AAWSAP DIRD Traversable Wormholes Stargates and Negative Energy April 6 2010
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Defense
Intelligence
Reference
Document
Acquisition Threat Support
6 April 2010
!COD : 1 December 2009
DI A-08 - 1004-004
Traversable Wormholes,
Stargates, and Negative
Energy
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Traversable Wormholes, Stargates, and Negative Energy
Prepared by:
Acquisition support Division (DW0-3)
Defense Warning Office
Directorate for Analysis
Defense Intelligence Agency
MP Person 58
Administrative Note
COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized.
This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining to
this document should be addressed tolAAP Person 1 I AAWSA Program
Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington,
DC 20340-5100.
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Contents
I. Summary .............................................................................................................v
II. A Brief Review of Transversable Wormholes and the Stargate Solution ............ 1
A. Traversable Wormholes ................................................................................. 1
B. The "Stargate" Solution ................................................................................. 4
c. What a Wormhole Looks Like in the Real World ............................................. 7
III. The General Relativistic Definition of Exotic Matter and the Energy Conditions 9
A. Examples of Exotic or "Negative" Energy Found in Nature ........................... 10
B. Generating Negative Energy in the Lab ........................................................ 11
1. Static Radial Electric & Magnetic Fields .................................................... 11
2. Squeezed Quantum Vacuum ..................................................................... 12
3. Gravitationally Squeezed Electromagnetic ZPF ......................................... 16
4. Vacuum Field Stress: Negative Energy from the Casimir Effect ................ 18
5. Dynamical Casimir Effect: Moving Mirrors ................................................ 20
6. Casimir Effect: Negative Energy for Traversable Wormholes .................... 20
IV. Constructing a Traversable Wormhole is not Easy .......................................... 21
A. Negative Energy Requirements and Energy Condition Violations ................. 21
B. Physical Constraints on Negative Energy ..................................................... 22
c. Observing Negative Energy in the Lab .......................................................... 25
V. Conclusion: The Way Forward .......................................................................... 26
VI. References...................................................................................................... 29
Figures
Figure 1. Intra-Universe Wormhole as a Hyperspace Shortcut Through
Conventional Space .................................................................................vi
Figure 2. Inter-Universe Wormhole (top} and Intra-Universe Wormhole (bottom}.3
Figure 3. Diagram of a Simultaneous View of Two Remote Compact Regions, n1
and n2, of Minkowski Space Used to Create the Wormhole Throat on ...... 5
Figure 4. The Same Diagram as in Figure 3 Except as Viewed by an Observer
Sitting in Region n1 Who Looks Through the Wormhole Throat and Sees
Remote Region n2 on the Other Side....................................................... 5
Figure 5. A Thin Shell of (Localized} Mass-Energy Possessing Two Principal Radii
of Curvature, p1 and p2 ............................................................................ 6
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Figure 6. A Spherically Symmetric Traversable Wormhole Observed in Space........ 7
Figure 7. A Stargate ............................................................................................... 8
Figure 8. A Stargate in Times Square ..................................................................... 9
Figure 9. Conceptual Squeezed Light Negative Energy Generator ........................ 14
Figure 10. Sodium Chamber Negative Energy Separator ...................................... 15
Figure 11. Alternative Conceptual Squeezed Light Negative Energy Generator .... 15
Figure 12. Schematic of the Casimir Effect ........................................................... 18
Tables
Table 1. Substantial Gravitational Squeezing Occurs for Vacuum ZPF ................. 18
Table 2. Negative Equivalent Mass Required for Traversable Wormhole .............. 22
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Traversable Wormholes, Stargates, and Negative Energy
I. Summary
Implementation of faster-than-light (FTL) interstellar travel via traversable
wormholes generally requires the engineering of spacetime into very
specialized local geometries. The analysis of these via Einstein's General
Theory of Relativity, plus the resultant equations of state, demonstrates that
such geometries require the use of "exotic" matter. It has been claimed that
since such matter violates the energy conditions, FTL spacetimes are not
plausible. However, it has been shown that this is a spurious issue. The
identification, magnitude, and production of exotic matter are seen to be a key
technical challenge, however. These issues are reviewed and summarized, and
an assessment on the present state of their resolution is provided.
In 1985 CalTech physicists M. Morris and K. Thorne discovered the principle of
traversable wormholes based on Einstein's General Theory of Relativity
(published in 1915). Morris and Thorne (Reference 1) and Morris et al.
(Reference 2) did this as an academic exercise at the request of Carl Sagan,
who had completed the draft of his novel Contact. This little exercise led to the
development of two new cottage industries in spacetime physics research: the
study of traversable wormholes and the study of time machines. Wormholes
are hyperspace tunnels through spacetime connecting either remote regions
within our universe or two different universes; they even connect different
dimensions and different times. Space travelers would enter one side of the
tunnel and exit the other, passing through the throat along the way. The
travelers would move at~ c (c is the speed of light, 3 x 108 m/s) through the
wormhole and therefore not violate Special Relativity, but external observers
would view the travelers as traversing multi-light-year distances through
space at FTL speed; Figure 1 illustrates this effect. A "stargate" is a special
class of traversable wormhole solutions to Einstein's general relativistic field
equation that possesses very simple physics and flat entry and exit openings.
Traversable wormholes are unlike the well-known, non-traversable Einstein
Rosen Bridges or Schwarzschild wormholes that are formed from collapsed
stellar matter (that is, black holes) or spherically symmetric vacuum regions.
Black holes are collapsed stars that have all their mass concentrated at an
infinitesimal point where the induced gravitational field crushes all matter and
spacetime. However, even Einstein-Rosen bridges can be made traversable by
an infinitesimal tweaking of their spacetime metric. In the case of black holes,
the singularity of collapsed matter, along with its crushing gravity field, totally
blocks the way through the tunnel. A traversable wormhole does not have a
singularity blocking the tunnel or any crushing gravity field. Explorers would
enter one side of the tunnel, travel through the throat, and exit the other side.
Traversable wormholes also do not possess an event horizon, a region of high
gravitational field strength separating the inside space surrounding the black
hole's singularity from the outside universe. Once you go through a black
hole's event horizon, you can never come back out because you will have to
attain FTL speed to escape it. Not even light can escape from an event horizon.
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Figure 1. Intra-Universe Wormhole as a Hyperspace Shortcut Through Conventional Space
Traversable wormholes are creatures of classical general relativity theory
allowing for very comfortable travel through the Cosmic Neighborhood. But
from the viewpoint of modern physics, the Cosmic Neighborhood can
encompass other universes, other space dimensions, and other times beyond
the four-dimensional spacetime realm. Mankind has certainly not discovered
all of the universe's facets and will need to continue to construct new
experiments and technology in order to verify (or not) these undiscovered
facets. Wormholes can possess normal or backward (in special cases) motion
through time and normal or nonexistent gravitational stresses on space
travelers, and their entry/exit openings (or throats) are spherically shaped,
flat, cubic shaped, polyhedral shaped, generic shaped, and so forth.
