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’
2
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.
3
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).
4
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-
5
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)
6
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)
7
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).
References
Anchordoqui A., Romero, G. E., Torres, D. F., and Andruchow, I., Mod.
Phys. Lett. 14, 791 (1999)
Birrell, N. D., and Davies, P. C. W., Quantum fields in curved spacetime,
Cambridge, Cambridge University Press, 1982.
Epstein, H., Glaser, V., and Jaffe, A., Nuovo Cimento 36, 1016 (1965).
Flamm L., Physikalische Zeitschrift 17, 448 (1916)
Flanagan, E. E., and Wald, R. M., Phys. Rev. D 54, 6233 (1996).
Friedman, J. L., Schleich, K., and Witt, D. M., Phys. Rev. Lett. 71, 1486
(1993).
Hochberg, D., Popov, A., and Sushkov, S., Phys. Rev. Lett. 78, 2050 (1997).
Krasnikov, S., Phys. Rev. D 57, 4760 (1998).
Krasnikov, S., Eprint gr-qc 9909016.
Khatsymovsky, V., Phys. Lett. B 399, 215 (1997a).
Khatsymovsky, V., in Proceedings of the II Int. Conference on QFT and
Gravity, TGPU Publishing, Tomsk, 1997b.
Morris, M. S. and Thorne, K. S., Am. J. Phys. 56, 395 (1988).
Page, D. N., Phys. Rev. D 25, 1499 (1982).
Sushkov, S. V., Phys. Lett. A 164, 33 (1992).
Taylor, B. E., William A. Hiscock W. A., and Anderson P. R., Phys. Rev. D
55, 6116 (1997)
Visser, M., Lorentzian wormholes — from Einstein to Hawking, New York,
AIP Press, 1995.
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Wormhole Induction Propulsion (WHIP)

Eric W. Davis, Ph.D.

National Institute for Discovery Science
1515 E. Tropicana Ave., Suite 400
Las Vegas, Nevada 89119

 

ABSTRACT

Space flight by means of wormholes is described whereby the traditional rocket propulsion approach can be abandoned in favor of a new paradigm involving the manipulation of spacetime. Maccone (1995) extended Levi-Civita’s 1917 magnetic gravity solution to the Morris and Thorne (1988) wormhole solution and claimed that static homogeneous magnetic/electric fields can create spacetime curvature manifesting itself as a traversable wormhole. Furthermore, Maccone showed that the speed of light through this curvature region is slowed by the magnetic (or electric) induced gravitational field there. Maccone’s analysis immediately suggests a way to perform laboratory experiments whereby one could apply a powerful static homogeneous magnetic field in a vacuum, thereby creating spacetime curvature, and measure the speed of a light beam through it. Magnetic fields employed in this scenario must achieve magnitudes > 10¹⁰ Tesla in order for measurable effects to appear. Current magnetic induction technology is limited to static fields of ~ several x 10³ Tesla. However, destructive chemical (implosive/explosive) magnetic field generation technology has reached peak rate-of-rise field strengths of ~ 10⁹ Tesla/sec. It is proposed that this technology be exploited to take advantage of the high rate-of-rise field strengths to create and measure spacetime curvature in the lab.

INTRODUCTION

Rapid interplanetary and interstellar space flight by means of spacetime wormholes is possible, in principle, whereby the traditional rocket propulsion approach can be abandoned in favor of a new paradigm involving the use of spacetime manipulation. In this scheme, the light speed barrier becomes irrelevant and spacecraft no longer need to carry large mass fractions of traditional chemical or nuclear propellants and related infrastructure over distances larger than several astronomical units (AU). Travel time over very large distances will be reduced by orders of magnitude. Einstein published his General Theory of Relativity (GTR) in 1915. In 1917, physicist Tullio Levi-Civita read a paper before the Academy of Rome about creating artificial gravitational fields (spacetime curvature) by virtue of static homogeneous magnetic or electric fields as a solution to the GTR equations. This paper went largely unnoticed. In 1988, Morris and Thorne published an exact solution to the GTR equations which describe the creation of traversable wormholes in spacetime by virtue of exotic (mass-energy r c² < stress-energy t ) matter-energy fields (see figures 1 and 2). Visser (1995) has extended and added to the knowledge base of this research. The essential features of these solutions are that wormholes possess a traversable throat in which there is no horizon or singularity. For the purpose of this study, we also impose the additional constraint that travel through the wormhole is causal, although, this is not a necessary constraint in general. When these properties are employed together with the GTR field equations, it becomes necessary to introduce an exotic material in the wormhole’s throat which generates its spacetime curvature.

Maccone (1995) extended and matched Levi-Civita’s solution to the Morris and Thorne solution and claimed that the earlier describes a wormhole in spacetime. More specifically, Maccone claims that static homogeneous magnetic/electric fields with cylindrical symmetry can create spacetime curvature which manifests itself as a traversable wormhole. Although the claim of inducing spacetime curvature is correct, Levi-Civita’s metric solution is not a wormhole. A near-term lab experiment based on Maccone’s analysis will be discussed. It is my intent to introduce a new space propulsion concept which employs the creation of traversable wormholes by virtue of ultrahigh magnetic fields in conjunction with exotic matter-energy fields. I call this propulsion concept "Wormhole Induction Propulsion" or WHIP. It is speculated that future WHIP spacecraft could deploy ultrahigh magnetic fields along with exotic matter- energy fields (e.g. radial electric or magnetic fields, Casimir energy field, etc.) in space to create a wormhole and then apply conventional space propulsion to move through the throat to reach the other side in a matter of minutes or days, whence the spacecraft emerges several AU’s or light-years away from its starting point. The requirement for conventional propulsion in WHIP spacecraft would be strictly limited by the need for short travel through the wormhole throat as well as for orbital maneuvering near distant worlds. The integrated system comprising the magnetic induction/exotic field wormhole and conventional propulsion units could be called WHIPIT or "Wormhole Induction Propulsion Integrated Technology."

THEORETICAL BRIEF

Levi-Civita’s spacetime metric for a static uniform magnetic field was originally conceived by Pauli (1981):

 (1),

where and are integration constants which are determined by appropriate boundary conditions and … are Cartesian coordinates (… = space,  = time) with orthographic projection. The important parameter in (1) is:

 (2)

which measures the radius of spacetime curvature induced by a homogeneous magnetic field with cylindrical symmetry (axis, ) 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).

At the center of this region (), this becomes:

(4),

for 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).

Equations (2), (4) and (5) are formulae to use for creating and detecting spacetime curvature in the lab.

TECHNICAL ISSUES

Traversable wormholes are creatures of classical GTR and represent non-trivial topology change in the spacetime manifold. This makes mathematicians cringe because it raises the question of whether topology can change or fluctuate to accommodate wormhole creation. Black holes and naked singularities are also creatures of GTR representing non-trivial topology change in spacetime, yet they are accepted by the astrophysics and mathematical communities — the former by Hubble Space Telescope discoveries and the latter by theoretical arguments due to Kip Thorne, Stephen Hawking, Roger Penrose and others. The Bohm-Aharonov effect is another example which owes its existence to non-trivial topology change in the manifold. The topology change (censorship) theorems discussed in Visser (1995) make precise mathematical statements about the "mathematician’s topology" (topology of spacetime is fixed!), however, Visser correctly points out that this is a mathematical abstraction. In fact, Visser (1990) proved that the existence of an everywhere Lorentzian metric in spacetime is not a sufficient condition to prevent topology change. Furthermore, Visser (1990, 1995) elaborates that physical probes are not sensitive to this mathematical abstraction, but instead they typically couple to the geometrical features of space. Visser (1990) also showed that it is possible for geometrical effects to mimic the effects of topology change. Topology is too limited a tool to accurately characterize a generic traversable wormhole; in general one needs geometric information to detect the presence of a wormhole, or more precisely to locate the wormhole throat (Visser, private communication, 1997).

Landis (1997) has made technical criticisms of Maccone’s (1995) work suggesting that the Levi-Civita metric in the presence of a uniform magnetic field does not form a wormhole within the Morris and Thorne (1988) framework. While the latter view is correct, the technical arguments are not accurate or complete. Changing the coordinate system from Cartesian to cylindrical (x¹ = rcosj , x² = rsinj , x³ = z, let x⁴ = t) puts equation (1) into the form (Maccone, 1995):

(6).

This is a cleaner form, but what is the Levi-Civita metric really? We can find out from making a change of (radial) variable by letting r = asinq , dr = acosq dq and substituting these into equation (6):

 (7),

where a is the constant radius defined by equation (2). The spatial part of (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.

In addition, directing attention on the behavior of wormhole geometry at asymptotic infinity is not too profitable. Visser (private communication, 1997; Hochberg and Visser, 1997) demonstrates that it is only the behavior near the wormhole throat that is critical to understanding what is going on, and that a generic throat can be defined without having to make all the symmetry assumptions and without assuming the existence of an asymptotically flat spacetime to embed the wormhole in. One only needs to know the generic features of the geometry near the throat in order to guarantee violations of the null energy condition (NEC; see Hawking and Ellis, 1973) for certain open regions near the throat (Visser, private communication, 1997). There are general theorems of differential geometry that guarantee that there must be NEC violations (meaning exotic matter-energy is present) at a wormhole throat. In view of this, however, it is known that static radial electric or magnetic fields are borderline exotic when threading a wormhole if their tension were infinitesimally larger, for a given energy density (Herrmann, 1989; Hawking and Ellis, 1973). Other exotic (energy condition violating) matter-energy fields are known to be squeezed states of the electromagnetic field, Casimir (electromagnetic zero-point) energy and other quantum fields/states/effects. With respect to creating wormholes, these have the unfortunate reputation of alarming physicists. This is unfounded since all the energy condition hypotheses have been experimentally tested in the laboratory and experimentally shown to be false — 25 years before their formulation (Visser, 1990 and references cited therein). Violating the energy conditions commits no offense against nature.

EXPERIMENTAL APPROACH

Table I below shows the radius of curvature generated by a range of magnetic field strengths via equation (2). Equations (2), (4) and (5) suggest a way to perform a laboratory experiment whereby one could apply a powerful static homogeneous (cylindrically symmetric) magnetic field in a vacuum, thereby creating spacetime curvature in principle, and measure the speed of a light beam through it. A measurable slowing of c in this arrangement would demonstrate that a curvature effect has been created in the experiment. The achievable precision in measuring this

Table I. Radius of Spacetime Curvature Induced by B-Field

 

B ( x 3.484 Tesla)	a (meters)
1	1018 (105.7 ly)
102	1016 (1.06 ly)
103	1015 (0.11 ly)
105	1013 (66.7 AU)
107	1011 (0.67 AU)
109	109 (1.44 Solar Radii)
1012	106 (0.16 Earth Radii)
1015	103
1018	1

 

ly = light-year, AU = Astronomical Unit

would be c - v(0) or c² - v²(0) as seen from equation (5). Electric fields could also be used to create the same effect, however, the field strengths required to accomplish the same radius of curvature or slowing of c is seventeen times larger than magnetic field strengths (Maccone, 1995).

