Notes on wormhole existence in scalar-tensor and F(R) gravity
ABSTRACT Some recent papers have claimed the existence of static, spherically symmetric wormhole solutions to gravitational field equations in the absence of ghost (or phantom) degrees of freedom. We show that in some such cases the solutions in question are actually not of wormhole nature while in cases where a wormhole is obtained, the effective gravitational constant G_eff is negative in some region of space, i.e., the graviton becomes a ghost. In particular, it is confirmed that there are no vacuum wormhole solutions of the Brans-Dicke theory with zero potential and the coupling constant \omega > -3/2, except for the case \omega = 0; in the latter case, G_eff < 0 in the region beyond the throat. The same is true for wormhole solutions of F(R) gravity: special wormhole solutions are only possible if F(R) contains an extremum at which G_eff changes its sign. Comment: 7 two-column pages, no figures, to appear in Grav. Cosmol. A misprint corrected, references updated
Article: Living in the RED
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ABSTRACT: It is shown that the static spherically symmetric solutions to the Brans-Dicke theory of gravitation give rise either to a naked singularity if the post Newtonian parameter gamma1.Physical review D: Particles and fields 03/1995; 51(4):2011-2013.
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ABSTRACT: The role of Mach's principle in physics is discussed in relation to the equivalence principle. The difficulties encountered in attempting to incorporate Mach's principle into general relativity are discussed. A modified relativistic theory of gravitation, apparently compatible with Mach's principle, is developed.Physical Review - PHYS REV X. 01/1962; 125(6):2194-2201.
arXiv:1005.3262v3 [gr-qc] 8 Jun 2010
Notes on wormhole existence in scalar-tensor and F(R) gravity
K.A. Bronnikova,b,1, M.V. Skvortsovab, and A.A. Starobinskyc
aCenter of Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow 119361, Russia
bInstitute of Gravitation and Cosmology, PFUR, 6 Miklukho-Maklaya St., Moscow 117198, Russia
cLandau Institute for Theoretical Physics of RAS, 2 Kosygina St., Moscow 119334, Russia
Some recent papers have claimed the existence of static, spherically symmetric wormhole solutions to grav-
itational field equations in the absence of ghost (or phantom) degrees of freedom. We show that in some
such cases the solutions in question are actually not of wormhole nature while in cases where a wormhole
is obtained, the effective gravitational constant Geff is negative in some region of space, i.e., the graviton
becomes a ghost. In particular, it is confirmed that there are no vacuum wormhole solutions of the Brans-
Dicke theory with zero potential and the coupling constant ω > −3/2, except for the case ω = 0; in the
latter case, Geff < 0 in the region beyond the throat. The same is true for wormhole solutions of F(R)
gravity: special wormhole solutions are only possible if F(R) contains an extremum at which Geff changes
It is well known that to build a static traversable
wormhole in general relativity it is necessary to in-
voke a matter source of gravity that violates the
Null Energy Condition (NEC), at least in the neigh-
borhood of the wormhole throat . With dynamic
wormholes the situation is more complex: first, the
notion of a wormhole throat is then less evident
and even admits different definitions [2,3]; second,
a dynamic wormhole may exist not eternally but
only in a certain time interval, and in this case the
requirements to its matter source may be weak-
ened . In what follows, we will restrict ourselves
to static wormhole space-times.
The nonexistence theorem for static wormholes
in the presence of any matter respecting the NEC
was recently generalized  to the class of the-
ories of gravity whose space-times are related to
that of general relativity by a conformal mapping.
This class includes theories without ghost fields
even though many of them admit NEC violation.
The generalization occurs under certain conditions.
Thus, for a scalar-tensor theory (STT) of gravity,
formulated in a space-time MJ(the Jordan frame)
with the metric gµν using the Lagrangian
f(Φ)R + h(Φ)gµνΦ,µΦ,ν− 2U(Φ)
(R is the Ricci scalar, Lmis the matter Lagrangian,
f , h and U are arbitrary functions), the above
theorem holds if the non-minimal coupling func-
tion f(Φ) is everywhere positive (in other words,
the graviton is not a ghost) and also
l(Φ) := fh +3
(the Φ field itself is not a ghost).2The latter con-
dition becomes evident if one performs the stan-
dard conformal mapping to the Einstein frame (the
space-time ME with the metric gµν) such that
after which the Lagrangian (1) transforms to
2LE= (signf)[R + εgµνφµφ,ν] − 2|f|−2U
where ε = signl(Φ), bars mark quantities obtained
from or with gµν, indices are raised and lowered
with gµνand gµν, and the Φ and φ fields are re-
The Brans-Dicke (BD) STT is the special case
of (1) corresponding to
f(Φ) = Φ,
l(Φ) = ω + 3/2,
h(Φ) = ω/Φ,
ω = const. (6)
2Our conventions are: the metric signature (+ − − −);
the curvature tensor Rσµρν = ∂νΓσ
that the Ricci scalar R > 0 for de Sitter space-time and the
matter-dominated cosmological epoch; the system of units
8πG = c = 1.
