physrevd.82.124034
TRANSCRIPT
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Wormhole solutions to Hor ˇava gravity
Marcelo Botta Cantcheff,* Nicolas E. Grandi,† and Mauricio Sturla‡
IFLP - CONICET and Departamento de Fı sica - UNLP CC67 CP1900 La Plata, Argentina(Received 10 June 2009; revised manuscript received 18 November 2010; published 20 December 2010)
We present wormhole solutions to Horava nonrelativistic gravity theory in vacuum. We show that, if the
parameter is set to one, traversable wormholes connecting two asymptotically de Sitter or anti-de Sitter
regions exist. In the case of arbitrary , the asymptotic regions have a more complicated metric with
constant curvature. We also show that, when the detailed balance condition is violated softly, traversable
and asymptotically Minkowski, de Sitter or anti-de Sitter wormholes exist.
DOI: 10.1103/PhysRevD.82.124034 PACS numbers: 04.60.Àm, 04.70.Às, 04.70.Bw
I. INTRODUCTION
A. Wormholes
In general relativity, a wormhole solution is often de-
fined as a solution of Einstein equations with nontrivial
topology that interpolates between two asymptotically flat
spacetimes. A slightly more general definition replaces the
asymptotically flat regions by other vacua as anti-de Sitter(AdS) or de Sitter spacetimes, modeling gravitational sol-
itons. A wormhole is said to be traversable whenever
matter can travel from one asymptotic region to the other
by passing through the throat. The classical example of
a wormhole is the Einstein-Rosen bridge [1] which com-
bines Schwarzschild black-hole and white-hole models,
and it is not traversable. There are no other traversable
wormhole solutions in standard gravity in vacuum, nor
in the presence of physically acceptable matter sources
(see e.g. [2–4]).
It is a widely accepted point of view that a correct
theory of quantum gravity will modify our understanding
of spacetime structure at small scales. In particular, it isgenerally assumed that general relativity is a theory that
describes large scales, and that at the UV it should be
modified by quantum effects. In this context, it seems
natural to wonder whether small wormholes in vacuum
could be supported by such quantum corrections, withoutrequiring the addition of any exotic matter. Indeed, in
theories that correct general relativity at short distances,
like Einstein-Gauss-Bonnet, such wormholes have been
shown to exist [5,6,8,9]. They would be created at small
scales, and would be invisible at the scales at which
Einstein gravity is recovered.
B. Hor ˇava-Lifshitz gravity
Recently, a power counting renormalizable nonrelativ-
istic theory of gravity was proposed by Horava [10]. Since
then, a lot of work has been done on the subject. Formal
developments were presented in [11–18], some spherically
symmetric solutions were presented in [19–24], toroidal
solutions were found in [25], gravitational waves were
studied in [26,27], cosmological implications were inves-
tigated in [28–35], and interesting features of field theory
in curved space and black-hole physics were presented
in [36–38,41,42].
A state of the theory is defined by a four-dimensional
manifold M equipped with a three-dimensional foliationF , with a pseudo-Riemannian structure defined by an
Euclidean three-dimensional metric in each slice of the
foliation gijð ~ x; tÞ, a shift vector N ið ~ x; tÞ, and a lapse func-
tion N ð ~ x; tÞ. This structure can be encoded in the Arnowitt-
Deser-Misner decomposed metric
ds2 ¼ ÀN 2ð ~ x; tÞdt2 þ gijð ~ x; tÞðdxi þ N ið ~ x; tÞdtÞðdx j
þ N jð ~ x; tÞdtÞ: (1)
The dynamics for the set ðM ;F ; gij ; N i ; N Þ is defined as
being gauge invariant with respect to foliation-preserving
diffeomorphisms, and having a UV fixed point at z
¼3,
where z is defined as the scaling dimension of time ascompared to that of space directions ½ ~ x ¼ À1, ½t ¼ Àz.
This choice leads to power counting renormalizability of
the theory in the UV. To the resulting action one may add
relevant deformations by operators of lower dimensions,
that lead the theory to a IR fixed point with z ¼ 1, in which
symmetry between space and time is restored, and thus a
general relativistic theory may emerge [43,45]. This is
close in spirit to the idea of recovering four-dimensional
Einstein gravity as a sort of macroscopic limit (z ¼ 1) of a
collection of three-dimensional theories [44] whose quan-
tization is better understood [46].