From Table I, it is apparent that laboratory magnetic field strengths would need to be > 10⁹ - 10¹⁰ Tesla so that a significant radius of curvature and slowing of c can be measured. Experiments employing chemical explosive/implosive magnetic technologies would be an ideal arrangement for this. The limit of magnetic field generation for chemical explosives/implosives is ~ several x 10³ Tesla and the quantum limit for ordinary metals is ~ 50,000 Tesla. Explosion/implosion work done by Russian (MC-1 generator, ISTC grant), Los Alamos National Lab (ATLAS), National High Magnetic Field Lab and Sandia National Lab (SATURN) investigators have employed magnetic solenoids of good homogeneity with lengths of ~ 10 cm, having peak rate-of-rise of field of ~ 10⁹ Tesla/sec where a few nanoseconds is spent at 1000 Tesla, and which is long enough for a good measurement of c (J. Solem, private communication, 1997). Further, with picosecond pulses, c could be measured to a part in 10² or 10³. At 1000 Tesla, c² - v²(0) » 0 m²/sec² and the radius of curvature is 0.368 light-years. If the peak rate-of-rise of field (~ 10⁹ Tesla/sec) can be used, then a radius of curvature £ several x 10⁶ km can be generated along with c² - v²(0) ³ several x 10⁴ m²/sec².

It will be necessary to consider advancing the state-of-art of magnetic induction technologies in order to reach static field strengths that are > 10⁹ - 10¹⁰ Tesla. Extremely sensitive measurements of c at the one part in 10⁶ or 10⁷ level may be necessary for laboratory experiments involving field strengths of ~ 10⁹ Tesla. Magnetic induction technologies based on nuclear explosives/implosives may need to be seriously considered in order to achieve large magnitude results. An order of magnitude calculation indicates that magnetic fields generated by nuclear pulsed energy methods could be magnified to (brief) static values of ³ 10⁹ Tesla by factors of the nuclear-to-chemical binding energy ratio (³ 10⁶). Other experimental methods employing CW lasers, repetitive-pulse free electron lasers, neutron beam-pumped UO₂ lasers, pulsed laser-plasma interactions or pulsed hot (theta pinch) plasmas either generate insufficient magnetic field strengths for our purposes or cannot generate them at all within their operating modes (see also Table II).

Table II. Current High and Ultrahigh Magnetic Field Generation Technologies

Magnetic Field Strength (Tesla)	Field Generation Technology
10 - 300	Superconductivity, Hybrid Magnets, Pulsed Magnetsa
360	Magnetic flux compression by electromagnetic forcea
400	One-turn coil connected to strong laser produced plasmaa
~ 103	High powered pulsed lasersa
1000 - 3000	Magnetic flux compression by chemical explosionb
102 - 105	White Dwarf starsc
107 - 109	Neutron starsc
³ 109	Magnetic flux compression by nuclear explosiona

^a D. Judd, private communication, 1997

^b J. Solem, private communication, 1997

^c S. Stephens, private communication, 1995

WHIP SPACECRAFT CONCEPT

WHIP spacecraft will have multifunction integrated technology for propulsion. The Wormhole Induction Propulsion Integrated Technology (WHIPIT) would entail two modes. The first mode is an advanced conventional system (chemical, nuclear fission/fusion, ion/plasma, antimatter, etc.) which would provide propulsion through the wormhole throat, orbital maneuvering capability near stellar or planetary bodies, and spacecraft attitude control and orbit corrections. An important system driver affecting mission performance and cost is the overall propellant mass-fraction required for this mode. A desirable constraint limiting this to acceptable (low) levels should be that an advanced conventional system would regenerate its onboard fuel supply internally or that it obtain and process its fuel supply from the situ space environment. Other important constraints and/or performance requirements to consider for this propulsion mode would include specific impulse, thrust, energy conversion schemes, etc. Further discussion of these is beyond the scope of this paper and is left for the reader to explore on their own.

The second WHIPIT mode is the stardrive component. This would provide the necessary propulsion to rapidly move the spacecraft over interplanetary or interstellar distances through a traversable wormhole. The system would generate a static, cylindrically symmetric ultrahigh magnetic field to create a hypercylinder curvature envelope (gravity well) near the spacecraft to pre-stress space into a pseudo-wormhole configuration. The radius of the hypercylinder envelope should be no smaller than the largest linear dimension of the spacecraft. As the spacecraft is gravitated into the envelope, the field-generator system then changes the cylindrical magnetic field into a radial configuration while giving it a tension that is greater than its energy density. A traversable wormhole throat is then induced near the spacecraft where the hypercylinder and throat geometries are patched together (see figure 3). The conventional propulsion mode then kicks on to nudge the spacecraft through the throat and send its occupants on their way to adventure. This scenario would apply if ultrahigh electric fields were employed instead. If optimization of wormhole throat (geometry) creation and hyperspace tunneling distance requires a fully exotic energy field to thread the throat, then the propulsion system would need to be capable of generating and deploying a Casimir (or other exotic) energy field. Although ultrahigh magnetic/electric and exotic field generation schemes are speculative, further discussion is beyond the scope of this paper and will be left for future work. A hypothetical WHIP spacecraft concept is depicted in Figure 4.

CONCLUSIONS

A candidate for breakthrough propulsion physics has been identified in the form of a traversable wormhole created by virtue of ultrahigh magnetic or electric fields with an additional exotic energy component. Maccone (1995) claimed that cylindrically symmetric ultrahigh magnetic (electric) fields can create a traversable wormhole in the Morris and Thorne (1988) framework. It has been shown that this is incorrect. Instead, a hypercylinder curvature effect having a position dependent gravitational potential is induced. This effect can be used to create a wormhole by patching the hypercylinder envelope to a throat that is induced by either radially stressing the ultrahigh field or employing additional exotic energy. Maccone correctly showed that the speed of light through the hypercylinder region is slowed by the magnetic induced gravitational field there. This suggests a way to perform laboratory experiments whereby one could apply an ultrahigh magnetic field in a vacuum, thereby creating a hypercylinder curvature effect, and measure the speed of a light beam through it. While chemical explosive/implosive magnetic induction technology has achieved static field strengths of ~ several x 10³ Tesla, the peak rate-of-rise of field is ~ 10⁹ Tesla/sec. Field strengths > 10⁹ - 10¹⁰ Tesla would need to be generated to impart a measurable slowing of light speed in this scenario. It is proposed that the peak rate-of-rise of field be exploited as a means to achieve this goal in the near-term. Magnetic induction technologies based on nuclear explosives/implosives may need to be considered in order to achieve results of larger magnitude. A Wormhole Induction Propulsion system has been introduced to exploit the possibilities of traversable wormholes.

ACKNOWLEDGEMENTS

I wish to thank Marc Millis for allowing me to use WHIP and WHIPIT which he coined at the February, 1997 NASA Breakthrough Propulsion Physics Regional Brainstorming Workshop at Austin, TX. My gratitude to Matt Visser for his many valuable suggestions and comments on this work. I also thank Dean Judd, Johndale Solem, George Hathaway and John Alexander for their technical contributions and remarks. This research is partially supported by the National Institute for Discovery Science. ( Figure 4: Hypothetical WHIP spacecraft concept. / Figure 1: Embedded space representation of a Morris and Thorne (1988) traversable wormhole.
 

Figure 3: Hypothetical view of two wormhole mouths patched to a hypercylinder curvature envelope. The small (large) configuration results from the radius of curvature induced by a larger (smaller) ultrahigh magnetic field.

Figure 2: What a wormhole mouth might look like to space travelers.

REFERENCES

Maccone, C. (1995) "Interstellar Travel Through Magnetic Wormholes", JBIS, Vol. 48, No. 11, pp. 453-458.

Levi-Civita, T. (1917) "Realtà fisica di alcuni spazi normali del Bianchi", Rendiconti della Reale Accademia dei Lincei, Series 5, Vol. 26, pp. 519-533.

Morris, M. and Thorne, K. (1988) "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", Am. J. Phys., Vol. 56, No. 5, pp. 395-412.

Visser, M. (1995) Lorentzian Wormholes - From Einstein to Hawking, AIP Press, Woodbury, NY.

Pauli, W. (1981) Theory of Relativity, Dover reprint, New York, pp. 171-172.

Visser, M. (1990) "Wormholes, baby universes, and causality", Phys. Rev. D, Vol. 41, No. 4, pp. 1116-1124.

Landis, G. (1997) "Magnetic Wormholes And The Levi-Civita Solution To The Einstein Equation", JBIS, Vol. 50, No. 4, pp. 155-157.

Hochberg, D. and Visser, M. (1997) "Geometric structure of the generic static traversable wormhole throat", LANL Abstract gr-qc/9704082, to appear in Phys. Rev. D.

Hawking, S. W. and Ellis, G. F. R. (1973) The Large-Scale Structure of Space-Time, Cambridge Univ. Press, Cambridge, pp. 88-91 and pp. 95-96.

Herrmann, F. (1989) "Energy density and stress: A new approach to teaching electromagnetism", Am. J. Phys., Vol. 57, No. 8, pp. 707-714.

Prepared for the Proceedings of the NASA Breakthrough Propulsion Physics Workshop, NASA Lewis Research Center, Cleveland, Ohio (Aug. 12-14, 1997). To appear in the NASA Proceedings, 1998.