Why consider wormholes for travel through space, time, and other dimensions?
All standard space propulsion engineering is based on Newton's three laws of
motion, which is dependent on the expenditure of propellant to induce thrust
generating momentum transfer on a spacecraft. Many investigators have
proposed interstellar propulsion schemes based on a variety of nuclear (fission,
fusion, and pulsed) rockets, electric (ion or plasma) rockets, matter-
antimatter annihilation rockets, solar or laser sails, fusion or laser ramjets,
interstellar ion scoops, beamed energy propulsion (sails, rockets, and ramjets),
and so forth. Many of these modes either have been experimentally tested at
one time or another in our recent history or remain as theoretical proposals,
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but all are based on Newtonian mechanics. The limiting speed of space flight,
based on any of these modes, is the speed of light. It is important to point out
that for the interstellar travel application, Newtonian rocket propulsion modes
suffer from enormous mass ratios> 105 - 10 100 (depending on the specific
impulse) for spacecraft cruise velocities > 0.0Sc, if the travel time is
constrained to within 100 years for a one-way interstellar voyage. If the cruise
velocity is increased to sub-relativistic, near-relativistic, or even ultra
relativistic speeds and thus reduces the one-way travel time, then the mass
ratio increases ( exponentially!). The mass ratio is the initial spacecraft mass
(payload +structure+ propellant) at launch divided by the final spacecraft
mass (payload + structure) at "burnout." The large ratios given above show
that Newtonian rockets consist mostly of propellant in order to propel the
propellant, along with a given tiny payload, through interstellar space. The
specific impulse is a measure of rocket propulsion system efficiency: how
much impulse (thrust multiplied by time) is produced per unit of mass of
propellant expenditure. It is desired that rocket propulsion systems possess a
very high specific impulse in order to reduce the mass ratio, and hence
propellant mass requirement, to reasonable levels.
The non-traditional propulsion modes (sails, ramjets, beamed power, etc.)
have different efficiencies and constraints, but they are all still dependent on
Newtonian mechanics, even though their mass ratio and specific impulse
characteristics are slightly improved over that of the traditional modes. But all
traditional and non-traditional propulsion modes come with a great cost in
interstellar voyage travel time. At non-relativistic and sub-relativistic cruise
speeds, it will take explorers several human lifetimes to reach stellar
destinations. At low relativistic to ultra-relativistic cruise speeds, the travel
time will be reduced to hours, days, weeks, months, or years. However, at
these cruise speeds, relativistic time dilation will kick in, and the returning
interstellar voyagers will find that decades to thousands of years have elapsed
on Earth since their launch date and that their families and culture no longer
exist or are unrecognizable. This is an undesirable outcome for any interstellar
voyage. Furthermore, traditional Newtonian propulsion cannot transcend time
or spacetime dimensions or universes.
The solution to this problem is to dispense entirely with long interstellar
voyage times or the undesirable outcome of relativistic time dilation. Explorers
could deploy a wormhole-stargate near the Earth's surface, in Earth's orbit, or
anywhere in the solar system they like and just pass through the "stargate"
and come out the other side in remote spacetime within seconds, moving
through the throat at low cruise speeds (30 mph!) and with no time dilation
effects. Explorers could travel through the wormhole-stargates in small scout
ships or send probes unencumbered by either enormous propellant mass
ratios or extensive life support provisions. Effective travel time through the
Cosmic Neighborhood via stargates would become irrelevant but could be
estimated to be many times or thousands of times the speed of light. Explorers
could spend all day investigating the remote spacetime location and then
return home through the stargate in time to have dinner with their families. If
explorers were to really push the envelope, they would design their stargate
so they could return from their voyage in time to wave goodbye to themselves
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as they see themselves depart on their journey. This is no longer recognized in
classical general relativity physics as a time paradox issue. It is very easy to
build a time machine, given a traversable wormhole. But time travel via
wormhole is beyond the scope of this paper. Suffice it to say that classical
general relativity theory is seriously infested with time machines; the theory
both allows for and demands time travel in order to preserve self-consistency
of dynamic spacetime solutions for just about every problem ever studied.
Implementation of FTL interstellar travel via traversable wormholes generally
requires the engineering of spacetime into very specialized local geometries.
Analysis of these via the general relativistic field equation, plus the resultant
source matter equations of state, demonstrates that such geometries require
the use of "exotic" matter in order to produce the requisite FTL spacetime
modification. Exotic matter is generally defined by general relativity physics to
be matter that possesses (renormalized} negative energy density (sometimes
negative stress-tension = outward pressure, aka gravitational repulsion or
antigravity}. This term is very misunderstood and misapplied by the non
general-relativity community. This misconception can be cleared up by
defining what negative energy is and where it can be found in nature and by
reviewing the proposed experimental concepts for generating negative energy
in the laboratory. In addition, it has been claimed that FTL spacetimes are not
plausible because exotic matter violates the general relativistic energy
conditions. However, this has been shown to be a spurious issue. The
identification, magnitude, and production of exotic matter are seen as key
technical challenges, however. FTL spacetimes also possess features that
challenge the notions of causality, and quantum effects allegedly place
constraints on them. These issues are reviewed and summarized, and an
assessment on the present state of their resolution is provided.
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II. A Brief Review of Transversable Wormholes and the
Stargate Solution
How does one study the physics of FTL spacetimes with in the framework of general
relativity t heory? When studying spacetime physics, the normal philosophy is to take
the general relativistic field equation, add some form of matter, make simplifying
assumptions, and then solve to deduce what the geometry of spacetime will be. 1 This is
very difficult to do because there are ten nonlinear second-order partial differential
equations with four redundancies (arbitrary choice of spacetime coordinates) and four
constraints (stress-energy conservation). There is a tremendous body of research that
takes exactly this approach, either analytically or numerically. However, this is not the
best strategy for understanding wormhole spacetimes. The appropriate strategy is to
decide beforehand on a definition of the traversable wormhole t hat you desire and
decide what the spacetime geometry shou ld look like. Given the desired geometry, use
the general relativistic field equation to calculate the distribution of matter required to
set up this geometry. Then one needs to assess whether the required distribution of
matter is physically reasonable and whether it violates any basic rules of physics, etc.
The following sections briefly outline the key results for traversable wormholes.
