold only along achronal any setting where curvesdeformed both to starting the and boundary.by ending Such the at curves aforementioned the topological would boundary classically censorship cannot results be of be forbidden [ smoothly fromexactly being compute causal quantum stress-energy tensor expectation values for general free bulk might call torsion wormholes withback-reaction, e.g. and thus whether ora not choice it makes of the periodic wormholescalar or traversable, is fields, anti-periodic controlled boundary for by conditions instance,or under it to the depends on ofon whether this the the quotient spacetime back-reacts on the to renders the wormholes traversable. the covering space Hawking states for any quantum fields.wormhole topologies This context having includes fundamental many group examples with familiar tary constructions. Inindependent [ asymptotically flat four-dimensionalto wormhole. address many This non-perturbativeused issues. approach a allowed perturbative The framework [ contrasting to approach argueof that of such standard [ classical local wormhole quantum fields on have a Hartle-Hawking-like broad states class whose stress-energy in with the ER = EPR [ general backgrounds in whichother; causal e.g., curves when can both mouths travelically are from AdS embedded region one in of wormhole the spacetime. mouth same This to asymptotically expectation flat the was or recently asymptot- verified by two complemen- negative, but the nonlocalother boundary more coupling quickly than transmit they signals couldis travel satisfied. from through one Similar the setups wormhole. boundary have been Inrotating to this studied AdS the for sense AdS the AANEC a holographic dual of certain quantum teleportation protocols, they are of broader interest boundary causality in the context of AdS/CFT [ Jafferis, and Wall (GJW) [ otherwise disconnected asymptotically AdS a bulk perturbation ofto run a between scalar the quantum two field boundaries. whose The back-reaction averaged allows null causal energy curves along these trajectories is curves; i.e., along onlyin the fastest causal with curves non-contractibleANEC connecting closed to two fail. spacelike events Indeed, curves, [ it thesecond is this Casmir law achronal effect ANEC [ (AANEC) causes thattravel follows the through from the the generalized wormhole than to go around. Similar conclusions follow from requiring JHEP11(2019)037 ] 2 Z 23 how the the horizon ]. Again, we 2.2 measures the . We flesh out on the horizon M 23 , which in par- i 2 Z V M / i ∆ ) 2.3 1 kk S T − h h × . This setting is useful as 1 S by discussing the extension . 4 × 2.4 = (rBTZ , we review the construction of 2 M for all fermion masses, suggesting that . The integrated T ]. We then recall in section M i 23 in the torsion wormholes of [ kk quotient of the rotating Ba˜nados-Teitelboim- T , so that h . But the average is non-zero when the black m – 3 – 1 2 of BTZ and BTZ φ S Z 2 Z quotient, so on the physical spacetime is a , providing an analytic expression for the null stress- ˜ for a spinor field of arbitrary mass. The above quotient M/ 2 ] involved quantum fields in a Hartle-Hawking-like state 3 Z M = to vanish on the BTZ horizon for any quantum field. This is the black hole temperature and 23 M the requires the expected null-null component of ˜ M T M i ˜ to be independent of M can be non-zero. Nevertheless, the simplicity of BTZ can be -Wormholes kk i 2 T ] (rBTZ) times with a brief review of certain asymptotically AdS (or asymp- V h 0 and that bulk spinors alone would suffice to yield an eternally Z M i -wormholes studied in [ ∆ 26 2 h , kk → , where 2.1 Z . Working our way all the way up to spin-2 would allow understanding ] and find geometries useful for our consideration of spinor fields. In i T T d quotients 25 h ) T V 23 2 ] satisfying these properties is called the Kaluza-Klein zero-brane X ∆ Z h 23 × T diverges as i The outline of the paper is as follows. In section V ∆ the stress-energy tensor symmetry is then broken byexpectation the value used to provide ancan analytic expression then for be evaluated numerically and combined with first-order perturbation theory to 2.1 Review ofThe AdS exactly solvable models ofpropagating [ on the Killing symmetry of BTZstate preserves the on BTZ the Hartle-Hawking covering state. space As a result, in that stress-energy tensor of quantum scalar fieldsThe can be extension computed of exactly in such theseticular backgrounds. computations describes to further fermions properties isexample required from outlined for [ in fermions section to(KKZBO) yield spacetime non-trivial and results. is discussed The in detail in section 2 Preliminaries and review We begin intotically section AdS h traversable wormhole in thatof limit. our We conclude results in topropagators section in other AdS higher spinof particles the effect which, of like linearized gravitons the on spinor, such wormholes; have this exactly remains a soluble goal for future work. depend on the BTZhole angular rotates, coordinate andwe is compute negative withexpectation the value appropriate of choice the time-advancescalar of governing case, traversability periodicity. we of find the Following wormhole. [ As in the wormholes from [ the relevant case theZanelli spacetime black hole [ the details of thisenergy case tensor on in the section horizonbreaks of rotational symmetry, and we find the sign of the integrated null energy to generally curved spacetime computations involving higher spincomplications. fields Here can we lead take to a significantreaction technical first from step Weyl in fermions thisfind of direction that by any stress-energy computing mass tensor the expectation quantum back- values can be computed exactly. scalar fields. The study of more general quantum fields is clearly of interest, though in JHEP11(2019)037 ] 2 × 7→ Z , π and ) (2.1) [0 ), but ). , π ∈ ]. ∞ θ , 26 θ, φ [ = 0 ( , where the ], it has the U, V, φ, θ is the length 2 ) or of tensor . In particu- parameterize V π −∞ φ ] is a quotient φ 23 :( + /J V 2.1 ) 3 + 2 27 ) = 1 πr J as well as on the φ S with below. Note that and and 1 . θ, φ ∼ 1 θ ( S V  × cease to be isometries. U geon [ φ 2 U 2 = 2 J dφ P 2 U ) R and takes values in ( ) where below, even without rotation, UV 1 ). Here the quotient is smooth φ J π ); see figure − 2.3 π + (1 to act on both + U, V, φ , the expected null time delay of a 2 + i when r 3 . However, as noted in [ V induced by back-reaction from quantum 3 J φ + ∆ contains a non-contractible cycle with h ∞ = constant for any 1 is the identity and the quotient contains a is the identity and the quotient contains a V, U, φ V, U, φ, θ ( = ( φ 2 2 /J 2 3 J ], J dUdV U . For example, the 7→ – 4 – 7→ 2 ` , π ) ) X 4 [0 associated with the internal − ∈ θ ). The KKZBO is more complicated than the above ) is global AdS 2 ). Another interesting related spacetime is the Kaluza- θ gives a singular spacetime with what may be called an θ ) 0 indicates that the wormhole becomes traversable, as U, V, φ φ ( . Again, ( 1 2.1 U, V, φ, θ V V :( homotopy that is not deformable to the boundary and is < S V homotopy that is not deformable to the boundary and can UV 1 :( 1 i 2 2 = and ending at J 2 Z V Z ) = ) = J U θ ∆ (1 + φ ( h ( . These singularities are localized at −∞ 1 U = U S = 2 U ds for any ). As before, the quotient is smooth but singularities arise when it is Kaluza- ] for any π π now acts on the angle , π + 2 [0 J ∈ φ, θ can be obtained by allowing the isometry is the AdS length scale (which we will often take to be one) and 2 ] studied the Kaluza-Klein zero-brane orbifold (KKZBO) given by ) under the isometry φ − 3 ` = 0 or 23 We are interested in finding the quantity Both of the above quotients are non-orientable, but an orientable spacetime (BTZ A variety of interesting spacetimes can be constructed as quotients of ( Without rotation, the BTZ metric can be written in the form 2.1 /J is the identity and that the quotient BTZ φ ) 1 2 1 V, U, the fact that the KKZBOto is study orientable fermions. makes it a much more interesting contextgeodesic starting in at which fields. A negative value now associated with theat zero branes. It canexamples be as represented it by breaks theadvantage closed translational of extending curves symmetry to in rotating blackIn holes particular, where the both KKZBO admitsquantum an fields interesting can extreme limit become in large. which As back-reaction reviewed from in section ( Klein reduced on the may be calledthree-dimensional zero-brane spacetime. Once asnon-contractible again, they cycle represent with point-like defects in the resulting non-contractible cycle with be represented by the closed curve S lar, [ Klein end-of-the-world brane geometryisometry (hereafter KKEOW) (BTZ BTZ factor. In particular, but Kaluza-Klein reduction on end-of-the-world brane at products with some otherof simple factor ( J homotopy that is notcurve deformable to the boundary and can be represented by the closed Here of the BTZ horizon. Wethe use Kruskal-like null coordinates directions. ( Thegives metric the ( global BTZ black hole when one makes the identification and techniques. In this sectionconsider we only set non-rotating the BTZ. of the black hole to zero and compute the back-reaction. We begin by reviewing simple examples of the above geometries JHEP11(2019)037 1 2 ], S Z not 14 / × ) (2.3) (2.2) θ 1 X S in non- × i V below. and/or ∆ h φ geon and 2.4 2 . As shown in [ P i R V . ∆ 2 h φ 2 = for the m . The extension to rotating 1 V U in the quotient (BTZ S µν . ∂ g i to vanish in the Hartle-Hawking i 1 2 i kk = on suppressed angles kk T kk − T h µ J h T ∂ φ h µ σ k dU φ∂ is non-zero