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INTERSTELLAR TRAVEL BY MEANS OF WORMHOLE INDUCTION PROPULSION (WHIP) Eric W. DavisNational Institute for Discovery Science1515 E. Tropicana Ave., Suite 400Las Vegas, Nevada(702) 798-1700, FAX (702) 798-1970, E-mail: nids@anv.netAbstractSpace flight by means of wormholes is described whereby the traditional rocket propulsion approach can be abandoned in favor of a new paradigm involving the manipulation of spacetime. Maccone (1995) extended Levi-Civita’s 1917 magnetic gravity solution to the Morris and Thorne (1988) wormhole solution and claimed that static homogeneous magnetic/electric fields can create spacetime curvature manifesting itself as a traversable wormhole. Furthermore, Maccone showed that the speed of light through this curvature region is slowed by the magnetic (or electric) induced gravitational field there. Maccone’s analysis immediately suggests a way to perform laboratory experiments whereby one could apply a powerful static homogeneous magnetic field in a vacuum, thereby creating spacetime curvature, and measure the speed of a light beam through it. Magnetic fields employed in this scenario must achieve magnitudes > 1010Tesla in order for measurable effects to appear. Current magnetic induction technology is limited to static fields of several x 103Tesla. However, destructive chemical (implosive/explosive) magnetic field generation technology has reached peak rate-of-rise field strengths of 109Tesla/s. It is proposed that this technology be exploited to take advantage of the high rate-of-rise field strengths to create and measure spacetime curvature in the lab.INTRODUCTIONRapid interplanetary and interstellar space flight by means of spacetime wormholes is possible, in principle, whereby the traditional rocket propulsion approach can be abandoned in favor of a new paradigm involving the use of spacetime manipulation. In this scheme, the light speed barrier becomes irrelevant and spacecraft no longer need to carry large mass fractions of traditional chemical or nuclear propellants and related infrastructure over distances larger than several astronomical units (AU). Travel time over very large distances will be reduced by orders of magnitude. Einstein published his General Theory of Relativity (GTR) in 1915. In 1917, physicist Tullio Levi-Civita read a paper before the Academy of Rome about creating artificial gravitational fields (spacetime curvature) by virtue of static homogeneous magnetic or electric fields as a solution to the GTR equations. This paper went largely unnoticed. In 1988, Morris and Thorne published an exact solution to the GTR equations which describe the creation of traversable wormholes in spacetime by virtue of exotic (mass-energy ρc2< stress-energy τ) matter-energy fields. Visser (1995) has extended and added to the knowledge base of this research. The essential features of these solutions are that wormholes possess a traversable throat in which there is no horizon or singularity. For the purpose of this study, we also impose the additional constraint that travel through the wormhole is causal, although, this is not a necessary constraint in general. When these properties are employed together with the GTR field equations, it becomes necessary to introduce an exotic material in the wormhole’s throat which generates its spacetime curvature.Maccone (1995) extended and matched Levi-Civita’s solution to the Morris and Thorne solution and claimed that the earlier describes a wormhole in spacetime. More specifically, Maccone claims that static homogeneous magnetic/electric fields with cylindrical symmetry can create spacetime curvature which manifests itself as atraversable wormhole. Although the claim of inducing spacetime curvature is correct, Levi-Civita’s metric solution is not a wormhole. A near-term lab experiment based on Maccone’s analysis will be discussed. It is my intent to introduce a new space propulsion concept which employs the creation of traversable wormholes by virtue of ultrahigh magnetic fields in conjunction with exotic matter-energy fields. I call this propulsion concept “Wormhole Induction Propulsion” or WHIP. It is speculated that future WHIP spacecraft could deploy ultrahigh magnetic fields along with exotic matter- energy fields (e.g. radial electric or magnetic fields, Casimir energy field, etc.) in space to create a wormhole and then apply conventional space propulsion to move through the throat to reach the other side
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2in a matter of minutes or days, whence the spacecraft emerges several AU’s or light-years away from its starting point. The requirement for conventional propulsion in WHIP spacecraft would be strictly limited by the need for short travel through the wormhole throat as well as for orbital maneuvering near distant worlds. The integrated system comprising the magnetic induction/exotic field wormhole and conventional propulsion units could be called WHIPIT or “Wormhole Induction Propulsion Integrated Technology.”THEORETICAL BRIEFLevi-Civita’s spacetime metric for a static uniform magnetic field was originally conceived by Pauli (1981):[]( )( )[]dsdxdxdxccdxx dxx dxaxxxaxa21 22 23 21224 21122 221 22 233=+++−++−+−()()()expexp()()( )(), (1)where c1and c2are integration constants which are determined by appropriate boundary conditions and x1…x4are Cartesian coordinates (x1…x3= space, x4= time) with orthographic projection. The important parameter in (1) is:( )a cBxBG==−−+−2 4118101234840 10πµ.,(2)which measures the radius of spacetime curvature induced by a homogeneous magnetic field with cylindrical symmetry (axis, xz3=) about the direction of the field. From the coefficient of dx4in (1), Maccone derived the “speed of light function” which gives the gravitationally induced variation of light speed within the curvature region:( )[]( )[]( )v zcLaLaza( )expexpcosh=+−2121.(3)At the center of this region (z = 0), this becomes:( )[]( )[]( )[]( )[]vccLaLaLBKLBK( )expexpexpexp0 21212121=+=+−−,(4)for 0 < <<La,where ( )K cG=−2 4012πµ=+34840 1018.xis the radius of curvature constant. 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 = 3x108m/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:BKLccvv=i#?i#?i#?i#?i#>i#?±−20022ln( )( ). (5)Equations (2), (4) and (5) are formulae to use for creating and detecting spacetime curvature in the lab.Technical IssuesTraversable wormholes are creatures of classical GTR and represent non-trivial topology change in the spacetime manifold. This makes mathematicians cringe because it raises the question of whether topology can change or fluctuate to accommodate wormhole creation. Black holes and naked singularities are also creatures of GTR representing non-trivial topology change in spacetime, yet they are accepted by the astrophysics and mathematical communities - the former by Hubble Space Telescope discoveries and the latter by theoretical arguments due to Kip
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3Thorne, Stephen Hawking, Roger Penrose and others. The Bohm-Aharonov effect is another example which owes its existence to non-trivial topology change in the manifold. The topology change (censorship) theorems discussed in Visser (1995) make precise mathematical statements about the “mathematician’s topology” (topology of spacetime is fixed!), however, Visser correctly points out that this is a mathematical abstraction. In fact, Visser (1990) proved that the existence of an everywhere Lorentzian metric in spacetime is not a sufficient condition to prevent topology change. Furthermore, Visser (1990 and 1995) elaborates that physical probes are not sensitive to this mathematical abstraction, but instead they typically couple to the geometrical features of space. Visser (1990) also showed that it is possible for geometrical effects to mimic the effects of topology change. Topology is too limited a tool to accurately characterize a generic traversable wormhole; in general one needs geometric information to detect the presence of a wormhole, or more precisely to locate the wormhole throat (Visser 1997).Landis (1997) has made technical criticisms of Maccone’s (1995) work suggesting that the Levi-Civita metric in the presence of a uniform magnetic field does not form a wormhole within the Morris and Thorne (1988) framework. While the latter view is correct, the technical arguments are not accurate or complete. Changing the coordinate system from Cartesian to cylindrical (x1= rcosI•, x2= rsinI•, x3= z, let x4= t) puts equation (1) into the form (Maccone 1995):( )( )[]( )dsccdtdrr ddzzazara2122212222122= −++ −++−−expexpI•. (6)This is a cleaner form, but what is the Levi-Civita metric really? We can find out from making a change of (radial) variable by letting r = asinθ, dr = acosθdθ and substituting these into equation (6):( )( )[][]dsccdta dddzzaza2122222222= −++++−expexpsinθθ I•, (7)where ais the constant radius defined by equation (2). The spatial part of (7),[]da dddzσθθ I•222222=++sin, is recognized as the three-metric of a hypercylinder S2 x ℜ. 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.In addition, directing attention on the behavior of wormhole geometry at asymptotic infinity is not too profitable. Visser (1997) and Hochberg and Visser (1997) demonstrates that it is only the behavior near the wormhole throat that is critical to understanding what is going on, and that a generic throat can be defined without having to make all the symmetry assumptions and without assuming the existence of an asymp totically flat spacetime to embed the wormhole in. One only needs to know the generic features of the geometry near the throat in order to guarantee violations of the null energy condition (NEC) (Hawking and Ellis 1973) for certain open regions near the throat (Visser 1997). There are general theorems of differential geometry that guarantee that there must be NEC violations (meaning exotic matter-energy is present) at a wormhole throat. In view of this, however, it is known that static radial electric ormagnetic fields are borderline exotic when threading a wormhole if their tension were infinitesimally larger, for a given energy density (Herrmann 1989 and Hawking and Ellis 1973). Other exotic (energy condition violating) matter-energy fields are known to be squeezed states of the electromagnetic field, Casimir (electromagnetic zero-point) energy and other quantum fields/states/effects. With respect to creating wormholes, these have the unfortunate reputation of alarming physicists. This is unfounded since all the energy condition hypotheses have been experimentally tested in the laboratory and experimentally shown to be false - 25 years before their formulation (Visser 1990). Violating the energy conditions commits no offense against nature.