A. TRAVERSABLE WORMHOLES
Traversable wormholes represent a class of exact metric solutions of the general
relativistic field equation. The solutions are "exact" in the sense that no approximations
requiring a plethora of physical assumptions have to be made to derive the appropriate
spacetime geometry. To define a stable traversable wormhole one needs to define t he
desirable physical requirements it is to have in order to achieve the desired FTL travel
benefit. The desired requirements are the following (Reference 1, 3):
• Travel time through the wormhole tunnel or throat should be ::; 1 year as seen by
both the travelers and outside static observers.
• Proper time as measured by travelers should not be dilated by relativistic effects.
• The gravitational acceleration and tidal-gravity accelerations between different parts
of the travelers' body shou ld be ::; 1 go (go is the acceleration of gravity near t he
Earth's surface, 9.81 m/s 2) when going through the wormhole.
• Travel speed through the tunnel/throat should be < c.
• Travelers (made of ord inary matter) must not couple strong ly to the material that
generates the wormhole curvature; the wormhole must be threaded by a vacuum
tube t hrough which the travelers can move.
• There is no event horizon at the wormhole throat.
1 The Einstein field equation is: Gµv"' R,,., - [(1/2) g,,., R] = -(SrcG/c4)Tµ, ,, where G,,, is the Einstein curvature tensor,
R,,., is the Ricci curvature tensor, R "' Rµµ (the trace of Rµ,,) is the Ricci scalar curvature, Tµv is the stress-energy
momentum tensor (a matrix quantity that encodes the density and flux of a matter source's energy and
momentum), G is Newton's universal gravitation constant (6.673 x 10-11 Nm 2/kg 2 ), and c is the speed of light. In
simplest terms, this relation states that gravity is a manifestation of the spacetime curvature ( Gµv) induced by a
source of matter (T,..). The Greek indices (µ, v = 0...3) denote spacetime coordinates, xo...x, , such that x1 ...x, =
space coordinates and xo"' time coordinate.
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• There is no singularity of infinitely collapsed matter residing at the wormhole throat.
These requirements then lead us to define a spherically symmetric Lorentzian
spacetime metric, ds2,2 that prescribes the required traversable wormhole geometry
(Reference 1, 3):
ds2 = - e2lf!<rl c 2 dt2 +[1 -b(r)/ rr' dr2 + r 2 d02 (1)
where standard spherical-polar coordinates are used (r: 2nr = circumference; 0 s 0 s n;
0 s rps 2n), tis time (-oo < t < oo), d<E>2 = dff- + sin2 Bdq}, r/1,.r) is the freely specifiable
redshift function that defines the proper t ime lapse through the wormhole throat, and
b(r) is the freely specifiable shape function that defines the wormhole throat's spatial
(hypersurface) geometry. The throat is spherically shaped. There are a large number of
variations of Equation ( 1), which define traversable wormholes having different
properties. The reader should consult (Reference 3) for further details . By inserting
Equation (1) into the Einstein field equation and cranking through the math, one can
derive the density and flux of energy and momentum (a.k.a. pressure) encoded by T,,v
for the source of matter that is required to produce the traversable wormhole. The
results show that the source of matter must have zero or negative energy density
and/or an outward radial tension (negative pressure) that is larger than the magnitude
of the energy density (Reference 1-3). Travelers moving through the throat at very
high speed will tend to measure a negative energy density. These exotic properties are
required to create and thread open the wormhole, and stabilize it against collapse (see
Section III for more details).
The technical description of a trip through a spherically symmetric traversable
wormhole is simply given by the proper time and/or the proper distance of travel
through its throat as measured by space travelers, while the (radial) travel velocity
through the throat is v = v(r) < c. The proper time of travel as measured by space
travelers going through the wormhole is given by /J., = f(yv) - 1dA, where y = [1 -
( v/c)2J - 112 and the integration (over the element of proper distance, d1c) is taken from
the wormhole entrance to its exit. The proper distance of travel as measured by the
space travelers is /J.1c = v!J.,. Remote static observers watching the space travelers go
through the wormhole will measure their travel time to be M = f(ve<P('"l)- 1d). and their
travel distance will be t,,.). = vM, where the integration is taken over the same limits as
before.
2 A spacetime metric, ds 2 , is a Lorentz-invariant distance function between any two points in spacetime that is
defined by ds 2 = gµ,dX1'dx '', where 9w is the metric tensor which is a 4x4 matrix that encodes the geometry of
spacetime and dxµ is the infinitesimal coordinate separation between two points.
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Figure 2 shows two diagrams
representing the embedded space
(Flamm diagram) representation of
Equation (1), which depicts the
geometry of an equatorial ( 0 = rc/2)
slice through space at a specific
moment of time (t = const). The top of
Figure 2 shows the embedding diagram
for a traversable wormhole that
connects two different universes (i.e.,
an inter-universe wormhole). The
bottom diagram in the figure is an intra-
universe wormhole with a throat that
connects two distant regions of our own
universe. These diagrams serve to aide
in visualizing traversable wormhole
geometry and are merely a geometrical
exaggeration.
There was originally one other criterion
for defining a traversable wormhole,
which was that it must be embedded
within the surround ing (asymptotically) Figure 2. Inter-Universe Wormhole (top) and Intra
Universe Wormhole (bottom).
flat spacetime. However, Hochberg and
Visser (Reference 4) proved that it is only the behavior near the wormhole throat that is
critical to understanding the physics, 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 in which to embed the wormhole. Therefore, 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 Section III for further
detail) for certain open regions near the throat. So one is free to place our wormhole
anywhere in spacetime because it is only the geometry and physics near the throat that
matters for any analysis. This fact led to the development of a number of different
traversable wormhole throat designs that are cubic shaped, polyhedral shaped, flat-face
shaped, generic shaped, etc. The reader should consult (Reference 3) for a complete
technical review of the various types (and shapes) of traversable wormhole solutions
found in general relativity theory.
One knows that one needs exotic or negative energy to create and thread open a
traversable wormhole. So in this regard, one asks what kind of wormhole one can make
with less effort. To answer this question one can relate the local wormhole geometry to
the global topolog ical invariant of the spacetime via the Gauss-Bonnet Theorem
(Reference 5). In the Gauss-Bonnet Theorem the local wormhole geometry is quantified
by the energy density, U (in geometrodynamic units, TJ = G = c = 1), threading the
wormhole throat plus a spatial curvature constant (for the throat). The global
topological invariant of spacetime is quantified by the Euler Number, xe, which is itself
defined in terms of the genus, g, representing the number of handles (or throats or
tunnels) a wormhole can be assigned. These two topological quantities are related via
xe = 2(1 - g). Therefore, the (static) wormhole Gauss-Bonnet relation is given by U :;;
xe/4 or U :;; (1 - g)/2 (Reference 5). (The case for dynamic traversable wormholes has
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results that are similar to the static case.) This relation will help to decide if a
traversable wormhole having one throat, or two or more throats should be built and at
what energy cost this will incur.