µρ−..., Rµν = Rσµσν, so
Let us recall that the NEC reads Tµνkµkν≥ 0,
where Tµνis the stress-energy tensor (SET) of mat-
ter and kµis an arbitrary null vector with respect
to the metric gµν. The mapping (3) transforms
the SET according to Tµν= |f|−1Tµν while kµre-
mains a null vector in the metric gµν. Thus if the
NEC holds in MJ, it also holds in ME, and vice
versa, and it means, in particular, that wormholes
are absent in ME.
Assuming that matter in ME does respect the
NEC, the above theorem is proved in  (following
the idea expressed earlier in ) using the fact that
if both f(Φ) and l(Φ) are smooth and positive ev-
erywhere, including limiting points, the mapping
(3) transfers a flat spatial infinity in one frame to
a flat spatial infinity in the other. Therefore, if we
suppose that there is an asymptotically flat worm-
hole in MJ, its each flat infinity has a counterpart
in ME, the whole manifold ME is smooth, and we
obtain a wormhole there, contrary to what was as-
sumed. We conclude that static and asymptotically
flat wormholes are absent in the Jordan frame as
A special case of this situation is connected with
matter concentrated on a thin shell. Accordingly,
as was explicitly shown in , in any STT with a
non-ghost scalar field, in any thin-shell wormholes
built from two identical regions of static, spheri-
cally symmetric space-times, the shell has negative
surface energy density, thus violating both null and
weak energy conditions.
In , a possible wormhole behavior of static,
spherically symmetric configurations was also con-
sidered in STT under weakened requirements, al-
lowing f(Φ) to reach zero or even become negative.
It has turned out that if f only reaches zero, twice
asymptotically flat wormhole solutions in the Jor-
dan frame can exist but only in exceptional cases:
(i) the corresponding Einstein-frame solution must
comprise an extreme black hole, whose double hori-
zon is then mapped to the second spatial infinity
in MJ; it is not possible with vacuum solutions
but can happen with nonzero electric or magnetic
fields;3(ii) additional fine tuning is necessary to
avoid a solid angle deficit or excess at this second
infinity, and (iii) the theory itself should be very
3Such an example has been found in . In a more general
context, inclusion of an electric field as a source of gravity
in STT enlarges the number of classes of solutions but, just
as is the case with vacuum solutions, wormholes can exist
either with ε = −1 or in special cases with Geff somewhere
becoming infinite or negative .
Rather a wide (although still special) class of
wormhole solutions exists in theories where a tran-
sition to f < 0 is allowed, so that the manifold
ME is mapped according to (3) to only a part of
the manifold MJ(the conformal continuation phe-
nomenon [10,11]). However, previous studies have
shown that such solutions are generically unsta-
ble under spherically symmetric perturbations ,
the instability appearing due to a negative pole of
the effective potential at the transition surface to
f < 0. The existence of this pole still does not
guarantee an instability, and a further study of
non-linear dynamical evolution is necessary; but
even if such wormholes might exist, their “remote
mouths” would be located in anti-gravitational re-
gions with f < 0. So, they cannot connect different
parts of our Universe but could only be bridges to
other universes (if any) with very unusual physics.
In addition, it is known that a generic space-like
anisotropic curvature singularity arises dynami-
cally if f → 0 , and it is unclear how to avoid
Some recent publications contain results on
static, spherically symmetric wormhole existence
which seem to contradict these conclusions.
1. It is claimed that there are vacuum wormhole
solutions in the BD theory corresponding to
the coupling constant ω in the non-ghost in-
terval (−3/2,−4/3) .
2. A similar claim is made for the BD theory
with ω in a larger non-ghost interval includ-
ing the value ω = 0 .
3. It is asserted that vacuum BD wormholes ex-
ist for ω < −2 but nothing is said about the
range −2 ≤ ω < −3/2 .