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4EXPERIMENTAL APPROACHTable 1 below shows the radius of curvature generated by a range of magnetic field strengths via equation (2). Equations (2), (4) and (5) suggest a way to perform a laboratory experiment whereby one could apply a powerful static homogeneous (cylindrically symmetric) magnetic field in a vacuum, thereby creating spacetime curvature in principle, and measure the speed of a light beam through it. A measurable slowing of c in this arrangement would demonstrate that a curvature effect has been created in the experiment. The achievable precision in measuring this TABLE 1. Radius of Spacetime Curvature Induced by B-Field.B ( x 3.484 Tesla)a (m)11018(105.7 ly)1021016(1.06 ly)1031015(0.11 ly)1051013(66.7 AU)1071011(0.67 AU)109109(1.44 Solar Radii)1012106(0.16 Earth Radii)101510310181would be c - v(0) or c2- v2(0) as seen from equation (5). Electric fields could also be used to create the same effect, however, the field strengths required to accomplish the same radius of curvature or slowing of c is seventeen times larger than magnetic field strengths (Maccone 1995).From Table 1, it is apparent that laboratory magnetic field strengths would need to be > 109- 1010Tesla so that a significant radius of curvature and slowing of c can be measured. Experiments employing chemicalexplosive/implosive magnetic technologies would be an ideal arrangement for this. The limit of magnetic field generation for chemical explosives/implosives is ∼ several x 103Tesla and the quantum limit for ordinary metals is ∼50,000 Tesla. Explosion/implosion work done by Russian (MC-1 generator, ISTC grant), Los Alamos National Lab (ATLAS), National High Magnetic Field Lab and Sandia National Lab (SATURN) investigators have employed magnetic solenoids of good homogeneity with lengths of ∼ 10 cm, having peak rate-of-rise of field of ∼ 109Tesla/swhere a few nanoseconds is spent at 1000 Tesla, and which is long enough for a good measurement of c (Solem 1997). Further, with picosecond pulses, c could be measured to a part in 102or 103. At 1000 Tesla, c2- v2(0) ≈ 0 m2/s2and the radius of curvature is 0.368 light-years. If the peak rate-of-rise of field ( 109Tesla/s) can be used, then a radius of curvature ≤ several x 106km can be generated along with c2- v2(0) ≥ several x 104m2/s2. In general, we can use Table 1 and equation (4) to see that for B ≤ 108Tesla we obtain c2- v2(0) ≈ 0 m2/s2, and for B ∼ 109- 1018Tesla we obtain c2- v2(0) ∼ 104- 1014m2/s2. It can be seen from this result that the use of ultrahigh magnetic fields will be necessary to obtain measurable consequences of any spacetime effects which may occur in a laboratory experiment.It will be necessary to consider advancing the state-of-art of magnetic induction technologies in order to reach static field strengths that are > 109- 1010Tesla. Extremely sensitive measurements of c at the one part in 106or 107level may be necessary for laboratory experiments involving field strengths of ∼ 109Tesla. Magnetic induction technologies based on nuclear explosives/implosives may need to be seriously considered in order to achieve large magnitude results. An order of magnitude calculation indicates that magnetic fields generated by nuclear pulsed energy methods could be magnified to (brief) static values of ≥ 109Tesla by factors of the nuclear-to-chemical binding energy ratio (≥ 106). Other experimental methods employing CW lasers, repetitive-pulse free electron lasers, neutron beam-pumped UO2lasers, pulsed laser-plasma interactions or pulsed hot (zeta pinch) plasmas generate insufficient magnetic field strengths for our purpose. Table 2 shows the currently available (high and ultrahigh) magnetic field generation technologies. However, it will be shown in the next section that several technologies listed in Table 2 are more than adequate for generating magnetic fields which could create and stabilize very large wormholes.
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5TABLE 2. Current High and Ultrahigh Magnetic Field Generation Technologies.Magnetic Field Strength (Tesla)Field Generation Technology10 - 300Superconductivity, Hybrid Magnets, Pulsed Magnets360Magnetic flux compression by electromagnetic force400One-turn coil connected to strong laser produced plasma∼ 103High powered pulsed lasers1000 - 3000Magnetic flux compression by chemical explosion102- 105White Dwarf stars107- 109Neutron stars≥ 109Magnetic flux compression by nuclear explosionWHIP SPACECRAFT CONCEPTWHIP spacecraft will have multifunction integrated technology for propulsion. The Wormhole Induction Propulsion Integrated Technology (WHIPIT) would entail two modes. The first mode is an advanced conventional system (chemical, nuclear fission/fusion, ion/plasma, antimatter, etc.) which would provide propulsion through the wormhole throat, orbital maneuvering capability near stellar or planetary bodies, and spacecraft attitude control and orbit corrections. An important system driver affecting mission performance and cost is the overall propellant mass-fraction required for this mode. A desirable constraint limiting this to acceptable (low) levels should be that an advanced conventional system would regenerate its onboard fuel supply internally or that it obtain and process its fuel supply from the in situ space environment. Other important constraints and/or performance requirements to consider for this propulsion mode would include specific impulse, thrust, energy conversion schemes, etc. Further discussion of these is beyond the scope of this paper and is left for the reader to explore on their own.The second WHIPIT mode is the stardrive component. This would provide the necessary propulsion to rapidly move the spacecraft over interplanetary or interstellar distances through a traversable wormhole. The system would generate a static, cylindrically symmetric ultrahigh magnetic field to create a hypercylinder curvature envelope (gravity well) near the spacecraft to pre-stress space into a pseudo-wormhole configuration. The radius of the hypercylinder envelope should be no smaller than the largest linear dimension of the spacecraft. As the spacecraft is gravitated into the envelope, the field-generator system then changes the cylindrical magnetic field into a radial configuration while giving it a tension that is greater than its energy density. A traversable wormhole throat is then induced near the spacecraft where the hypercylinder and throat geometries are patched together. The conventional propulsion mode then kicks on to nudge the spacecraft through the throat and send its occupants on their way to adventure. This scenario would apply if ultrahigh electric fields were employed instead. After a wormhole is created, it will be important to stabilize it against collapse by threading it with matter or energy fields of stupendous negative (outward) tension. If b denotes the size of the wormhole throat (minimum radius), then the tension (outward radial pressure) at the throat must be at least (Morris and Thorne 1988):()( )τπ=≈−−850 104 2140102Gc bxmb.. (8)Table 3 below shows the tension required to induce and stabilize a range of wormhole throat sizes. By inspecting Table 3, it becomes apparent that the calculated tensions are indeed stupendous. One can see that for wormhole throats smaller than 0.11 ly, the required tension will be greater than 5.0 x 1012N/m2which exceeds the tensile strength of steel or tungsten (∼ several x 1011N/m2). Indeed, for a 1000 m wormhole throat, the required tension of 5.0 x 1036N/m2has the same magnitude as the pressure at the center of the most massive neutron star. However, if we make a very large wormhole of 1.0 ly in size, then we can use non-material fields to do the job. The outward tension required to open and stabilize a 1.0 ly wormhole is 5.59 x 1010N/m2. This is achievable by threading the wormhole throat with a magnetic field of only 118.5 Tesla. One can see in Table 2 that magnetic generation technologies based on superconductivity, hybrid or pulsed magnets can easily achieve this field strength. For magnetic field strengths of ∼
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6102- 103Tesla, the corresponding magnetic field tension and wormhole size is ∼ 1010- 1012N/m2and ∼ 1 - 10-1ly, respectively. It is apparent from Table 2 that there are technologies which can meet these requirements.TABLE 3. Wormhole Throat Size Induced by Applied Tension.b (m)τ( x 5.0 N/m2)1018(105.7 ly)1061016(1.06 ly)10101015(0.11 ly)10121013(66.7 AU)10161011(0.67 AU)1020109(1.44 Solar Radii)1024106(0.16 Earth Radii)1030103103611042If optimization of wormhole throat (geometry) creation and hyperspace tunneling distance requires a fully exotic energy field to thread the throat, then the propulsion system would need to be capable of generating and deploying a Casimir (or other exotic) energy field. Although exotic field generation schemes are speculative, further discussion is beyond the scope of this paper and will be left for future work.CONCLUSIONSA candidate for breakthrough propulsion physics has been identified in the form of a traversable wormhole created by virtue of ultrahigh magnetic or electric fields with an additional exotic energy component. Maccone (1995) claimed that cylindrically symmetric ultrahigh magnetic (electric) fields can create a traversable wormhole in the Morris and Thorne (1988) framework. It has been shown that this is incorrect. Instead, a hypercylinder curvature effect having a position dependent gravitational potential is induced. This effect can be used to create a wormhole by patching the hypercylinder envelope to a throat that is induced by either radially stressing the ultrahigh field or employing additional exotic energy. Maccone correctly showed that the speed of light through the hypercylinder region is slowed by the magnetic induced gravitational field there. This suggests a way to perform laboratory experiments whereby one could apply an ultrahigh magnetic field in a vacuum, thereby creating a hypercylinder curvature effect, and measure the speed of a light beam through it. While chemical explosive/implosive magnetic induction technology has achieved static field strengths of ∼ several x 103Tesla, the peak rate-of-rise of field is ∼ 109Tesla/s. Field strengths > 109- 1010Tesla would need to be generated to impart a measurable slowing of light speed in this scenario. It is proposed that the peak rate-of-rise of field be exploited as a means to achieve this goal in the near-term. Magnetic induction technologies based on nuclear explosives/implosives may need to be considered in order to achieve results of larger magnitude. Further, it has been shown that it is possible, in the Morris and Thorne (1988) framework, to create very large (∼ 10-1- 1 ly) traversable wormholes in principle. These large wormholes require fields with outward tensions of < 1013N/m2to create and stabilize them against collapse. High magnetic field strengths of ∼102- 103Tesla were shown to be adequate for inducing the required tension. There are several magnetic field generation technologies available which are capable of attaining these field strengths. A Wormhole Induction Propulsion system has been introduced to exploit the possibilities of traversable wormholes that are created by high or ultrahigh magnetic induction systems.AcknowledgmentsI wish to thank Marc Millis for allowing me to use WHIP and WHIPIT which he coined at the February, 1997 NASA Breakthrough Propulsion Physics Regional Brainstorming Workshop at Austin, TX. My gratitude to Matt Visser for his many valuable suggestions and comments on this work. I also thank Alan Holt, Dean Judd, Johndale Solem, George Hathaway and John Alexander for their technical contributions and remarks. This research is partially supported by the National Institute for Discovery Science.
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7ReferencesHawking, S. W. and G. F. R. Ellis (1973) The Large-Scale Structure of Space-Time, Cambridge Univ. Press, Cambridge, 4:88-91 and 95-96.Herrmann, F. (1989) “Energy Density and Stress: A New Approach to Teaching Electromagnetism,” Am. J. Phys., 57:707-714.Hochberg, D. and M. Visser (1997) “Geometric Structure of the Generic Static Traversable Wormhole Throat,” LANL Abstract gr-qc/9704082, to appear in Phys. Rev. D.Landis, G. (1997) “Magnetic Wormholes And The Levi-Civita Solution To The Einstein Equation,” JBIS, 50:155-157.Levi-Civita, T. (1917) “Realtà fisica di alcuni spazi normali del Bianchi”, in Rendiconti della Reale Accademia dei Lincei, Series 5, 26:519-533.Maccone, C. (1995) “Interstellar Travel Through Magnetic Wormholes,” JBIS, 48:453-458.Morris, M. and K. Thorne (1988) “Wormholes in Spacetime and their use for Interstellar Travel: A Tool for TeachingGeneral Relativity,” Am. J. Phys., 56:395-412.Pauli, W. (1981) Theory of Relativity, Dover reprint, New York, 171-172.Solem, J. (1997) Personal Communication, Los Alamos National Lab, Los Alamos, NM, February, 1997.Visser, M. (1990) “Wormholes, baby universes, and causality,” Phys. Rev. D, 41:1116-1124.Visser, M. (1995) Lorentzian Wormholes - From Einstein to Hawking, AIP Press, Woodbury, NY.Visser, M. (1997) Personal Communication, Washington University, St. Louis, MO, July, 1997.------------------------------------------------------------Nomenclature-------------------------------------------------------------a: Radius of Spacetime Curvature (m)B: Magnetic Field Intensity (Tesla)c: Speed of Light in Vacuumv(z): Speed of Light Function (m/s)(3.00 x 108m/s)L: Length of Magnetic Solenoid (m)G: Universal Gravitational ConstantK: Radius of Curvature Constant(6.673 x 10-11Nm2/kg2)(3.4840 x 1018Tesla•meter)µ0: Permeability of Free Spacely: Light-Year(4πx 10-7H/m)(9.46 x 1015m)AU: Astronomical Unit τ: Outward Field Tension (N/m2)(1.50 x 1011m) b: Wormhole Throat Radius (m)-------------------------------------------------------------------------------------------------------------------------------------------

