Multi-mouth Traversable Wormholes

Abstract

We describe the construction of traversable wormholes with multiple mouths in four spacetime dimensions and discuss associated quantum entanglement. Our wormholes may be traversed between any pair of mouths. In particular, in the three-mouth case they have fundamental group F2F_2 (the free group on two generators). By contrast, connecting three regions A,B,C in pairs (AB, BC, and AC) using three separate wormholes would give fundamental group F3F_3. Our solutions are asymptotically flat up to the presence of possible magnetic fluxes or cosmic strings that extend to infinity. The construction begins with a two-mouth traversable wormhole supported by backreaction from quantum fields. Inserting a sufficiently small black hole into its throat preserves traversability between the original two mouths. This black hole is taken to be the mouth of another wormhole connecting the original throat to a new distant region of spacetime. Making the new wormhole traversable in a manner similar to the original two-mouth wormhole provides the desired causal connections. From a dual field theory point of view, when AdS asymptotics are added to our construction, multiparty entanglement may play an important role in the traversability of the resulting wormhole.

Full text

Multi-mouth Traversable Wormholes
Roberto Emparan,a,b Brianna Grado-White,c,d Donald Marolf,cand Marija
Tomaševićb,e
aInstitució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, E-08010
Barcelona, Spain
bDepartament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de
Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
cDepartment of Physics, University of California, Santa Barbara, CA 93106, USA
dMartin Fisher School of Physics, Brandeis University, Waltham MA, USA
eKavli Institute for Theoretical Physics University of California, Santa Barbara, CA 93106
E-mail: emparan@ub.edu,bgradowhite@brandeis.edu,
marolf@physics.ucsb.edu,mtomasevic@icc.ub.edu
Abstract: We describe the construction of traversable wormholes with multiple mouths in
four spacetime dimensions and discuss associated quantum entanglement. Our wormholes
may be traversed between any pair of mouths. In particular, in the three-mouth case they
have fundamental group F2(the free group on two generators). By contrast, connecting
three regions A, B, C in pairs (AB,BC, and AC) using three separate wormholes would
give fundamental group F3. Our solutions are asymptotically flat up to the presence of
possible magnetic fluxes or cosmic strings that extend to infinity. The construction begins
with a two-mouth traversable wormhole supported by backreaction from quantum fields.
Inserting a sufficiently small black hole into its throat preserves traversability between
the original two mouths. This black hole is taken to be the mouth of another wormhole
connecting the original throat to a new distant region of spacetime. Making the new
wormhole traversable in a manner similar to the original two-mouth wormhole provides the
desired causal connections. From a dual field theory point of view, when AdS asymptotics
are added to our construction, multiparty entanglement may play an important role in the
traversability of the resulting wormhole.
arXiv:2012.07821v2 [hep-th] 2 Apr 2021

Contents
1 Introduction 1
2 Review of two-mouth traversable wormholes 3
3 Gravitational construction 6
3.1 Multi-mouth wormhole construction guide 7
3.2 Size limits on the third mouth 9
3.3 Lowering the small mouth down the throat 12
3.4 Signaling between mouths 13
4 Discussion 14
A Matched asymptotics construction 18
B Mass at a height in the throat 23
C Multi-mouth wormholes from a perturbed bifurcate horizon 23
1 Introduction
Under certain natural assumptions, topological censorship theorems forbid constructions of
traversable wormholes in classical general relativity [1,2]. In globally hyperbolic spacetimes
obeying the null curvature condition, such theorems require causal curves to be deformable
to curves that lie entirely in the boundary of the spacetime, and that these curves can
be made to remain causal throughout the deformation. Recently, however, it was shown
that well-controlled quantum effects could be used to violate the null energy condition in
a manner allowing the construction of traversable wormholes [3–9] that circumvent these
theorems; see also [10,11] for studies of the dynamical production of such traversable
wormholes.
Quantum effects in gravity are typically difficult to control unless they are in some sense
small. For this reason, the above constructions of traversable wormholes can be thought
of as starting with background spacetimes that contain an almost traversable wormhole
that can be rendered traversable with small corrections. In classical solutions satisfying
the null energy condition, this generally requires the background to contain a bifurcate
horizon having no causal shadow12; see Fig. 1. Naively then, it might seem as if traversable
1A causal shadow is defined as a bulk region which is causally disconnected from the boundary, see [12]
for more details.
2Though the wormholes of [5] are not explicitly written in this form, [9] gave a similar construction
which could be written as a perturbation around a bifurcate horizon.
– 1 –

wormholes are constrained to connect only two regions of spacetime having a single mouth
in each region, as backgrounds with more interesting connectivity require some sort of finite
causal shadow, and this in turn necessitates a larger amount of negative energy to make
the wormhole traversable.
Figure 1:Left: A bifurcate horizon in a two-sided asymptotically flat spacetime. Right: A
spacetime with a causal shadow (shaded in purple).
Nevertheless, we show below that constructions with higher connectivity can still
be controlled. Our analysis begins with the more familiar two-mouth asymptotically flat
wormholes of [5], enhanced by including a large number Nfof four-dimensional massless
fermions. We then perturb this solution by adding a small black hole to the bottom of
the wormhole throat. Wormholes are very fragile, and semiclassical black holes have large
masses in Planck units, so one may worry that the insertion of this small black hole could
destroy traversability. However, the extreme redshift deep in the wormhole throat allows
semiclassical black holes to sit in the bottom and leave traversability intact. Indeed, we
show that one can actively pass a small black hole through a wormhole mouth and place it
at the bottom of the throat without choking it.
We take this small black hole to contain an additional wormhole that connects to
another distant region of spacetime. This new wormhole can then be made traversable
with further quantum effects in a manner similar to the original, two-mouth wormhole.
The resulting spacetime then has fundamental group F2, the free group on two generators.
This differs from the fundamental group F3that would be obtained by adding three separate
two-mouth wormholes connecting three distant regions of spacetime A, B, C in pairs AB,
BC, and AC; see Fig. 2.
The above construction also has interesting implications for the quantum states of
wormholes. First, the ability to add a small black hole to a two-mouth traversable wormhole
indicates additional traversable excited states beyond those anticipated in the analyses of
[4,5]. Second, at least when embedded in AdS/CFT, our three-mouth traversable wormhole
appears to involve a new entanglement structure different from the TFD-like entanglement
associated with two-mouth wormholes.
The paper is organized as follows. In Sec. 2, we review the construction of four-
dimensional asymptotically flat traversable wormholes of [5]. We describe the gravitational
construction of our multi-mouth traversable wormholes in Sec. 3. We conclude with a
brief discussion in Sec. 4, focusing on quantum states and entanglement. The details of
the matched asymptotics expansion for the construction that employs the wormholes of [5]
– 2 –