4. Wormhole solutions are found in some F(R)
theories of gravity  which are equivalent
to the BD theory with ω = 0 and a nonzero
The purpose of this paper is to clarify the situ-
ation in all these cases. To this end, we begin the
next section with designating the conditions under
which a static, spherically symmetric metric is said
to describe a wormhole. Then we explicitly write
the vacuum solution of the general massless STT
and specify its properties for the BD theory, thus
covering items 1–3 above. Section 3 is devoted to
the properties of wormhole solutions in F(R) theo-
ries. Our previous conclusions  are confirmed in
all these cases. Section 4 contains some remarks of
2 Wormholes in scalar-tensor grav-
2.1 The wormhole notion
The general static, spherically symmetric metric
can be written as
ds2= e2γ(u)dt2− e2α(u)du2− r2(u)dΩ2.
where u is an arbitrary radial coordinate and
dΩ2= (dθ2+ sin2θdϕ2) is the linear element on a
unit sphere. The metric (7) is asymptotically flat
as u tends to some value u0 (finite or infinite) if
r → ∞,γ ≈ γ0+ γ1/r + o(1/r),
where the prime denotes d/du, γ0, γ1= const, and
the last condition in (8) is the requirement of a cor-
rect circumference to radius ratio for large circles.
We will say that the metric (7) describes a
wormhole if it regular in some range (u1, u2) of
the radial coordinate, does not contain horizons
(that is, γ(u) and α(u) are finite) in this range,
and is asymptotically flat both as u → u1 and as
u → u2. The existence of two large r asymptotic
regions inevitably means that there is at least one
regular minimum of the function r(u), called a
We thus consider only globally regular config-
urations. One could admit more general regular
asymptotic behaviors, e.g., anti-de Sitter, but for
our present discussion it is sufficient to be restricted
to asymptotic flatness. We thus also discard possi-
ble wormholes in which r(u) reaches large (as com-
pared with the throat radius) but finite values of r
at one or both sides, as happens, e.g., when a worm-
hole connects two closed worlds. Our arguments
could be easily modified to include such cases. An-
other case of interest which is not covered by our
definition and deserves a separate study is that of
a cosmological horizon located far enough from the
throat, as is the case in some known models [17,18].
2.2Vacuum solutions of a general STT
Let us now consider the theory (1) assuming f ≥ 0,
U ≡ 0 and Lm= 0. Then in the Einstein frame we
have a massless, minimally coupled scalar field as
the only source of gravity. The static, spherically
symmetric solutions to the field equations are well
known in this case: these are the Fisher solution
of 1948 ) if ε = +1 and the Bergmann–Leipnik
solution of 1957 ) (sometimes called the “anti-
Fisher” solution) in case ε = −1. Both solutions
were repeatedly rediscovered afterwards. Let us re-
produce them in the simplest joint form, follow-
Two combinations of the Einstein equations for
the metric (7) and φ = φ(u) read R0
2= 0. Choosing the harmonic radial coordinate
u, such that α(u) = γ(u)+2lnr(u), we easily solve
these equations.Indeed, the first of them reads
simply γ′′= 0, while the second one is written as
β′′+ γ′′= e2(β+γ). Solving them, we have
0= 0 and R0
γ = −mu,
e−β−γ= s(k,u) :=
k > 0,
k = 0,
k < 0.
where k and m are integration constants; two
more integration constants have been suppressed
by choosing the zero point of u and the scale along
the time axis.As a result, the metric has the
(note that flat spatial infinity here corresponds to
u = 0, and m has the meaning of the Schwarzschild
mass). Moreover, the scalar field equation in this
gauge reads φ′′= 0, hence
φ = Cu,C = const (the scalar charge) (11)
without loss of generality. Lastly, due to the
component of the Einstein equations (the con-
straint equation), the integration constants are re-
2k2signk = 2m2+ εC2.(12)
Eqs.(10), (11) describe the Fisher solution in
the case ε = +1, hence k > 0 due to (12); in the
case ε = −1, they give the anti-Fisher solution, in
which k can be arbitrary. These solutions are si-
multaneously the Einstein-frame vacuum solutions
of any STT (1) with U(Φ) = 0 and signl(Φ) = ε.
Considering (10) as gµνdxµdxνand applying the
mapping (3), we easily obtain the solution in the
Jordan frame MJ.