and spacelike in all cases, though factors (just ρ ∞ takes the form 1 ∂ 0 Jx S Z φ , so that the right alone may be used to represent ρσ g N V – 5 – ) covering space, compute back-reaction on the geon, KKEOW, and KKZBO geometries, with null µν ↔ g πG X 2 1 2 U P × = 4 R − also act on i φ ν J V ∆ φ∂ h µ ∂ ) is related to the integrated null stress-energy tensor along the = , but we will in fact compute X X µν × which exchanges T for a null horizon × J ν back to the (BTZ on the horizon and its image k i ), and take the quotient of the result to obtain a self-consistent semi- µ x ) indicated. The left region (dashed lines) is the image of the right (solid lines) k kk 2.2 T µν h T isometry U, V 2 = 0 this is a result of the additional action of ≡ Z . Conformal diagram of the V kk = T The stress-energy tensor of a scalar field As noted in the introduction, symmetry requires U between a point metric using ( classical solution. BTZ and to broken rotational symmetry will be reviewedstate below on in BTZ or section BTZ where the symmetry isthe broken. resulting At linear order in perturbation theory we may then lift in the linearized approximation androtating in BTZ the (or presence BTZ ofhorizon rotational through symmetry, Here for shown in this figure. a null geodesic firedboundary from at a the finite left future time boundary with in (on average) the coordinate distant past then arrives at the right Figure 1 coordinates ( under the the quotient. The desired isometries for the KKEOW ands KKZBO geometries) not shown in the figure. The geodesic proper JHEP11(2019)037 ) − (2.8) (2.6) (2.7) (2.4) (2.5) : ], the cor- J 23 is a coincident i kk ). Terms involving , geon, the KKEOW T h ∆ 2 Jx − ( 1 ˜ φ RP  2 / . 1  ˜

1 M ) and ) i 0 . x ) ], − . The pointwise contribution to x to be negative for one choice of ( i (  ˜ 2 φ ) 29 Jx φ M Z ( ]. The scalar two-point function on φφ U 2 i h φ Jx ` ∂ ( 2 23 ) kk U + ˜ φ x ∂ T m ( ) h Z  x φ  is explicitly finite. Non-zero contributions ( ) U 2 φ 1 + x ∂ / ( M U 1

i ˜ p 0 φ ∂ − x – 6 –   kk  1 → T 2  h x h 1 , and ∆ is given by the same formula as the scaling − √ = 2 = lim , 2 2 i ≡ and its image under the relevant isometry Z M ) ∆ = 1 R i ) is the half squared-geodesic distance in an associated ) 0 x ) ) is thus a sum of four terms involving coincident limits ]. It may be written in the form ˜ x ( π M x ( 1 cycle of the quotient. ( 4 28 kk φ = 0, one will find 2.3 x, x and thus vanish by the symmetry noted above. Thus all 2 T kk ( with an appropriately scaled linear combination of the scalar h Z T σ M ˜ ) = h i ) corresponds to a choice of periodic (+) or anti-periodic ( , we may write , M M Z 2 U kk ( ∂ 3 2.5 T /` ], for linear quantum fields correlations functions in this state may h ) 0 = 23 AdS µ G ∂ x, x ( µ σ k is known exactly [ 1 + 3 ≡ ) on the covering space ) on the quotient x x Z ( ( . ˜ φ φ The stress-energy tensor ( As reviewed in [ M where four dimensional embedding space of the associated operator in any dual CFT [ We now review further explicitempty AdS scalar results from [ or symmetry imposes periodic or anti-periodicrect scalars. choice turns As out verifiedspacetime, to and by be the detailed periodic KKZBO computations boundary geometry. conditions in for [ 2.2 the Explicit calculation potentially divergent terms vanish and come only from the two cross terms. These contribute equally and give We can now see that creating a traversable wormhole is simple, as unless some coincidence boundary conditions on the of two point functionscoincident involving points all in possible the pairings coveringHartle-Hawking of space state are on proportional to the stress-energy tensor in the field The choice of sign in ( where the computation istry to be done in the Hartle-Hawking state onbe the quotient computed geome- using thefield method of images. In particular, one may identify the quantum limit of derivatives actingthe on null the stress-energy two-point function tensorfunction. can Since thus be determined by derivatives of the scalar Green’s Contracting with the null vectors eliminates all but the first term, so JHEP11(2019)037 th . i n . )  (2.9) 2 ) (2.11) (2.12) 0 1) VU − + from the 0 (∆ 8∆ + 4∆ th term in the  geon. The full − n n,p 2 UV  i 2 ) , so the sum over U . Since the three- 0 (6 1 kk π P . 2 φ produces a tower of S T R U U h + − R 1 + ) + 2( 1 ` 0 = φ S φ p V ( 0 ) + ) will still prove useful in 0 3 from the 2 = + V U U to recast the BTZ Green’s ] only the result in the limit r i 0 2.11 3 φ  kk 23 -independent result , + 8 T 2 n h 2 ) (2.10) p in our Kruskal-like coordinates is )(1 + n 2∆ + ∆ 2 U 2 with radius , Z = 0 and  2 ( − ] for associated results in the context UV 1 1) cosh 1 3 R 1 + S V S ` 23 of contributions (2 − 4 R 0 p = AdS p (1 + U  0 . The additional isometry to produce the , so the back-reaction does indeed render V U G 0 n 2 U − Z + i U πn  + 4 ∈ 2 kk X n covering space such that ` . – 7 – 1)( T 2 3 + 2 1 h 1 + with 0 m h − S ) = φ π s ∆ Z ( ) is known exactly, it is useful to write the full four- = − UV 1 = ( 0 n + 1)  `  ) BTZ φ 2 2.10 n 2 ) / clearly diverges, the result ( eff 0 G 5 U ) n m V 2 0 and + (2 U U 2 mode on the 1 + to denote the contribution to φ , numerically integrating the full expression for periodic scalars , Kaluza-Klein (KK) reduction on the /` √ th n + = ) m ) over i 2 p r (1 + 0 0 n + ` 3 )(1 + φ kk is quite complicated, so we recall from [ U T 2.11 ( x, x h i πU ( UV U σ kk 32 wormholes. ) choice can also be of interest; see [ , and by extension the two-point function on the T 3 2 − h denotes a choice of boundary conditions at AdS infinity. We will always is the mode number on the internal (1 + ], the half-squared geodesic distance in ) and the Z 1 + 1 + 2 AdS Z   23 ≡ G ), for periodic scalars one finds in this limit the ) = 2.10 0. Using ∈ . For general 0 = n 3 p Z n 2.10 i → geon means that we are concerned with cases where x, x The calculation for the KKEOW geometry is slightly more involved. For any given By substituting the expression for the embedding space geodesic distance, we can From [ ( kk 2 σ /` T P h + where dimensional two-point function ( dimensional stress-energy tensor asterm in a ( sum over four-dimensional mass massive three-dimensional fields with effective masses given by While the sum ofsection ( yields a strictly negativethe result wormhole traversable. for each sum ( R images is a sum over calculate expression for r function as a sum over images in the AdS where We can use the fact that BTZ is a quotient space of AdS take the (+) boundaryHowever, condition the below, ( as thisof choice the is above always consistent with unitarity. where the JHEP11(2019)037 i ) 2 3 + ,p B J r kk =0 T n . For 2.13 h i (2.14) (2.13) , each φ B kk coincide to avoid in the ψ T h U n,p − i using Dirichlet ] for details. and kk p . 23 T the action of ) A h = 0 modes. For ψ U ( , π n O ; see [ = 0 now depends on + N , they are related by the i φ 3 µ kk γ ∆ T = 0 contributions h − n 2 representation and from the action of the isometry below, with the understanding π p 6 A a B 1) γ . , both of which form a representation − 2 i / γ 2∆ + 3∆ 1) 2 = ( and ] also found more complicated examples − − d − =0 ( d i 23 2 a A + ipπ I γ Q e 2  3 if we also impose a cutoff at small is + in the 2 − – 8 – = 0. Indeed, such terms are independent of = d ], or as choice of N . As a result, the expressions for i − 3 U i γ 30 2.14 [ J πU 1  . | ≤ − =0 ab 64 i Y p 8∆ + 4∆ d φ | = η terms and Abel summation for the constant term. One may thus − 1 . For numerical calculations, we can simply choose some p − 3 = 2 d . /U γ } a B + b γ is negative for all periodic scalars of the type discussed above. is finite and continuous. However, at 3 − , γ i and 1 a 1 ] that the explicit terms above vanish when summed over n,p πU kk varies with = 3 γ i T { 23 i 32 a h A kk /U γ V T = h ∆ h ,p =0 n i geon, spinors can be well-defined only when both representations are present. as a sum of explicitly finite terms. kk 2 i T h each term and sum over modes with kk RP T h N , π Since the two spinor representations differ by the sign of all The KKZBO computations are similar but the integrated Such contributions include a factor of of . These alternating signs play an important role, as the It was also shown in [ θ 2 6= 0 regularization for the 1 In particular, our theorycorresponding to on a the different geon representation, but must which contains are exchanged two when spinorη one fields traverses a rewrite that the sign onrepresentation, the with right-hand-side of action of the three-dimensional parityreversing operator, isometry which we of may 3Dlike take the to Minkowski be space. any orientation- As a result, on a non-orientable spacetime We shall simply denote the two choices as We now consider how thefermion fields. above To constructions begin, should recallsentations be that in spin generalized odd groups dimensional to admit spacetimes. two accomodatedistinct inequivalent One possible fundamental bulk may repre- definitions think of of thisof choice the as arising Clifford from algebra two the simple scalars discussed here,negative periodic along boundary all conditions horizon makewhere generators, the the though integrated sign [ of 2.3 Subtleties of spinor representations Again, the integrated φ coincides with the KKEOW actionsthere of as well. In particular, corresponding care is required for the Since the four-dimensional stateenergy is tensor Hadamard can and have the onlymust quotient integrable cancel singularities is when and smooth, summed the the over large non-integrable full terms stress- in ( possible issues from incomplete cancellations of such terms at finite on all have non-integrable singularitiesand at give JHEP11(2019)037 ) i ) of x ˆ J 1 2.16 J (2.15) (2.16) (2.17) ( taking ) more B ˆ J ψ ) contains ) , 2.16 ) generates x (  stress-energy ) is an operator . Thus ( A 2.15 . The the lack x 2.17 geon, which we ψ ψ x ( B . One may also ˜ h 2 i M ˜ M A  ˜ RP ψ B 1 ˆ ψ J ] ) stress-energy tensor on h ). 