Figure 2:Left: A two-dimensional analogue of our spatial topology has two handles. The actual
three-dimensional space has fundamental group F2, the free group on two generators. Right: A
space with three wormholes connecting regions A, B, C in pairs AB,BC ,AC has three handles. In
three dimensions, the fundamental group would be F3.
are given in the Appendix A. In Appendix C, we discuss the construction of multi-mouth
traversable wormholes starting from the asymptotically flat two-mouth wormholes of [9].
2 Review of two-mouth traversable wormholes
We review here the construction of two-mouth traversable wormholes in four-dimensional
asymptotically flat space of [5]. As described in Appendix C, the implementation of our
construction of multi-mouth wormholes using the two-mouth asymptotically flat traversable
wormholes of [9] proves to be more difficult.
Building a traversable wormhole requires some source of average negative null energy,
so that RdλTabkakb<0for knull and λan affine parameter. Intuitively, this is because
null rays moving into the wormhole throat initially converge, but need to diverge to exit
the other side. This focusing/defocusing of light rays is controlled by the null energy
through the Raychaudhuri equation, with positive null energy causing null rays to focus
and negative null energy causing them to defocus. Classical matter, which obeys the null
energy condition Tabkakb>0, is thus insufficient to construct a traversable wormhole.
Quantum effects, however, can give rise to negative null energy (e.g. Casimir energy) when
certain boundary conditions are imposed on quantum fields, as can happen in the presence
of non-trivial topology.
The wormholes of [5] start with near-extremal magnetically charged Reissner-Nordström
(RN) black holes and take the near-horizon limit, where the metric becomes
ds2=r2
e−(ρ2
r−1)dτ2
r+dρ2
r
ρ2
r−1+dΩ2.(2.1)
Here reis the horizon radius of extremal RN black holes of charge Q(quantized and
dimensionless), which is
re=√πlP
Q
g,(2.2)
where gis the coupling constant of the U(1) gauge field and
lP=pGN(2.3)
– 3 –

is the Planck length. The coordinates ρrand τrare functions of the usual rand tcoordinates
of RN. This near-horizon metric is AdS2×S2with the AdS2factor presented in Rindler
coordinates.
Note, however, that the S2factor has constant size in (2.1). To make a traversable
wormhole that connects different portions of asymptotically flat space, the metric must
be modified to allow the size of the S2to vary, so that the black hole spacetime can be
sewn onto the asymptotically flat region. One may take this variation to be slow, with the
perturbation away from AdS2×S2being of the form
ds2=r2
e−(1 + ρ2+γ)dτ2+dρ2
1 + ρ2+γ+ (1 + φ)dΩ2,(2.4)
where φand γare small functions that, respectively, encode the changing size of the sphere
and a perturbation of the AdS2factor. Now τand rare global coordinates for the AdS2
factor, which make it appear easy to send causal signals from one side to the other.
From the Raychaudhuri equation, spacetimes of the form (2.4) satisfying the null energy
condition must have φ(ρ)monotonic. But connecting the throat region to the asymptotic
regions at both ends requires that the spheres S2grow at both the wormhole mouths, and
therefore φmust grow in both directions at large |ρ|. Completing the construction in a
solution of Einstein-Hilbert gravity thus requires the introduction of negative energy.
The construction of [5] creates negative Casimir energy by using the magnetic field of
the black hole and a massless, charged fermion field. The magnetic field creates localized
Landau levels near each field line, which gives a large number Qof effective 1+1 dimensional
massless fermions. As shown in Fig. 3, field lines that loop through the wormhole yield 1+1
dimensional theories on S1×R. Since constant φyields the exact solution (2.1) with
vanishing stress-energy, a small negative stress-energy suffices to allow growth of φat large
positive and negative ρso long as the negative stress-energy threads the entire wormhole
and this growth is correspondingly slow.
Figure 3:The traversable wormhole of [5]. Magnetic field lines thread the wormhole, while
fermions localize into their lowest Landau level near each field line. For field lines that form closed
loops (for example, in blue), this creates effective 1+1 dimensional massless theories on S1×R
whose Casimir energy makes the wormhole traversable.
Ref. [5] showed that the energy of such wormholes differs from the energy of two
– 4 –

disconnected extremal black holes by the amount3
E=r3
e
GN`2−NfQ
6π
π` +dout −1
4`,(2.5)
where π` is equal to the “length” of the throat in an appropriate conformally rescaled metric,
dout is the separation between mouths in the exterior region, and Qis the integer magnetic
charge of the black holes.
The first term in (2.5) is the energy above extremality of a near-extremal RN black
hole, and although it is a classical contribution, it can be small when `is large. It can thus
be offset by the second, quantum contribution from the Casimir energy. Minimizing (2.5)
determines the equilibrium wormhole length. When dout `one gets
`=16r3
e
GNNfQ∼Q2lP
Nf
(2.6)
so that the wormhole binding energy is
Emin =−NfQ
16`=−N2
fg3
256π3/2QlP
.(2.7)
This solution requires that dout Q2lP/Nf, but it is also possible to find configurations
with dout Q2lP/Nf, in which the balance is achieved between the two quantum terms
in (2.5) with the classical energy being negligible. In this case energy minimization gives
`=dout/π. The binding energy is still Emin ∼ −NfQ/` but this is now much smaller than
in (2.7), since `is much larger than in (2.6).
We emphasize that the magnetic field lines must form closed loops in order to generate
Casimir energy. This requires that both wormhole mouths be placed in the same asymptotic
region of spacetime. As a result, the mouths attract each other gravitationally. As long as
the initial separation dout between the mouths is sufficiently large, the wormhole will remain
open for long enough to cross it before collapsing – the time to merger is ∼d3/2
out (which is
just the time for two point masses to collide starting from rest), while the the transit time
along the throat is parametrically O(dout). However, additional structure can be added to
create longer-lived traversable wormhole solutions. This can be achieved by introducing
an external magnetic field (in GR, this would be a Melvin flux tube [13]) tuned to keep
the mouths apart, or by attaching cosmic strings that pull them (exact solutions exist for
both mechanisms [14,15]). Alternatively, instead of balancing them into exact (though
unstable) equilibrium, one can set the mouths into a long-lived Keplerian orbit around each
other, as proposed in [5]. Though this orbiting will cause the wormhole mouths to radiate
gravitational and electromagnetic waves and hence eventually coalesce, the time scale for
this to happen is d3
out, and so again much longer than the time needed to traverse the
wormhole when the wormhole mouths are sufficiently far apart. Additionally, as noted in
[9], it is possible to create a more complicated stable solution by anchoring cosmic strings to
some stable spherical shell at a finite distance. This approach uses several strings to attach
3This assumes a simplified model for fermion propagation in the exterior region; see [5] for details.
– 5 –

each black hole to the shell, with each string anchored to a different location on the shell.
Stability arises from the fact that the angles between the strings depend on the location of
the black holes.
It is expected that the wormhole throat must be longer than the distance between
the wormhole mouths, though the wormholes of [9] approximately saturate this bound in
certain limits. In d > 4this is a sharp bound that follows from, for example, the Generalized
Second Law [16], or in AdS/CFT, from boundary causality [17]. These statements prohibit
wormholes from being the fastest causal curves between distant points, and indeed the
travel time through wormholes of [5] is longer by a factor of order unity.4
3 Gravitational construction
Given the two-mouth wormhole described above, our idea for constructing multi-mouth
traversable wormholes is simple: place a small, near-extremal black hole in the throat
of a larger-mouth wormhole, and extend it into a wormhole with another small mouth
in the same asymptotic region as the larger mouths. Technically, the insertion of the
two small mouths in the initial large wormhole solution is a straightforward (if possibly
tedious) problem of matched asymptotic expansions: the small mouths can be treated
as perturbations of, respectively, the throat and the asymptotic region, while the effects
of the latter on the mouths are incorporated as tidal perturbations of the near-extremal
Reissner-Nordström black hole. In Appendix A, we explain how to obtain the solution to
this perturbation problem. For our purposes here, we simply need the lowest order in the
matched asymptotic expansion, in which the backreaction of the mouths is neglected. We
will only go beyond this order in Sec. 3.2, where we calculate how the insertion of the small
mouth modifies the energetics in the backreacted solution.
Matching the geometry of the small mouths onto the background spacetime is thus a
generic and unproblematic part of the construction, but there are other aspects that must be
dealt with more carefully. One still fairly simple question is that of mechanical equilibrium
(and possibly stability) of the new configuration. Actually, this arises at the first order
in the matched asymptotic expansions, as we explain in Appendix A. Another problem is
how to achieve the negative energies that make the throats traversable. The answers to
these questions vary depending on the details of the model we choose – in other words,
on the tools of which we avail ourselves for the construction. We may restrict ourselves to
working within the same theory as [5], with only fields and matter available in the Standard
Model (specifically, a Maxwell field and light fermions electrically coupled to it, in addition
to gravity) – or the Beyond-the-Standard-Model dark sector of [20] – to construct our
multiboundary traversable wormholes. Or, instead, we may resort to a larger set of tools,
4However, in d= 4 asymptotically flat spacetimes, the Shapiro time-delay associated with the wormhole
mouths means that the fastest causal curve between two distant points always lies far from the center of
mass. Thus, the sharp bounds mentioned above are always trivially satisfied, and a sharp, local bound
is lacking for wormhole transit times. However, it may be possible to derive sharper local bounds by
considering either the quantum focusing conjecture [18], or by considering short wormhole’s tendency to
form time machines [19].
– 6 –