In particular, in the BD theory we have accord-
ing to (5)
f(Φ) = Φ = Φ0exp(φ/
|ω + 3/2|),
where φ = Cu and we can put Φ0= 1 without loss
2.3 The BD theory, ω > −3/2, ε = +1.
For k > 0, which is always true if ε = +1 (leaving
aside the trivial case k = 0 with flat metric), it
is helpful to apply the coordinate transformation
e−2ku= 1 − 2k/x, after which the Jordan-frame
solution in the BD theory takes the form
Φ = Pξ,
Padt2− P−adx2− P1−ax2dΩ2?
P(x) ≡ 1 − 2k/x,
where we have redefined the integration constants:
ξ = −C/(2k
and the relation (12) passes on to
ω + 3/2),a = m/k < 1, (15)
(2ω + 3)ξ2= 1 − a2.
The index J is used to stress that it is the Jordan-
frame metric. It is the so-called Brans class I so-
lution , written with the explicitly separated
conformal factor 1/Φ in the metric; in the square
brackets in (14) we have the Fisher metric.4
Now, can the solution (14) describe wormholes?
To answer this question, we notice that the solution
is defined and is regular in the range 2k < x < ∞
and is asymptotically flat as x → ∞. It is thus
sufficient to check if the other end, x = 2k, can
be another flat infinity or a regular surface beyond
which the solution could be continued.
The quantity gtt= Pa−ξis finite and non-zero
at x = 2k (i.e., P = 0) only if a = ξ, i.e., in the
special solution in which gtt ≡ 1. On the other
hand, r2= x2P1−a−ξis infinite at x = 2k if a +
4The form (14) of the solution coincides with the one
used in  if we re-denote a − ξ = A, a + ξ = −B. The
substitution x = y[1 + k/(2y)]2converts it to the isotropic
coordinates employed, e.g., in [14,22], and the constants C,λ
and B used there are related to ours by C = 2ξ/(a−ξ), λ =
1/(a − ξ), B = k/2.
ξ > 1 and is finite there if a + ξ = 1. Thus if
a+ξ > 1, the surface area of the sphere x = const
has a minimum (i.e., there is a throat) at some
intermediate point x = x1> 2k. (Note that this
is a would-be wormhole configuration considered
in [14,15] in the case a ?= ξ.) However, in all such
cases we have either a naked singularity at x = 2k
or a repulsive non-flat asymptotic unattainable for
Indeed, one can check that, as x → 2k (P →
0), the Kretschmann invariant K = RµµρσRµµρσ
of the metric (14) behaves as P2(a+ξ−2). Hence
in case a + ξ < 2 we have a naked singularity at
x = 2k. This happens irrespective of a being equal
to ξ or not, even though gtt= const when a = ξ
(in which case ω < 0). Thus all such would-be
wormhole configurations have naked singularities.5
For a+ξ ≥ 2 we have a finite (for a+ξ = 2) or
zero limit of K at x = 2k. The range of ξ required
for that is
2 −√−6ω − 8
2ω + 4
− 3/2 < ω < −11/8,
2 −√−6ω − 8
2ω + 4
− 11/8 ≤ ω ≤ −4/3.
Thus the maximum possible value of ω in this case
is ω = −4/3, achieved at a = 1/2, ξ = 3/2. How-
ever, since a < 1 and therefore ξ > a, we inevitably
have gtt = Pa−ξ→ ∞, i.e., this asymptotic is
repulsive and inaccessible to timelike geodesics.
Moreover, the limit x → 2k is characterized by an
infinite proper distance along the radial direction
integral of the Riemann tensor components in an
orthonormal frame diverges as x → 2k, just as it
happens in the case of usual “strong” singularities,
which also indicates the absence of a globally reg-
ular behavior there (see also the discussion in 
in this connection). Summarizing, this configura-
tion cannot be called a wormhole according to our
definition. Though, one certainly cannot exclude
that its part containing a throat might be used for
obtaining a wormhole by, say, a cut-and-paste pro-
cedure with some asymptotically flat space-time.
≤ ξ ≤
√2ω + 3,
≤ ξ ≤2 +√−6ω − 8
2ω + 4
However, the iterated
5A similar situation occurs in Einstein gravity when one
attempts to construct a would-be wormhole solution sup-
ported by the phantom Chaplygin gas  or the phantom
generalized Chaplygin gas : a curvature singularity arises
at a finite value of the spherical radius r.
Let us note that throats appear at all values of
ω > −3/2, even very large ones. Indeed, suppose,
e.g., ξ = 2(1−a), so that a+ξ = 2−a > 1. Then
by (16) we have
2ω + 3 =
1 + a
4(1 − a),
which can be made arbitrarily large by taking a
close enough to unity. More generically, as follows
from Eq.(16), a + ξ > 1 if 0 < ξ < (ω + 2)−1.