The following is the result of our analysis for traversable wormholes having:
• 1-handle/throat (i.e., flat torus or spherical wormhole topology) giving g = 1, thus
Xe = 0, and so U s 0
• 2-handles/throats giving g = 2, thus xe = -2, and so U s -1/2
• 3-handles/throats giving g = 3, thus xe = -4, and so U s -1; and so on.
It is clear from this that as the number of wormhole handles/throats increases the
amount of negative energy required to create the wormhole will grow larger in
magnitude. This is an undesirable demand on any putative negative energy generator.
It is clear then that item (a) defines the most desirable engineering solution one can
hope for: a 1-handle/throat traversable wormhole that will require zero or (arbitrarily)
little negative energy to create. The magnitude of energy condition violations and the
amount of negative energy required to build a traversable wormhole will be addressed.
B. THE "STARGATE" SOLUTION
It is a straightforward exercise to design a real "stargate" from wormhole physics. A
stargate is essentially a traversable wormhole with a flat-face shape for the throat as
opposed to the spherical-shaped throat of the Morris and Thorne wormhole as discussed
in the previous section. A traveler going through a stargate will simply be shunted into
another remote spacetime region within our universe or into another universe.
The flat-face traversable wormhole solution is derived from the thin shell (a.k.a.
junction condition or surface layer) formalism of the Einstein field equation (Reference
6, 7). The procedure is to take two copies of flat Minkowski space and remove from
each identical regions of the form n x 1.R, where n is a three-dimensional compact
spacelike hypersurface and 9, is a timelike line (time axis). Then identify these two
incomplete spacetimes along the timelike boundaries an x ~H. The resulting spacetime is
geodesically complete and possesses two asymptotically flat regions connected by a
traversable wormhole. The throat of the wormhole is just the junction an, which is a
two-d imensional space-l ike hypersurface, at which the two original Minkowski spaces
are identified (see Figures 3 and 4) .
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_a_,
Figure 3. Diagram of a Simultaneous View of Two Remote Compact Regions, Il1 and Il2, of Minkowski
Space Used to Create the Wormhole Throat an (time is suppressed in this diagram)
.n.,_
Figure 4. The Same Diagram as in Figure 3 Except as Viewed by an Observer Sitting in Region n1 Who
Looks Through the Wormhole Throat an and Sees Remote Region n2 (dotted area inside the circle) on
the Other Side
It is a standard result of the thin shell formalism that the Einstein field equation may be
cast in terms of the surface stress-energy tensor S ;i of a th in shell of matter (or mass
energy) localized inside t he wormhole throat an (Reference 8):
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4
S = - - c- (K' -8'K k) (2)
J 4rrG i J k
where the second fundamental form K ;i is a matrix that represents the extrinsic
curvature of an (telling how the wormhole throat is curved with respect to the
enveloping four-dimensional spacetime), Sii is the three-dimensional unit matrix, and
K \ is t he trace (sum of diagona l matrix elements) of KV K ;i is a diagonal matrix
having the two principal radii of curvature, p1 and p2, of the thi n shell as its components
(see Figure 5). S ;i may be interpreted in terms of the thin shell's surface energy density
cr and principal surface tensions, S1 and S2, which are also diagonal matrix components.
thin shell of mass-energy pl
Figure 5. A Thin Shell of (Localized) Mass-Energy Possessing Two Principal Radii of Curvature, p1 and
p2.
Equation (2) is solved and the components of S ;i are found to be (Reference 8):
(3a)
3 The Latin indices (i, j, k = 0... 2) denote three-dimensional hypersurface coordinates, ><° ...x2, such that x 1, x 2 "'
space coordinates and><° = time coordinate.
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(3b)
(3c)
These are the Einstein field equations for a traversable wormhole that is produced by a
thin shell of localized matter. Equations (3a-c) imply that (for ~ a convex
hypersurface) one is dealing with negative surface energy density and negative surface
tensions. This is exotic matter! The negative surface tension ( = positive outward
pressure, a.k.a. gravitational repulsion) is required to keep the throat open and stable
against collapse. To make this thin shell wormhole entirely flat requires that one
chooses the throat oD. to have at least one fl at face (picture the thin shell in Figure 5
becoming flat). On that face the two principal radii of curvature become p1 = p2 = oo as
required by standard three-dimensional geometry; therefore, substituting this
requirement into Equations (3a-c) gives:
(4)
which is a remarkable result. This
means that a traveler encountering and
going through such a wormhole
stargate will feel no tidal gravitational
forces and see no exotic matter
threading the throat. A traveler stepping
through the throat will simply be
shunted into another remote spacetime
region or into another universe (note:
the Einstein field equation does not fix
the spacetime topology, so it is possible
that wormholes are inter-universe as
well as intra-universe tunnels).
Therefore, one can construct a stargate
by generating a thin shell or surface
layer of exotic matter much like a thin
film of soap stretched across a loop of
wire.
C. WHAT A WORMHOLE LOOKS
LIKE IN THE REAL WORLD
The exotic matter threading a
traversable wormhole throat produces
repulsive gravity, which will then deflect Figure 6. ASpherically Symmetric Traversable
Wormhole Observed in Space
light rays going through and around it.
The entrance to the spherically symmetric Morris & Thorne wormhole looks like a
sphere that contains the mirror image of a whole other universe or remote region
within our own universe, incredibly shrunken and distorted (see Figure 6). This is an
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example of the topological inversion manifested in wormhole geometry. The spherical
wormhole entrance/exit (a.k.a. the throat) is called a hypersphere because it is the
hyperspace surface of our four-dimensional spacetime. If one were to travel through
the wormhole and look back at it from the other side, then one would see a sphere (the
entry way back home) that seemed to contain the whole original universe or home
region of space near Earth (within your universe). This would look just like a glass
Christmas tree ornament, which is just a spherical mirror that reflects, in principle, the
entire universe around it.
A flat-faced wormhole, or stargate, which is also a hypersurface, would not distort the
mirror image of the remote space reg ion or other universe seen through it because the
negative surface energy density and negative surface tensions of the exotic matter
threading its throat is zero as seen and felt by light and matter passing through it
(recall Equation (4)). See Figures 7 and 8.
Figure 7. A Stargate (adapted from Reference 9)
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Figure 8. A Stargate in Times Square
If a small wormhole (three or more dimensional) were to begin to appear or even bump
into our local space, one would perceive t his process as the occurrence of an unusually
bright spot in the sky. Blue and red Doppler shifting of this bright spot would manifest
when t he intersection of the wormhole with our local space grows or recedes,
respectively.