Interstellar travel through spacetime manipulation
Paulo Alexandre dos Santos
Department of Electronics - University of Aveiro, Portugal
August 23, 2004
Abstract
The basic requirements for an effective superluminal solution are outlined.
The discussion starts by talking about ____ holes and wormholes stating its
disadvantages and pathologies. Then an important solution discovered ten years
ago by Miguel Alcubierre is discussed, namely the Warp drive solution.
Recently the idea of a Manyfold Universe was proposed to explain dark matter.
The authors of this theory are Nima Arkani-Hamed(A), Savas Dimopoulos(D),
Gia Dvali(D) and Nemanja Kaloper(K). Their work became known as the ADDK
proposal. Here it will be shown how this geometrical construction of the universe
allow for spacetime shortcuts, thus enabling connecting distant parts of the ordinary
4D Einstein universe.
1 Introduction
In stories and in science fiction movies we often have audio and video communication
between people set apart by enormous intergalactic distances. We also have interstellar
travel where immense distances are crossed in just a few hours. Moreover, people travel
quietly and undisturbed by any discomfort or distress. They don’t feel any harmful
acceleration, nor do they have to use any tight spacesuit. They can simply pretend to
be at home with all of its commodities and enjoy the ride!
This kind of scenario would be the ultimate accomplishment in exploratory space
programs. We know that even traveling between very close planets, such as between
1
CERN-OPEN-2004-023
23/08/2004
Earth and the planet Mars becomes prohibitive, due to the excessive round-trip delay
time. The delay takes into account the low speed at which we can move in space.
Of course, if we could travel even at only a tenth of the speed of light, the journey
would take approximately 2 hours, something not that far fetched considering current
and developing technologies. Nevertheless, that speed would still not be adequate for
reaching even the limits of the nearest galaxy!
Fortunately, General Relativity (GR) theory opens new directions to address this
problem. There are solutions to the Einstein equations that allow for velocities above
the speed of light, which is considered the speed limit for matter according to the Special
Relativity (SR) theory. So, because old physical laws must be valid even in more up to
date theories, locally the speed limit will always be c, even in GR.
2 ____ holes and Wormholes
Some solutions to the Einstein equations predict not only ____ holes, but also
white holes that spit out what enters inside the ____ hole to another spacetime zone.
Nevertheless, the physical laws do not reduce simply to GR, and quantum corrections
predict that white holes cannot exist due to instability, especially owed to the second
law of Thermodynamics. Besides, if all matter ceased to exist in the expected place
where the ____ hole is, how would its gravity effects be felt?
However, there are those who believe that in some cases a white hole can be formed
in order to connect two faraway space regions and, who knows, even time! Michael
Morris and Kip Thorne studied some of the aspects of how to use ____ holes and
wormholes to travel faster than light [1]. Schwarzschild solutions are known to describe
gravity for any symmetric, non rotating, non charged massive body. They also describe
____ holes and even wormholes with those same three characteristics. Unfortunately
for the Schwarzschild wormholes, the results were not that enthusiastic: tidal gravitational
forces (inhomogeneities of gravity) at the throat would generate enormous
relative accelerations between the head and feet of a person falling inside; second a
Schwarzschild wormhole has a very rapid expansion and recontraction of its throat that
2
even moving at the speed of light, one cannot pass all the way through the wormhole;
thirdly, a Schwarzschild wormhole has a past horizon, which like that of a white hole
is unstable against small perturbations – these tremendously accelerates the sealing off
of the wormhole, making it even more impossible to get through.
However, wormholes can connect two different universes or different parts of our
universe and both cases are described by the same solution of the Einstein field equations.
Only their topologies differ, but the topology of the solutions are not constrained
by the Einstein equations. Recently, it has been proposed that gravity might propagate
through special wormholes with very small throats that connect two different parts of
our universe using extra dimensions in a theory referred to as the Manyfold Universe,
although normal matter cannot [2]. This is because normal matter and fields are too
”big” to enter these tiny throats in the higher dimensional spacetime.
Although Schwarzschild solutions are not suitable for FTL travel, there are other solutions
that were later found which have a better behavior like the Kerr solution. There
are other solutions, of course, but unfortunately they are believed not to exist freely in
nature. Of course this does not exclude the fact that a very advanced civilization might
be able to engineer those metrics in practical terms. Beside tidal forces, one of the major
problems with wormholes seems to be the fact that enormous amounts of negative
energy are required in order to keep the wormhole open. Depending on the geometry
of the wormhole, more severe problems can occur such as the wormhole collapse and
lethal X-rays jets. In fact, the first evidence for ____ holes was the detection of this
extremely high energetic X-ray radiation that is formed when matter falls inescapably
into the singularity.
Thus, we conclude that the exposed above is not the best for FTL travel.
3 Warp drive
Warp drive is a construction based in GR found by the Mexican physicist Miguel
Alcubierre in 1994 [3], that in theory would allow to surpass the speed of light. In this
case, phenomenal energies would be needed at least in the initial formulation made by
3
Alcubierre. Later work done by Chris Van Den Broeck, Loup, Waite, Halerewicz and
Krasnikov showed that energy requirements could be considerably lowered [4, 5, 6, 7] (at
least in theory). The major accomplishment was done by Krasnikov that lowered the
amount of energy needed to 10-5g, although there is not any clue on how to generate a
Planck sized bubble. The way warpdrive works can be easily described. According to
GR, space can be deformed even deliberately. This is the principle of metric engineering
as was already suggested in the former section. Of course, it is not easy to visualize
space being deformed since we live inside it, but it is possible by an analogy imagining
an elastic surface, for example the surface of a balloon. In this analogy, space is the
surface of the balloon; the three dimensional space around it ”doesn’t exist”, being
useful just for better visualization.
General relativity forbids speeds greater than the speed of light relative to space
but doesn’t impose any limit to the warping speed of space, i.e., the speed at which
spacetime can be deformed. Unfortunately, the warp drive needs negative energy and
lots of it. Physicists usually work only with positive energy and this includes antimatter.
Negative energy is defined as an energy density below the vacuum energy. It
is also described as being generated by some exotic matter. Only in certain particular
setups has negative energy been produced. These include the Casimir effect and certain
experiments with lasers. But the energy obtained in these experiments is extremely low.
Even for the most optimistic calculations, considering the lowest energy needed for the
warp drive, means that it’s impossible to obtain the necessary energy by any of these
methods.
3.1 Warp drive basics
The concept idea behind warp drive is to take a piece of space around the spaceship
and drag it with an arbitrary large velocity relatively to the surrounding space. A
bubble is generated that encloses this piece of space around the spaceship and, of course,
the spaceship itself. This bubble can accelerate relatively to a certain observer, with
any desired acceleration; however the acceleration is not felt inside the ship because
4
it is at rest relatively to the surrounding space inside the bubble. This would be an
advantage, because in order to accelerate to speeds thousands of times greater than
light, acceleration would need to be very low and require a lot of time to attain the
maximum speed, otherwise people would not bare the acceleration of many G’s.
The warp drive metric considering displacement in the x axis is defined by
ds2 = -dt2 + (dx - vsf(rs)dt)2 + dy2 + dz2 (3.1)
where vs, rs and the form function f are given by
vs(t) =
dxs(t)
dt
(3.2)
rs(t) =