In order to have a control on the number of terms arising
as possible potential terms, one may impose the so-called
detailed balance condition: the potential term in the action
for 3þ 1 dimensional nonrelativistic gravity is built from
the square of the functional derivative of a suitable action
for Euclidean three-dimensional gravity (here three-
dimensional indices are contracted with the inverse
DeWitt metric). Condensed matter experience on this
kind of construction tells us that the higher dimensional
theory satisfying the detailed balance condition inherits
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the quantum properties of the lower dimensional one. It has
to be noted that the construction still works even when the
detailed balance condition is broken softly, in the sense of
adding relevant operators of a dimension lower than that of
the operators appearing at the short distance fixed point
z ¼ 3. In the UV, the theory still satisfies detailed balance,
while in the IR, the theory still flows to a z ¼ 1 fixed point.
We will not go through the above described steps in
more detail, but state the resulting action together with thedeformations that will be relevant to our purposes.
The interested reader can refer to the original paper [10].
The action for nonrelativistic gravity satisfying the detailed
balance condition can be written as
S ¼Z ffiffiffi
gp
N ðL0 þL1Þ ; (2)
where we have defined the Lagrangians
L0 ¼2
2ðK ijK ij À K 2Þ þ 22ðÃW R À 3Ã2
W Þ8ð1À 3Þ ;
L1 ¼ 22ð1À 4Þ32ð1À 3Þ R2 À 2
2w4
Cij À w2
2Rij
Â
Cij À w2
2Rij
:
(3)
Here , , , and w are arbitrary couplings. The dynamics
in the infrared is controlled by L0 and then, if ¼ 1,
general relativity is recovered. On the other hand, in the
UV the terms in L1 become dominant, and the anisotropy
between space and time is explicit.
To define soft violations of the detailed balance condi-
tion, we may add to the above Lagrangians a new term
with the form
L 2 ¼26
8ð1À 3ÞR; (4)
or we can distort the relative weight of L0 andL1. For our
purposes, it will be enough to write the deformed action as
S ¼Z ffiffiffi
gp
N ðL0 þ ð1À 21ÞL1Þ þ 2
2LÞ2 ; (5)
the detailed balance condition is recovered for 1 ¼2 ¼ 0. More general soft violations are also possible.
Since Horava-Lifshitz theory realizes a field theory
model for quantum gravity that is power counting renor-
malizable in the UV, one might expect that its classical
solutions differ from those of general relativity at small
scales, providing a mechanism to avoid singularities. For
example, it may have vacuum wormholes of the kind
described above, only perceivable at microscopic scales
and hidden at large scales. The purpose of this paper is to
show that exact solutions with this behavior exist in the
Horava model.
II. CONSTRUCTING SOLUTIONS WITHREFLEXION SYMMETRY
We are going to focus our work on the simplest case of
static spherically symmetric wormholes. Under this as-
sumption, the pseudo-Riemannian structure of Horava
gravity can be written in terms of the metric
ds2
¼ À~N 2
ðÞdt2
þ1
~ f ðÞd2
þ ðro þ 2Þ2d22 ; À1 < <1: (6)
This metric has two asymptotic regions ! Æ1 con-
nected at ¼ 0 by a S2 throat of minimal radius ro. We
will also require the additional condition of Z 2 reflection
symmetry with respect to the throat of the wormhole,
namely,
~ f ðÞ ¼ ~ f ðÀÞ and ~N 2ðÞ ¼ ~N 2ðÀÞ: (7)
As we will see, this last condition simplifies the construc-
tion of smooth wormhole solutions in vacuum.
We will consider the simplest way of constructingwormhole solutions proposed by Einstein and Rosen [1].