But, as we have seen, in all such cases wormholes
are not obtained.
Of interest is the special case a = ξ = 1/2
ω = 0, the only case where the sphere x = 2k is
regular. The metric (14) then acquires the so-called
spatial Schwarzschild form
J= dt2− dx2/(1 − 2k/x) − x2dΩ2
= dt2− 4(2k + y2)dy2− (2k + y2)2dΩ2,(18)
where y2= x−2k. It is a wormhole metric, defined
in the range y ∈ R. The initial range x > 2k (in
which the manifold ME is defined) corresponds to
either y > 0 or y < 0. The Jordan-frame manifold
MJ consists of two regions y > 0 and y < 0, each
of them mapped into MEaccording to (3), plus the
regular sphere y = 0, the throat.
Thus the only wormhole solution in this family
exists for ω = 0 due to the conformal continuation
phenomenon [10,11]. The transition sphere corre-
sponds to Φ = 0; in the whole range of y we have
Φ = y/?y2+ 2k, so that at y < 0, beyond the
throat, the quantity f(Φ) = Φ in the Lagrangian
(1) is negative, which means that the effective grav-
itational constant Geff is negative there.
The wormhole nature of this solution for ω = 0
was pointed out in 1996 in , where its multidi-
mensional generalization was also indicated.
2.4General STT, ε = −1
If ε = −1 (so l(Φ) < 0 and ω < −3/2) there are
three branches in the vacuum solution (10)–(12) ac-
cording to the sign of k, see (9). They correspond
to Brans classes II-IV . Of interest for us is the
case k < 0 (see [10,27] for more complete descrip-
tions). Then, as is easily verified, the metric (10) in
ME [where now s(k,u) = k−1sinku] has two flat
spatial infinities at u = 0 and u = π/|k| (choos-
ing this half-wave of the sine function without loss
of generality). It is the anti-Fisher wormhole, re-
peatedly described beginning with the papers 
and . And it is also evident that the wormhole
nature of this solution is preserved in any STT (1)
in which the function f(Φ) is smooth and positive
in the range 0 ≤ u ≤ π/|k|. This fact was already
pointed out in 1973 . The BD theory is just a
special case of such a theory, see Eq.(13) where, as
before, φ = Cu. Let us note that the value ω = −2
is not distinguished in any way: wormhole solutions
do appear for any ω < −3/2, in the whole ghost
range of the BD theory.
For full clarity, let us write this BD solution in a
form in which its wormhole nature is more obvious.
Substituting |k|u = cot−1(x/|k|), we obtain
+ (k2+ x2)dΩ2],
1 + m2/k2= σ2|2ω + 3|,
where x ranges over the whole real axis.
u ∈ (0,π/|k|), the exponentials in (19) do not affect
the qualitative nature of the metric.
J= e−(2m+σ)udt2− e(2m−σ)u[dx2
σ := −C/(2|k|
|ω + 3/2|),
3 Wormholes in F(R) gravity
In the above examples, we only discussed massless
field configurations, with U(Φ) ≡ 0. However, the
theorem proved in  concerns the general case of
the theory (1), with any potentials U(Φ). It there-
fore encompasses not only STT but also the metric
F(R) theories of gravity which are known to be
equivalent to the BD theory with ω = 0 and po-
tentials U(Φ) whose form depends on the choice
of the function F(R) (see, e.g., the reviews [29,30]
and references therein). Indeed, the field equations
µ+ (∇µ∇ν− δν
µ= 0 (20)
(FR := dF/dR) obtained by variation of the La-
2F(R) + Lm
(where FRR := d2F/dR2is not identically zero)
with respect to the metric are equivalent to those
due to the Lagrangian (1) with
f(Φ) = FR,h(Φ) = 0,
2U(Φ) = RFR− F, (22)
which is nothing else but the BD theory with ω =
0, whose form (6) is obtained by introducing the
parametrization f(Φ) = Φ. It should be stressed
that the two equivalent forms of the theory corre-
spond to the same (Jordan) conformal frame and
employ the same metric gµν. Thus the condition
f(Φ) > 0 takes the form dF/dR > 0 in F(R) grav-
ity. The function l(Φ) (2) cannot become negative,
however, it can reach zero at exceptional values of
R where FRR= 0 while f = FR?= 0. In the latter
case, the parametrization f(Φ) = Φ is no longer
applicable since f(Φ) loses monotonicity.