as did [9] (using e.g. cosmic strings as may appear in, say, grand unified theories), and aim
at a ‘proof of principle’ that such wormholes are possible with reasonable matter and field
content, e.g., satisfying basic energy conditions, and possibly within the landscape of string
theory. Allowing only Standard Model tools of course makes the task more difficult.
3.1 Multi-mouth wormhole construction guide
We begin by asking how equilibrium can be achieved when one introduces a new wormhole
mouth into the throat of the wormholes constructed in [5]. If near-extremal magnetic RN
solutions approximately describe all three mouths in a magnetic field background, then
equilibrium should not be hard to achieve. A uniform magnetic field can be approximated
by the field in between two large, static magnetic sources (even nonlinearly in GR [21]).
The third mouth can thus be thought of as sitting in a uniform, static magnetic field. This
will push the third mouth to one side, but the deep gravitational well can be used to make
the forces on this mouth balance at a finite displacement. In Appendix A, we work out how
this problem is solved when constructing the backreacted solution. Configurations with a
small black hole in the throat in equilibrium can thus be found.
However, from a purely mechanical perspective, perhaps the simplest possibility is to let
the small black hole be charged under a different U(1) gauge field than the bigger mouths.
This of course introduces physics beyond the Standard Model. Equilibrium configurations
can then be found that preserve the natural reflection symmetry of the two-mouth solution,
with the new source sitting in equilibrium at the bottom of the throat.
We must also consider the external mouth to which the mouth in the throat connects.
This third external mouth will face stability issues similar to those described for the two-
mouth wormhole above, which can then be resolved in similar ways. In particular, even if
the configuration is unstable, it can still be sufficiently long-lived to allow the throats to
be traversed so long as the additional black hole remains small. This is so even though the
addition of the small mouth in the throat will increase the time required to traverse the
larger wormhole due to a Shapiro-like time delay that we will analyze in Sec. 3.4.
Having described the mechanics involved with adding the third mouth, let us now
move on to the problem of achieving negative Casimir energies that thread the associated
wormhole. The effective two-dimensional massless fermions used to build the original two-
mouth wormhole will still travel along magnetic field lines, which form loops along the
non-contractible cycles of the wormhole and thus provide negative Casimir energies. Some
of these non-contractible field lines will thread the third mouth and hold it open as desired.
We get more varied possibilities if we enlarge our toolbox beyond the Standard Model.
For instance, still using the magnetic line mechanism of [5], we can allow for three U(1)
gauge fields, and three flavors of fermions electrically coupled to each of the gauge fields.
Then, with each pair of the mouths having opposite magnetic charges under one of the
U(1)’s,5the fermions travel along field lines in an independent manner.
5More precisely, the two big mouths can have charges (Q1, Q2,0),(−Q1,0, Q3)and the small one
(0,−Q2,−Q3), with |Q1|  |Q2|,|Q3|. This also allows easily for symmetric equilibrium positions for
the small mouth.
– 7 –

Cosmic strings with zero modes traveling along loops of string provide the requisite
Casimir energy in a simpler manner. We may use it in a hybrid fashion, by adding the
third mouth to the magnetic-line model of [5] and thread it with two cosmic strings, each
separately linked to the two big mouths; or else, if that hybrid is deemed too ugly to regard,
replace the fermions in the original two mouths with a cosmic string as in [9] in addition
to the two new cosmic strings, one along each new cycle. An explicit example is given in
Fig. 4.
Figure 4:Left: A three-mouth wormhole, held in mechanical equilibrium by cosmic strings.
The cosmic strings needed for Casimir energy are omitted here for illustrational clarity, but would
wrap the compact cycles around each pair of mouths. We take the small mouth to be charged
under a different U(1) symmetry than the previous black holes, with charge qe. This is done to
maintain the symmetry of the solution. Field lines from this small mouth in the throat will flow
through the wormhole and exit through the large mouths, giving them each a charge qe/2. The
other small mouth is placed equidistant between the larger mouths, but off the axis connecting
them, to avoid overlapping with the compact strings that are needed to generate Casimir energy
(otherwise, the compact string would run down this small wormhole mouth, and not directly around
the non-contractible cycle of the wormhole handle). We can then add two more cosmic strings that
run in from infinity along the y−axis. The tension of these strings is equal, and can be chosen
to compensate for the magnetic and gravitational forces. The tension of the original string that
stretches along the x−axis can then be increased to compensate for the additional force on the large
wormhole mouths from the small wormhole mouth. Right: The top view.
Finally, notice that the assumption about a hierarchy of mouth sizes, i.e., very different
charge numbers in the big and small mouths, can be achieved within the Standard Model,
while keeping the mouth geometries semiclassical, given the large separation between the
electroweak and Planck scales. However, we will see later that the energetics of the throat
demand that the number of flavors Nfbe large in order to allow the insertion of the small
mouth. Following [5], the Standard Model may provide as many as Nf= 54.
Thus, while many potential constructions are possible, the cosmic string method of
[9], possibly augmented with additional gauge fields, serves to prove that it is possible to
construct multi-mouth wormholes sufficiently long-lived to be traversable. Their existence
within the Standard Model, following the methods of [5], also seems likely, even if its
detailed investigation is more complicated.
– 8 –

3.2 Size limits on the third mouth
As described in [5], the traversability property is extremely fragile, as it can be destroyed
by perturbations of a small-but-finite size. We should therefore study more carefully just
how large the third mouth can be. In particular, we should note that the semiclassical
analysis used above requires the third mouth to be larger than the Planck length, indeed,
larger than pNflP, since a very large number of species Nfrestricts the validity of the
semiclassical description to length scales >pNflP. We should understand the conditions
under which these constraints can be satisfied.
As reviewed in Sec. 2, from the standpoint of the Raychaudhuri equation, the key point
is that the positive mass of the small mouth creates a focusing effect within the wormhole
throat that counteracts the defocusing effect of the negative Casimir energy. If the focusing
effect of the small mouth is too large, the topological censorship arguments of [1] require
the throat to collapse and for traversability to be destroyed. We expect that this places
an upper bound on the mass of the small black hole such that its energy, as measured
from outside the wormhole, does not exceed the binding energy of the wormhole. We now
perform an analysis that confirms this expectation precisely.
We start with the solution for the wormhole interior as described in [5], which we then
perturb with a localized source for the small black hole deep inside the throat. As we
have seen, the geometry in this region is described by a metric of the form of (2.4), where
the small functions γand φwill be treated to linear order. The Einstein equations are
sourced by the Maxwell field stress-energy tensor, the Casimir energy from the fermions,
and the small wormhole mouth. The former two contributions were computed in [5], and
for the latter we introduce a number of useful simplifications.6First, since the additional
wormhole mouth is small we can understand its backreaction by treating it as a localized,
delta-function mass source. This is a codimension-three source, and although it is possible
to perturbatively solve for its backreaction, we will simplify the problem further. We are
interested in the effect of the source on the overall size of the S2along the throat – which
is controlled by the scalar field φin (2.4) – and we can smear the source over the S2, so it
acts as a codimension-one, domain wall defect on the throat. This is a good approximation
for studying the gravitational effect of the small mouth at distances in the throat larger
than re. With this simplification, the geometry varies only along the throat direction ρ,
and then φand γare obtained by solving ordinary differential equations.
It will be sufficient to solve the (ττ )equation. To linear order in φand γthe Einstein
tensor takes the form
Gττ =γ−(1 + ρ2)(−1 + ρφ0+ (1 + ρ2)φ00)−(1 + ρ2)φ . (3.1)
The corresponding stress tensor consists of magnetic energy density, fermion Casimir energy,
and localized mass source,
Tττ =Tmag
ττ +Tfer
ττ +Tδ
τ τ ,(3.2)
6See Appendix Afor how this relates to the construction of matched asymptotics.
– 9 –

