arXiv:gr-qc/0003092v1 22 Mar 2000
Toward a Traversable Wormhole
S. V. Krasnikov∗
February 4, 2008
Abstract
In this talk I discuss pertinence of the wormholes to the problem
of circumventing the light speed barrier and present a specific class
of wormholes. The wormholes of this class are static and have arbitrarily
wide throats, which makes them traversable. The matter
necessary for these spacetimes to be solutions of the Einstein equations
is shown to consist of two components, one of which satisfies
the Weak energy condition and the other is produced by vacuum fluctuations
of neutrino, electromagnetic (in dimensional regularization),
and/or massless scalar (conformally coupled) fields.
Wormholes and their application to hyper-fast
travel
Wormholes are geometrical structures connecting two more or less flat regions
of a spacetime. This of course is not a rigorous definition, but, strange
though it may seem, there is no commonly accepted rigorous definition of the
wormhole yet. Normally, however, by a wormhole a spacetime is understood
resembling that obtained by the following manipulation:
1. Two open balls are removed each from a piece of approximately flat
3-space (the vicinities of thus obtained holes we shall call mouths of
the wormhole);
Email: redish@pulkovo.spb.su
1
2. The boundaries (2-spheres) of the holes are glued together, and the
junction is smoothed. In the process of smoothing a kind of tube arises
interpolating the spheres. We shall call this tube the tunnel and its
narrowest part the throat.
The resulting object (its two-dimensional version to be precise) is depicted
in Fig. 1. If in the course of evolution the spacetime surrounding such an object
remains approximately flat (which may not be the case, since flatness of
each 3-dimensional section does not guarantee that the 4-dimensional space
formed by them is also flat) we shall call the object a wormhole. Wormholes
arise in a natural way in general relativity. Even one of the oldest and
best-studied solutions of the Einstein equations — the Schwarzschild spacetime
— contains a wormhole, which was found at least 80 years ago (Flamm,
1916). This wormhole (also known as the Einstein-Rosen bridge) connects
two asymptotically flat regions (‘two universes’), but being non-static is useless
in getting from one of them to the other (see below).
Depending on how the vicinities of the mouths are extended to the full
spacetime the wormholes fall into two categories (Visser, 1995): It may happen
that the mouths cannot be connected by any curve except those going
through the tunnel (as it takes place in the Einstein-Rosen bridge). Such
wormholes are called inter-universe. A simplest static spherically symmetric
inter-universe wormhole can be described (Morris, 1988) by a manifold R2×
S2 endowed with the metric
ds2 = −e2dt2 + dr2/(1 − b/r) + r2(d2 + sin2 d2), (1)
where r ∈ (−∞,∞) (note this possibility of negative r , it is the characteristic
feature of the wormholes), (r) → 0 and b(r)/r → 0, when r → ±∞.
Alternatively as shown in Fig. 1 it may happen that there are curves from
one mouth to another lying outside the wormhole. Such a wormhole connects
distant parts of a ‘single’ universe and is called intra-universe. Though intrauniverse
wormholes are in a sense more interesting most papers deal with
inter-universe ones, since they are simpler. It does not matter much, however.
The distant regions of the ‘universes’ are taken to be approximately flat.
And it is usually implied that given an inter-universe wormhole we can as
well build an intra-universe one by simply gluing these distant regions in an
appropriate way.
It is stable intra-universe wormholes that are often used for interstellar
travel in science fiction (even though they are sometimes called ‘black holes’
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Figure 1: The sketch of a wormhole with the mouths in motion. One dimension
(corresponding to the coordinate ) is omitted. The ways in which the
upper and the lower parts are glued at t = 0 and at t = 1 are depicted by
thin solid lines and by dashed lines respectively. Though the geometry of
the wormhole does not change, the distance (as measured in the outer, flat
space) between mouths increases with time.
there). Science fiction (especially Sagan’s novel Contact) apparently acted
back on science and in 1988 Morris and Thorne pioneered investigations
(Morris, 1988) of what they called traversable wormholes — wormholes that
can be (at least in principle) traversed by a human being. It is essential in
what follows that to be traversable a wormhole should satisfy at least the
following conditions:
(C1). It should be sufficiently stable. For example the Einstein-Rosen bridge
connects two asymptotically flat regions (and so it is a wormhole), but
it is not traversable — the throat collapses so fast that nothing (at
least nothing moving with v ≤ c) can pass through it.