31 ). For fields of non-zero m A preserves the vielbein and ) + − 2.16 2.5 ˆ evaluated at x J µ ( ˜ reverses orientation and must requires one to first find some . ψ if B D ) ˆ 1 ˜ J ψ µ ˆ B x J J ( Jx  iγ gives the ( 2 i 1 B x √ ˜ denote the gamma matrices for each ψ and ψ j ). Since the Lagrangian ( ≡ h x to define the appropriate extension x + 1 ) µ A,B A J x γ ( ) + ( 2.4 ψ ) x B gives the similarly-vanishing B ( ˜ ψ m ˜ x ψ – 9 – 1 gives canonical normalization of  − J , ψ 2 ˜ µ ψ 1 fields. Let us consider first the expectation value of  √ ) D will again be related by ( x µ A B ( ≡ ˜ M iγ i ) ) will then satisfy the of motion on the quotient ( B below, any isometry can be extended to a map x A and ˜ ( ψ 3 ψ 1 ψ and  . In other words, in this context we may write ( ˆ 2.16 A J ) . In particular, the spinor field B h e M ψ Jx ) + x in the Hartle-Hawking state on the covering space ( and . The important point here is that = det( A 1 i A ˜ ) ψ L x ψ /J  ( ). We may thus write ) is built from the operator 2 A . The ansatz ( x 1 ). ψ ( √ Jx J ) ] in the scalar case, the Lagrangian on the BTZ covering space must also take  ( to spinors at ) is the vielbein determinant and satisfies the corresponding equation on the covering space x ˜ B e ψ ≡ 1 23 2.15 ˜ x ˜ ψ ψ J ) 1 ( x J ( B stress-energy tensor, which is a quadratic composite operator much like the stress-  . We will use these conditions in section A ψ a h We now consider the implications for stress-energy tensors on the Let us now consider the analogue of the scalar relation ( ψ A γ when and, as discussed in section four terms. The termthe involving covering coincident space points which at vanishesthe by term the involving same coincident symmetrytensor. described points above at for The scalars, finaland and two terms are cross-terms built from the correlators where no interactions, the fullstress-energy tensors stress-energy for tensor the isthe of course aenergy tensor sum of of a separately scalar field. conserved As in the scalar case, using the first in ( represent as BTZ thus interchange explicitly as M check that canonical normalizationis of the desiredHartle-Hawking method-of-images states ansatz. on As for scalars, correlation functions in spinor Since the equation ofthe motion is linear, itthe is isometry preserved by way to identify theJ corresponding two tangentspinors spaces. at It isbuilt natural from to use the isometry fermion representation. In termsscalars in of [ a method-of-imagesthe construction form paralleling ( that for spin, adding the field operators at distinct points fermion corresponding to either representationwell-defined alone, theory. this In AB this doublet context, of we fermions may yields use a the Lagrangian [ where det( non-contractible curve that reverses orientation. While it is inconsistent to have a single JHEP11(2019)037 . ) part 2 (2.18) θ dφ  2 − , and thus 1 S stress-energy below. R UV r A ) and the 2.4 + 4 2 ) 2.18 fields in each doublet geon above, and the full UV of radius 2 1 AB geon, the − ] found that back-reaction P S 2 R ) (2.19) (1 23 π 2 + RP r preserves (  + stress-energy tensor by the same . The cross-terms in the stress- 3 2 + J cross-correlators vanish identically. B J φ, θ dφ ) with an quotient of a spacetime with a Killing − ) AB 2 Z 2.18 V, U, V dU ( ) part of − → – 10 – ) UdV U, V, φ ( − `r U, V, φ, θ + 4 ) means that such :( 3 , the KKZBO spacetime can be defined in the presence of J discussed above. In particular, the 2.15 . thus remains zero and first-order back-reaction does not render 1 . Defining 2.1 dUdV 2 S 2 V AB ` dθ 4 1 − 2 S ( R 2 ) , it is clear that the ( UV 1 2.1 ]. We briefly review these results here for later use in studying back-reaction 22 (1 + , ) takes the form 17 = To begin, recall that the rotating BTZ (rBTZ) metric in our Kruskal-like coordinates The upshot of this discussion is that in any One might ask if fermionic back-reaction is more interesting on the KKEOW spacetime. 2 ds U, V, φ with line element as in section preserves the metric on ( We are interested in the Cartesian product of ( from matter sources becomesmay large. thus naturally Perturbations that createwith render an [ the eternally wormhole traversablefrom traversable wormhole the stress-energy in tensor this of limit our Weyl in fermions. agreement ization of our discussion thus far to this2.4 spacetime is reviewed The in section KKZBOAs spacetime and mentioned back-reaction inrotation. section In particular, it admits an extreme limit where [ opposite chirality, which vanish in Hartle-Hawking state. symmetry (in any spacetimeenergy tensors dimension) on the the horizon contributionwe will focus of vanish on unless spinors the the orientable to quotient KKZBO expected is spacetime orientable. in stress- For the this remainder reason of this work. The general- terms of each pair, the discussionstress-energy proceeds tensor precisely again vanishes. as for From the the even-dimensional four spinors dimensional admit perspective a the conserved pointare is notion that of exchanged chirality, and by that theenergy the two tensor orientation-reversing chiralities are isometry then built from two-point functions between Weyl spinor components of There the full spacetimeClifford algebra is and four-dimensional, it is soa invariant single there under orientation-reversal. Weyl is fermion. Thus onlyreduction we to However, need one three as only representation , consider for ofin where the the one pairs scalar obtains like case a theare one set fields related may of uncoupled proceed by free reversal by fields dimensional of that orientation come on the three-dimensional (orbifold) base space. In So despite the breakingtensor continues of to the vanish on Killingargument. the symmetry The horizon, on as shift does ∆ the the the wormhole traversable. of an interaction term in ( JHEP11(2019)037 , , − ) = r 3 2 − r will ˜ 2 + − M r (2.22) (2.23) (2.21) (2.20) 3 2 2 + ` ] found r ( π 23 = KKZBO = . , , which is just M 2 ) =0 0 q q φ 2 ( H ` d 3 + i 3d KKZBO , to act on spinor fields. are then a subleading . We will find similar i − d kk 1 + 3 2 + still far enough from kk T φ i S r r iqr h T 2 2 h ) kk + × ` 0 r T 2 φ − h . It is thus useful to focus on 2 − , the problem can be Kaluza- − − r 1 r φ S ( − that we will compute in section ], this invalidates our perturbative = of rBTZ dφdU , . This lift and quotient procedure 2 + 2 3 3 2 23 + 0 r + J J r r π/ π/ dUH H π 2 0 1 − 2 Z ) is that the zero-mode KKZBO dφ = computed on the quotient i = . ∞ π π 0 ) to rotating BTZ and to sources that break 2 2 − q kk − ` r Z 2.21 M Z T i + h 2.2 . Since the temperature of rotating BTZ is + − – 11 – kk r φ ∞ πr KKZBO ,H 2 2 + −∞ T , compute the associated back-reaction on ). In the scalar case, the numerics in [ N 2 i q ` r h 1 Z . As noted in [ G ], the correct result is S H kk 4 ) − N = ) is usefully described as a sum over Fourier modes of T 0 2.20 0 r 23 h 2 × φ ) over φ ` T = ) remain finite at − πG -circle to write → of φ 2 − ( φ , but it also suggests that a full non-perturbative analysis 1 2.20 + iq φ − 2.21 = ( r e S r average H q kk i ) to be approximately independent of = rBTZ X V ˜ ∆ over the dUh M 2.23 h ) = 0 i T ∞ φ smaller than any classical scale, but with 0 V Z − − ∆ 2 r h φ ` very close to 1 ( 6= 0 modes in ( 4 − + q H r + = , and then quotient the result again by r below by extending the KKZBO isometry 1 V S ∆ 3.1 × Perhaps the most interesting feature of ( We thus require the generalization of ( To understand the implications of the behavior below. 3 Spinors and stress-energyWe tensors now turn on to the thesection KKZBO fermion spacetime details for stress-energy tensors on the KKZBO. We begin in where the factor ofof 4 integration is relative to associatedthe those with extreme in making limit ( of use of ( symemtries to change the limits the average of the time delay ( we may average analysis for could produce a static, eternallycases traversable with wormhole. We willthat thus be our most perturbative interested treatment in effect remains as valid. the Variations with diverges in the extreme limit where the Green’s function the form since we preserve rotationalKlein reduced symmetry to on studying back-reaction thedirectly on internal three-dimensional compute rBTZ. the Indeed, effective section the three-dimensional integral over stress-energy the internal rotational symmetry. As shown in [ we must understand thewithout back-reactions of spin, such one a mayto source lift the on the the covering geometry. source space rBTZ As in thegives case the same result as computing back-reaction directly on the KKZBO spacetime. And JHEP11(2019)037 1) , not (3.3) (3.2) (3.1) (3 ) thus of this O b µ 1) being e j ˜ 1) gauge rows and , a b , j (3 (3 V , , which has O = a µ O e      leave invariant ) as the action as a spacetime a µ ) and defines the ) and j a µ 3.2 3). Choosing the je e ab UV , U η 2 , 1 must satisfy = 2 , (1 + . As a result, in addition j ˜ 1 } a b S Γ and the point at which the R = (0 , a ) 0 µ (such that ( a Γ j { j. UV ˜ b 1), one may use any vielbein that 00 0 0 , Γ − 1 a b      , can differ only by an internal j (1 1 as desired. It is easy to check that this 1 1 is given by treating + , a µ − j 1 r 1 e ) a µ ◦ − j ˜ e ) − 3 ), it also exchanges the 1 V J = ( and on − that satisfies ). 