which are given by
Tmag
ττ =−1
4g2gτ τ F2=1
8πGN
((1 + ρ2)(1 −2φ) + γ),
Tfer
ττ =−α
8πGN
, T δ
ττ =β
4πGN
δ(ρ).
(3.3)
We have placed the source at the center of the throat ρ= 0. In the next section we will
consider off-center positions (the off-center displacement created by the electric interaction
between the charged black holes is discussed in Appendix A). Furthermore we have defined7
α=GNQNf
4πr2
e
, β =GNm
re
,(3.4)
with mthe mass of the small black hole. We have smeared it over the sphere S2of radius
re, so we can trust the solution for |ρ|larger than one, but not near ρ= 0.
The Einstein (ττ )equation now involves only φand not γ, and takes the form
(1 + ρ2)φ00 +ρφ0−φ=α
1 + ρ2−2β δ(ρ).(3.5)
We solve this as
φ(ρ) = α(1 + ρarctan ρ)−β|ρ|.(3.6)
The contribution β|ρ|from the small black hole is the gravitational potential that a mass
creates in 1+1 gravity, which is how gravity along the throat behaves over scales larger
than its thickness re. As anticipated, we see the focusing effect of the mass m, which
makes φdecrease as |ρ|grows. If βis large enough, then at distances |ρ|>1(where our
approximations hold) this effect could overcome the defocusing of the negative Casimir
energy and the throat would close, as φmust increase towards the wormhole exits in order
to connect to the asymptotic regions. Therefore we will limit βto values such that φgrows
for large |ρ|. This gives
β < π
2α , (3.7)
which tells us that the maximum mass of the small black hole that we can put in the
wormhole is
m < NfQ
8re
.(3.8)
This mis locally measured in the vicinity of the small black hole, deep within the
throat, while the energy as measured outside the wormhole is redshifted by a factor of re/`,
giving
Ebh <NfQ
16`,(3.9)
that is,
Ebh <|Emin|(3.10)
7In order to get the dimensions of the source term correctly, bear in mind that ρis dimensionless and
physical lengths are in units of re.
– 10 –

where Emin is the energy gap between traversable and non-traversable wormholes that we
saw earlier in (2.7).
Writing the constraint (3.8) in the form
m < 1
8√πgNfmP,(3.11)
where the Planck mass is mP= 1/lP, we see that at weak coupling gwe need Nf1for
the small mouth to remain semiclassical, with mmP. Indeed the actual semiclassicality
condition when many fermion species are present, namely, mpNfmP, can also be
satisfied for Nf1.
The bound on the size of the small mouth may be even more stringent than (3.10),
since we expect that the radius of the small mouth cannot exceed the throat radius, that
is, we must have
GNm<re.(3.12)
This bound will be more stringent than (3.8) whenever
re<rNfQ
4πlP,(3.13)
or equivalently when
Q < g2Nf
4π2.(3.14)
We need Q/g 1, and actually Q/g pNf, in order for the black holes to be semiclassically
valid (see (2.2)), and the coupling will naturally be g.1. Still, (3.14) allows situations
where, due to the presence of a large number of fermions Nf, the small black hole reaches
its maximal size while the original throat is still far from collapsing. The reason is that Nf
enhances the binding energy of the wormhole while not affecting the classical size relations
(3.12). Conversely, when (3.14) does not hold, the binding energy is sufficiently small, such
that a black hole much smaller than the throat radius can nevertheless be heavy enough to
overwhelm the negative Casimir energy, and thus collapse the wormhole. In this case, the
approximations that lead to (3.10) hold well and the bound can be regarded as accurate.
From this analysis we conclude that, if traversability is to be preserved, the addition
of the third mouth should not lift the energy of the system above that of two disconnected
large extremal black holes. More generally, we expect that multi-mouth wormholes always
have lower energy than a collection of disconnected extremal black holes.
One may also ask about applying our construction to the perturbative two-mouth
wormholes of [9]. In that case, the wormhole only remains open for a short time, and so
can only be traversed by causal curves that start early enough at past null infinity. The
addition of the small mouth makes this restriction even more stringent. We analyze it in
Appendix C, concluding that the bound on the mass of the small mouth is stronger than
(3.10) by an additional factor of `p/re. Correspondingly, larger numbers of fermions Nf
are thus required for the third mouth to enter the semiclassical regime while preserving
traversability of the original throat.
– 11 –

3.3 Lowering the small mouth down the throat
The previous subsection dealt with configurations with a small mouth that is already at
the bottom of the wormhole throat. Now we want to investigate if it is indeed possible to
lower a mass mto that place, starting from a position near the big wormhole exit. Since
the wormhole is fragile, we carefully lower the mass in a Geroch-like adiabatic process.
Standing at the wormhole mouth, we attach the mass to a string which we slowly release
into the wormhole, so that the system is in equilibrium at every moment while the mass
is lowered. In contrast to the conventional Geroch process, when the small mass reaches
the bottom of the wormhole it will be at its equilibrium position, and it will remain there
when we remove the string. However, one may worry that this process could destroy the
wormhole. We have already seen that when the mass msits at the bottom of the wormhole,
its energy cannot be larger than the bound (3.7) without collapsing the throat. We want
to make sure that the energy that we are dropping into the wormhole as we lower the mass
remains sufficiently small throughout the entire process.
We thus generalize our previous study to now hold the mass at an arbitrary height
ρ=ρ0≥0. Then the stress tensor of the (smeared) mass is
Tδ
ττ =β
4πGN1 + ρ2
0δ(ρ−ρ0),(3.15)
where βis the same parameter for the mass mas in (3.4). The factor (1 + ρ2
0)accounts for
the fact that, since we keep fixed the black hole mass mas measured locally at ρ=ρ0, the
energy conjugate to the time τvaries with the redshift along the wormhole tube.
To counterbalance the gravitational potential that pulls down on the mass, we employ
a cosmic string whose tension pulls it upwards. Since the mass is smeared on S2, the string
will also have to be smeared, so the stress tensor is a step function of the form
8πGN(Tττ)string = 8πGN(Tρρ)string =−T
4πr2
e
Θ(ρ−ρ0).(3.16)
The string tension Twill be determined by solving the Einstein equations. The energy
equation (ττ )now becomes
(1 + ρ2)φ00 +ρφ0−φ=α
1 + ρ2−2β δ(ρ−ρ0)−T
4πΘ(ρ−ρ0).(3.17)
When we require that the solution is continuous at ρ=ρ0we find that the tension must
take the value
T=8πρ0
1 + ρ2
0
β . (3.18)
This tension remains finite all throughout the lowering process. It vanishes for ρ0→ ∞ and
ρ0→0, which is as expected since these correspond to the beginning of the process and to
the moment when we release the mass at its final equilibrium position. The solution for φ
is readily found to be8
φ(ρ) = α(1 + ρarctan ρ)−2β
1 + ρ2
0
(Θ(ρ−ρ0)(ρ−ρ0)−kρ).(3.19)
8See Appendix Bfor the complete solution to Einstein’s equations.
– 12 –