(C2). It should be macroscopic. Wormholes are often discussed [see (Hochberg,
1997), for example] with the radius of the throat of order of the
Plank length. Such a wormhole might be observable (in particular,
owing to its gravitational field), but it is not obvious (and it is a long
way from being obvious, since the analysis would inevitably involve
quantum gravity) that any signal at all can be transmitted through its
tunnel. Anyway such a wormhole is impassable for a spaceship.
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Should a traversable wormhole be found it could be utilized in interstellar
travel in the most obvious way. Suppose a traveler (say, Ellie from the abovementioned
novel) wants to fly from the Earth to Vega. One could think
that the trip (there and back) will take at least 52 years (by the terrestrial
clocks) even if she moves at a nearly light speed. But if there is a wormhole
connecting the vicinities of the Earth and Vega she can take a short-cut by
flying through it and thus make the round trip to Vega in (almost) no time.
Note, however, that such a use of a wormhole would have had nothing to do
with circumventing the light barrier. Indeed, suppose that Ellie’s start to
Vega is appointed on a moment t = 0. Our concern is with the time interval
tE in which she will return to the Earth. Suppose that we know (from
astronomical observations, theoretical calculations, etc.) that if in t = 0 she
(instead of flying herself) just emit a photon from the Earth, this photon
after reaching Vega (and, say, reflecting from it) will return back at best in
a time interval tp. If we find a wormhole from the Earth to Vega, it would
only mean that tp actually is small, or in other words that Vega is actually
far closer to the Earth than we think now. But what can be done if tp
is large (one would hardly expect that traversable wormholes can be found
for any star we would like to fly to)? That is where the need in hyper-fast
transport comes from. In other words, the problem of circumventing the
light barrier (in connection with interstellar travel) lies in the question: how
to reach a remote (i. e. with the large tp) star and to return back sooner
than a photon would have made it (i. e. in tE < tp)? It makes sense to
call a spaceship faster-than-light (or hyper-fast) if it solves this prolem. A
possible way of creating hyperfast transport lies also in the use of traversable
wormholes (Krasnikov, 1998). Suppose that a traveler finds (or builds) a
traversable wormhole with both mouths located near the Earth and suppose
that she can move the mouths (see Fig.1) at will without serious damage
to the geometry of the tunnel (which we take to be negligibly short). Then
she can fly to Vega taking one of the mouths with her. Moving (almost) at
the speed of light she will reach Vega (almost) instantaneously by her clocks.
In doing so she rests with respect to the Earth insofar as the distance is
measured through the wormhole. Therefore her clocks remain synchronous
with those on the Earth as far as this fact is checked by experiments confined
to the wormhole. So, if she return through the wormhole she will arrive back
to the Earth almost immediately after she will have left it (with tE a‰? tp).
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Remark 1. The above arguments are very close to those showing that a
wormhole can be transformed into a time machine (Morris, 1988), which is
quite natural since the described procedure is in fact the first stage of such
transformation. For, suppose that we move the mouth back to the Earth
reducing thus the distance between the mouths (in the ambient space) by
26 light years. Accordingly tE would lessen by ≈26 yr and (being initially
very small) would turn negative. The wormhole thus would enable a traveller
to return before he have started. Fortunately, tE ≈ 0 would fit us and we
need not consider the complications (possible quantum instability, paradoxes,
etc.) connected with the emergence of thus appearing time machine.
Remark 2. Actually two different worlds were involved in our consideration.
The geometry of the world where only a photon was emitted differs from
that of the world where the wormhole mouth was moved. A photon emitted
in t = 0 in the latter case would return in some tp′ < tE . Thus what
makes the wormhole-based transport hyper-fast is changing (in the causal
way) the geometry of the world so that to make tp′ < tE a‰? tp.