3.1 + = a ` 3 a µ 1 } – 12 – 3 e J U a ˆ − 3.1 J 3      ( row. We note that this combined operation is = diag( Γ (Γ)] J − { isometry ) in ( j = r , it would be more useful for our method-of-images φ that leaves the vielbein invariant. Since the isometry ab b 3 a − η 3 j ˆ = [ J labels the rows. J ) and an internal index ) 2.3 acts on spacetime indices a µ we take V Γ 3 1 J − U, V, φ, θ S `` ` 0 0 0 U ( × . U, V, φ, θ − r . Any two such vielbeins are related by an internal − . Thus 3.4 = ( a µν      g µ = UV studies a simplifying limit in preparation for more complete numerical 1 ab η that preserves the vielbein ( 1 + b ν 3.3 then sets up the calculation of the desired stress-energy tensor components e 1) transformation on covectors. There is then a corresponding action 3 ˆ , a µ = J labels the columns and e (3 a µ 3.2 e a O To describe the corresponding action on spinors, one should view ( However, as noted in section The natural action of the isometry preserves the metric, the vielbeins 3 transformation on spinors, up tothe a sign double to cover be of discussedthe the below four-dimensional associated associated with Clifford spin group, algebra four-dimensional defined spinor by representation. requiring The that spinor-space defines a of an transformation. Choosing theallows right us such to transformation define ais vielbein-preserving the case for changes the sign ofa all symmetry entries of in the the vielbein. construction to have an operation J where vector field for each field is evaluated but hasto no further acting action on on the the internal arguments index ( spinor fields are notspace tensors, to and each area point typically spacetime in defined index spacetime. bymetric attaching on an This the internal is internalsatisfies done to by be choosingtransformation. a For rBTZ vielbein and section calculations in section 3.1 Spinor extensionsAny of isometry the has a natural action on tensor fields via the associated diffeomorphism. But Section JHEP11(2019)037 ] ≡ = in in ) is 33 ν j ˜ , B µν (3.6) (3.7) (3.8) (3.4) (3.5) ). In D g 3.3 32 . This 3 3.5 and and A ab µ ω so that ( b , we can choose a Γ a a b 1 − 2). Since each three- ) as defined by ( , j , 1 3 ) ˆ . , J 3 = ( ,  Γ takes a block diagonal form  b , aµ a 2 Γ ψ 1) to the associated spin group and e = (0 ) a , b Γ λ ν µ 1 j to express the four-dimensional δ a − γ − , ) 1 = 1 + d ψ S ( j ˜ Γ . for ν a ( aν O and those defined by spinors invariant − e Γ a , D 3 1 λ µ γ 0 ˆ µν δ . J − ), so we may take the corresponding extra 2 ) matter  (Γ δg j ˜ γ ) + ( 1 3.1 δS → ψ g 2 λ a ) iγ ) ν 3 − √ matter – 13 – δe = D Γ µ , j ˜ δS = ( 2 ) γ Γ ( e µν one finds ( , on spinors is ambiguous up to an overall sign implies ψ 1 1 T †  Γ j ˜ 3 i ˆ det( , 2 J 0 = − = ) when the vielbein appears in the action only through 1 ), the Lagrangian is best written in terms of the vielbein so = − : (Γ ) µν is the with spin connection 3.6 a j ˜ T µν Γ that are invariant under ab 2.15 i T b a ) for a spinor field yields the Belinfante stress-energy tensor [ -translation used to construct rBTZ as a quotient of AdS j 1 = ( Σ φ kk S 2 3.7 ab µ T Our convention in this work will be to use Γ h × ω 1 3 1 2 Γ i + = µ ) reduces to ( . This tensor is both symmetric and conserved. ∂ j ˜ ab 3.7 . In either representation we thus find ≡ . As a result, any use of the terms periodic and anti-periodic spinors is generally a Σ 3 on spinor indices to be trivial. ˆ µ , where the subscripts denote the two spinor representations labelled ab J µ . Evaluating (  2.3 j D ω ν b − a B 1 2 e 0 γ The fact that the action of Since we will use Kaluza-Klein reduction on the This case is like what one often sees in flat space, where the discrete isometry is a special case of a ab − a A 3 0 η γ µ µ a and setting where ∂ continuous isometry. Continuous isometries liftcan uniquely be from used to define natural notions of periodic and anti-periodic spinors. Note that ( e For spinor fields, asthat in this ( source is defined via action 3.2 Computing For bosonic fields one defines the source for the Einstein by under choice of convention as these termschoice can of become this well-defined sign. only aftercontrast, making consider an the arbitrary latter isometry already preserves the vielbein ( that there are two naturalby notions spinors of on periodic rBTZ spinors on the quotient space: those defined terms of thedimensional three-dimensional spinor gamma representation matrices is invariantbasis under for the the action four-dimensional of representation any in Γ which each Γ section satisfied as desired. fields in terms of three-dimensional fields on a BTZ orbifold, it is useful to express Noting that JHEP11(2019)037 . 3 = x J (we µ (3.9) n 1 , and 0 (3.10) (3.11) S x ,  isometry and ) . Recall also µ 1 for a fermion 3 over propaga- x γ ˆ on the horizon S J ) 3 0 p × i BTZ AdS kk i x, x S ) 3 T . 0 h Z J } on the internal x (  ∈ ( ) 0 n β x ipθ ¯ 3 ψ BTZ e P ) propagators can be written S ] in the Euclidean case. In )] (3.12) 0 x 0 = 3 ( ν , x, J , and the tangent vector ( 34 ( α s  D ) ψ x, x on the horizon must again vanish j h ˜ BTZ x BTZ ( S x + S i )]Λ( k ˜ j ˜ s ψ ) ) = ( 3 0 0 ) and taking expectation values in the ˆ J x / kD 3 h j ˜ / nβ  3.8 x, x ( 0 x, J ) + Tr ) + in the covering space. As for scalars, in the α β ( { x into Fourier modes s , which for ( ( x x ) is a periodic spinor under our ˜ ψ 3 – 14 – 3 α ψ Im BTZ BTZ J  S S 3.9 ) 2 x, J A,B ν X 1 on spinors of the appropriate rBTZ ) = [ √ 0 D 3 ˆ +1 µ J p ( ≡ and at x, x γ ) 1) (  x . This construction and the relevant formulae are reviewed x 3 − ( x ). As noted earlier, this reduction yields three-dimensional Tr at ψ AdS 1 = ( S S 2.12 A,B X p by decomposing ). Inserting this into ( indicates that the right-hand-side adds togeher the contributions 0 i x × 3 x 3 ) of the associated geodesic distance kk is further given by an image sum → s J lim T x ( ( A,B h ˜ i ψ 2 , β j p BTZ ) s S 1) ( we may write the associated 3D stress-energy tensor on rBTZ , following the same procedure as used by [ α − ) = p i above, we can use the method of images to compute the expectation value . x B Hartle-Hawking state again yields 4 terms. Two of these are the expectation 3 ( kk = ( denotes a covariant derivative on the second argument that acts on the associated T 1 i ) to this geodesic at 3.1 0 0 h p ˜ S ψ ν i 3 ˆ D J × µν much as for scalars. The critical relation is x, x h T ( i Using the maximal symmetry of this spacetime, the AdS For each However, again as for scalars it will be useful to Kaluza-Klein reduce our spinor to a Having understood the action h s µ kk ∂ T ∂x in terms of thetwo spinor functions propagator Λ along the geodesic connecting in appendix The propagator tors in AdS spinor indices and Σ from the two three-dimensionalenergy representations. tensor can The be null-null rewritten component of this stress- Hawking state where that are again givenspinors by in ( both representations. in terms ofof the the 3D appropriate spinor effective propagator mass and choice of spinor representation in the BTZ Hartle- terms associated with distinctfull points four-dimensional stress-energy tensor such terms are manifestly non-singulartower for of all spinors on AdS assume periodicity on this circle). The resulting three-dimensional spinors have masses where we have chosen signsthat such that ( rBTZ values of by symmetry in the Hartle-Hawking state. We thus need only compute the remaining cross- in section h JHEP11(2019)037 n ), . ϕ . s (3.14) (3.13) B.19 ) must and   can also ) 1) frame  n 0 , 3 ) s φ 0 3.10 isometry to so that the for all φ /J to its image − ) 3 , ` 3 1 1 − ), and ( φ x `  ( S ) φ n ( − × ) = s r + s , this is a rotation ( B.17 r  β points along the in- to AdS ), one then finds the  A global coordinates. In  3 ), ( x ` 3 n 3.1 2 s ) sinh 0 = (rBTZ B.14 1) cosh ˜ ) and the relevant geodesics, coth M UV must continue to hold in this . In particular, both − ` 0 1 3.1 0 2 − j ˜ x , the contribution to ( 0 V x 0 − ). U n and VU d ) in the associated auxiliary frame and ) for can be written and rotation in AdS ds x 1)( B.9 when the geodesic from 3 whose lift to AdS isometries to map the geodesic segment n +2( isometries, diffeomorphisms, and internal 3.5 π intersects the axis of this rotation at the − 2 3.11  3 x 3 on spinor indices of our extended isometry 0  ) X x ) under the group generated by both an rBTZ onto the bifurcation of the (auxiliary) µ j ˜ 0 UV 0 k 3 φ ( 1) gauge, and thus on our choice of vielbein. in AdS x µ ,  between – 15 – 0 n into − ` ( 3 φ 1 µ ( x, x k X + 1) − µ ) is given by combining ( 0 r , we may choosing the lift of ) s n ( n V  0 ) holds in our auxiliary non-rotating BTZ SO(2 = rBTZ) and a ϕ α ( U , in terms of our rBTZ coordinates the explicit form of the Z 1 Z 3.2 ∈ / and then use AdS A X 3 n ) cosh p 0 3 . In particular, we may fix an auxiliary (say, nonrotating) BTZ + 1)( 1) so as to intersect the associated rotation axis at its midpoint, and − 3 VU n, x UV ( 4( + 0 −  1 − to the relevant image point and the choice of representations then yields UV ) = to the relevant image point ) plane. For each n x x 2 x ( is the unit-normalized vector at cosh . p + 2( ` ) then follows from the second line of ( µ i n ,X ) s 1 is the geodesic distance in AdS ( n kk is lifted to AdS X β T ) = ϕ n 0 0 h s 1) gauge transformations and using such transformations to separately simplify the x , As reviewed in appendix To proceed further, it is useful to note that the quotient However, it remains to find an expression for the spinor parallel propagator Λ. This x, x is unchanged. Noting that ( ( s -translation (for which AdS 3 ˆ point where depend on geodesic distance between two