The integration constant kdepends on details of the physics of the lowering, and in
particular on ρ0, but we expect it to vary in the range
0≤k≤1/2.(3.20)
When the mass is at ρ0→ ∞, we expect to have k= 0, which corresponds to just having
an additional mass mat one mouth and no strings, without any effect at the other mouth
at ρ→ −∞. Instead, when the mass reaches the bottom at ρ= 0 we will have k= 1/2,
which is the fully symmetric solution that we obtained in (3.6).
Earlier we saw that the condition that the wormhole remains open is that φgrows for
large |ρ|. In the solution (3.19) this requires that
2β
1 + ρ2
0
(1 −k)<π
2α . (3.21)
This bound becomes very weak when ρ0is large. When ρ0→0, so that k→1/2, we recover
the previous bound (3.7). Without the detailed dependence of kon ρ0we cannot know for
certain whether a more stringent condition occurs at some finite ρ06= 0. Nevertheless, it is
clear that when
β < C π
2α , (3.22)
that is,
m<CNfQ
8re
=C
8√πgNfmP,(3.23)
with Ca number ∈[1/2,1], then we can safely lower the full mass mto the bottom without
collapsing the wormhole.
Since the object being lowered can be a semiclassical black hole of mass mif Nf1,
this result also implies that information can be safely transmitted through the wormhole
in single batches of the order of the black hole entropy 4π(m/mP)2. It may be interesting
to explore further how this type of analysis constrains the rates of information transfer
through wormholes.
3.4 Signaling between mouths
Suppose that two parties, Aand B, use the original two-mouth wormhole to exchange
messages. What are the consequences of inserting a third, small mouth operated by c?
From the gravitational perspective, there are two different kinds of effects. First, the
message sent by A(a particle or a wave) may be partly absorbed by the small mouth and
thus be received by cand not B. The wormhole has then become a leaky pipeline. The
absorption probability is proportional to the area of the small mouth, and thus to c, and
can also have a dependence on the small mouth’s angular position in the S2of the large
throat.
Due to the relation between traversing a wormhole and quantum teleportation, these
effects will have counterparts in the entanglement structure of the three-mouth wormhole.
In the absence of specific realizations it is difficult to be precise, but some qualitative
features are plausibly realized. The leakiness of the line will likely appear as soon as
– 13 –

a channel for a third party is added, with losses plausibly proportional to the number
of degrees of freedom that cholds. The angular dependence requires a more detailed
understanding. In a wormhole where the throat geometry is well approximated by AdS2×
S2, any dual description will contain a sector modelled by quantum-mechanical degrees
of freedom charged under an approximate SO(3) (or SU (2)) symmetry group. Then, A
and Bcan control the angular position on the sphere of the messages they exchange by
selecting qubits with appropriately chosen charge distributions. Having information about
this angular position is essential if Aand Bintend to communicate efficiently with c: the
subset of degrees of freedom of their many-qubit system that hold the entanglement with c
must carry appropriate SO(3) charges.
A second effect is due to the Shapiro time delay that the signal will experience as it
travels in the vicinity of the small mouth within the throat. That is, if the small mouth
is placed in the throat geometry (2.4), then the signal that Asends to be Bwill take an
additional time to arrive. As measured by an observer in the throat, this delay is given by
the familiar Shapiro formula
δt ∼2mlog4`2
b2,(3.24)
where we have used the AdS2scale `as an infrared cutoff and where bis the distance of
closest approach of the signal to the small mouth. This distance bmay be translated into an
angular difference between the positions in S2of the mouth and the initial signal. However,
the delay measured by Aand Bis much larger due to the strong redshift or order ∼`/re
at the bottom of the AdS2throat. Here we might suppose that the increased travel-time
can be correlated with an increased complexity in decoding the teleported message.
4 Discussion
In the above work, we presented a construction of a multi-mouth traversable wormhole.
This was done by starting with a two-mouth traversable wormhole of the form described in
[5]. We then perturbed this solution by adding a small black hole in the throat of the larger
wormhole. So long as the total energy remains below the energy of a pair of disconnected
extremal black holes, the original wormhole will remain traversable. With large enough
numbers Nfof bulk fermion fields, this can be accomplished for small black holes far bigger
than the Planck length lP, and even also bigger than the effective cutoff length pNflP, so
that the term ‘black hole’ is appropriate. This small black hole can be placed in mechanical
equilibrium through the proper placement of cosmic strings. Additional compact cosmic
strings can be used to make all mouths traversable.
To maintain traversability, the small black hole in the throat must have an energy
below the gap Egap described in [5] that separates traversable wormholes from black holes.
It achieves this in part due to the strong redshift in the throat, so that the black hole’s
energy is much smaller than that of a similarly-sized black hole in the region outside the
wormhole. Nevertheless, we saw in Sec. 3.3 that such a black hole can be (carefully)
lowered into the wormhole throat from outside without causing the wormhole to collapse.
In particular, it shows that there can be configurations in which the total mass contained
– 14 –

within the wormhole is quite far above the energy associated with separate black holes and
in which the wormhole remains traversable, so long as this energy is not localized too deep
within the wormhole throat.
A rather different construction of a traversable three-mouth wormhole was recently
described in [22]. That analysis began with a non-traversable three-boundary wormhole
asymptotic to AdS3and added boundary interactions similar to those in the original work
by Gao, Jafferis, and Wall [3]. In particular, by taking a limit where the horizons that
shroud the original non-traversable wormhole become both very large and very hot, much
as in [23], the causal shadow becomes exponentially thin along large regions of the horizon.
In fact, in such regions the wormhole geometry becomes exponentially close to that of the
BTZ version of the Einstein-Rosen bridge. It is thus straightforward to apply a local version
of the analysis of [3] to show that appropriate boundary interactions can make the wormhole
traversable between any two boundaries.
The fact that the analysis of [22] largely reduces to that of [3] is associated with the fact
that the entanglement structure of the non-traversable three-boundary wormhole reduces in
this this limit (and in the relevant regions of the boundary) to the entanglement structure
of the thermofield double. To be specific, in the region where the causal shadow separating
boundaries Aand Bbecomes very thin, the corresponding parts of the dual field theory
on boundaries Aand Bare exponentially well approximated by a thermofield double state
[23]. In particular, neglecting exponentially small corrections we may say that this part of
boundary Ais entangled only with boundary Band has no entanglement with C. In this
sense, the traversability of the three-mouth wormhole constructed in [22] is associated only
with bipartite entanglement; thus, multipartite entanglement plays no role.
In contrast, multiparty entanglement seems likely to play an important role in the
three-mouth traversable wormhole constructed in the current work. To make the discussion
precise, we consider an asymptotically AdS version of our construction in which each of
the three mouths is associated with a separate asymptotic region (the negative energy then
comes not from fermion loops but from operator insertions at the mouths as in [3,4,22]).
Our three-mouth wormhole then becomes a three-boundary wormhole. In the limit where
the AdS scale `is large compared with the radii of the throats, the local geometry of the
throats will be identical to that of the asymptotically flat case described in the main text.
To argue for the possible importance of multiparty entanglement in our case, let us
first recall from [24] that multiparty entanglement may be quantified by considering the
entanglement wedge WAB of the joint AB system and computing
M3:= 2EW(AB)−I(A:B),(4.1)
where EW(AB)is the entanglement wedge cross section entropy [25] and I(A:B)is the
mutual information between Aand B. In particular, EW(AB)is 1/4Gtimes the area
A(ΣAB
min)of the minimal surface homologous to both Aand Bwithin the entanglement
wedge, where the homology condition now allows the homology surfaces to have additional
boundaries at finite boundaries of WAB , the entanglement wedge of AB. In particular, the
surface ΣAB
min will generally intersect the minimal surface ΣCthat computes the entropy of
– 15 –