(x - xs(t))2 + y2 + z21
2 (3.3)
f(rs) =
tanh ((rs + R)) - tanh ((rs - R))
2 tanh (R)
(3.4)
The center of the perturbation corresponds to the ship position xs and the speed
with which an observer watches the spaceship move is given by vs. R is the radius
of the bubble and can be interpreted as being inversely proportional to the bubble
wall thickness. If the wall is very thin, i.e., if is very large, the form function rapidly
approaches a top hat function,
lim
!1
f(r) =

1, r 2 [0,R]
0, r 2 (R,1)
(3.5)
3.1.1 Inside the bubble
We see that in the ship position, we have coordinates (xs(t), 0, 0) and f(0) = 1 which
from 3.1 implies that
-d2 = -dt2 + (dxs - vsdt)2
d = dt (3.6)
5
We observe that even if the ship speed is tremendously astronomic vs = dxs
dt , the relativistic
effect of time dilatation is not present. The spacetime is simply equivalent to
one where the ship is at rest. This is clearly a great advantage because in this theory
the acceleration is not felt inside the ship as stated earlier. So the bubble can accelerate
at will and it is the warp bubble that has to handle all the problems from the rapidly
changing speed.
3.1.2 Outside the bubble
Outside the warp-bubble, the top hat function is simply null f(r > R) = 0. So
the metric reduces to the Minkowski metric. This is also good because it shows that
outside the bubble, spacetime is not distorted, nor was it inside the bubble. So, the
real spacetime warping will occur between the walls of the bubble.
3.1.3 The null paths
Null paths allows us to calculate the velocity of photons if they are sent by the ship.
In this case we have f(rs) = 1 and we get
-dt2 + (dx - vsf(rs)dt)2 = 0
(dx - vsf(rs)dt) = dt
dx
dt
= 1 + vsf(rs) (3.7)
Take in mind that inside the bubble f(rs) = 1, which gives for the speed of the photon
the result v = 1 + vs, while outside the bubble we have a flat spacetime and v = c = 1
as expected. This will be useful in the next subsection.
3.2 Pathologies
Unfortunately several problems with warp drive were diagnosed by many scientists,
such as Hiscock and Pfenning [8, 9]. More recently, in 2002, these problems lead a
6
Portuguese scientist called Jos´e Nat´ario, to write an article in which he demonstrated
that pathologies were enough to realize its unpractical design [10].
First, as we already mentioned, we would need negative energy and lots of it. Although
we mentioned several attempts to lower the required energy, we believe that
the approach done in [9] is the most realistic one. Second, there is the problem with
horizons; this is probably the worst problem because even if we could generate the required
negative energy to create the warp field, it would still not be possible to control
the turn on or shutdown of the bubble. The bubble wall has a certain thickness that
is the space between two spherical surfaces. The spherical surface closer to the ship
is at rest relatively to the ship, but the other is moving faster than the speed of light
relatively to the ship. So, if the ship sends a control signal towards the outer surface
it stops somewhere inside the bubble wall. This point where the signal stops is part of
the so called horizon. We can understand this by looking to equation (3.7) and noting
that f(rs) decreases from 1 to 0 along the bubble region. The speed of the photon
depends on the form function f and it decreases along the bubble wall because f is
multiplying vs, whereas the bubble always moves with speed vs. Note that outside the
bubble f(rs) = 0 and the speed of the photon decreases from 1 + vs to 1 as already
noticed. But since vs is much greater than 1, the photon is outrunned by the inner
portion of the bubble, never reaching the external regions. This is how the horizon
problem arises. Note also that if the spherical ring had an inter radial length of zero,
the horizon problem would no longer exist. That would be the case where we used
the top hat function. But that would mean a discontinuity in spacetime at distance R
from the center of perturbation and nature does not allow discontinuities. This way it
is impossible to turn off the bubble once we arrive at the destination, since not even
a photon could reach the outer layers of the bubble. Thus, it would be necessary for
someone from the outside to turn the bubble off, in order to leave at the proper stop.
Third, there is the problem with Doppler blueshifts, which is almost as bad as the
horizons. Due to the Doppler effect, a ship traveling at thousands of times the speed
of light when colliding with photons will suffer severe impacts with enormous amounts
7
of energy. These collisions may destroy the warp field and consequently the ship (see
page 11 of [11]). Note also that the equation found by Hiscock et. al. and Nat´ario to
describe these impacts is exactly our equation 3.7 (see page 8 in [10] and equation 26
in [11]).
The classical frequency Doppler shift is given by
f = f0
v ± vO
v vS
(3.8)
where vO is the observer velocity and vS is the source velocity. If a photon is sent from
a star and collides with the ship at vs = 1000c, we have that vS = 0 and
f = f0
c + vs
c
= 1001f0 (3.9)
So we observe that for static sources and large speeds (vs c), the frequency shift
is equal to roughly the original frequency times the speed of the bubble divided by c
(f f0vs
c ).
3.3 Conclusions regarding the warp drive metric
One of the major problems with Alcubierre’s metric is the appearance of horizons,
i.e., a photon sent to the outer wall which is moving faster than light never reaches it.
This happens because in the ship’s frame we have d = dt and the Lorentz transform
is canceled. This has the advantage of synchronizing the ship time with an outside
observer at rest and permits the use of the classic Doppler shift formula.
We outlined so far in this report the severe problems of horizons and Doppler blue
shifts because we know that even if we could discover a new process to create a low
energy density and physically affordable warpdrive bubble, these two problems would
still remain due to the geometrical nature of the spacetime structure. Note also that
in a real space flight a ship at FTL speeds will likely collide with space dust, asteroids,
meteors and these would pose even greater problems.
8
4 Introducing extra dimensions
From the point of view of conventional physics, we know from the previous section
that practical FTL travel is almost impossible even being optimistic. Recent explanations
regarding the cosmological horizon problem [12] and dark matter [2] involve the
use of extra dimensions, thus making our three spatial plus one time dimensions a subset
of an even larger spacetime structure. These extra dimensions introduce ”tachyonic”
degrees of freedom allowing the creation of spacetime shortcuts in which FTL travel
may someday be ”possible”, assuming of course these extra dimensions exist (see the
section about the Manyfold Universe).
4.1 Manyfold Universe
In 1933, the Swiss astronomer Fritz Zwicky measured the speed of rotation of the
galaxy spiral arms. Others taking the same measurements later and with better equipment
confirmed Zwicky’s results. The spiral arms of the galaxies are rotating faster
than the expected velocity, considering the amount of visible stars and ____ holes.
Astronomers started wondering about invisible forms of matter that could possibly
generate such gravitational field. These fields are so powerful that raise the speed of
rotation by about seven times the average expected angular velocity.
This invisible form of matter was called ”dark matter”, because although it interacts
with ordinary matter raising the rotation speed of the spiral arms, this form of matter
was never detected [13]. Many theorists tried to explain this ”strange” behavior for
the unknown form of matter that at the same time interacts gravitationally with the
matter of the visible universe. However it evaded all the attempts to detect it. NASA
launched a space telescope able to detect in the universe this ”exotic” form of matter
with null results. Axions, WIMPs (Weak Interacting Massive Particles), neutralinos,
photinos and many other particles which were proposed to explain dark matter seem
unlikely to have physical existence.
Recently, Nimas-Arkani Hamed, Savas Dimopolous, Gia Dvali and Nemanja Kaloper
9
developed a model that can explain the dark matter. Their model is known as the
Manyfold Universe.
In the Manyfold universe, it is proposed that our world is a brane folded many times
inside sub-millimeter extra dimensions. The folding produces many connected parallel
branes or folds with identical microphysics – a Manyfold. Nearby matter on other folds
can be detected gravitationally as dark matter, since the light it emits takes a long time
to reach us traveling around the fold. For example, if something located at millions
of light years from us starts emitting gravitational waves, they will be felt here almost
immediately, while light will take millions of years to get to us.
Note that this will only happen between folds and there is not curvature associated
with the folding. In this scenario and using the wormhole analogy, this is equivalent to
having a quantum spacetime foam of wormholes between the folds. This seems plausible
because Quantum Mechanics (QM) with the Heisenberg uncertainty leads us to think
that in principle some singularities in classical theories like GR can be avoided, because
they smear out into Planck length distances. If this happens, then gravity might indeed
propagate through these wormholes.
Nevertheless, we are really more interested in having a throat with a larger circumference
in order to let matter and, of course, a spaceship to cross all the way between
the folds creating a path through the other spatial dimension(s). Although equivalent
to the wormhole case, in the Manyfold the metric has a higher number of dimensions
and so it can better model the topology of connecting two folds.
Maybe we can enlarge the throat by creating and connecting several of these wormholes
in parallel, thus creating a three dimensional link between folds.
Before proceeding the study of the expansion of the extra dimensions, allowing the
spacetime shortcut for a spaceship, we briefly outline the so-called braneworld model.
Almost all braneworld models predict that gravity propagate in extra dimensions,
while Standard Model (SM) fields are retained in the brane, due to the smallness of the
extra dimensions. For example, the most energetic photons observed have a wavelength
of 10-18m, while the extra dimensions are on the Planck scale 10-35m.
10
This could explain how geometry can act as a trapping mechanism keeping ordinary
matter in 4D spacetime, since gravity acts at lengths compatible with Planck scales.
If we want to create a shortcut in spacetime able to connect distant regions of the
universe using an extra dimension, we must enlarge the size of the extra dimension from
the Plank size to a macroscopical one, say 50m at least.
Recently, some researchers in Holland developed an equation to find the size of the
extra dimension in function of some known physical constants. The equation is given
by [14, 15]
R =
r
40G~2
e2c2 (4.1)
Note that 0 is the electric permittivity of the vacuum, G is the gravitational constant
in ordinary 3+1 spacetime, ~ is the Planck constant and e is the electron charge.
If we place the known values of these constant we arrive at 1.9×10-34m, compatible
with the Planck length. This equation was obtained considering the propagation of
gravity in one extra dimension and this study concludes that such a dimension must be
extremely small. This is compatible with the Manyfold Universe model. Now we have
a clear picture about how the Manyfold Universe works.
If we could enlarge by a still unknown physical process one of these constants, we
could possibly enlarge the size of the extra dimensions, at least locally. This would
allow the other interactions to enter this small extra dimension creating a spacetime
shortcut.
Note also that the size of the extra dimension depends only on the constants of our
physical dimension and this suggests a way of enlarging it.
We choose to work with the electric permeability of the vacuum, because eletromagnetic
interactions are a well understood phenomema and easier to control [16].
We must consider that changing the 0 implies changing the behavior of electromagnetism
and perhaps affecting the molecular or atomic structure of a macroscopic body.
Then if we want to enlarge the extra dimension to allow the passage of the macroscopic
body to the higher dimensional spacetime we must create a ”geometric manipulation
11
of spacetime” [16] that will alter 0 in the neighborhood of the body (not on the spot
where the body is) but in the spacetime region where the body resides 0 must remain
unchanged.
4.2 The Kalbermann-Halevi solution of the Einstein field equations
We can choose a simple metric to illustrate the expected geometric behavior. One
suitable ansatz seems to be the Kalbermann-Halevi ansatz [17]. This ansatz models
exactly the behavior of dark matter: a star being born at 326 million light years away
casts its gravity into our neighborhood within 2.5 × 10-10 seconds, but during the first
326 million years of its infancy the star will remain invisible until the arrival of the first
photons. During this ”invisibility time” it will be noticed as dark matter. This does not
means that gravity propagates at ”superluminal” speeds but that it simply connects
causally disconnected paths through a shortcut. Traveling through the ”throat”, gravity
can reach the other fold very quickly since it can be just a millimeter away through the
throat. On the other hand, light and massive particles will be guided through ordinary
spacetime and must traverse the longer way.
The Kalbermann-Halevi ansatz is given by the following ansatz
ds2 = a(R)dt2 - b(R)dR2 - c(R)dl2 (4.2)
The energy density G00 in the Kalbermann-Halevi ansatz is given by
G00 =
3a
4b2c

db
dR
dc
dR - 2
d2c
dR2 b

(4.3)
An example given by Kalbermann that gives the results mentioned earlier about
traveling 326 million light years in 2.5 × 10-10 seconds1 is
ds2 = (1 + kR2)dt2 - b(R)dR2 - e-kR2/2dl2 (4.4)
1Solve for the null path
12
but we leave b(R) as a generic function of R.
We must find the expression for b(R) that will provide a zero energy density, G00 = 0,
and from this expression we will deduce the one for low energy densities. Then b(R) is
given by
b(R) = (kR)2e-kR2
(4.5)
Note that the ansatz 4.4 using 4.5 reduces to the Minkowski metric for R = 0 as
expected. A desired but realistic G00 will then be given by G00 = 1
k . The k in all
these equations is given by k = 1062(pp 7 in [17]) and if we make G00 = 1
k we would
have a very low energy density (10-62) ideal for our ”superluminal” ansatz. Note that
integrating the energy density over the bulk volume2 in these conditions, the energy
needed for this process would still be low even considering that the bulk volume will be
raised by a factor of 1037 times.
We have a and c fixed and explicitly written (4.4) with b a variable. In this case
the expression for b will be close to the expression given in equation 4.5 since G00 will
become very low, near almost zero.
Using a low b(R), for instance the one given by equation 4.5 but keeping in mind a
general expression, we will compute the null path as follows

dl
dt
2
= (1 + kR2)ekR2/2
"
1 -
b(R)
1 + kR2

dR
dt
2
#
(4.6)
Using a high k value, for instance k = 1062, then b(R) 0 and b(R)
1+kR2 0. The
equation 4.6 will be approximately
v =
dl
dt
p1 + kR2e
kR2
4 (4.7)
Now it is clear how Kalbermann arrived at those extreme values (see page 8 near
equation 8 in [17]).
2since the brane dimensions do not enter in the expression for G00
13
4.3 Conclusion
We have seen that warp drive metrics are not suited for FTL travel due to the severe
pathologies mentioned above. The Manyfold Universe seems a promising field of research
as a FTL solution. However, the enlarging process of the extra dimension is an
unknown process and would probably require ultra high density matter fields still out
of reach to our technological know how. Nevertheless the natural explanation for dark
matter using simple geometry seems compelling and to accept this as way of allowing
SM fields to overcome the local speed of light looks the next logical step.
References
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for interstellar travel: A tool for teaching general relativity. American Journal of
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14
[7] Krasnikov, S. The quantum inequalities do not forbid spacetime shortcuts.
Phys.Rev. D67 (2003), (104013), 2003.
[8] Hiscock, W. A. Quantum effects in the alcubierre warp-drive spacetime. Classical
and Quantum Gravity, 14(11):L183–L188, November 1997.
[9] Pfenning, M. J. and Ford, L. H. The unphysical nature of ‘warp drive’. Classical
and Quantum Gravity, 14(7):1743–1751, July 1997.
[10] Natario, J. M. Warp drive with zero expansion. Classical and Quantum Gravity,
19(6):1157–1166, March 2002.
[11] Clark, C. and Hiscock W. H. and Larson S. L. Null geodesics in the alcubierre
warp-drive spacetime: the view from the bridge. Classical and Quantum Gravity,
16(12):3965–3972, December 1999.
[12] Chung, D. and Freese; K. Can geodesics in extra dimensions solve the cosmological
horizon problem. Physical Review D, 62, 2000.
[13] Nasa chandra report on dark matter. http://chandra.harvard.edu/resources/faq/dmatter/dmatter-
9.html, July 27 2004.
[14] Fernando Loup. Macroscopic spacetime shortcuts in the manyfold universe.
http://cdsweb.cern.ch/search.py?recid=739597.
[15] Fernando Loup. A more concise formalism for kalbermann-halevi
ansatz in the macroscopic spacetime shortcuts in the manyfold universe.
http://cdsweb.cern.ch/search.py?recid=782869.
[16] Millis M. Challenge to create the space drive. Journal of Propulsion and Power
(AIAA), 13(5):577–682, September 1997.
[17] German Kalbermann. Communication through an extra dimension.
Int.J.Mod.Phys. arXiv.org@gr-qc/9910063, A(15):3197–3206, 2000.
15

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Interstellar travel through spacetime manipulation
Paulo Alexandre dos Santos
Department of Electronics - University of Aveiro, Portugal

August 23, 2004

Abstract

The basic requirements for an effective superluminal solution are outlined.
The discussion starts by talking about ____ holes and wormholes stating its
disadvantages and pathologies. Then an important solution discovered ten years
ago by Miguel Alcubierre is discussed, namely the Warp drive solution.
Recently the idea of a Manyfold Universe was proposed to explain dark matter.
The authors of this theory are Nima Arkani-Hamed(A), Savas Dimopoulos(D),
Gia Dvali(D) and Nemanja Kaloper(K). Their work became known as the ADDK
proposal. Here it will be shown how this geometrical construction of the universe
allow for spacetime shortcuts, thus enabling connecting distant parts of the ordinary
4D Einstein universe.