points where ( finitesimal generator of rotations of preserve the relevant axis soJ that the action as well as inframe as the well. physical Computing each one,summing contribution over to the ( expression ( in the ( geodesic from midpoint of the geodesic, andbirfucation we surface may of then our choose auxiliary theaxis BTZ isometry followed moving coordinates by the to a geodesic be translation to just along the a it. rotation In around particular, the we same may choose this AdS spinor parallel propagator Λ to be trivial along suchbe geodesics, generated with by Λ( taking theφ quotient of AdS terms of the standard embedding coordinates reviewed in appendix be invariant under the combinedSO(2 action of AdS computations for each on AdS running from BTZ horizon. Using the (non-rotating version of the) vielbein ( whence of course dependsRather on than our compute choice the ofit result is SO(2 simpler directly to for proceed the by choice noting ( that, for each particular, the explicit form of JHEP11(2019)037 . x − ρ are φ ( , and ,p (3.16) (3.17) (3.15) ). + . This U r =0 p i , so that n . Writing 2 i = θ 3.13 p ρ 0 , or for any kk = 0. This is T V U cos h 2 1 + θ φ ρ p  ) 3 sin = 0, p ρ  ( ) dθ ) 2 3 V p  ( ) φ R m 3 2 ` = − ρ r = 0 we find 0 m ( 2 ) is summed over = 0) for any U n O − θ 6 + 4 6= 0 the terms + 8 +  = 0 = 2 3.16 = 0 term near + ρ θ r θρ is the circumference of the circle n 3 + exp h ]. = 0. It is useful to understand this cos ) 1 + 2 πρ cos p = 0 (sin 23 θ φ ( )) supplies an additional factor of U p thus raises no issues. Integrating over 3 2 − + φ ρ r r ρ = m ) φ sin ( 2 6= 0 and 2 + 2 to 1. p at U − ( O p ` dθ − 3   ) is separately integrable for each U 2 is the radial coordinate defined by either 1 φ R m ρ as above imposes +  `  (1 +  0 r 3.16 φ θ at fixed 2 x 2 + 6 – 16 – 2 − ` ρ 1 + = 0, or 2 ` 4 r  ) 2 U yields 3 2 ρ ρ 2 cos p θ  ρ θ ρdρdφ − ) πρ ; in other words, 2 + vanishes in this limit. For  0 p ) also vanish, and all other terms give finite results due 8 1 + cosh ∝ ( ρ sin x µ  2 3  about this axis.  k ρ (1 + 3 sin 1 1 0 exp ) µ m to 3.17 5 is quite complicated, we will compute the details of stress- − − x φ 2 = ϕ = x i + πρ U dUdφ φ r or ,p ` 8 kk + cosh cosh 3 + 4 r + T ] found the expressions to simplify greatly in the limit x =0 ` ` h  n ) is singular at 23 i + φ coordinate system whose rotation axis orthogonally intersects the ≡ sinh ( ` + on the horizon and ) = kk r φ 3 0, which in particular holds for the 3.16 T − x  h r 2 3 → U, φ and sinh = 0). Integrating over − ( ` s θ θ ) 0 also yields a finite result since sinh Ue φ cos r U = ` − ρ . The result simply vanishes for any ,p φ = ρ ( = , simplifying the result to and below. However, it is useful to first discuss certain general features of ( =0 defined as before and where we have set = 0 (cos − φ ρ n U in a global AdS i φ ρ − φ U However, in our case the singularity in ( The expression ( In the scalar case, [ 0 3.4 ` kk − x /`, r r = T at ) 0 0 h − at small φ both integral of the explicit terms into ( the fact that the measure e singularity since, as described above,most the an four-dimensional integrable stress-energy singularity tensor at canfinite, this have point. so at any For non-integrableis singularity precisely must what cancel occurs when in ( the scalar case studied in [ with tion φ even more true in our case, as defined by rotating either 3.3 General features Since the general form of energy tensor profiles and the associated back-reaction on the metric numerically in sec- where or midpoint of the geodesic from φ Note that choosing JHEP11(2019)037 i on kk 3 T ˆ h J (3.20) (3.18) (3.19) on the N ). One might , ) 3 2.11 . m ) ( and the average over m of the isometry O ( over the full horizon is j ˜ φ ) yields O + i . Indeed, for non-rotating 2 3 + kk + . 3.16 m 2 ) r T 2 3 h m m 2 m /U 2 − (1 −  2  O 2 ρ ρ 2 ] found cases where the sign of the integrated / and the symmetry of non-rotating BTZ 3 ) + 23 ) 1 + ) gives fixed, which yields µ φ 2 1 + ) ϕ + ρ p 2 r m p 2.11 ρ . Since in that case all other factors in + ]. We will phrase our results in terms of the the spinor result ( µ + ρ in detail. For this task we resort to numerics k (1 + 3 ρ 23  – 17 – µ 3 with i  (1 + ) m φ n 3 φ − kk πρ U r ϕ − . As above, our results are for the effective three- T 8 3 = 4 sinh ( r ( πρ h 4 − = 0. We choose our cutoffs such that our answers do V 2 e − ,φ ,ψ ∆ e ) dU 2 , φ =0 → − φ =0 R n n U + µ i i r k i ∝ = 0 as well. kk = kk µ ) kk T T φ and U h h ,φ n T h i ϕ ) has many similarities to the scalar expression ( sinh ( =0 ` kk n U i T h 3.16 kk = T in the bosonic expressions ( h ,ψ This odd-parity behavior turns out to arise from the non-rotating limit m requires ( =0 4 n 0 regardless of whether a similar limit is taken for φ i 1 + kk → T ` h ) → . This stands in marked contrast to the scalar results whose signs are generally → − 0 φ φ φ − The expression ( Though with specially engineered boundary conditions [ φ 4 ( → − -independent. − number of Kaluza-Klein modes over(integrable) which we singularity sum at and alsonot regulate the change functions significantly near when the these cutoffs are altered. scalar stress-energy depends on 3.4 Numerical results It remains to study and follow the samedimensionless basic quantity strategy as indimensional [ Kaluza-Klein-reduced stress-energy tensor. We impose a cutoff under are even, the integrated stress-energythe full becomes horizon an must odd vanish.find function But below of this that symmetry is for brokennon-zero. non-zero by angular rotation and, velocity indeed, the we average will of It is interesting thatφ the above expressionsφ for our fermionr field all changeBTZ signs there is under no preferred sign of the unit-vector of view). However,similar the differences other in discrepancies the reflect limit of the large difference in kinematics. We find The fact that the denominatorsconsider differ a by single a real factor scalar of and a 4 four-component may spinor be (from ascribed the to four-dimensional the point fact that we while ∆ expect a particularly simplepation relation numbers between are the small two andvs. in quantum fermionic effects the are statistics large suppressed is mass somains unimportant. limit that different where the But in occu- choice the that of kinematic limit, bosonic fermionic structure associated indices. with of In the the non-trivial particular, expressions action at re- large JHEP11(2019)037 ) ], 4 3 at − 23 ], at φ with . Fig- = 0, a  23 . Both 2 in the 3 φ of which / U = and 10. N U √ = 1 U and )). As in [ − = 2 r 1 ψ ϕ S , suggesting that 2.23 + 1)) (3.21) + r `/R N ( . In the scalar case, over both → (see ( we integrate the stress- + = 1 and U r i and a scalar field at extremality in figure shows the details of the − + U, m r i r kk ( ψ 3 T f kk h T average 1.4 h = 50, with i +1 . We thus find a large average numerical + N V  N r h i dU are shown in figures 1) T kk R i − for a given 1.2 T h kk φ . We choose our spinors to be periodic T h dU 02, the sum over BTZ images has been done ` )) + ( . <  p . The physical importance of this quantity 1.0 R for a spinor field . We have chosen | ( − = 0 r U U |  AdS ∂ ` = U, m – 18 – ] by requiring some KK modes to have the ( 0.8 ( = 0 in the extreme limit + f 23 r plots the integral of µ p ∂ → . As opposed to the scalar contribution, the spinor 5 µ 1) k − 0.6 − T r ( . As the spinor stress-energy tensor vanishes at , it is proportional to N . We will interpolate between the origin and = as U − with N . Notably, the spinor contribution is everywhere positive, as X 2.4 = + φ 0.4 p >  r U /T ). Figure 1 | 1. Here the cutoff = . U vs. | 2.8 ∝ and i 0.2 = 0 for small φ kk = 1 in units of inverse φ i T h 〉 m kk ` numerical average kk i T i h ℓ〈T kk V 0.2 0.1 h T -0.3 -0.1 -0.2 h such that we get a positive overall contribution to the stress-energy tensor. . Plot of 3 ˆ J shows the contributions to the stress-energy tensor for a spinor and scalar of the same = 3, and the sum over KK modes has been performed to We also numerically calculate the contributions to Some results for the dimensionless quantity In particular, our numerical expressions are computed via 2 n a non-perturbative treatment may lead to an eternally traversable wormhole as in [ boundary condition in ( extremality as a function ofis the that, as radius discussed inwe section find numerically that thistime-advance function is independent of linear interpolation from thethe origin scalar was case. used, while a constant interpolation wasfor used various for values of contribution generically changes signthis phenomenon at must different be engineered [ ure mass at a particular valueopposed of to the scalarinterpolation contribution of which changes sign. Figure have an additional sign change.energy In tensor computing only over a linear approximation forand the integrate spinor the and interpolating a functionunder for constant for the scalar, as shown in figure where we have added an extra term so as to sum over an even number of terms, Figure 2 particles have mass same units, as wellto as JHEP11(2019)037 . The φ U and 0.04 = 3. − r n 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 = + 0.03 r 02 for the spinor and . = 0 1.5  0.02 1, the sum over Kaluza-Klein . = U = 0 1.0  0.01 at various values of ϕ 〉 ψ 0.5 kk 0.00 ℓ〈T -0.280 -0.285 -0.290 -0.295 U 10. The cutoff – 19 – ϕ 0.0 ψ √ 0.04 〉 = kk 1 S ℓ〈T 0.03 -0.5 `/R = 20, and the sum over BTZ images was performed to 0.02 at extremality for a spinor -1.0 = 1 and N compares the relative size of this quantity at extremality for one i m kk 6 , the spinor contribution to the stress-energy tensor is less than that T h m ` 0.01 -1.5 .