Let us first take a static spherically symmetric solution
written in the standard form
ds2 ¼ ÀN 2ðrÞdt2 þ 1
f ðrÞ dr2 þ r2d22 ; 0 < r < 1:
(8)
This may or may not represent a wormhole, and it may
have singularities and/or event horizons. In the present
paper we will concentrate in the solutions of this kindpresented in [19,20]. Such solutions are defined by the set
S ðM ; f ðrÞ ; N ðrÞÞ ; 0 < r < 1: (9)
Since N iðrÞ ¼ 0 we have omitted explicit mention of the
shift vector. The foliation F is implicit in the form of the
metric (8) as being defined by constant t surfaces. We will
take the restriction of M to Mþ whose boundary is a
sphere of radius ro, namely,
Sþ ðMþ ; f þðrÞ ; N þðrÞÞ ; ro < r < 1 ; (10)
where f þðrÞ and N þðrÞ stand for f ðrÞ and N ðrÞ in the
restricted domain r > ro. By defining the new variable through r ro þ 2, 2 Rþ
0one may write
Sþ ¼ ðMþ ; ~ f þðÞ ; ~N þðÞÞ ; 0 < < 1 ; (11)
where ~N þðÞ ¼ N þðro þ 2Þ and ~ f þðÞ ¼ f þðro þ2Þ=42. Now one may define the reflected set coordinat-
ized with 2 RÀ0
as
SÀ ðMÀ ; ~ f ÀðÞ ; ~N ÀðÞÞ ; À1 < < 0 ; (12)
where ~N ÀðÞ ¼ N þðro þ 2Þ ¼ ~N þðÀÞ and ~ f ÀðÞ ¼ f þðro þ 2Þ=42 ¼ ~ f þðÀÞ. HereMÀ is a copy of Mþ.
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So the extended manifold may be finally defined by
joining Mþ and MÀ at their boundaries at ¼ 0, with
the foliation extended accordingly.
~S ð ~M ; ~ f ðÞ ; ~N ðÞÞ ; À1 < < 1 ; (13)
where ~N ðzÞ ¼ N þðro þ 2Þ, ~ f ðzÞ ¼ f þðro þ 2Þ=42.
Note that these functions satisfy (7) and that the resulting
metric has the form (6).
Following Einstein and Rosen [1], if the equations of
motion are singularity free for all values of the independent
variables, we are dealing with a solution of the field
equations which is regular in the region À1 < < 1.
This corresponds to a spacetime composed by two identi-
cal parts > 0 and < 0 with two asymptotic regions
! Æ1 joined by a hypersurface ¼ 0 of minimal
radius ro and topology S2 Â R.
That the resulting set ~S is a solution of the theory at any
point Þ 0 is made evident by changing variables back to
r > ro, what results in the original solution Sþ. On the
other hand, the change of variables is singular at ¼ 0
(r ¼ ro), which implies that special attention should bepaid to the equations of motion there. As shown in the
Appendix, in order to get a smooth solution of the vacuum
equations when completing the space with the point
r ¼ ro, it is sufficient to impose the condition
f ;rðroÞ ¼ 0: (14)
In other words, the reflection point has to be chosen as a
stationary point of the function f ðrÞ. This is a sufficient
condition for the extended set ~S to be a solution of the
source free equations of motion. Even if for the special
case ¼ 1 more general solutions can be constructed (see
Appendix), in the present paper, for simplicity, we willstick to solutions satisfying the above condition.
Moreover, we will show in the next section that condition(14) exactly fulfills the bound to avoid naked singularities
through the formation of a traversable wormhole. In other
words, there is a class of solutions (6) that contain a naked
singularity in ¼ 1 Horava gravity [15], but they precisely
correspond to those that may extended to a traversable
wormhole geometry according to the condition (14).
Let us finally discuss the causal behavior at the throat
and the condition of traversability in the present frame-
work. Since the starting solution (6) may have event hori-
zons, located at points rh at which
N ðrhÞ ¼ 0: (15)
The reflected solution will satisfy 8: ~N ðÞ Þ 0 when-
ever the stationary radius fulfills ro > rh. On the other
hand, the vanishing of the component g at ¼ 0 is a
coordinate singularity, that does not prevent traversability.
This is guaranteed by the existence of a continuous choice
of the future light cone [3]. In our case the timelike unit
vector ua ¼ ~N À1rat will be smoothly defined among the
bridge whenever ro > rh [47]. Therefore, to get a travers-
able wormhole, a sufficient condition is
ro > rhþ ; (16)
where rhþ is the position of the outmost horizon.
III. WORMHOLE SOLUTIONS TOHORıAVA GRAVITY
A. Asymptotically flat wormholes
A spherically symmetric solution with a flat asymptotic
region was found in [20] by setting 1 ¼ 0, 2 ¼ 1 and
¼ 1 in (5). This theory has a Minkowski vacuum, and it
is natural to look for spherically symmetric solutions thatapproaches that vacuum at infinity. As it was shown there,
such solutions exist and take the form
N 2 ¼ f ¼ 1þ !r2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi
rð!2r3 þ 4!M Þq
; (17)
where M is an integration constant and ! ¼ 162=2.