III. The General Relativistic Definition of Exotic Matter and
the Energy Conditions
This section will consider the physics of the exotic matter that is required to build
traversable wormholes. What exactly is "exotic" matter? In classical physics the energy
density of all observed forms of matter (fields) is non-negative. What is exotic about
the type of matter that must be used to generate traversable wormhole spacetime is
that it must have negative energy density and/or negative flux (Reference 10). The
energy density is "negative" in the sense that the configuration of matter fields one
must deploy to generate and thread a traversable wormhole throat must have an
energy density, pE ( = pc 2 , where pis the rest-mass density), that is less than or equal
to its pressures/tensions, Pi (Reference 1, 3). 4 In many cases, these equations of state
are also known to possess an energy density that is algebraically negative, i.e., the
energy density and flux are less than zero. It is on the basis of these conditions that
all Latin indices (e.g., i, j, k = 1...3) that are affixed to physical quantities
4 From this point forward in the text,
denote the usual 3-dimensional space coord inates, x 1 . .. x 3 , indicating the spatial components of vector or tensor
quantit ies.
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one can call this material property "exotic." The condition for ordinary, classical (non
exotic) forms of matter that all are familiar with in nature is that PE > P i and/or PE ~ 0.
These conditions represent two examples of what are variously called the "standard"
energy conditions: Weak Energy Condition (WEC: PE~ 0, PE+ Pi ~ 0), Null Energy
Condition (NEC: PE + Pi~ 0), Dominant Energy Condition (DEC), and Strong Energy
Condition (SEC). These energy conditions forbid negative energy density between
material objects to occur in nature, but they are mere hypotheses. Hawking and Ellis
(Reference 11) formulated the energy conditions in order to establish a series of
mathematical hypotheses governing the behavior of collapsed-matter singularities in
their study of cosmology and black hole physics. More specifically, classical general
relativity allows one to prove lots of general theorems about the behavior of matter in
gravitational fields. The impact or implications of the DEC or SEC will not be considered
because they add no new information beyond the WEC and NEC.
The bad news is that real physical matter is not "reasonable" because the energy
conditions are in general violated by semiclassical quantum effects (occurring at order
ri) (Reference 3). 5 More specifically, quantum effects generically violate the average
NEC (ANEC). Furthermore, it was discovered in 1965 that quantum field theory has the
remarkable property of allowing states of matter contain ing local regions of negative
energy density or negative fluxes (Reference 12). This violates the WEC, which
postulates that the local energy density is non-negative for all observers. And there are
also general theorems of differential geometry that guarantee that there must be a
violation of one, some, or all of the energy conditions (meaning exotic matter is
present) for all traversable wormhole spacetimes. With respect to creating traversable
wormhole spacetimes, "negative energy" has the unfortunate reputatio n 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 (Reference 13).
Further investigation into this technical issue showed that violations of the energy
conditions are widespread for all forms of both "reasonable" classical and quantum
matter (Reference 14-18). Furthermore, Visser (Reference 3) showed that all (generic)
spacetime geometries violate all the energy conditions. So the condition that PE > Pi
and/or PE ~ 0 must be obeyed by all forms of matter in nature is spurious. Violating the
energy conditions commits no offense against nature. Negative energy has been
produced in the laboratory and this will be discussed in the following sections.
A. EXAMPLES OF EXOTIC OR "NEGATIVE" ENERGY FOUND IN
NATURE
The exotic (energy condition-violating) fields that are known to occur in nature are:
• Static, radially-dependent electric or magnetic fields. These are borderline exotic, if
their tension were infinitesimally larger, for a given energy density (Reference 11,
19).
• Squeezed quantum vacuum states: electromagnetic and other (non-Maxwellian)
quantum fields (Reference 1, 20).
5 Planck's reduced constant, lJ = 1.055 x 10- 34 J-s.
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• Gravitationally squeezed vacuum electromagnetic zero-point fluctuations (Reference
21).
• Casimir effect, i.e., the Casim ir vacuum in flat, curved, and topological spaces
(Reference 22-28).
• Other quantum fields/states/effects. In general, the local energy density in quantum
field theory can be negative due to quantum coherence effects (Reference 12).
Other examples that have been studied are Dirac field states: the superposition of
two single particle electron states and the superposition of two multi-electron
positron states (Reference 29, 30). In the former (latter), the energy densities can
be negative when two single (multi-) particle states have the same number of
electrons (electrons and positrons) or when one state has one more electron
( electron-positron pair) than the other.
Cosmological inflation (Reference 3), cosmological particle production (Reference 3),
classical scalar fields (Reference 3), the conformal anomaly (Reference 3), and
gravitational vacuum polarization (Reference 14-17) are among many other examples
that also violate the energy conditions. Since the laws of quantum field theory place no
strong restrictions on negative energies and fluxes, then it might be possible to produce
exotic phenomena such as faster-than - light travel (Reference 31-33), traversable
wormholes (Reference 1-3), violations of the second law of thermodynamics (Reference
34, 35), and time machines (Reference 2, 3, 36). There are several other exotic
phenomena made possible by the effects of negative energy, but they lie outside the
scope of the present study. This section will review the previously listed items 1 thru 4
and examine their applicability and technical maturity. Dirac field states are currently
under study by investigators. Also, the issue of capturing and storing negative energy is
not considered in what follows because free-space negative energy sources appear to
be a more desirable option for inducing traversable wormholes than stored negative
energy, and because there is very little technical literature that addresses how to
capture and store negative energy (see, e.g., Reference 10). The issue of capturing and
storing negative energy will be left for future investigations.
B. GENERATING NEGATIVE ENERGY IN THE LAB
1. Static Radial Electric & Magnetic Fields
It is beyond the scope of this study to include all the technical configurations by which
one can generate static, radially-dependent electric or magnetic fields. Suffice it to say
that ultrahigh-intensity tabletop lasers have been used to generate extreme electric and
magnetic field strengths in the lab. Ultrahigh-intensity lasers use the chirped-pu lse
amplification (CPA) technique to boost the total output beam power. All laser systems
simply repackage energy as a coherent package of optical power, but CPA lasers
repackage the laser pulse itself during the amplification process. In typical high-power
short-pulse laser systems, it is the peak intensity, not the energy or the fluence, which
causes pulse distortion or laser damage. However, the CPA laser dissects a laser pulse
according to its frequency components, and reorders it into a time-stretched lower
peak-intensity pulse of the same energy (Reference 37-39). This benign pulse can then
be amplified safely to high energy, and then only afterwards reconstituted as a very
short pulse of enormous peak power - a pulse which could never itself have passed
safely through the laser system. Made more tractable in this way, the pulse can be
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amplified to substantial energies (with orders of magnitude greater peak power)
without encountering intensity-related problems.
The extreme output beam power, fields and physical conditions that have been
achieved by ultrahigh-intensity tabletop lasers are (Reference 39):
• Power Intensity "' 10 19 to 10 30 W/m 2 (10 34 W/m 2 using SLAC as a booster).