boundary C, and indeed will split it into two parts ΣA
Cand ΣB
C. As a result, since SAB =SC
for our wormhole, we may rewrite M3in the form
4GM3=A(ΣAB
min) + A(ΣA
C)−A(ΣA)+A(ΣAB
min) + A(ΣB
C)−A(ΣB),(4.2)
where ΣA,ΣBare the minimal surfaces homologous to boundaries Aand Bin the usual
sense. The right hand side of (4.2) is now manifestly positive since, for example, ΣAis
homologous to ΣAB
min ∪ΣA
Cand is also by definition minimal within that homology class;
see Fig. 5. Furthermore, in the limit used in the main text in which the AdS scale `and
the radii of the large mouths are much larger than the radius of the third small mouth, it
is clear that the third small mouth sets the only scale in the problem. Thus in that case,
dimensional analysis guarantees that 4GM3will be first order in the area of the third small
mouth, or in other words, that M3is first order in the corresponding entropy: M3∼SC.
Figure 5:A three-mouth wormhole. ΣAB
min is the minimal surface homologous to both Aand B
that stays within the entanglement wedge of AB (orange). ΣA
C(dark blue) and ΣA
C(purple) are
the portions of ΣC, split by ΣAB
min, such that ΣAB
min ∪ΣA
Cis homologous to Aand ΣAB
min ∪ΣB
Cis
homologous to B.
This shows that our construction applies in the limit where the multi-mouth wormhole
has significant multiparty entanglement. The remaining question is thus whether this
multiparty entanglement plays an important role in our wormhole’s traversability. While
a complete analysis of this question is beyond the scope of the current work, the further
remarks below appear to point in this direction.
Let us briefly consider the locations of ΣA,ΣB, and ΣC. In [23,26], it was found that
narrowing of the entanglement shadow region between these three surfaces, so that the
separation between some two of these surfaces becomes small relative to their distance to the
third, was indicative of a region of mostly bipartite entanglement between the corresponding
boundaries. In contrast, regions where the distance between the various entangling surfaces
is roughly the same between each pair of surfaces might naturally be taken as a signal
of tripartite entanglement. In particular, [26] associated large amounts of multipartite
– 16 –

entanglement to certain AdS black holes whose temperature was small compared to the
AdS scale, while [23] showed that states dual to hot black holes are well-approximated by
sewing together various copies of |TFDistates. See Fig. 6below.
Figure 6:Left: A hot AdS3three-mouth wormhole. The entanglement shadow becomes very
narrow in regions where pairs of extremal surfaces approach each other, indicating regions of
strong bipartite entanglement. Right: A cold AdS3three-mouth wormhole. At any point on
one entangling surface, the distance to the other two entangling surfaces is roughly the same. This
suggests strongly tripartite entanglement.
Recall also that our analysis of back-reaction suggested that one mouth (C) must
remain small relative to the other two. We thus assume that this is so. Before we add in
C, the extremal surfaces associated with Aand Bcoincide and lie at the bottom of the
AB throat. In the limit where Cis much smaller than Aand B, it will have little effect on
the geometry far from C. Thus the extremal surfaces associated with Aand Bwill remain
close over most of their area, and in particular at the top of the wormhole in Fig. 7below.
This indicates that there is still strong two-party entanglement between Aand B, as one
would expect since SCSA, SB.
On the other hand, the extremal surface associated with Cwill remain close to the
bottom of the small wormhole throat. Since the only scale in the problem in the region
near Cis the size of ΣC, the separation in this region between any two minimal surfaces
will be comparable to the scale of ΣCitself. The fact that e.g. the separation in this region
between ΣAand ΣBis comparable to the separation between ΣAand ΣCthen indicates
that multiparty entanglement plays an important role in this region of the geometry, and
thus presumably also in making this region traversable. More physically, one might rephrase
this remark by stating that a signal entering the wormhole through mouth Cmust then
find itself for some non-trivial amount of time in the entanglement shadow region which,
due to the entanglement with C, fails to be part of either of the entanglement wedges of A
or Balone. One thus expects that the three-party entanglement of the field theory dual is
required to describe propagation of the signal in this region.
The large bipartite entanglement between Aand Bis consistent with the idea that
Chas little effect on signals being sent between Aand B. But it would be interesting to
consider quantum mechanical duals in more detail, as well as the quantum teleportation
– 17 –

Figure 7:The extremal surfaces associated with the mouths of a multi-mouth wormhole in
asymptotically flat space are depicted, with one mouth Cmuch smaller than the other two, Aand
B. In this limit, the extremal surfaces associated with Aand Bremain close together over most
of their area, and in particular much closer than their distance to the extremal surface associated
with C, which sits at the bottom of the small wormhole throat.
protocols associated with traversing the wormhole in the bulk in analogy with the discussions
of e.g. [3,27–29]. In particular, if the entanglement of Cwith Aand Bis indeed mostly of
the multi-party sort, then the dual description of sending a signal from Cto either one of A
or Bmust necessarily involve all three systems. While this idea may at first seem unfamiliar,
it is consistent with the fact that the asymptotically flat region of our gravitational solution
does in fact provide interactions between each pair of mouths AB,AC, and BC.
Acknowledgments
We thank Chris Akers and Sergio Hernández-Cuenca for useful discussions. RE and MT
were supported by ERC Advanced Grant GravBHs-692951, MEC grant FPA2016-76005-
C2-2-P, and AGAUR grant 2017-SGR 754. BG-W was supported by a University of
California, Santa Barbara central fellowship. DM was supported in part by NSF grant
PHY1801805 and by funds from the University of California. MT was also supported
in part by the Heising-Simons Foundation, the Simons Foundation, and National Science
Foundation Grant No. NSF PHY-1748958. This work was partly carried out while RE,
DM, and MT were participants in a KITP program, where their research was supported in
part by the National Science Foundation under Grant No. NSF PHY-1748958 to the KITP.
A Matched asymptotics construction
At a technical level, the insertion of the small mouth in the initial wormhole throat can be
carried out as a typical construction of matched asymptotics. The throat characteristic size
reis much larger than the small mouth radius, GNm, so we consider a “near zone” around the
small mouth where rre, and a “far zone” in the throat where rGNm. In the far zone
the small mouth is well approximated as a pointlike source whose effect can be incorporated
as a small perturbation of the throat. In the near zone, the mouth is approximated as a near-
extremal RN black hole, which is not exactly round nor asymptotically flat, since it is tidally
distorted by the curvature of the throat. The two zones overlap where GNmrre,
and this allows to match the geometries and the parameters in them, so as to build a
– 18 –