This solution has two event horizons at radius
rhÆ ¼ M
1Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1À 1
2!M 2
s ; (18)
which disappear leaving a naked singularity whenever
2!M 2 < 1: (19)
To find a wormhole solution we will apply to (17) the
above described reflection technique. As discussed above,
this entails to look for an stationary point of the spherically
symmetric solution
f ;rðroÞ ¼ 2!ro À 2!2r3o þ !M
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi roð!2
r3
o þ 4!M Þp ¼ 0: (20)
By multiplying by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi
roð!2r3o þ 4!M Þp
, taking the square
and rewriting in terms of the variable x ¼ !2r3, this equa-
tion can be solved, resulting in
ro ¼
M
2!
1=3
; (21)
thus we can build a wormhole by taking a solution of the
form (17) with positive M , and reflecting it at the value
of ro given by (21).
In terms of this, the traversability condition (16) may be
written as
M
2!
1=3
> M
1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1À 1
2!M 2
s ; (22)
and can be solved by writing x ¼ ð2!M 2ÞÀ1=3 and solving
for x, getting
2!M 2 < 1 ; (23)
which is exactly (19). This implies that, in order to build a
traversable wormhole, we should start with the solution
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that has naked singularity. Note that after reflection the
singularity disappears, leaving us with a completely regu-
lar wormhole.
B. Asymptotically (A)dS wormholes
If the detailed balance condition is satisfied 1 ¼2 ¼ 0, a spherically symmetric solution with AdS asymp-totic behavior was found in [19], for the particular case in
which ¼ 1. They have the form
N 2 ¼ f ¼ 1þ ðrÞ2 À ffiffiffiffiffiffi
rp
; (24)
where ¼ ffiffiffiffiffiffiffiffiffi ÀÃp
and is a constant of integration.
A horizon will be present at the value of r satisfying
N 2ðrhÞ ¼ 0, which can be rewritten as ( x ¼ rh)
x4 þ 2 x2 À 2 x þ 1 ¼ 0: (25)
This polynomial has at most two real roots, as can be seen
from the fact that its second derivative is always positive.Its discriminant reads
Á ¼ 4
ð256À 274
Þ ; (26)
and it vanishes whenever there is a double root. This
implies that the two horizons will join into a single one
when ¼ Æ4=33=4. For smaller jj’s there is no event
horizon but a naked singularity. This can also be seen inFig. 1.
To find the reflection point to build our wormhole, we
write the derivative as
f ;rðroÞ ¼ 22ro À
2 ffiffiffiffiffiffiffiffi
ro
p ¼ 0: (27)
This condition uniquely defines the point at which the
solution must be reflected as
ro ¼1
4
2=3
: (28)
As mentioned before, in order to ensure traversablity, the
reflection point must be at the right of the outer horizon
ro > rhþ. This cannot be satisfied for any value of ro, as
can be seen in Fig. 1. In consequence, as it happened for the
asymptotically flat case, we should build our wormhole
from the solution without horizons jj < 4=33=4
, whichin turn implies ro < 1=
ffiffiffi 3
p . The wormhole obtained in
this way is traversable and asymptotically (A)dS by
construction.
If the detailed balance condition is relaxed to 1 Þ 0,
2 ¼ 0, a more general spherically symmetric solution
with an (A)dS asymptotic behavior has also been found
with ¼ 1 [19]. It takes the form
N 2 ¼ f ¼ 1þ 2r2
1À 21
À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi
rð214r3 þ 4M ð1À 2
1ÞÞ
q 1À 2
1
;
(29)
where ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ÀÃW p
and M a constant of integration.
Despite of its similarity with (17), this solution takes an
(A)dS form asymptotically, as can be easily verified. If
21
< 1 the solution has a curvature singularity at the origin.
On the other hand if 21
> 1, the square root becomes
complex bellow a finite radius r3c ¼ 4M ð21À 1Þ=2
13.