• Peak Power Pulse :-: : 103 fs.
• Electric field, E"' 10 14 to 10 18 V/m [note: compare this with the critical quantum
electrodynamic (QED) vacuum breakdown E-field intensity, Ee = 2me2c3/rie"' 10 18
V/m, defined by the total rest-energy of an electron-positron pair created from the
vacuum divided by the electron's Compton wavelength] 6 ,
• Magnetic field, B "' several x 106 Tesla (note: the critical QED vacuum breakdown B
field intensity is Be = Eclc"' 10 10 Tesla).
• Ponderomotive Acceleration of Electrons "' 10 17 to 10 30 go (go is the acceleration of
gravity near the Earth's surface, 9.81 m/s 2 ).
• Light Pressure"' 109 to 10 15 bars.
• Plasma Temperatures > 10 10 K.
The vigilant reader might assert that the electric and magnetic fields generated by
ultrahigh-intensity lasers are not static. But in fact, these fields are static over the
duration of the pulse-width while at peak intensity. The data above illustrates that
ultrahigh-intensity lasers can generate an electric field energy density ~
10 16 to 10 28
J/m 3 and a magnetic field energy density ~ 10 19 J/m 3 . However, there remains the
problem of engineering this type of experiment because classical electromagnetic
theory states that every observer associated with the experiment will see a non
negative energy density that is oc E2 + B2 , where E and Bare measured in an observer's
reference frame. It is not known how to increase the tension in these fields using
current physics, but some new physics may provide an answer. Th is technical problem
must be left for future investigation.
2. Squeezed Quantum Vacuum
Substantial theoretical and experimental work has shown that in many quantum
systems the limits to measurement precision imposed by the quantum vacuum zero
point fluctuations (ZPF) can be breached by decreasing the noise in one observable (or
measurable quantity) at the expense of increasing the noise in the conjugate
observable; at the same time the variations in the first observable, say the energy, are
reduced below the ZPF such that the energy becomes "negative." "Squeezing" is thus
the control of quantum fluctuations and corresponding uncertainties, whereby one can
squeeze/reduce the variance of one (physically important) observable quantity provided
the variance in the (physically unimportant) conjugate variable is stretched/increased.
The squeezed quantity possesses an unusually low variance, meaning less variance
than would be expected on the basis of the equipartition theorem. One can in principle
6 Electron mass, m e = 9.11 x 10-3 i kg; electron charge, e = 1.602 x 10- 19 C.
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exploit quantum squeezing to extract energy from one place in the ordinary vacuum at
the expense of accumulating excess energy elsewhere (Reference 1).
The squeezed state of the electromagnetic field is a primary example of a quantum field
that has negative energy density and negative energy flux. Such a state became a
physical reality in the laboratory as a result of the nonlinear-optics technique of
"squeezing," i.e., of moving some of the quantum-fluctuations of laser light out of the
cos[co(t - z/c)] part of the beam and into the sin[co(t - z/c)] part (Reference 20, 40-
44). 7 The observable that gets squeezed will have its fluctuations reduced below the
vacuum ZPF. The act of squeezing transforms the phase space circular noise profile
characteristic of the vacuum into an ellipse, whose semimajor and sem iminor axes are
given by unequal quadrature uncertainties (of the quantized electromagnetic field
harmonic oscillator operators). This applies to coherent states in general, and the usual
vacuum is also a coherent state with eigenvalue zero. As this ellipse rotates about the
origin with angular frequency ro, these unequal quadrature uncertainties manifest
themselves in the electromagnetic field oscillator energy by periodic occurrences, which
are separated by one quarter cycle, of both smaller and larger fluctuations compared to
the unsqueezed vacuum.
Morris and Thorne (Reference 1) and Caves (Reference 45) point out that if one
squeezes the vacuum, i.e., if one puts vacuum rather than laser light into the input port
of a squeezing device, then one gets at the output an electromagnetic field with weaker
fluctuations and thus less energy density than the vacuum at locations where cos 2 [co(t -
=
z/c)] 1 and sin 2 [co(t - z/c)] < < 1; but with greater fluctuations and thus greater
energy density than the vacuum at locations where cos 2 [co(t - z/c)J < < 1 and sin 2 [co(t -
z/c)] =1. Since the vacuum is defined to have vanishing energy density, any region
with less energy density than the vacuum actually has a negative (renormalized)
expectation value for the energy density. Therefore, a squeezed vacuum state consists
of a traveling electromagnetic wave that oscillates back and forth between negative
energy density and positive energy density, but has positive time-averaged energy
density.
For the squeezed electromagnetic vacuum state, the energy density pE-sqvac is given by
(Reference 46):
PE-sqvac =( 2 1/" } inh s[sinh s + cosh scos ( 2w(t- z I c) + 8)] (J / m 3) (5)
where L3 is the volume of a large box with sides of length L (i.e., the quantum field is
placed in a box with periodic boundary conditions), 1; is the squeezed state amplitude
(giving a measure of the mean photon number in a squeezed state), and 8 is the phase
of squeezing. Equation (5) shows that pE-sqvac falls below zero once every cycle when the
condition cash 1; > sinh I; is met. It turns out that this is always true for every nonzero
value of I;, so pE-sqvac becomes negative at some point in the cycle for a general
squeezed vacuum state. On another note, when a quantum state is close to a squeezed
vacuum state, there will almost always be some negative energy densities present.
Negative energy can be generated by an array of ultrahigh-intensity lasers using an
ultra-fast rotating mirror system (Reference 47). In this scheme a laser beam is passed
7 w is the angular frequency of light, I is time, and z denotes the z- axis direction of beam propagation.
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through an optical cavity resonator made of a lithium niobate (LiNbQ3) crystal that is
shaped like a cylinder with rounded si lvered ends to reflect light. The resonator will act
to produce a secondary lower frequency light beam in wh ich the pattern of photons is
rearranged into pairs. The squeezed light beam emerg ing from the resonator will
contain pu lses of negative energy interspersed with pulses of positive energy.
In this concept both the negative and positive energy pulses are 10- 15 second ~
duration. In principle a set of rapidly rotating mirrors could be arranged to separate the
positive and negative energy pulses from each other. The light beam would be set to
strike each mirror surface at a very shallow angle while the rotation wou ld ensure that
the negative energy pulses would be reflected at a slightly different ang le from the
positive energy pulses. A small spatial separation of the two different energy pulses
would occur at some distance from the rotating mirror. Another system of mirrors
would be needed to redirect the negative energy pulses to an isolated location and
concentrate them there. See Figure 9 for an illustration of this concept.
+
+ ).