global solution for the throat (which then is itself matched to the exterior at ρ1, i.e., at
distances rein the throat).
The construction is thus reduced to solving two different problems of gravitational
perturbations, in the near and far zones. In the following, we describe how these problems
are set up in the different zones, and how they allow us to obtain the balance of forces
inside the throat. The explicit backreacted solution in the region near the throat exit at
ρ1is explained in Sec. 3.2.
Near zone
We want to insert a small RN black hole in the AdS2×S2throat geometry
ds2=r2
e−(1 + ρ2)dτ2+dρ2
1 + ρ2+dΩ2,(A.1)
which is threaded with magnetic field lines with gauge potential
Aφ=Q
2(1 −cos θ).(A.2)
For our purposes here, we neglect the small corrections to the geometry γand φ(2.4).
We will also consider that the insertion is an extremal RN black hole, neglecting that it is
slightly above extremality. Both effects can be well controlled parametrically and can be
added if desired by slightly perturbing the construction below.
The near zone is the geometry within a region of size rearound a point at some
value ρ=ρ0along the throat and at an arbitrary position in the S2. We allow for a finite
displacement ρ06= 0 away from the throat bottom. As we will see, this plays an important
role when all the mouths are charged under the same U(1): the resulting gravitational pull
towards the bottom can compensate for the magnetic force that the field (A.2) exerts on
the small mouth, and stabilize its position.9
We introduce near-zone cylindrical coordinates (¯
t, ¯
x)=(¯
t, ¯z, ¯σ, φ), aligned with the
throat and centered on the chosen point, such that
¯
t=req1 + ρ2
0τ , ¯z=re
p1 + ρ2
0
(ρ−ρ0),(A.3)
and ¯σis a stereographical polar coordinate on the S2of radius re,
¯σ= 2retan θ
2,(A.4)
so that
r2
edΩ2=d¯σ2+ ¯σ2dφ2
1 + ¯σ2
4r2
e2.(A.5)
9This displacement is a different effect than discussed in Sec. 3.3. Both can be treated separately, and
combined if desired.
– 19 –

Then, when |¯
x|  re, (A.1) becomes
ds2' − 1+2A¯z+¯z2
r2
ed¯
t2+1−2A¯z+1−4A2r2
e¯z2
r2
ed¯z2
+1−¯σ2
2r2
ed¯σ2+ ¯σ2dφ2+O¯
x
re3
,(A.6)
and the magnetic potential
Aφ=Q
4
¯σ2
r2
e 1−¯σ2
4r2
e
+O¯
x
re3!.(A.7)
We have introduced a parameter
A=ρ0
rep1 + ρ2
0
,(A.8)
which is to be interpreted as an acceleration, since the terms linear in ¯zin the metric
correspond to the gravitational potential that accelerates a particle at ρ=ρ0towards the
throat bottom at ρ= 0. The almost uniform magnetic field along the ¯zdirection is
B=Q
2r2
e
.(A.9)
In the limit where re→ ∞ keeping ¯
xfinite, we recover Minkowski space in (A.6). The
corrections for small ¯
x/reaccount for the modification of the near-zone due to the curvature
of the throat. They give rise to tidal distortions on any object localized within this zone.
More concretely, when we insert an extremal RN black hole, the metric will take the form
ds2' − 1+2A¯z+¯z2
r2
eH−2(¯
x)d¯
t2+1−2A¯z+1−4A2r2
e¯z2
r2
eH2(¯
x)d¯z2
+1−¯σ2
2r2
eH2(¯
x)d¯σ2+ ¯σ2dφ2
+GNm
re
h(1)
µν (¯
x) + (GNm)2
r2
e
h(2)
µν (¯
x)d¯xµd¯xν+O¯
x
re3
,(A.10)
and the magnetic potential
Aφ=Q
4
¯σ2
r2
e1−¯σ2
4r2
e+q
21−¯z
√¯z2+ ¯σ2+GNm
re
a(1)
φ+(GNm)2
r2
e
a(2)
φ+O¯
x
re3
.(A.11)
Here
H(¯
x) = 1 + GNm
|¯
x|= 1 + GNm
√¯z2+ ¯σ2(A.12)
has been introduced such that, when re→ ∞ keeping ¯
xand mfinite, we recover the
extremal RN black hole of mass mand (integer) charge
q=glP
√πm(A.13)
– 20 –

in isotropic cylindrical coordinates. When reis large but finite, the functions h(1,2)
µν and a(1,2)
φ
account for the small distortion and polarization that the tidal field and the magnetic field
induce on the black hole. These functions are determined by solving the Einstein equations
for a perturbation of the RN black hole, with the condition that the solution remains regular
at the black hole horizon, and that it asymptotes to (A.6) when |¯
x|  GNm.
The tidal fields in the asymptotic region are of two kinds: the leading effect is the linear,
dipole field ∼A¯zthat, as we mentioned, acts as an accelerating gravitational potential. We
will presently obtain the solution for the corresponding perturbation h(1)
µν , and in this way
fix Aso that the gravitational pull towards the bottom exactly balances the magnetic force
of the background field (A.7) towards the oppositely charged mouth.
The subleading terms, quadratic in ¯
x/re, represent a quadrupolar tidal field. When
ρ0= 0, i.e., A= 0, the corresponding perturbations h(2)
µν also appear in the calculation
of the Love numbers of a black hole as a measure of its deformability in linear response
theory. This problem has been solved for RN black holes in [30], where the explicit results
for the metric perturbation can be found. When A6= 0 the perturbations h(2)
µν depend on
the solution for h(1)
µν and the problem becomes more complicated. Since we do not need
them, we will not pursue their calculation further in this article.
Force balance
The following analysis is, as we have indicated, relevant only when there is an interaction
between the magnetic charge of the small mouth and the magnetic field in the throat. In
models with several U(1)’s these forces can be avoided and the small mouth then sits at
equilibrium at ρ=ρ0= 0 with A= 0. However, when all the magnetic charges are in the
same gauge field, we expect to be able to equilibrate their forces against the gravitation in
the throat.
In order to simplify the analysis, we can use the equivalence principle in the region
|¯
x|  A−1around the small mouth to trade the tidal gravitational field for an acceleration.
This transforms the problem into that of a magnetic black hole uniformly accelerated by an
external magnetic field. An exact solution in the Einstein-Maxwell theory for this system
was given by Ernst in [31]. It combines the features of the C-metric for an accelerating
black hole with those of a magnetic Melvin flux tube. The solution has been much further
studied following [32], so we will be brief.
We are interested in the particular case of a RN black hole of mass mand charge q,
immersed in a magnetic field Band moving with acceleration A. This will be an adequate
approximation to our system when the black hole is small, in the sense that GNm, qlP
(BlP)−1, A−1, and within distances away from the black hole smaller than A−1. This
region excludes the acceleration horizon of the Ernst solution, which is not present in the
situation of interest to us. One can easily show that the horizon of the black hole is extremal
(degenerate) when q=glP
√πm1 + O(GNmA)2.
In this limit, the absence of conical singularities at the axis of rotational symmetry is
equivalent to the requirement that Newton’s law is obeyed [31],
qB 'mA(1 + O(mA)2).(A.14)
– 21 –