To find the event horizons, we rewrite the condition
f ðrhÞ ¼ 0 as ( x ¼ rh)
x4 þ 2 x2 À 4Mx þ ð1À 21Þ ¼ 0: (30)
Again, this polynomial in x has a second derivative that is
everywhere positive regardless of the values of M and 1,
implying that it may have at most two real roots that can beidentified with two event horizons. These event horizons
will coincide when the roots become a single double root,
namely, when the discriminant vanishes
À 256ð27M 4 þ 2ð921À 8ÞM 2 þ 4
1ð2
1À 1ÞÞ ¼ 0 ; (31)
or in other words when
M 2Æ ¼ 1
27ð8À 92
1Æ ð4À 32
1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4À 32
1
q Þ: (32)
Here we see that in order to have a horizon we need
21
< 4=3. Under this assumption, the value of M 2À is al-
ways negative, implying an imaginary value of M À thatshould be discarded. Regarding M 2þ, it is always positive,
determining a bound that M 2 should satisfy in order to have
a horizon. Such a bound read M 2 > M 2þ, or more explicitly
M 2 > 1
27ð8À 92
1Æ ð4À 32
1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4À 32
1
q Þ: (33)
For values of M satisfying this inequality, the discriminant
is negative and two horizons exists. On the other hand for
M 2 < M 2þ, there is no event horizon.
1
o
2 3 4
1
2
x
x o
x h
x h
FIG. 1 (color online). In this plot the relation between the
horizons and the reflection point is shown, for the case ¼ 1,
1 ¼ 2 ¼ 0. As can be seen in the plot, the radius of the outer
horizon, when it is present, is always larger than that of the
reflection point, implying that a traversable wormhole must be
constructed from a solution with a naked singularity.
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To build an asymptotically (A)dS wormhole, we reflect
the above solution at the stationary point r ¼ ro defined by
f ;rðroÞ ¼ 22ro
1À 21
À 221
4r3o þM ð1À 21Þ
ð1À 21Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi
roð214r3o þ 4M ð1À 21ÞÞ
q ¼ 0:
(34)
This is solved by
roÆ ¼M 1=3
1À 2
21
Æ 1
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4
21
À 3
s 1=3
: (35)
Again the condition 21
< 4=3 shows up, now to ensure that
a stationary point exists at which the solution can be
reflected. For 1 < 21
< 4=3, the value roþ is always
smaller than the value rc at which the solution becomes
complex. This implies that no wormhole can be used to
cure this problem, suggesting that only small violations of
the detailed balance condition 2
1 < 1 should be accepted.In the case 2 < 1, it can be checked that roþ is never
larger than rhþ. In other words, to build a traversable
wormhole one should start with a solution without horizon,
i.e. one satisfying M 2 < M 2þ. Such solutions contain a
naked singularity at the origin that disappears after reflec-
tion. This is analogous to what happened for the cases
studied in the previous sections. It is interesting to note
(see the Appendix) that a more general class of wormhole
solutions could be constructed for the previous cases in
which ¼ 1, without imposing the condition f ;rðroÞ ¼ 0.
We do not discuss it here because they do not present a
mechanism to avoid naked singularities.
C. Exotic asymptotic behavior
Spherically symmetric solutions of the action satisfying
the detailed balance condition 1 ¼ 2 ¼ 0 were found
for arbitrary in [19]. They take the form
f ¼ 1þ ðrÞ2 À ðrÞ2Æ ffiffiffiffiffiffiffiffiffi 6À2p =À1 ; (36)
N 2 ¼ ðrÞÀ2ð1þ3Æ2 ffiffiffiffiffiffiffiffiffi 6À2p
=À1Þ f ðrÞ ; (37)
where ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ÀÃW
p and is a constant of integration.
There are two solutions for each value of , defined by
the choice of the sign in front of the square root.To build a Z 2 symmetric solution (that we will loosely
call a wormhole, even if, as we will see, its asymptotic
behavior is nontrivial) we take the derivative and look for
a stationary point
f ;rðroÞ ¼ 22ro À 2Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 À 2p
À 1
 ðroÞð2Æ ffiffiffiffiffiffiffiffiffi 6À2p =À1ÞÀ1 ¼ 0 ; (38)
N 2 ;rðroÞ ¼ À21þ 3 Æ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 À 2
p
À 1
 ðroÞÀ2ð1þ3Æ2 ffiffiffiffiffiffiffiffiffi 6À2p =À1ÞÀ1 f ðroÞ ¼ 0: (39)
This equations can be simplified to ( x ¼ ro)
2 x2 ¼ 2 Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 À 2p
À1
x2Æ
ffiffiffiffiffiffiffiffiffi 6À2p
=À1 ; (40)
ð1þ 3 Æ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p Þ f ðroÞ ¼ 0: (41)
We see from the second equation that either the value of satisfies
1þ 3Æ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 À 2
p ¼ 0 ; (42)
that is solved only by ¼ 1 when the minus sign is chosen,
corresponding to the AdS case already analyzed in the
previous section; or the solutions must be reflected at the
horizon f ðroÞ ¼ 0. Even if this may result in a nontravers-
able wormhole, we will study the possibility of a solution
of this form. In this last case equations read
2 Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 À 2p
À 1x2Æ ffiffiffiffiffiffiffiffiffi 6À2p
=À1 ¼ 2 x2 ; (43)
1þ x2 À x2Æ ffiffiffiffiffiffiffiffiffi 6À2p =À1 ¼ 0 ; (44)
and can be simplified to
2 Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 À 2p
À 1x2Æ ffiffiffiffiffiffiffiffiffi 6À2p
=À1 ¼ 2 x2 ;
2 Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p
À2Ç ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 À 2p ¼
x2:
(45)
The second equation has no solution if we choose the plus
sign. On the other hand, for the minus sign, two possibil-
ities arise from the second line, either
2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 À 2
p > 0 and À 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p > 0 ; (46)
or
2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 À 2
p < 0 and À 2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p < 0: (47)
The first is solved by > 1, the second instead is solved
by 1=2 < < 1.