Thus we have seen that a traversable wormhole can possibly be used as a
means of ‘superluminal’ communication. True, a number of serious problems
must be solved before. First of all, where to get a wormhole? At the moment
no good recipe is known how to make a new wormhole. So it is worthwhile
to look for ‘relic’ wormholes born simultaneously with the Universe. Note
that though we are not used to wormholes and we do not meet them in our
everyday life this does not mean by itself that they are an exotic rarity in
nature (and much less that they do not exist at all). At present there are
no observational limits on their abundance [see (Anchordoqui, 1999) though]
and so it well may be that there are 10 (or, say, 106) times as many wormholes
as stars. However, so far we have not observed any. So, this issue remains
open and all we can do for the present is to find out whether or not wormholes
are allowed by known physics.
Can traversable wormholes exist?
Evolution of the spacetime geometry (and in particular evolution of a wormhole)
in general relativity is determined via the Einstein equations by properties
of the matter filling the spacetime. This circumstance may turn out
to be fatal for wormholes if the requirements imposed on the matter by con-
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ditions (C1,C2) are unrealistic or conflicting. That the problem is grave
became clear from the very beginning: it was shown (Morris, 1988), see also
(Friedman, 1993), that under very general assumptions the matter filling a
wormhole must violate the Weak Energy Condition (WEC). The WEC is the
requirement that the energy density of the matter be positive in any reference
system. For a diagonal stress-energy tensor Tik the WEC may be written as
WEC : T00 ≥ 0, T00 + Tii ≥ 0, i = 1, 2, 3 (2)
Classical matter always satisfies theWEC (hence the name ‘exotic’ for matter
violating it). So, a wormhole can be traversable only if it is stabilized by some
quantum effects. Candidate effects are known, indeed [quantum effects can
violate any local energy condition (Epstein, 1965)]. Moreover, owing to the
non-trivial topology a wormhole is just a place where one would expect WEC
violations due to fluctuations of quantum fields (Khatsymovsky, 1997a). So,
the idea appeared (Sushkov, 1992) to seek a wormhole with such a geometry
that the stress-energy tensor produced by vacuum polarization is exactly the
one necessary for maintaining the wormhole. An example of such a wormhole
(it is a Morris-Thorne spacetime filled with the scalar non-minimally coupled
field) was offered in (Hochberg, 1997). Unfortunately, the diameter of the
wormhole’s throat was found to be of the Plank scale, that is the wormhole
is non-traversable. The situation considered in (Hochberg, 1997) is of course
very special (a specific type of wormholes, a specific field, etc.). However
arguments were cited [based on the analysis of another energetic condition,
the so called ANEC (Averaged Null Energy Condition)] suggesting that the
same is true in the general case as well (Flanagan, 1996, see also the literature
cited there). So an impression has been formed that conditions (C1) and (C2)
are incompatible, and TWs are thus impossible.
Yes, it seems they can
The question we are interested in is whether such macroscopic wormholes
exist that they can be maintained by the exotic matter produced by the
quantum effects. To put it more mathematically let us first separate out the
contribution TQ
ik of the ‘zero-point energy’ to the total stress-energy tensor:
Tik = TQ
ik + TC
ik . (3)
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In semiclassical gravity it is deemed that for a field in a quantum state | i
(in particular, | i may be a vacuum state) TQ
ik = h |Tik| i, where Tik is
the corresponding operator, and there are recipes for finding TQ
ik for given
field, metric, and quantum state [see, for example, (Birrel, 1982)]. So, in
formula (3) TQ
ik and Tik are determined by the geometry of the wormhole and
the question can be reformulated as follows: do such macroscopic wormholes
exist that the term TC
ik describes usual non-exotic matter, or in other words
that TC
ik satisfies the Weak Energy Condition, which now can be written as
G00 − 8TQ
00 ≥ 0, (G00 + Gii) − 8(TQ
00 + TQ
ii ) ≥ 0, i = 1, 2, 3. (4)
(we used the formulas (2,3) here)? One of the main problem in the search
for the answer is that the relevant mathematics is complicated and unwieldy.
A possible way to obviate this impediment is to calculate TQ
ik numerically
(Hochberg, 1997; Taylor 1997) using some approximation. However, the
correctness of this approximation is in doubt (Khatsymovsky, 1997b), so we
shall not follow this path. Instead we shall study a wormhole with such a
metric that relevant expressions take the form simple enough to allow the
analytical treatment.