For an extremal black hole this is
B=√π
glP
A(1 + O(mA)2).(A.15)
If we use that the magnetic field Balong the wormhole throat is given by (A.9), then we
find that the equilibrium position, ρ0, of the small mouth in the wormhole is given by
ρ0
p1 + ρ2
0
=glP
√π
Q
2re
=g2
2π,(A.16)
that is,
ρ0=g2
2π
1
q1−g4
4π2
.(A.17)
For fixed g,ρ0is independent of the wormhole charge. The physical distance reρ0along
the wormhole grows linearly with Q, as expected.
It is now possible to find the near-zone geometry with the leading order backreaction
from the tidal acceleration. To do so, we expand the Ernst solution within distances A−1
of the black hole, keeping only terms up to linear order in A. Doing this, we obtain the
metric10
ds2' − (1 + 2A¯z)H−2(¯
x)d¯
t2+ (1 −2A¯z)H2(¯
x)d¯z2
+1−2GNmA ¯z
√¯z2+ ¯σ2H2(¯
x)d¯σ2+ ¯σ2dφ2(A.18)
where H(¯
x)is the same as in (A.12) and the magnetic potential is
Aφ=q
21−¯z
√¯z2+ ¯σ2+B
2¯σ2−(GNm)21 + ¯z2
¯z2+ ¯σ2.(A.19)
These give the perturbations h(1)
µν and a(1) in (A.10) and (A.11).
Far and very far zones
Having explained in detail the set up for the near-zone construction, our description of the
far-zone analysis will be much more succinct. The far zone geometry is (A.1), or more
precisely, (2.4). Here the small near-extremal RN solution is inserted as a localized delta-
function source of stress-energy and magnetic charge. These create a small distortion of the
throat geometry (2.4), which must be solved as a perturbation of the Einstein equations.
It is clear that a solution to the problem exists, and although its explicit form is likely to
be complicated, fortunately we do not need it.
We can obtain useful information more easily if we restrict ourselves to the asymptotic
region of the far zone, where ρ1– the ‘very far zone’. Recall that the throat is measured
in units of re, so these are distances re, where the throat joins the exterior of the original
10The usual C-metric (x, y )coordinates are transformed into (¯z, ¯σ)as y−1=A(√¯z2+ ¯σ2+m)and
x=¯z
√¯z2+¯σ2−mA/2.
– 22 –

wormhole. At these large distances from the small mouth, we cannot resolve the precise
location in S2of the delta-source. Then, we can appropriately smear its position over the
entire S2, and regard it as a codimension-one defect along the ρdirection (notice that
having ρρ0also allows us to neglect ρ0).
In this manner, we can obtain the effect that the insertion of the small mouth has on
the junction of the throat to the exterior mouth. This analysis is presented in Sec. 3.2,
where we use it to determine how big mcan be before making such a junction impossible.
B Mass at a height in the throat
In Sec. 3.3 we presented the solution for the φperturbation when we have a mass m
(smeared on the S2) at a height ρ=ρ0in the throat of the wormhole, suspended from a
cosmic string. Here we give the solution to the entire set of Einstein’s equations. The (ρρ)
equation
ρφ0−φ=−α
1 + ρ2−T
4πΘ(ρ−ρ0),(B.1)
is automatically satisfied by (3.19) with tension Tgiven by (3.18). The remaining equation,
along the sphere directions, is
γ00 + (1 + ρ2)φ00 + 2ρφ0+ 4φ= 0 ,(B.2)
which is solved by
γ=−α(ρ2+ 3)ρarctan ρ+ρ2−ln1 + ρ2
+β
1 + ρ2
0|ρ−ρ0|(1 + ρ2)+2ρ0(ρ−ρ0)2Θ(ρ0−ρ)−3ρ0ρ2+ (1 −2k)ρ3.(B.3)
The condition (3.21) ensures that γdecreases at large |ρ|. This criterion could also have
been taken as the condition for being able to match the wormhole to the exterior mouth
and thus keep it open.
If we set ρ0= 0 and k= 1/2we obtain the solution for γfor the configurations in
Sec. 3.2.
C Multi-mouth wormholes from a perturbed bifurcate horizon
We can also consider using the wormholes of [6,8,9], which perturb around a bifurcate
Killing horizon. These wormholes will generally be strongly time-dependent, and can be
traversed by curves only if they leave past null infinity at sufficiently early times. This fact
leads to a more stringent bound on the size of the smaller wormhole mouth.
This construction starts with a classical background containing a pair of maximally-
extended, charged RN-like black holes, as in Fig. 8. These black holes are held apart by
cosmic strings that run off to infinity. An additional compact cosmic string wraps the
horizon – and will be used to generate the Casimir energy needed for traversability. We can
then quotient this background geometry by identifying a point with the one obtained by
swapping the black hole mouths and the asymptotic regions (see Fig. 8). This quotienting
– 23 –

Figure 8:Left: The quotient that creates a wormhole similar to Fig. 3starts with two maximally
extended black holes, held apart by cosmic strings that run off to infinity (black). A point is
identified with the point given by swapping the black hole mouths and the asymptotic regions, e.g.
it can be thought of as identifying a point with its πrotation about the black dot in the figure.
Right: The result of this quotient gives a wormhole with two mouths in the same asymptotic
region. Quantum fluctuations of the light blue compact string will generate negative energy.
results in a wormhole with two mouths in the same asymptotic region, with the compact
cosmic string becoming a shorter compact cosmic string, and the cosmic strings stretching
to infinity becoming a single string that goes through the wormhole mouth and stretches
to infinity on either side.
Since the cosmic strings lie along the horizon, the classical cosmic string stress tensor,
which is proportional to the induced metric, will not contribute to hTkkiM. Quantum
fluctuations of the string will contribute, however. In [9], these fluctuations were modelled
as 1+1 dimensional massless free scalar fields. For the string stretching to infinity, the
points identified in the quotient will lie on two distinct non-compact strings in the covering
space. The fluctuations on two different strings will be uncorrelated, and so the quantum
fluctuations of the non-compact string will not contribute to RhTkkiMdλ. However, the
contributions from quantum fluctuations of the compact string are non-zero, and give rise
to the Casimir energy needed for traversability.
In the extremal limit of the RN black holes of the classical backgrounds, the back-
reaction becomes large, and so the wormhole remains open for longer times; however, our
ability to treat the back-reaction perturbatively will break down. At finite temperature,
the fact that the wormholes are open for a finite amount of time limits the size of the black
hole that can be placed in the throat. We compute that effect below.
We note that from [9],
∆V=r+
r−r+−r−
r−−1−(r−/r+)2
e2κ+r+Z+∞
−∞
dUhkk ,(C.1)
where ZdUhkk (Ω) = 8πG ZdΩ0H(Ω,Ω0)ZdU hTkki(Ω0),(C.2)
ZhTU U idU =−e−κ+d/2cκ+
16r2
+
,(C.3)
– 24 –

and
H=X
j
Ym=0,j(Ω)Hmj , Hmj =r2j+ 1
4π
2r2
+
2κ+r++j(j+ 1) (C.4)
for Ym=0,j(Ω) = q2j+1
4πPj(cos θ)are standard scalar spherical harmonics on S2with vanishing
azimuthal quantum number.
Putting this all together, we find
∆V=−GNcπκ+
2
1
r+r−r+−r−
r−−1−(r−/r+)2
eκ+(2r+−d/2) ZdΩ0H(Ω,Ω0).(C.5)
For concreteness, we can choose a geodesic at θ=π/2, and keep just the lowest term in
C.4, which dominates at small κ+:
∆V=−GNcπ
2
1
r−r+−r−
r−−1−(r−/r+)2
eκ+(2r+−d/2).(C.6)
where cis the central charge associated with quantum fluctuations of the compact cosmic
string, where we’ve chosen a geodesic through θ=π/2, and where Uand Vare defined
such that the metric on the bifurcation surface is
ds2=r−
r+r+−r−
r−1+(r−/r+)2
e−2κ+r+dUdV (C.7)
Then, if the time (3.24) became longer than this available crossing time, a signal would be
delayed by the presence of the third mouth for too long to make it across the wormhole.
Considering the case that b∼re/2, this leads to a bound
∆v=δt r−
r+r+−r−
r−1+(r−/r+)2
e−2κ+r+!3
2
.∆V(C.8)
which then gives
m.cπ
4 log 4d2
r2
+
r1/2
+
r3/2
−r+−r−
r−−3/2−3/2(r−/r+)2
eκ+(3r+−d/2) (C.9)
or, mGN.`2
p
r−
, which is suppressed relative to the construction in the Appendix 3.2 by an
additional factor of `p/r−, making it more difficult to create a semiclassical black hole in
the throat.
– 25 –

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