We conclude that, when > 1=2, the second equation in
(45) provides a value of ro at which the solution can be
reflected. The first equation on the other hand, determines
the value of that is compatible with that ro. In other
words, wormhole solutions can be built from the solution
with the minus sign whenever > 1=2, by reflecting the
exterior region at the horizon.
To explore the asymptotic behavior of the resulting
wormhole, we take the r ! 1 limit. We see that the second
term in f ðrÞ in (36) will dominate whenever
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2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p
À 1< 2 ; (48)
which is always satisfied for > 1=2. Regarding N 2ðrÞin (37), we see that for large r it behaves as a power law
with exponent
n¼ À
21þ 3 À 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6À 2
p
À 1 þ2 ; (49)
which is positive and smaller than 4 for 1=2 < < 3 and
negative for > 3. The (A)dS case n ¼ 2 corresponds to
¼ 1 and has been studied in the previous sections.
This results in an asymptotic metric with the form
ds2 ¼ ÀðrÞndt2 þ dr2
ðrÞ2 þ r2d22: (50)
The scalar curvature is asymptotically constant and finite
R ¼ÃW
2 ð12þ 4n þ n
2
Þ þ2
r2 ;
! ÃW
2ð12þ 4nþ n2Þ ; (51)
and it never vanishes as a function of n. As can be easily
seen, the asymptotic isometry group is SOð3Þ Â R, the first
factor representing the rotations on the sphere, while the
second refers to time translations.
IV. CONCLUDING REMARKS
In this work we have constructed wormhole solutions of
Horava gravity theory in vacuum by reflection of knownspherically symmetric metrics. For both the asymptotically
Minkowski or (A)dS cases, for the condition to have purely
exterior solutions, namely, wormholes constructed with
regions exterior to the horizon of the starting solution,
that we imposed in order to ensure traversability, forces
us to choose the parameters in the starting spherically
symmetric solutions that imply a naked singularity. Afterreflection the singularity disappears, leaving us with a Z 2and spherically symmetric wormhole, where two asymp-
totic regions are connected through a topologically S2
neck.
Under the detailed balance condition, the ¼
1 case
provides a wormhole with de Sitter or AdS asymptotic
regions. A similar solution is obtained when the detailed
balance condition is softly broken by a deformation of the
kind parameterized by 1, whenever 21
< 1. In the case
21
> 1 no wormhole exists, and the solution becomes
complex at a finite distance to the origin, which suggests
that only small violations of the detailed balance condition
21
< 1 should be allowed. On the other hand, if 2 is turned
on, the resulting asymptotic regions correspond to flat
Minkowskian spacetime. On the general > 1=2 case,
the asymptotic regions have constant curvature, and isome-
try group SOð3Þ Â R.
The fact that a traversable wormhole can be constructed
when the starting solution has no horizon suggest the
following interpretation: if we start with value of the
integration constant (or M ) for which a horizon exists,
an outside observer cannot say whether the solution
represents a microscopic wormhole hidden inside the hori-zon, or a black hole with a censored singularity at the
center. When varying , the horizon shrinks and at some
point it disappears. From the macroscopic point of view,
described by a relativistic theory of gravity, the singularity
would become naked violating the cosmic censorship
principle. But from the microscopic nonrelativistic point
of view, the wormhole provides a mechanism to avoid
it. Therefore, we may observe a sort of complementarity
between these two mechanisms to censure the
singularities.