1 Introduction

In stories and in science fiction movies we often have audio and video communication
between people set apart by enormous intergalactic distances. We also have interstellar
travel where immense distances are crossed in just a few hours. Moreover, people travel
quietly and undisturbed by any discomfort or distress. They don’t feel any harmful
acceleration, nor do they have to use any tight spacesuit. They can simply pretend to
be at home with all of its commodities and enjoy the ride!
This kind of scenario would be the ultimate accomplishment in exploratory space
programs. We know that even traveling between very close planets, such as between
1

CERN-OPEN-2004-023

23/08/2004

Earth and the planet Mars becomes prohibitive, due to the excessive round-trip delay
time. The delay takes into account the low speed at which we can move in space.
Of course, if we could travel even at only a tenth of the speed of light, the journey
would take approximately 2 hours, something not that far fetched considering current
and developing technologies. Nevertheless, that speed would still not be adequate for
reaching even the limits of the nearest galaxy!
Fortunately, General Relativity (GR) theory opens new directions to address this
problem. There are solutions to the Einstein equations that allow for velocities above
the speed of light, which is considered the speed limit for matter according to the Special
Relativity (SR) theory. So, because old physical laws must be valid even in more up to
date theories, locally the speed limit will always be c, even in GR.
2 ____ holes and Wormholes
Some solutions to the Einstein equations predict not only ____ holes, but also
white holes that spit out what enters inside the ____ hole to another spacetime zone.
Nevertheless, the physical laws do not reduce simply to GR, and quantum corrections
predict that white holes cannot exist due to instability, especially owed to the second
law of Thermodynamics. Besides, if all matter ceased to exist in the expected place
where the ____ hole is, how would its gravity effects be felt?
However, there are those who believe that in some cases a white hole can be formed
in order to connect two faraway space regions and, who knows, even time! Michael
Morris and Kip Thorne studied some of the aspects of how to use ____ holes and
wormholes to travel faster than light [1]. Schwarzschild solutions are known to describe
gravity for any symmetric, non rotating, non charged massive body. They also describe
____ holes and even wormholes with those same three characteristics. Unfortunately
for the Schwarzschild wormholes, the results were not that enthusiastic: tidal gravitational
forces (inhomogeneities of gravity) at the throat would generate enormous
relative accelerations between the head and feet of a person falling inside; second a
Schwarzschild wormhole has a very rapid expansion and recontraction of its throat that

2

even moving at the speed of light, one cannot pass all the way through the wormhole;
thirdly, a Schwarzschild wormhole has a past horizon, which like that of a white hole
is unstable against small perturbations – these tremendously accelerates the sealing off
of the wormhole, making it even more impossible to get through.
However, wormholes can connect two different universes or different parts of our
universe and both cases are described by the same solution of the Einstein field equations.
Only their topologies differ, but the topology of the solutions are not constrained
by the Einstein equations. Recently, it has been proposed that gravity might propagate
through special wormholes with very small throats that connect two different parts of
our universe using extra dimensions in a theory referred to as the Manyfold Universe,
although normal matter cannot [2]. This is because normal matter and fields are too
”big” to enter these tiny throats in the higher dimensional spacetime.
Although Schwarzschild solutions are not suitable for FTL travel, there are other solutions
that were later found which have a better behavior like the Kerr solution. There
are other solutions, of course, but unfortunately they are believed not to exist freely in
nature. Of course this does not exclude the fact that a very advanced civilization might
be able to engineer those metrics in practical terms. Beside tidal forces, one of the major
problems with wormholes seems to be the fact that enormous amounts of negative
energy are required in order to keep the wormhole open. Depending on the geometry
of the wormhole, more severe problems can occur such as the wormhole collapse and
lethal X-rays jets. In fact, the first evidence for ____ holes was the detection of this
extremely high energetic X-ray radiation that is formed when matter falls inescapably
into the singularity.
Thus, we conclude that the exposed above is not the best for FTL travel.

3 Warp drive

Warp drive is a construction based in GR found by the Mexican physicist Miguel
Alcubierre in 1994 [3], that in theory would allow to surpass the speed of light. In this
case, phenomenal energies would be needed at least in the initial formulation made by

3

 Alcubierre. Later work done by Chris Van Den Broeck, Loup, Waite, Halerewicz and
Krasnikov showed that energy requirements could be considerably lowered [4, 5, 6, 7] (at
least in theory). The major accomplishment was done by Krasnikov that lowered the
amount of energy needed to 10-5g, although there is not any clue on how to generate a
Planck sized bubble. The way warpdrive works can be easily described. According to
GR, space can be deformed even deliberately. This is the principle of metric engineering
as was already suggested in the former section. Of course, it is not easy to visualize
space being deformed since we live inside it, but it is possible by an analogy imagining
an elastic surface, for example the surface of a balloon. In this analogy, space is the
surface of the balloon; the three dimensional space around it ”doesn’t exist”, being
useful just for better visualization.
General relativity forbids speeds greater than the speed of light relative to space
but doesn’t impose any limit to the warping speed of space, i.e., the speed at which
spacetime can be deformed. Unfortunately, the warp drive needs negative energy and
lots of it. Physicists usually work only with positive energy and this includes antimatter.
Negative energy is defined as an energy density below the vacuum energy. It
is also described as being generated by some exotic matter. Only in certain particular
setups has negative energy been produced. These include the Casimir effect and certain
experiments with lasers. But the energy obtained in these experiments is extremely low.
Even for the most optimistic calculations, considering the lowest energy needed for the
warp drive, means that it’s impossible to obtain the necessary energy by any of these
methods.

3.1 Warp drive basics

The concept idea behind warp drive is to take a piece of space around the spaceship
and drag it with an arbitrary large velocity relatively to the surrounding space. A
bubble is generated that encloses this piece of space around the spaceship and, of course,
the spaceship itself. This bubble can accelerate relatively to a certain observer, with
any desired acceleration; however the acceleration is not felt inside the ship because

4

it is at rest relatively to the surrounding space inside the bubble. This would be an
advantage, because in order to accelerate to speeds thousands of times greater than
light, acceleration would need to be very low and require a lot of time to attain the
maximum speed, otherwise people would not bare the acceleration of many G’s.
The warp drive metric considering displacement in the x axis is defined by
ds2 = -dt2 + (dx - vsf(rs)dt)2 + dy2 + dz2 (3.1)

where vs, rs and the form function f are given by

vs(t) =dxs(t)dt                                            (3.2)

rs(t) =(x - xs(t))2 + y2 + z21

2 (3.3)
f(rs) =
tanh ((rs + R)) - tanh ((rs - R))
2 tanh (R)
(3.4)
The center of the perturbation corresponds to the ship position xs and the speed
with which an observer watches the spaceship move is given by vs. R is the radius
of the bubble and can be interpreted as being inversely proportional to the bubble
wall thickness. If the wall is very thin, i.e., if is very large, the form function rapidly
approaches a top hat function,
lim
!1
f(r) =

1, r 2 [0,R]
0, r 2 (R,1)
(3.5)
3.1.1 Inside the bubble
We see that in the ship position, we have coordinates (xs(t), 0, 0) and f(0) = 1 which
from 3.1 implies that
-d2 = -dt2 + (dxs - vsdt)2
d = dt (3.6)
5
We observe that even if the ship speed is tremendously astronomic vs = dxs
dt , the relativistic
effect of time dilatation is not present. The spacetime is simply equivalent to
one where the ship is at rest. This is clearly a great advantage because in this theory
the acceleration is not felt inside the ship as stated earlier. So the bubble can accelerate
at will and it is the warp bubble that has to handle all the problems from the rapidly
changing speed.
3.1.2 Outside the bubble
Outside the warp-bubble, the top hat function is simply null f(r > R) = 0. So
the metric reduces to the Minkowski metric. This is also good because it shows that
outside the bubble, spacetime is not distorted, nor was it inside the bubble. So, the
real spacetime warping will occur between the walls of the bubble.
3.1.3 The null paths
Null paths allows us to calculate the velocity of photons if they are sent by the ship.
In this case we have f(rs) = 1 and we get
-dt2 + (dx - vsf(rs)dt)2 = 0
(dx - vsf(rs)dt) = dt

dxdt= 1 + vsf(rs) (3.7)