2

0.4 0.2 0.0 1.0 0.8 0.6

+ -

ψ = r 〉 r kk 0.00 . Details of the interpolation between the origin and . Plot of ℓ〈T 0.01 0.05 0.04 0.03 0.02 to the mean.four-dimensional Figure complex scalar field andmentioned four before, four have the dimensional same realdimensional number scalar of mass fields, degrees which of as freedom.of For the all equivalent values scalar of fields, the three- owing to cancellations from the spinor kinematic structure. Figure 4 four dimensional mass modes was performed to least in the presence of some large parameter that controls quantum fluctuations relative Figure 3 scalar in figure JHEP11(2019)037 2) and , 1 , = (0 m = 5. The spinor is a vector field on the horizon. m n ϕ i 10 m=0 m=1 m=2 kk T . Relevant values of the h N at extremality for one four- , where µ - i 8 k kk =r µ T + ϕ h r + = 3. This quantity appears constant at extremality for n i 01. The sum over Kaluza-Klein modes . 6 kk dφdUr T 8 h 2 2 = 0 + π/ π/  − for the small value of R + ∞ 0 r 6 dφdUr – 20 – R 4 2 2 π/ π/ − R ∞ 4 0 R 2 ` known as the Kaluza-Klein zero-brane orbifold (KKZBO). 01, the sum over Kaluza-Klein modes modes was performed to includes an overall factor of 1 . ϕ 〉 quotient is associated with a sign that controls the periodicity i S = 10 with a cutoff kk 2 kk 2 = 0 1 × 〈T Z T S + h  are indicated above. /4∫r ψ `/R m 〉 ψ 〉 = 1000 and the sum over BTZ images was performed to kk . 0.10 0.05 0.20 0.15 kk 〈T , up to corrections at small m N + 〈T + + r ℓ∫r ∫r -0.10 -0.15 -0.20 -0.05 . Plot of the quantity . Plot of the ratio of the quantity quotient of rBTZ = 10. The cutoff 2 Z 1 The fact that spinor fields carry Lorentz indices leads to notable differences from the S = 200, and the sum over BTZ images was performed to of a As in the scalar case,of the the bulk field as well as the overallscalar sign case. in the The null spinor stress-energy contribution to the integrated stress-energy tensorwith is an always significantly equivalent less number than of that of degrees the of scalars freedom. 4 Conclusion We have studied the contributions of bulk spinors to the integrated null stress-energy tensor Figure 6 dimensional complex spinordimensional field mass and fourwas four-dimensional performed to real scalar fields of the same four- `/R N for all values of four-dimensional mass Figure 5 JHEP11(2019)037 ] ) ) − r 6 37 , were 2.21 = 3 -circle. 36 + , φ becomes r 18 , /T 1 15 ], the quantity ∝ for non-rotating 23 described above. φ φ average i V − h to be independent of becomes short-ranged, without i where AdS bulk supersymmetry -circle is correspondingly reduced V φ m ∆ h average i V is an odd function of h µ T ). As in the scalar case [ k µ ϕ – 21 – average ]. It also interesting that, again as in the scalar i 22 V h is odd as well and integrates to zero over the i kk = 1 bulk supersymmetry do not lead to special cancella- = 0, so that the time delay T h N is associated with the fact that the simplicity of the model in T and the average over the (anti-)symmetry is broken at non-zero angular velocity, the is manifestly due to the breaking of the BTZ Killing sym- µ φ φ k µ )) but to remain non-traversable when entered across the other ϕ → − , π KKZBO i φ (0 kk ∈ ], numerical results suggest quotient operation. This T quotient used to build KKZBO from extreme BTZ. Since this Killing h φ 2 23 2 Z 0 is nonzero at Z between bosons and fermions on the extreme KKZBO spacetime. This lack to the generator-dependent time delay still varies with i i < ] and can be used to compute the stress-energy of such fields on the KKZBO i kk kk 35 kk T T h h T h have opposite signs in the two representations and cancel. As a result, a more chiral average i A natural next step for further investigation would be to study fields of even higher spin, Such results in particular make clear that, even though extreme BTZ preserves certain While the above This overall factor of µ V k h µ -circle (say for spin-3/2 and spin-2reaction fields to would our wormhole then geometry.of allow It signals would passing one also through be our to interesting wormholesto as study to was understand study gravitino done the how for back-reaction and GJW differences wormholesinformation graviton in from that [ back- the can be scalar transmitted. case above affect limits on the amount of supersymmetry at the same level. and in particular toExpressions better for the understand vector what propagator forgiven kinematic both in factors massive and [ might massless arise particlesspacetime on in AdS in such direct cases. analogy to the computations performed here. Further extensions to of cancellations is especiallyrelates fermions manifest and at bosons largenon-vanishing of of mass nearly equal masses.metry It by should the notsymmetry be is a part surprise of the since supersymmetry the algebra on extreme BTZ, the quotient must break at extremality, though with valuesdue noticeably to smaller partial than cancellations in associated the scalar with case the (see variation figure insupersymmetries, sign models with with tions in T large. At least in therelative presence to of some the largeeternally mean, parameter traversable that this controls wormhole quantum suggests as fluctuations case in that studied [ a in [ non-perturbative treatment may lead to an have this property. sign of relative to the scalarthe average case. negative (and But thus choosing also the aforementioned sign correctly still makes half-circle. four dimensions means that bothalgebra possible enter three-dimensional with representations equal of weight. theϕ Terms Clifford that for each representation areconstruction not that proportional caused to the two representations to enter on a less equal footing would not BTZ, so in that caseAs the a spinor result, at leastrelating in the limit of largerotation black hole one radius where finds theφ the Green’s function wormhole ( to be traversable when entered across one half of the defined by the JHEP11(2019)037 is (A.4) (A.2) (A.1) 2 , 2 R are defined V and U   φ φ ` ` − − r r embedding functions become ), where .  2  2 3 sinh sinh φ φ ` ` − − dX  r  r V V t t U, V, φ + UV UV 2 2 )] (A.3) − − − ` − + − ` 2 1 ∗ r r )] t r t . In our new Kruskal-like coordinates, 2 ∗ U U 2 , but to describe a rotating black hole + 1 + 1 + + − ` , the AdS r − ` dX  − r t + r r t φ 2 2 − − − − + φ +  + − (  r r ` ` r + t + `r 2 κ φ φ − − 2 = r ( r . 2 2 ` ` . The embedding functions for this geometry − − κ − r r − φ  r φ  r r − dT ` ` + √ √ r φ – 22 – + 1 sinh 1 cosh r r − − + 2 2 − − 2 → 1 − − cosh sinh r r cosh cosh and − − = exp [ = exp [ φ α α α α dT cosh sinh 2 2 + + + V V r r √ √ √ √ U V − r ` ` ` ` UV UV √ √ + − = UV UV UV UV = = = =  U U 2 1 1 2 2 1 + 1 + κ 2 − − ` 2 T T ds X X   1 1 + 1 1 + ` ` ` ` = = = = = ∗ r 1 1 2 2 T T X X in Rindler coordinates. The metric in the embedding space 3 and 2 − r + r . For this problem, we would like a coordinate reparametrization that makes − 2 ` 2 + 2 − 2 − r r r − − 2 2 + = r r κ ≡ α and using the corotating coordinates where for the null directions manifest, soto we use be the coordinates ( The usual Rindler patchsolution has we a requires two horizon horizons are at given by A Geodesic length in Kruskal-likeWe start coordinates with AdS We would like to thanksions. Zicao Fu, This Brianna work Grado-White, wasfrom and supported the Aron in University Wall part for of by useful California. NSF discus- grant PHY1801805 and in part by funds Acknowledgments JHEP11(2019)037 . .  (B.1) (A.8) (A.5) (A.7) 2   dφ ) ) 0 2 − φ . −   ` ) ) 0 φ 0 UV r ( x φ = 1, as it should ( − 2 2 r + 4 − ` ]. , with the necessary n X 2  d ) φ ) ( 39 x H ( + 2 r UV X  ) sinh , as the geodesic norm at a − 0 0 ν ) + , satisfies (1 n 0 0 s UV 2 + timelike x ν µ pointed out in footnotes. The spacelike (A.6) µ r ( g ∂ − s 1 s 1) cosh d 0 − X ≡ − ) ) + ( = 0 . x µ VU ) , which takes vectors between the two ( ) 0 µ 0 n V geodesic, which are given by 0 1 0 ν µ n d g X U V dU +2( ) x, x 0 x, x = 4 can be found in [ ( ) − ( 0 −  1)( s s ) 0 d ) = 0, and the constraints on the bitensor of 0 0 µ µ x, x − x ( φ ∂ ∂ x, x ]. ν ( – 23 – ( σ 2 , this metric becomes the BTZ metric with n UdV − = = σ 35 1 ` T µ π ( ), one can also derive an expression for the geodesic UV ) 0 1 − φ µ ( − µ ∇ x ( −  n ( d µ n ] for the parallel propagator in − `r A.4 2 + 2 n r T cos cosh . In particular, 34 0 φ  ` ` ] + 4 ≡ x + 1) 0 ) + 38 ν 0 ∼ V ) = ) = x n 0 0 0 ( and µ 1 ) cosh U 0 φ dUdV 1 x ∇ T 2 x, x x, x 3 ) ` ( ( x Kruskal-like coordinates using the geodesic distance in the embed- 4 s s VU ( in appendix C of [ 3 1 + 1)( − 0 + T for timelike geodesics in ν ν  0 2 i g ) 2 UV 1 ` ( µν UV . T  h UV 1 1 M ) = − 0 2 + 2( ` in our AdS (1 + s = x, x cosh ( = ) is the same quantity that appears in the calculation of the scalar Green’s function 2 − ` σ 0 r 2 We start with defining the norms of the AdS With our embedding functions ( = + ds s x, x ( 2 + We also define the bitensortangent of spaces defined parallel at transport point is in the opposite direction of the geodesic length. We follow the derivation given infew [ changes requiredcalculation for of analytic continuation to AdS for a spacelike geodesic, B Spinor propagator in AdS As a consistency check, the norm of this geodesic, Substituting our Kruskal-likegeodesic embedding distance functions, in AdS we arrive at an expression for the The geodesic we willσ be concerned withand for is given our by calculations is the spacelike case. This distance ding spacetime. The result is [ Under an identification r The metric that results from these embedding functions is JHEP11(2019)037 . ) 0 µν g B.7 (B.5) (B.6) (B.7) (B.8) (B.2) (B.3) (B.4) → − 0 By invert- µν g to similarly- 6 . 0 ,  0 µ . ` s α β and  n δ ` s ) ). Inserting ( 0 µν s g x g ( and . csc csch β ) − − µ ` 0 1 ` 1 → − µ n √ x − ( n − ) µν δ = 0 as g − = . 0 ) and ) C 0 0 s x ν ν α β , C ( x, x ) = ) δ n n 0 0 α 0 0 ) to and ν 0 ) to be . )Λ( . µλ  γ x β x g and ) 0 x, x ` µ s g ) x, x 0 is a solution of the spacelike Dirac µ ) − n ν − B.2 x  γ ν Λ( i + ` s n ) √ x n 0 − 0 )( µ cot ( Ψ( )]Λ( 0 λ µ 0 x δ ν n ` s 1   n n ( α β ( 0 n ) + = β ν = ] by the transform x, x µν − coth ¯ 0 / mα γ nβ g 0 ψ , x A 0 35 ` ) 1 µ α β µν µν )( x  − )Λ( γ x g g ) ( 0 C ) + ( = ( )( C β α / n s – 24 – ) relating the derivatives of A C ( C + ψ A +  C = Λ( h α x, x C = = A ( + 0 A ( − 0 α ( ν S mβ ν ) A ) − 0 1 2 ) = n ( A 0 n ) = [ and x µ 0 − m 1 2 µ ]. − ( = 0 ∇ 1)( A α 0 ∇ − with respect to primed and unprimed coordinates are fixed x, x 34 ) Ψ x, x ( 0 νλ − 0 / ( ) = ) = C α β D 0 0 g α β are instead given by ( d S µ (  S + d 1 2 ∇ x, x x, x H A + Λ( Λ( , which transports a spinor from 0 0 0 1)( µ µ α β β D − D  d 5 ( + 1 2 and some functions of the geodesic distance + 0 µ α γ µ ) yields ], these functions in n   34 = B.6 The spinor propagator We also define two functions / n In the case of timelikeIn geodesics, [ these differ from [ 5 6 for into ( with appropriate short-distance singularities. Insymmetry the of AdS the vacuum it spacetime. will It share the must maximal therefore be of the form In principle, we couldthe define covariant a derivative, vierbein, andHowever, find solve as the a for exact system the scalar form ofgreatly case, of PDEs simplify making the the for use derivation gamma of the [ the matrices spinor maximal and parallel symmetry propagator. of AdSequation turns out to parallel propagator Λ The covariant derivatives of Λ by the parallel transport of the gamma matrices and ( ing the second equation and substituting the first, we find In order to properly treat spinors at two disconnected points, we also need to write a spinor indexed quantities For spacetimes with negative curvature, JHEP11(2019)037 .  ` from s 2 (B.9) (B.16) (B.13) (B.15) (B.10) (B.11) C 2 cos , and ≡ g ) . z s A − ( g ) δ s √ − ( δ m √ , or equivalently, in ], with ab, d − m 34 − and = ) = ,  d z α ` dz ) , we need to find the short s 0 ( 2 d  ) x γ ) (B.12) g 2 z . 2 ) in Mink  − − 1) 1) 2 . ms 2  ) 4 ` ( √ x d 1 ( − − 1 s 4 , cosh ( ): . Taking the trace of the above δ − b B.10 d − s s − 2 ( ( . This is a hypergeometric equation ≡ 2 = = 0 2 + d mδ d/ z α ` z − 2 a ) (B.14)  K ( = ) z 2 m Γ ( mβ mα d α − d/  α 2 − AC − − c − z 2 1 1 m s β α s + √ ) + 2 ) ) + ( 2 2 − z d/ C C ( 0 2 ≡ d/ – 25 – 0 C 2 − α )  γ d − + dz π 1 z π 1)( ) (  m A A 4 2 m z s z γ − −  d − 1)( 1)( d − ( ≈ − + + 1) − − (1 1 2 ) z 00 d d d s ( ( ( ) = + α ( s 2 2 1 2 α ( − ) = α m z + + ;  0 0 d + 1 ), which is just the solution of ( , we make the substitutions β α s ) becomes − d = 0 is given by 2 d ( 0 s α  a, b, c ( Aα B.10 H 1) ) + .( z ( − ) in AdS 00 ], one finds a second order equation for d γ ) is the modified Bessel function of the second kind. The series expansion of ) → ∞ 35 z z B.10 ( + ( ` 7 n 00 − K α ), with differential operator (1 z z ( For timelike geodesics, we would instead make the same substitutions as in [ γ 7 where here primes denote derivativesin with respect to to obtain B.2 Solution inTo AdS solve ( where this solution around The solution to this equation,in properly the normalized coincident such limit, that is the left and right sides agree B.1 Solution inIn Minkowski order space to find the properdistance normalization behavior of of the solutions inthe AdS limit Substituting the second equation intotable the 1 first and of using [ identities involving equation gives a set of two coupled differential equations where a prime indicates a derivative with respect to JHEP11(2019)037 , s (B.20) (B.21) (B.17) (B.18) (B.19) . We’ll ], so we i . There 34 µν T h , → ∞ , the null-null  z / k ) i 3.2 ) , and as . , x i ] 0 are , x µ z 0  x J 32 solution in [ 3 x  1 z h J λ ; d , ( Λ( . i H . ) S  requires the + sign. The 0 0  ) m` ) 0 1 ) k x | 2 ν ν ( D g but kept the sum over three- m` − The expectation value of the m` ψ  j ˜ µ  m` ) 2 d (  | p  1 8  x γ ( ) m` Γ( ]. 1 2 0 ) (B.22) ψ  )Γ 39 h m`,  ), as we don’t know anything about Γ( + Tr 0 , x 2 0 2 x, x d   m` m` / ( x j ˜ . As shown in section 2 0 ( 1) ) S x, x  0 0 Γ x ( S − m`, ν  x 2 d d S 3 ( = D =  π . In order for this stress-energy tensor to be Γ 0 x d Γ(1 − 2 d x, J γ , though large B − d ) ( | † 1 0  at − ) – 26 – ` S 0 2 1 ) 2 k m` F I  | and x, x 2 m` ` ( x, x ) s 2 / kD 2 ( ). For simplicity, we will choose the (+) boundary A S   + 1. This is the same as the ) = ) m` d S  0 ν ( 2 d 0  2.8  Tr − γ 2 d D λ 2 for timelike geodesics, see [ ( = µ x, x ( − 4 ∓ c A,B ≈ X γ z = 0 is , ) i 2 =  s s h ( λ  m` =  and Λ( Tr λ i 0 + α 0 x ) = ν ν 2 d µν are given by z δ → T ( lim x h  =  = i λ ν 2 , as this is allowed for any effective 3D fermion mass. We do so for all ) around γ b 0 s p k ν ν , ( µ = g  k i m` α in KKZBO µν i − i ≡ T 2 d h kk kk T T = ), but we can still calculate observables such as h h 0 a For a similar calculation in AdS 8 x, x dimensional fermion representations real, we expect that thethe second fact trace that is related to the first by complex conjugation. Using component of the stress-energy tensor in the quotient KKZBO spacetime is where we’ve suppressed the sum over Kaluza-Klein modes where a primed indexordinate on and the action covariant from derivativepoints, denotes the namely a left. derivative The on parallel the propagators second become co- trivial at coincident We still don’t have aΛ( closed form expression for only carry out theBelinfante calculation stress-energy for tensor our in quotient terms of spacetime. the spinor propagator is [ The choice ofanalogous sign to in that these referencedcondition quantities for in corresponds all ( tocomputations a in the choice main of text. boundary conditions B.3 The above expression must matchso with the the coefficients Minkowski solution in the limit of vanishing Both are allowed forexpansion sufficiently of small can proceed as follows.are two We independent want solutions, solutions which that up decay to as overall a normalizations power of with JHEP11(2019)037 .  ) x (B.26) (B.27) (B.28) (B.23) (B.24) (B.25) 3 ) terms, s ( ), as they x, J β s ( Λ( α  . / n  )  i ) x † 3 A / k ∗ 0 . In these coordinates, . i  −   γ 2 † ϕ ) x, J / k ) / k / k . x ) C ) † ρ∂ 3 ( dϕ Λ( ) , x  0 2 0 , x / 2 n , x β 0 ) 0 ρ . We therefore have geodesic x x = 0 / k x 3 3 x x, J x j ˜ + 3 ) with metric 1 µ + 3 J 3  . As the trace of the Hermitian 0 J ( J † ϕ Λ( 2 (  ( β x, J j S ˜ / n Tr ( S 0 S )) /` x, J  0 / k − 0  2 ( S γ 2 k x j t, ρ, ϕ ˜ k 0 0 ρ 3 +  S † † k k = D dρ D k ), we obtain j ˜  j ˜ D D ) Im j ˜ Tr )   † † s x, J 1 +  j j − ˜ ˜ ( / kD C B.5 h h j ˜ Tr β Tr +  = )Λ( Im + −  2 − / n 0  ` A ) Tr s ) dt ( 2 = s = Tr = Tr =  – 27 – jγ ˜ (  A 2  α 0 i coordinates ( 2 β j ˜ † γ Im ) − 3  0 + /` coth j ˜ 0 2 x ) C ) + 3 ρ ` 0 α A,B X ( s 1 and 2 x (  2 β 3 − † α  x, J − 1 + / k ( + (( / = k 0 x, J S j ˜ µ ∂ ) involves a path ordered integral along the geodesic in ques- i = ( ∂s 0 k − β  , we define the origin by the intersection of the timelike axis ) and substituting ( 0 D S kk 3   µ k = T Tr ) ) / kD x, x h / kγ  2 µ µ B.7  0 and a perpendicular unit vector / kk / kD n n j ˜ γ ds ρ µ µ Tr  Im h ∂ k k 2 Tr Tr 2 ( 2 ( /` A,B X  2 − − ) ρ 1 µ representations are distinguished by gamma matrices with opposite sign: Im = = 1+ n µ i B √ k . The sum over representations will cancel terms containing , we can choose global AdS A,B X ( kk 3 µ B = T γ − − h and µ − = = n A i = kk µ T A h For a timelike slicedefined of by AdS the isometrynorm and the geodesic itself as in figure In general, calculating Λ( tion, so it’s easiereven to choose for coordinates noncoincident such points.space that AdS the As parallel our propagator calculation becomes ultimately trivial takes place in the covering The γ involve an odd number ofas gamma they matrices, contain and an introduce even a number of factor gamma of matrices. 2 to The the stress-energy tensor is therefore the first, and the stress-energy tensor becomes Plugging in our ansatz ( where we’ve also used conjugate is the complex conjugate of the original trace, we have as expected. The sum of the two traces can then be represented as the imaginary part of the trace of the Hermitian conjugate of the first term becomes JHEP11(2019)037 = 128 ρ (B.33) (B.29) (B.30) (B.31) (B.32) Phys. Rev. , J. Math. , ) and Phys. Rev. and BTZ im- , p U, V, φ . ) s ( . β  ` . s 2  φ ` + r (1988) 395 ) = 0 coth 0  ` 1  . 56 2 / µ ϕ x, x 2 U sinh ρ / k ϕ − ` 3 i 2 µ ` Λ(  ` ρ + ∂ ∂s ik ∂ / ϕ. 2 φ/` . ρ φ/`  / 2 n ρ − n I i − r ) = Tr = 2 r 2 µ p  − = − ρ – 28 – ϕ e e / n ) = ) = j ˜ µ 0 0 / k k Amer. J. Phys. j = ˜ =  , )( x, x x, x U U µ n ϕ Tr n Λ( Λ( , we have Wormholes in spacetime and their use for interstellar travel: Causality and multiply connected space-time µ µ = =  k ), which permits any use, distribution and reproduction in φ ⇒ D µ µ − µ ` 4 ( n ϕ The particle problem in the general theory of relativity r n µ 2 µ − k k − = ].  i . ]. kk exp CC-BY 4.0 T ) along the geodesic: h 0 2 as SPIRE This article is distributed under the terms of the Creative Commons 2 U IN SPIRE γ [ x, x IN 1 + Ether flow through a drainhole: a particle model in general relativity [  (1973) 104 iγ φ ` + = 14 r j ˜  (1935) 73 2 . Explicitly, in terms of our Kruskal-like coordinates ( n (1962) 919 Phys. A tool for teaching general relativity 48 H.G. Ellis, M.S. Morris and K. S. Thorne, A. Einstein and N. Rosen, R.W. Fuller and J.A. Wheeler, sinh [3] [4] [1] [2] r References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. The fullages calculation involves a` sum over Kaluza-Klein modes This expression is manifestlycomponent vielbein of independent. the stress-energy The tensor final is expression therefore for the null-null isometry The trace inside of the stress-energy tensor therefore becomes covariance of Λ( Additionally, as the vielbein for this coordinate system is diagonal, we can rewrite the the spinor parallel propagator for our spacelike geodesic is trivial, as can be seen from the JHEP11(2019)037 ] , ]. , ]. 61 Int. J. JHEP , 71 , SPIRE , D 76 IN SPIRE ]. (2018) ][ IN [ ]. Fortsch. Phys. , D 98 SPIRE IN arXiv:1707.02270 Fortsch. Phys. SPIRE [ ][ ]. ]. Phys. Rev. , IN , ]. Phys. Rev. Lett. ][ ]. , SPIRE SPIRE Phys. Rev. IN IN ]. , Visualizing Interstellar’s SPIRE arXiv:1509.02659 ]. ][ ][ IN SPIRE [ [ arXiv:1804.00491 IN , ]. (2017) 025021 Topological censorship and higher ][ SPIRE SPIRE IN IN ]. ]. ][ gr-qc/9902061 arXiv:1807.07917 SPIRE D 96 ][ [ [ ]. IN (2016) 201 Solving the Schwarzian via the conformal ][ Topologically nontrivial configurations in the SPIRE SPIRE Wormholes, time machines and the weak energy ]. Rotating traversable wormholes in AdS gr-qc/9305017 Topological censorship IN ]. IN The black hole in three-dimensional space-time SPIRE Analytic self-gravitating Skyrmions, cosmological ]. [ Traversable wormholes in four dimensions ][ IN B 752 arXiv:1502.03809 Diving into traversable wormholes – 29 – ]. [ A perturbative perspective on self-supporting Traversable asymptotically flat wormholes with short Phys. Rev. arXiv:1708.03040 ][ SPIRE , , arXiv:0910.5751 ]. SPIRE Traversable wormholes via a double trace deformation IN SPIRE [ (1999) 104039 (2019) 045006 IN IN ][ [ SPIRE ][ hep-th/9204099 36 arXiv:1705.08408 IN [ Cool horizons for entangled black holes SPIRE [ (1995) 1872] [ (1988) 1446 limit of superconformal field theories and supergravity hep-th/9711200 ][ D 60 IN Phys. Lett. (2015) 486 Achronal averaged null energy condition ]. [ Eternal traversable wormhole , N ][ 75 61 Teleportation through the wormhole 83 -model system SPIRE σ (2010) 024038 arXiv:1608.05687 [ IN (2017) 136 (1992) 1849 [ arXiv:1704.05333 [ Phys. Rev. The large 08 (1999) 1113 , 69 D 81 arXiv:0705.3193 arXiv:1807.07239 arXiv:1908.03273 [ [ 38 Erratum ibid. Dear qubitzers, GR=QM , arXiv:1306.0533 Class. Quant. Grav. [ Am. J. Phys. Proving the achronal averaged null energy condition from the generalized second Phys. Rev. Lett. [ , (2017) 151 , JHEP , ]. , arXiv:1707.04354 12 [ Phys. Rev. (2018) 005 (2017) 1700034 , Einstein-nonlinear SPIRE d IN wormholes transit times Phys. Rev. Lett. 046016 arXiv:1807.04726 12 (2013) 781 65 bootstrap law Theor. Phys. JHEP 4 [ (2007) 064001 bounces and AdS wormholes (1993) 1486 genus black holes condition Wormhole Z. Fu, B. Grado-White and D. Marolf, M. Ba˜nados,C. Teitelboim and J. Zanelli, L. Susskind, J. Maldacena, A. Milekhin and F. Popov, Z. Fu, B. Grado-White and D. Marolf, E. Caceres, A.S. Misobuchi and M.-L. Xiao, J. Maldacena and L. Susskind, L. Susskind and Y. Zhao, J. Maldacena, D. Stanford and Z. Yang, T.G. Mertens, G.J. Turiaci and H.L. Verlinde, J. Maldacena and X.-L. Qi, J.M. Maldacena, P. Gao, D.L. Jafferis and A.C. Wall, N. Graham and K.D. Olum, A.C. Wall, F. Canfora, N. Dimakis and A. Paliathanasis, G.J. Galloway, K. Schleich, D.M. Witt and E. Woolgar, E. Ayon-Beato, F. Canfora and J. Zanelli, O. James, E. von Tunzelmann, P. Franklin and K.S. Thorne, J.L. Friedman, K. Schleich and D.M. Witt, M.S. Morris, K.S. Thorne and U. Yurtsever, [8] [9] [6] [7] [5] [24] [25] [21] [22] [23] [18] [19] [20] [15] [16] [17] [13] [14] [11] [12] [10] JHEP11(2019)037 , , , Nucl. , ] Phys. Rev. ]. , ]. SPIRE field theories, SPIRE IN IN N ][ ]. ][ Cambridge Graduate Texts in , , gr-qc/9302012 ]. Large Traversable wormholes in SPIRE IN [ SPIRE IN , Cambridge University Press, Geometry of the (2+1) black hole ][ ]. hep-th/9905111 , Ph.D. thesis, University of Sheffield, [ ]. (2013) 069902] [ arXiv:1906.10715 [ SPIRE ]. ]. , Cambridge University Press, Cambridge IN Conformal field theory SPIRE Fermions in odd space-time dimensions: Back D 88 ][ IN (2000) 183 ][ SPIRE SPIRE arXiv:1907.13140 – 30 – IN , IN hep-ph/0502089 (2019) 070 [ [ ][ 323 , Cambridge University Press, Cambridge U.K. (2007). Information transfer and black hole evaporation via 09 Supergravity Erratum ibid. [ (2005) 71 JHEP hep-th/9912059 (1986) 669 , Vector two point functions in maximally symmetric spaces [ Phys. Rept. 37 Entropies of scalar fields on three-dimensional black holes hep-th/9412144 Single exterior black holes and the AdS/CFT conjecture , [ 103 , Springer, Germany (1997). hep-th/9808081 [ (1993) 1506 (2000) 3021 Dirac fermions on rotating space-times String theory. Volume 2: superstring theory and beyond (1995) 340 Introduction to the ADS/CFT correspondence Spinor parallel propagator and Green’s function in maximally symmetric spaces D 48 Few Body Syst. A 33 ]. , (1999) 066002 B 447 SPIRE IN Cambridge U.K. (2015). Sheffield, U.K. (2014). traversable BTZ wormholes AdS and bounds on information transfer Contemporary Physics J. Phys. Commun. Math. Phys. to basics U.K. (2012). string theory and gravity Monographs on Mathematical Physics Phys. Rev. [ D 59 Phys. V.E. Ambru¸s, S. Hirano, Y. Lei and S. van Leuven, B. Freivogel, D.A. Galante, D. Nikolakopoulou and A. Rotundo, H. Nastase, W. Mueck, B. Allen and T. Jacobson, M. de Jesus Anguiano Galicia and A. Bashir, D. Freedman and A. Van Proeyen, P. Di Francesco, P. Mathieu, and D. Senechal, O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y.J. Oz, Polchinski, J. Louko and D. Marolf, I. Ichinose and Y. Satoh, M. Ba˜nados,M. Henneaux, C. Teitelboim and J. Zanelli, [39] [36] [37] [38] [34] [35] [31] [32] [33] [29] [30] [27] [28] [26]