+
+ r, Positive Energy
+ Pulses
Rotating Redirector
+
Mirror Sy&.em
Laser & -+-+-+-
LiNl:DJ
Resonator
•IIIIJ • •
Alternating Pulses of
Negative& Positive
Energy
Negative Energy Concentrated
Pulses Negative Energy
Figure 9. Conceptual Squeezed Light Negative Energy Generator
The rotating mirror system can actually be implemented via non-mechanical means. A
chamber of sodium gas is placed within the squeezing cavity and a laser beam is
directed through the gas. The beam is reflected back on itself by a mirror to form a
standing wave within the sodium chamber. This wave causes rapid variations in the
optical properties of the sodium thus causing rapid variations in the squeezed light so
that one can induce rapid reflections of pu lses by careful design (Reference 41). An
illustration of this is shown in Figure 10.
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Figure 10. Sodium Chamber Negative Energy Separator (Reference 41)
Another way to generate negative energy via squeezed light would be to manufacture
extremely reliable light pulses containing precisely one, two, three, etc., photons apiece
and combine them together to create squeezed states to order (Reference 47).
Superimposing many such states could theoretically produce bursts of intense negative
energy. See Figure 11 for a conceptual diagram of this concept. Photonic crystal
research has already demonstrated the feasibility of using photonic crystal waveguides
(mixing together the classical and quantum properties of optical materials) to engineer
light sources that produce beams containing precisely one, two, three, etc., photons.
For example, researchers at Melbourne University used a microwave oven to fuse a tiny
diamond, just 1/l000th of a millimeter long, onto an optical fiber, which could be used
to create a single photon beam of light (Reference 48, 49). The combining of different
beams containing different (finite integer) numbers of photons is already state-of-the
art practice via numerous optical beam combining methods that can readily be
extended to our application .
• • • •
•
•• •• •• •• •• ••
•••
••• ••• ••• •••
Figure 11 Alternative Conceptual Squeezed Light Negative Energy Generator
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Finally, Ries et al. (Reference SO) experimentally demonstrated the very first simple,
scalable squeezed vacuum source in the laboratory t hat consisted of a continuous -wave
diode laser and an atom ic rubidium vapor cell. The experimenta l tools one needs to
beg in exploring the generation of negative energy for the purpose of creating
traversable worm holes are just now becoming available.
3. Gravitationally Squeezed Electromagnetic ZPF
A natural source of negative energy comes from th e effect that gravitational fields (of
astronomical bodies) in space have upon the surround ing quantum vacuum. For
example, the gravitational field of the Earth produces a zone of negative energy around
it by dragging some of the virtua l quanta (a.k.a. vacuum ZPF) downward. This concept
was initially developed in the 1970s as a byproduct of studies on quantum field theory
in curved space (Reference 25). However, Hochberg and Kephart (Reference 21)
derived an important application of th is concept to the problem of creating and
stabilizing traversable wormholes. They showed that one can utilize the negative energy
densities, which arise from distortion of the vacuum ZPF due to the interaction with a
prescribed gravitational background, for providing a violation of the energy conditions.
The squeezed quantum states of quantum optics provide a natural form of matter
having negative energy density.
The analysis, via quantum optics, showed that gravitation itself provides the
mechanism for generating the squeezed vacuum states needed to support stable
traversable wormholes. The production of negative energy densities via a squeezed
vacuum is a necessary and unavoidable consequence of the interaction or coupling
between ordinary matter and gravity, and this defines what is meant by gravitationally
squeezed vacuum states. The magnitude of the gravitational squeezing of the vacuum
can be estimated from the quantum optics squeezing condition for given transverse
momentum and (equivalent) energy eigenvalues, j, of two electromagnetic ZPF field
modes, such that this condition is subject toj ➔ 0, and it is defined as (Reference 21):
(6)
where ,l is the ZPF mode wavelength, r is the radial distance from the center of the
astronomical body in question, Ro is the radius of the Earth (6.378 x 106 m), Mo is the
mass of the Earth (5,972 x 10 24 kg), Mis the mass of the astronomical body, and rs is
the Schwarzschild radius of the astronomical body. 8 Note that rs is only a convenient
radial distance parameter for any object under examination and so there is no black
hole collapse involved in this analysis. Any radial distance from the body in question
can be chosen to perform this analysis, but using rs makes the equation simpler in
form. Also note that Equation (6) contains an extra factor of two (compared to the j
derived in Reference 21) in order to account for the photon spin. The squeezing
condition plus Equation (6) simply states that substantial gravitational squeezing of the
vacuum occurs for those ZPF field modes with A~ Bnrs of the mass in question (whose
8r, = 2GM/c2 . According to general relativity theory, this is the critical radius at which a spherically symmetric
massive body becomes a black hole, i.e ., at which light is unable to escape from the body's surface.
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gravitational field is squeezing the vacuum). The corresponding local vacuum state
energy density is: pE-gsvac = -211h1c!A 4 .
The general result of the gravitational squeezing effect is that as the gravitational field
strength increases, the negative energy zone (surrounding the body) also increases in
strength. Table 1 shows when gravitational squeezing becomes important for sample
bodies and their associated pE-gsvac. The table shows that in the case of the Earth,
Jupiter and the Sun, the squeezing effect is extremely feeble because only ZPF mode
wavelengths above 0.2 m to 78 km are affected, each having very minute pE-gsvac. For a
solar mass black hole (radius of 2.95 km), the effect is still feeble because only ZPF
mode wavelengths above 78 km are affected. But note that Planck mass bodies will
have an enormously strong negative energy zone surrounding them because all ZPF
mode wavelengths above 8.50 x 10- 34 m will be squeezed, in other words, all
wavelengths of interest for vacuum fluctuations. Protons will have the strongest
negative energy zone in comparison because the squeezing effect includes all ZPF mode
wavelengths above 6.50 x 10- 53 m. Furthermore, a body smaller than a nuclear
diameter (;::: 10- 16 m) and containing the mass of a mountain (;::: 10 11 kg) has a fairly
strong negative energy zone because all ZPF mode wavelengths above 10- 15 m will be
squeezed. In each of these cases, the magnitude of the corresponding pE-gsvac is very
large.