As another related point, we may see that for instance
in the solution (24) the condition to have a well-defined
Hawking temperature [19] is exactly the opposite to theone that allows to form a traversable wormhole. This is
consistent with the fact that in the last case, the degrees of
freedom in both sides of the throat are causally connected,
and therefore, one should not take traces to evaluate the
asymptotic quantum amplitudes.A remark should be made about our definition of
traversability. In this paper we have constructed worm-
holes that are traversable for particles that move according
to the causal structure of the metric, namely, inside the
light cones. Since in the present context symmetry between
space and time is broken, one may imagine particles with a
more complicated causal behavior. For example, a Lifshitz
scalar has corrections to its dispersion relation that are
quartic in momenta. Depending on the sign of the correc-
tion, the field quanta may eventually follow worldlines that
travel outside the metric light cones. This may indeed
enlarge the set of solutions which are microscopically
traversable by such particles.
ACKNOWLEDGMENTS
The authors want to thank G. Giribet for valuable dis-
cussions and A. Kehagias for help and encouragement.
We are also grateful to G. Silva for providing a comfortable
work environment during the preparation of themanuscript.
APPENDIX
In the present Appendix we are going to show that the
spherically symmetric wormhole solutions built in this
paper, are in fact vacuum solutions of the equations of
motion. To that end, we start from the ansatz (6)
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ds2 ¼ À ~N 2ðÞdt2 þ 1
~ f ðÞ d2
þ ðro þ 2Þ2d22 ; À1 < <1: (A1)
As explained in Sec. II, away from ¼ 0 a solution to
Horava equations with this general form can be obtained
from the spherically symmetric metrics presented in[19,20], by identifying r
¼ro
þ2. We will prove that,
if ro is chosen to be a stationary point of f , the functionscan be extended with continuity at the matching point
¼ 0 (or r ¼ ro) to obtain a solution in À1 < < 1,
in such a way that the presence of any additional matter
shell is not needed.
To check whether a matter shell at ¼ 0 is required,
we will first write the equations of motion in the form
ðgravityÞ ¼ ðmatterÞðÞ ; (A2)
where in the left-hand side we include all the contributions
coming form the Horava action, while in the right-hand
side we include a -function source describing the hypo-
thetical matter shell. Then we will integrate on the variable
both sides of the above equation in the interval ðÀa ; bÞ:Z b
Àa
dðgravityÞ ¼ ðmatterÞj¼0: (A3)
Finally, we will take the a, b ! 0 limit. Only if the
resulting left-hand side is nonvanishing in that limit, will
we conclude that a nontrivial source must contribute to
the right-hand side.
It is interesting to observe that for all the solutions
considered in this work, the functions N ðÞ and
AðÞ 1=f ðro þ 2Þ satisfy the following relations:
N ðÞ ¼ !ðro þ 2Þ! ffiffiffiffiffiffiffiffiffiffi AðÞp ; (A4)
where ! ¼0
and ¼1
in the case ¼1
, and ! ¼À 1þ3þ2 ffiffiffiffiffiffiffiffiffi 6À2p À1 and an arbitrary constant, in the case
Þ 1. The reader should not confuse ! and with the
previously defined integration constants with the same
name, that will not be referred in the present Appendix.
On the other hand, it is not difficult to see that, for all the
solutions we have considered, it is possible to expand A0ðÞnear ¼ 0 as follows:
A0ðÞ ¼ c1 þ c33 þ O½5 ; (A5)
where c1 / f ;rðroÞ. This behavior at small is crucial in
order to prove our claim.
In what follows, we compute the different contributions
to (A3) coming from all the solutions to Horava gravity,
presented in this paper. We will evaluate the left-hand side
of (A3) using a general AðÞ with continuous first deriva-
tive, and N ðÞ with the general form expressed in (A4).
(i) The contribution to the time component of (A3)
results
ðmatterÞN j¼0 ¼ lima ;b!0
Z b
Àa
d
512ðro þ 2Þ4ð1ð2 À 1Þ þ ðro þ 2Þ2ð0Ãþ 24ÞÞ AðÞ3
þ 256ðro þ 2
Þ4
ð1 À 21þ ðro þ 2
Þ2
ð0Ãð3ðro þ 2
Þ2
ÃÀ 2Þ À 224
ÞÞ AðÞ4
À 256ðro þ 2Þ51AðÞ A0ðÞ À 32
ðro þ 2Þ61ð À 1Þ A0ðÞ2 À 256ðro þ 2Þ4 AðÞ2ð1ð2 À 1Þ
þ ðro þ 2Þððro þ 2Þ2ð0Ãþ 24Þ À 1Þ A0ðÞÞ
: (A6)
Plugging the expansion (A5) into (A6), we see that the integrand is completely regular, and therefore, the temporal
component of (A3) must have a vanishing right-hand side.