The Morris-Thorne wormhole is not the unique static spherically symmetric
wormhole (contrary to what can often be met in the literature). Consider
a spacetime R2× S2 with the metric:
ds2 =
2()[−d 2 + d2 + K2()(d2 + sin2 d2)], (5)
where
and K are smooth positive even functions, K = K0 cos /L at
∈ (−L, L), K0 ≡ K(0) and K is constant at large . The spacetime is
obviously spherically symmetric and static. To see that it has to do with
wormholes consider the case
∼
0 exp{Bx}, at large . (6)
The coordinate transformation
r ≡ B−1
0 expB, t ≡ Br, (7)
then brings the metric (5) in the region t < r into the form:
ds2 = −dt2 + 2t/r dtdr + [1 − (t/r)2]dr2 + (BK0r)2(d2 + sin2 d2). (8)
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It is obvious from (7) that as r grows the metric (5,8) becomes increasingly
flat (the gravitational forces corresponding to it fall as 1/r) in a layer |t| < T
(T is an arbitrary constant). This layer forms a neighborhood of the surface
= t = 0. But the spacetime is static (the metric does not depend on ). So,
the same is true for a vicinity of any surface = const . The spacetime can
be foliated into such surfaces. So this property (increasing flatness) holds in
the whole spacetime, which means that it is a wormhole, indeed. Its length
(the distance between mouths as measured through the tunnel) is of order of
0L and the radius of its throat R0 = min(
K).
The advantage of the metric (5) is that for the electro-magnetic, neutrino,
and massless conformally coupled scalar fields TQ
ik can be readily found (Page,
1982) in terms of
,K, and their derivatives [actually the expression contains
also one unknown term (the value of TQ
ik for
= 1), but the more detailed
analysis shows that for sufficiently large
this term can be neglected]. So, by
using this expression, calculating the Einstein tensor Gik for the metric (5)
and substituting the results into the system (4) we can recast it [the relevant
calculations are too laborious to be cited here (the use of the software package
GRtensorII can lighten the work significantly though)] into the form:
Ei ≥ 0 i = 0, 1, 2, 3, (9)
where Ei are some (quite complex, e. g. E0 contains 40 terms; fortunately
they are not all equally important) expressions containing
, K, and their
derivatives and depending on what field we consider. Thus if we restrict
ourselves to wormholes (5), then to answer the question formulated above all
we need is to find out whether such
exist that it
i). has appropriate asymptotic behavior [see (6)],
ii). satisfies (9) for some field,
iii). delivers sufficiently large R0.
It turns out (Krasnikov, 1999) that for all three fields listed above and for
arbitrarily large R0 such
do exist (an example is sketched in Fig. 2) and
so the answer is positive.
Acknowledgments
I am grateful to Prof. Grib for stimulating my studies in this field and to
Dr. Zapatrin for useful discussion.
8
W
x
L
W0
Figure 2: A conformal factor
satisfying requirements (i) — (iii).
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9
(1), 
and 
are integration constants which are determined by appropriate boundary conditions and 
… 
are Cartesian coordinates (
= space, 
(2) 
) about the direction of the field (G = universal gravitation constant, c = speed of light, B = magnetic field intensity in Tesla, m 0 = vacuum permeability - all in mks units). From the coefficient of 
in (1), Maccone derived the "speed of light function" which gives the gravitationally induced variation of light speed within the curvature region: 
(3). 
), this becomes: 
(4), 
where 
. Equation (4) is based on the assumption that the magnetic field is created by a solenoid of length L oriented along the z-axis, and that c = 3x108 m/sec at the solenoid’s ends (z = ± L/2), while at z = 0, c slows down according to (4) because of the presence of the artificially induced spacetime curvature. Further, Maccone inverted equation (4) and solved for B to get: 
(5). 
(7), 
, is recognized as the three-metric of a hypercylinder S2 x  . So equation (7) shows that Levi-Civita’s spacetime metric is simply a hypercylinder with a position dependent gravitational potential: no asymptotically flat region, no flared-out wormhole mouth and no wormhole throat. Maccone’s equations for the radial (hyperbolic) pressure, stress and energy density of the "magnetic wormhole" configuration are thus incorrect. 