Take in mind that inside the bubble f(rs) = 1, which gives for the speed of the photon
the result v = 1 + vs, while outside the bubble we have a flat spacetime and v = c = 1
as expected. This will be useful in the next subsection.
3.2 Pathologies
Unfortunately several problems with warp drive were diagnosed by many scientists,
such as Hiscock and Pfenning [8, 9]. More recently, in 2002, these problems lead a
6
Portuguese scientist called Jos´e Nat´ario, to write an article in which he demonstrated
that pathologies were enough to realize its unpractical design [10].
First, as we already mentioned, we would need negative energy and lots of it. Although
we mentioned several attempts to lower the required energy, we believe that
the approach done in [9] is the most realistic one. Second, there is the problem with
horizons; this is probably the worst problem because even if we could generate the required
negative energy to create the warp field, it would still not be possible to control
the turn on or shutdown of the bubble. The bubble wall has a certain thickness that
is the space between two spherical surfaces. The spherical surface closer to the ship
is at rest relatively to the ship, but the other is moving faster than the speed of light
relatively to the ship. So, if the ship sends a control signal towards the outer surface
it stops somewhere inside the bubble wall. This point where the signal stops is part of
the so called horizon. We can understand this by looking to equation (3.7) and noting
that f(rs) decreases from 1 to 0 along the bubble region. The speed of the photon
depends on the form function f and it decreases along the bubble wall because f is
multiplying vs, whereas the bubble always moves with speed vs. Note that outside the
bubble f(rs) = 0 and the speed of the photon decreases from 1 + vs to 1 as already
noticed. But since vs is much greater than 1, the photon is outrunned by the inner
portion of the bubble, never reaching the external regions. This is how the horizon
problem arises. Note also that if the spherical ring had an inter radial length of zero,
the horizon problem would no longer exist. That would be the case where we used
the top hat function. But that would mean a discontinuity in spacetime at distance R
from the center of perturbation and nature does not allow discontinuities. This way it
is impossible to turn off the bubble once we arrive at the destination, since not even
a photon could reach the outer layers of the bubble. Thus, it would be necessary for
someone from the outside to turn the bubble off, in order to leave at the proper stop.
Third, there is the problem with Doppler blueshifts, which is almost as bad as the
horizons. Due to the Doppler effect, a ship traveling at thousands of times the speed
of light when colliding with photons will suffer severe impacts with enormous amounts
7
of energy. These collisions may destroy the warp field and consequently the ship (see
page 11 of [11]). Note also that the equation found by Hiscock et. al. and Nat´ario to
describe these impacts is exactly our equation 3.7 (see page 8 in [10] and equation 26
in [11]).
The classical frequency Doppler shift is given by
f = f0
v ± vO
v vS
(3.8)
where vO is the observer velocity and vS is the source velocity. If a photon is sent from
a star and collides with the ship at vs = 1000c, we have that vS = 0 and
f = f0
c + vs
c
= 1001f0 (3.9)
So we observe that for static sources and large speeds (vs c), the frequency shift
is equal to roughly the original frequency times the speed of the bubble divided by c
(f f0vs
c ).
3.3 Conclusions regarding the warp drive metric
One of the major problems with Alcubierre’s metric is the appearance of horizons,
i.e., a photon sent to the outer wall which is moving faster than light never reaches it.
This happens because in the ship’s frame we have d = dt and the Lorentz transform
is canceled. This has the advantage of synchronizing the ship time with an outside
observer at rest and permits the use of the classic Doppler shift formula.
We outlined so far in this report the severe problems of horizons and Doppler blue
shifts because we know that even if we could discover a new process to create a low
energy density and physically affordable warpdrive bubble, these two problems would
still remain due to the geometrical nature of the spacetime structure. Note also that
in a real space flight a ship at FTL speeds will likely collide with space dust, asteroids,
meteors and these would pose even greater problems.
8
4 Introducing extra dimensions
From the point of view of conventional physics, we know from the previous section
that practical FTL travel is almost impossible even being optimistic. Recent explanations
regarding the cosmological horizon problem [12] and dark matter [2] involve the
use of extra dimensions, thus making our three spatial plus one time dimensions a subset
of an even larger spacetime structure. These extra dimensions introduce ”tachyonic”
degrees of freedom allowing the creation of spacetime shortcuts in which FTL travel
may someday be ”possible”, assuming of course these extra dimensions exist (see the
section about the Manyfold Universe).
4.1 Manyfold Universe
In 1933, the Swiss astronomer Fritz Zwicky measured the speed of rotation of the
galaxy spiral arms. Others taking the same measurements later and with better equipment
confirmed Zwicky’s results. The spiral arms of the galaxies are rotating faster
than the expected velocity, considering the amount of visible stars and ____ holes.
Astronomers started wondering about invisible forms of matter that could possibly
generate such gravitational field. These fields are so powerful that raise the speed of
rotation by about seven times the average expected angular velocity.
This invisible form of matter was called ”dark matter”, because although it interacts
with ordinary matter raising the rotation speed of the spiral arms, this form of matter
was never detected [13]. Many theorists tried to explain this ”strange” behavior for
the unknown form of matter that at the same time interacts gravitationally with the
matter of the visible universe. However it evaded all the attempts to detect it. NASA
launched a space telescope able to detect in the universe this ”exotic” form of matter
with null results. Axions, WIMPs (Weak Interacting Massive Particles), neutralinos,
photinos and many other particles which were proposed to explain dark matter seem
unlikely to have physical existence.
Recently, Nimas-Arkani Hamed, Savas Dimopolous, Gia Dvali and Nemanja Kaloper
9
developed a model that can explain the dark matter. Their model is known as the
Manyfold Universe.
In the Manyfold universe, it is proposed that our world is a brane folded many times
inside sub-millimeter extra dimensions. The folding produces many connected parallel
branes or folds with identical microphysics – a Manyfold. Nearby matter on other folds
can be detected gravitationally as dark matter, since the light it emits takes a long time
to reach us traveling around the fold. For example, if something located at millions
of light years from us starts emitting gravitational waves, they will be felt here almost
immediately, while light will take millions of years to get to us.
Note that this will only happen between folds and there is not curvature associated
with the folding. In this scenario and using the wormhole analogy, this is equivalent to
having a quantum spacetime foam of wormholes between the folds. This seems plausible
because Quantum Mechanics (QM) with the Heisenberg uncertainty leads us to think
that in principle some singularities in classical theories like GR can be avoided, because
they smear out into Planck length distances. If this happens, then gravity might indeed
propagate through these wormholes.
Nevertheless, we are really more interested in having a throat with a larger circumference
in order to let matter and, of course, a spaceship to cross all the way between
the folds creating a path through the other spatial dimension(s). Although equivalent
to the wormhole case, in the Manyfold the metric has a higher number of dimensions
and so it can better model the topology of connecting two folds.
Maybe we can enlarge the throat by creating and connecting several of these wormholes
in parallel, thus creating a three dimensional link between folds.
Before proceeding the study of the expansion of the extra dimensions, allowing the
spacetime shortcut for a spaceship, we briefly outline the so-called braneworld model.
Almost all braneworld models predict that gravity propagate in extra dimensions,
while Standard Model (SM) fields are retained in the brane, due to the smallness of the
extra dimensions. For example, the most energetic photons observed have a wavelength
of 10-18m, while the extra dimensions are on the Planck scale 10-35m.
10
This could explain how geometry can act as a trapping mechanism keeping ordinary
matter in 4D spacetime, since gravity acts at lengths compatible with Planck scales.
If we want to create a shortcut in spacetime able to connect distant regions of the
universe using an extra dimension, we must enlarge the size of the extra dimension from
the Plank size to a macroscopical one, say 50m at least.
Recently, some researchers in Holland developed an equation to find the size of the
extra dimension in function of some known physical constants. The equation is given
by [14, 15]
R =
r
40G~2
e2c2 (4.1)
Note that 0 is the electric permittivity of the vacuum, G is the gravitational constant
in ordinary 3+1 spacetime, ~ is the Planck constant and e is the electron charge.
If we place the known values of these constant we arrive at 1.9×10-34m, compatible
with the Planck length. This equation was obtained considering the propagation of
gravity in one extra dimension and this study concludes that such a dimension must be
extremely small. This is compatible with the Manyfold Universe model. Now we have
a clear picture about how the Manyfold Universe works.
If we could enlarge by a still unknown physical process one of these constants, we
could possibly enlarge the size of the extra dimensions, at least locally. This would
allow the other interactions to enter this small extra dimension creating a spacetime
shortcut.
Note also that the size of the extra dimension depends only on the constants of our
physical dimension and this suggests a way of enlarging it.
We choose to work with the electric permeability of the vacuum, because eletromagnetic
interactions are a well understood phenomema and easier to control [16].
We must consider that changing the 0 implies changing the behavior of electromagnetism
and perhaps affecting the molecular or atomic structure of a macroscopic body.
Then if we want to enlarge the extra dimension to allow the passage of the macroscopic
body to the higher dimensional spacetime we must create a ”geometric manipulation
11
of spacetime” [16] that will alter 0 in the neighborhood of the body (not on the spot
where the body is) but in the spacetime region where the body resides 0 must remain
unchanged.
4.2 The Kalbermann-Halevi solution of the Einstein field equations
We can choose a simple metric to illustrate the expected geometric behavior. One
suitable ansatz seems to be the Kalbermann-Halevi ansatz [17]. This ansatz models
exactly the behavior of dark matter: a star being born at 326 million light years away
casts its gravity into our neighborhood within 2.5 × 10-10 seconds, but during the first
326 million years of its infancy the star will remain invisible until the arrival of the first
photons. During this ”invisibility time” it will be noticed as dark matter. This does not
means that gravity propagates at ”superluminal” speeds but that it simply connects
causally disconnected paths through a shortcut. Traveling through the ”throat”, gravity
can reach the other fold very quickly since it can be just a millimeter away through the
throat. On the other hand, light and massive particles will be guided through ordinary
spacetime and must traverse the longer way.
The Kalbermann-Halevi ansatz is given by the following ansatz
ds2 = a(R)dt2 - b(R)dR2 - c(R)dl2 (4.2)
The energy density G00 in the Kalbermann-Halevi ansatz is given by
G00 =
3a
4b2c

db
dR
dc
dR - 2
d2c
dR2 b

(4.3)
An example given by Kalbermann that gives the results mentioned earlier about
traveling 326 million light years in 2.5 × 10-10 seconds1 is
ds2 = (1 + kR2)dt2 - b(R)dR2 - e-kR2/2dl2 (4.4)
1Solve for the null path
12
but we leave b(R) as a generic function of R.
We must find the expression for b(R) that will provide a zero energy density, G00 = 0,
and from this expression we will deduce the one for low energy densities. Then b(R) is
given by
b(R) = (kR)2e-kR2
(4.5)
Note that the ansatz 4.4 using 4.5 reduces to the Minkowski metric for R = 0 as
expected. A desired but realistic G00 will then be given by G00 = 1
k . The k in all
these equations is given by k = 1062(pp 7 in [17]) and if we make G00 = 1
k we would
have a very low energy density (10-62) ideal for our ”superluminal” ansatz. Note that
integrating the energy density over the bulk volume2 in these conditions, the energy
needed for this process would still be low even considering that the bulk volume will be
raised by a factor of 1037 times.
We have a and c fixed and explicitly written (4.4) with b a variable. In this case
the expression for b will be close to the expression given in equation 4.5 since G00 will
become very low, near almost zero.
Using a low b(R), for instance the one given by equation 4.5 but keeping in mind a
general expression, we will compute the null path as follows

dl
dt
2
= (1 + kR2)ekR2/2
"
1 -
b(R)
1 + kR2

dR
dt
2
#
(4.6)
Using a high k value, for instance k = 1062, then b(R) 0 and b(R)
1+kR2 0. The
equation 4.6 will be approximately

 

 

 

 

v =
dl
dt
p1 + kR2e
kR2
4 (4.7)
Now it is clear how Kalbermann arrived at those extreme values (see page 8 near
equation 8 in [17]).
2since the brane dimensions do not enter in the expression for G00
13
4.3 Conclusion
We have seen that warp drive metrics are not suited for FTL travel due to the severe
pathologies mentioned above. The Manyfold Universe seems a promising field of research
as a FTL solution. However, the enlarging process of the extra dimension is an
unknown process and would probably require ultra high density matter fields still out
of reach to our technological know how. Nevertheless the natural explanation for dark
matter using simple geometry seems compelling and to accept this as way of allowing
SM fields to overcome the local speed of light looks the next logical step.
References
[1] Morris, Michael S. and Thorne, Kip S. Wormholes in spacetime and their use
for interstellar travel: A tool for teaching general relativity. American Journal of
Physics, 56(5):395–416, May 1988.
[2] Nima Arkani-Hamed, Savas Dimopoulos, Nemanja Kaloper and Gia Dvali. Manyfold
universe. JHEP, December 2000.
[3] Alcubierre, Miguel. The warp drive: hyper-fast travel within general relativity.
Classical and Quantum Gravity, 11(5):L73–L77, May 1994.
[4] Broeck, C. Van Den. A ‘warp drive’ with more reasonable total energy requirements.
Classical and Quantum Gravity, 16(12):3973–3979, December 1999.
[5] F. Loup and D.Waite and E. Halerewicz, Jr. Reduced total energy requirements for
a modified alcubierre warp drive spacetime. arXiv.org@gr-qc, (0107097), November
2001.
[6] F. Loup and R. Held and D. Waite and E. Halerewicz, Jr. and M. Stabnok
and M. Kuntzman. A causally connected superluminal warp drive spacetime.
arXiv.org@gr-qc, (0202021), February 2002.
14
[7] Krasnikov, S. The quantum inequalities do not forbid spacetime shortcuts.
Phys.Rev. D67 (2003), (104013), 2003.
[8] Hiscock, W. A. Quantum effects in the alcubierre warp-drive spacetime. Classical
and Quantum Gravity, 14(11):L183–L188, November 1997.
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15

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The warp drive metric considering displacement in the x axis is defined by

ds^2 = -dt^2 + (dx - v_s f(r_s) dt)^2 + dy^2 + dz^2                 (3.1)

where v_s, r_s and the form function f are given by

v_s(t)=dx_s(t)/dt                                                                          (3.2)


r_s(t) = [(x-x_s (t))^2 + y^2 + z^2] ^ 1/2                                   (3.3)

f(r_s)=[tanh(sigma (r_s + R)) - tanh(sigma(r_s - R))]/[2 tanh(sigma R)]                                                                                                 (3.4)

 

 

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