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
[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.
[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