(ii) Once the integral of the regular terms is omitted since it vanishes trivially in the limit a, b ! 0, the nontrivial
contribution to the radial () component of (A3) results in
ðmatterÞj¼0 ¼ lima ;b!0BT0 þ Z b
Àad32ðro
þ2
Þ4þ!!
ffiffiffiffiffiffiffiffiffiffi AðÞp ðð8ðro þ
2
Þ1ð6À 7Þ AðÞ À 8ðro þ 2
ÞðÀ1
þ ðro þ 2Þ2ð0Ãþ 24ÞÞ AðÞ2Þ A0ðÞ À 11ðro þ 2Þ21ð À 1Þ A0ðÞ2Þ
; (A7)
where the boundary term BT0 reads
BT 0 ¼1
ð128ðro þ 2Þ6þ!!1ðÀ1þ Þ
ffiffiffiffiffiffiffiffiffiffi AðÞq
Þ A0ðÞjbÀa: (A8)
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Replacing the expansion (A5) into (A7), we see that the component of (A3) will have no contribution coming from the
integral in (A7). The possibility of a contribution coming from the boundary term BT0 will be analyzed bellow.
(iii) The angular contribution to (A3) results
ðmatterÞj¼0¼ lima ;b!0
BT1þBT2ÀBT3þ
Z b
Àa
d16ðro þ2Þ5þ!! A0ðÞ
2ffiffiffiffiffiffiffiffiffiffi AðÞp Â
99
4ðro þ2Þ21ðÀ 1Þ A0ðÞ2
Àðro þ2Þ1ð175À 193þ 9ðÀ 1Þ!Þ AðÞ A0ðÞþ 2AðÞ2ð81ð23À 19À 2!Þ
þ9ðro þ
2
Þððro þ2
Þ2
ð0Ãþ2
4
ÞÀ 1Þ A0ðÞÞ ; (A9)
where we have again omitted in the integral regular terms
that do not contribute in the a, b ! 0 limit. The bound-
ary terms that appear in (A9) have the form
BT 1 ¼32
2ðro þ 2Þ7þ!!1ð À 1Þ AðÞ3=2 A00ðÞjbÀa
;
(A10)
BT2 ¼ À 16
3ðro þ 2Þ6þ!!
ffiffiffiffiffiffiffiffiffiffi AðÞq
A0ðÞð21ð2ð17
À 15þ 2! À 2!Þ þ roð À 1ÞÞ AðÞ þ 42ððro
þ 2Þ2ð0Ãþ 24Þ À 1Þ AðÞ2ÞjbÀa ; (A11)
BT 3 ¼152
2ðro þ 2Þ7þ!!1ðÀ 1Þ
ffiffiffiffiffiffiffiffiffiffi AðÞq
A0ðÞ2jbÀa:
(A12)
Using the expansion (A5) it is straightforward to prove that
the integrand of (A9) is regular. Therefore, if there is any
nonvanishing contribution, it must come from the bound-
ary terms.
Equation (A11) shows that the leading order of BT2 near
¼ 0 is proportional to
AðÞ3=2 A0ðÞ3
: (A13)
The vanishing of this term is guaranteed by imposing thecondition c1 ¼ 0 in (A5), which may also be expressed as
f ;rðroÞ ¼ 0. In addition, this condition suffices to ensure
the vanishing of the remaining boundary terms BT0, BT1,
and BT3. In the limit a, b ! 0 it implies that the right-
hand side of Eq. (A9) is zero, and then all possible con-
tributions to the left-hand side of the component of
(A3) vanish.
The above analysis allows us to conclude that the
condition f ;rðroÞ ¼ 0 is sufficient to ensure that no matter
shell is needed to support the wormhole solution at ¼ 0.
Let us finally notice that, as explained in the end of Sec. II,
for ¼ 1 Eq. (A9) becomes trivially regular, and the
condition above is not necessary to obtain a regular
solution.
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