Hence, according to , the only opportunity
of obtaining wormhole solutions in the theory (21),
both vacuum and supported by matter respecting
the NEC, is connected with conformal continua-
tions through a sphere at which F(R) has an ex-
tremum reached at some value R = R0. As is the
case with STT, such spheres, being regular in the
Jordan frame, are singular in the Einstein frame.
After crossing such a sphere, one gets  to a
region with negative values of Φ in the BD for-
mulation of the same theory, which in turn cor-
respond to negative values of the effective grav-
itational constant Geff.
that the wormhole behavior is generic among such
special solutions. However, similarly to the case
f(Φ0) = 0 in STT, in dynamic solutions there ap-
pears a generic spacelike anisotropic curvature sin-
gularity where the Kretschmann invariant diverges
as R → R0 [33,34].
The opportunity of FRR= 0 while FR?= 0 at
some R = R0is also of interest. At such surfaces, a
weak singularity can arise at which the space-time
metric and Riemann tensor components are finite
(see  for a discussion of its structure in the cos-
mological case), but R generically behaves there as
R0+ λ√u − u0, λ = const if u = u0is the surface
at which R = R0, where u is the time coordinate in
the cosmological setting or a spatial coordinate in
a static space-time. One can check that the terms
with derivatives of FR in Eq.(20) remain finite at
u = u0 in this case. Still differential curvature in-
variants are infinite at such a surface, and there
is no analytic extension across it. Solutions with
λ = 0 which are regular at R = R0 are not ex-
cluded but require strong fine tuning of initial or
boundary conditions. However, in all such cases,
according to , globally regular wormhole solu-
tions cannot appear as long as FR is everywhere
Possible wormhole geometries in F(R) gravity
have been discussed in [16,32]. In , only con-
It has been shown 
ditions near a throat were discussed; we can once
again recall that our results , based on conformal
mappings, do not rule out the existence of throats.
In  a few specific examples of wormhole solu-
tions in F(R) gravity, supported by matter with
anisotropic pressure and without a fully specified
equation of state, were found. However, in the first
two examples, Eqs.(25), (31) of that paper, FR is
negative over some range of R for matter satisfy-
ing the weak energy condition. In the other two
examples, Eqs.(35) and (39) of , either FR be-
comes zero at the throat, so that Geffis infinite and
becomes negative beyond it, or the radial pressure
and/or density of matter diverges there. It is once
again confirmed that static wormhole-like solutions
can formally exist in F(R) theory but look rather
unrealistic (cf. ).
4 Concluding remarks
One can ask a natural question: why, in rather a
simple subject, quite a number of incorrect infer-
ences appear in the literature. In our view, one
reason is purely terminological: some authors call
wormholes anything that has a throat. Actually
their analysis reduces to proving its existence only.
Meanwhile, to prove that there is a wormhole it is
sufficient to find two (not necessarily flat) regular
asymptotic regions with growing r(u) (or a simi-
lar function if spherical symmetry is absent), and
the existence of its minimum is then evident even
without explicitly pointing it out.
Another reason for wrong or incomplete infer-
ences is in some cases a very clumsy parametriza-
tion used in BD (and other) solutions. For instance,
with the parameters C and λ mentioned in foot-
note 4 it is impossible to consider the case a = ξ
in the solution (14) which is of particular interest.
And one more oddity of some wormhole studies
is a persistent usage of the curvature coordinate r,
which is certainly convenient for solving the field
equations in many (but not all) cases but is two-
valued in the throat region. It really looks funny
when a solution is first written in terms of an ad-
missible coordinate u but is then transformed to r
with much effort in order to seek a throat using the
so-called shape function b(r). Meanwhile, it is suf-
ficient just to find a minimum of r(u). Using r as a
coordinate, one frequently remains restricted to one
half of the configuration while another half, even if
its metric is the same, can have different properties
(as, e.g., in the solution (18) where Geff< 0 beyond
the throat). In this way, one also sometimes loses
wormhole solutions which are asymmetric with re-
spect to the throat. Though, with certain care,
asymmetric wormholes can certainly be described
in the r parametrization as well .
We hope that these remarks can be of some use
for researchers and especially students working in
AAS was partially supported by the RFBR grant
08-02-00923 and by the Scientific Programme “As-
tronomy” of the Russian Academy of Sciences.
KB and MS acknowledge partial support from the
RFBR grant 09-02-00677a and the NPK MU grant
of the Peoples’ Friendship University of Russia.
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