However, the estimates for the wavelengths in Table 1 might be too small. Ford
(private communication, 2007) argues that Reference 21 is in error because spacetime
is flat on scales smaller than the local radius of curvature, which is defined by the
inverse square root of the typical Riemann curvature tensor component in a local
orthonormal frame, or Ac;::: (?c2/GM) 112 . According to Ford, only ZPF modes with A ;;;;: Ac
will be squeezed by the gravitational field. This leads to a different local vacuum state
energy density (for r >> rs) (Reference 15):
2n 2hc
PE-gsvac = - ~
2n 2hc
== - - l- 4 - (7)
c
2n 2hG2 M 2
(JI m 3 )
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Table 1. Substantial Gravitational Squeezing Occurs for Vacuum ZPF When )., ~
8 1trs
Mass of body (kg} rs (m} )., (m} PE-gsvac ( J / m 3 }
Su n = 2.00 x 10 30 2 .95 X 10 3 2': 78.0 X 10 3 - 1.69 X lQ-4 4
Ju piter = 1. 90 x 10 27 2.82 2': 74 - 2. 08 X lQ- 32
Ea rth = 5.98 x 10 24 8.87 X 10- 3 2': 0.23 - 2.23 X 10- 22
Typical mountain:::::: 10 11 : : : 10-16 2': 10-15 - 6 ,25 X 10 35
Pla nck mass= 2 .1 8 x 10-8 3. 23 X 10- 35 2': 8 .50 X 10- 34 -1. 20 X l O lOB
Prot on = 1. 67 x 10- 27 2.48 X 10- 54 2': 6 .50 X 10- 5) - 3,5 0 X 10 184
For example, near the surface of the Earth (r ~ Ro, M = Mo), 11.c:::::: 2.42 x 10 11 m and
hence, Equation (7) gives PE-gsvac:::::: - 1.82 x 10- 70 J/m 3. Compare these va lues with 11. 2':
0.23 m and PE-gsvac:::::: - 2.23 x 10- 22 J/m 3 in Table 1. The resolution of this disagreement
remains an open question.
One is presently unaware of any way to artificially generate gravitational squeezing of
the vacuum in the laboratory. This will be left for future investigation. However, it is
predicted to occur in the vicinity of astronom ical matter. Naturally occurring traversable
wormholes in the vicin ity of astronomical matter would therefore become possible.
4. Vacuum Field Stress: Negative Energy from the Casimir Effect
The Casim ir effect is by far the easiest
and most well known way to generate
negative energy in the lab. The Casimir
effect that is fami liar to most people is
the force that is associated with the
electromagnetic quantum vacu um
(Reference 51). This is an attractive
force that must ex ist between any two
neutral (uncha rged), parallel, flat,
conducting surfaces (e.g., metallic
plates) in a vacuum. Th is force has been
well measured and it can be attributed
to a minute imbalance in the vacuum
elect romagnetic zero-point energy
density inside the cavity between the
conducting su rfaces versus the vacuum
electromagnetic zero-point energy
density in t he free-space region outside
of the cavity (Reference 52-54 ). See
Figure 12 for an illustration of t his Figure 12. Schematic of the Casimir Effect
effect.
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It turns out that there are many different types of Casimir effects found in quantum
field theory (Reference 22-24, 28, 55). For example, if one introduces a single infinite
plane conductor into the Minkowski (flat spacetime) vacuum by bringing it adiabatically
from infinity so that whatever quantum fields are present suffer no excitation but
remain in their ground states, then the vacuum (electromagnetic) stresses induced by
the presence of the infinite plane conductor produces a Casimir effect. This result holds
equally well when two parallel plane conductors (with separation distance d) are
present, which gives rise to the familiar Casimir effect inside a cavity. Note that in both
cases, the spacetime manifold is made incomplete by the introduction of the plane
conductor boundary condition(s). The vacuum region put under stress by the presence
of the plane conductor(s) is called the Casimir vacuum. The generic expression for the
energy density of the Casimir effect is pcE = -A(ric)a4, where A= (,(D)/8rr.2 in spacetimes
of arbitrary dimension D (Reference 22-24). The appearance of the zeta-function (,(D)
is characteristic of expressions for vacuum stress-energy tensors, r::,; .
In our familiar
four-dimensional spacetime (D = 4), A= rc 2/720. To calculate r,:; for a given quantum
field is to calculate its associated Casimir effect.
Analogs of the Casimir effect also exist for fields other than the electromagnetic field.
When considering the vacuum state of other fields, one must consider boundary
conditions that are analogous to the perfect-conductor boundary conditions for the
electromagnetic field at the surfaces of the plates (Reference 22-24, 28). Other fields
are not electromagnetic in nature, that is to say they are non-Maxwellian, and so the
perfect-conductor boundary conditions do not apply to them. It turns out that complete
manifolds exhibit what is called the topological Casimir effect for any non-Maxwellian
fields. In order to define boundary conditions for other fields the conductor boundary
conditions are replaced and Minkowski spacetime by a manifold of the form 9t x r (i.e.,
a product space), where 9t is the real line defining the time dimension for this particular
product space and r is a flat three-dimensional manifold having any one of the
following topologies: 91 2 x S1, 9~ x T 2, T 3, 9t x K 2, etc., 9t being the real line that defines
any linear space dimension (e.g., 9t = line, 9t 2 = two-dimensional plane, etc.), T" being
the n-torus, K 2 the two-dimensional Klein bottle, S1 the circle, etc.
The case r = 9\ 2 x S1 has the closest resemblance to the electromagnetic Casimir effect,
the difference being that instead of imposing conductor boundary conditions, one
imposes periodic boundary conditions on some of the space coordinates in the three
dimensional manifold. When imposing this topological constraint on the field theoretic
calculation of the topological Casimir effect (for linear massless fields), one finds that
the generic expression for the energy density is also pcE = - A(11c)a4, where
A= ±dr (rr. 2/90), dr is the number of degrees of freedom (e.g., helicity states) per spatial
point, the plus sign holds for boson fields (giving a negative energy density) and the
negative sign for fermion fields (giving a positive energy density).
If one were to admit spin structure in the manifolds described above and the field is
spinorial, then there is another important subtlety that must be taken into account
when evaluating r::,; .
However, this introduces an additional complexity involving the
relationship between the spin structure and the global structure (i.e., the configuration
space or fibre bundle) of the field in question whereby the topology not only of the base
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manifold, but of the fibre bundle itself has an effect on r::,; . In addition to this, there
are (compactified) extra-space dimensional quantum field (i.e., D-Brane or "brane
world") analogs of the Casimir effect yet to be explored. But a detailed consideration of
these for producing traversable wormholes is beyond the scope of this report and will
be left for future investigation.
As a final note, the methods used to obtain the electromagnetic T~;; between parallel
plane conductors can also be used when the conductors are not parallel but are joined
together along a line of intersection. If the conductors have curved surfaces instead,
then one obtains results that are similar to the case of intersecting conductors. These
geometries have also been evaluated for the case of dielectric media. These particular
cases will not be considered further since there are technical subtleties involved that
complicate the calculations and application of the different approaches. This topic will
also be left for future investigation.
5. Dynamical Casimir Effect: Moving Mirrors
Negative energy can be created by a single moving reflecting (conducting) surface
(a.k.a. a moving mirror). A mirror moving with increasing acceleration generates a flux
of negative energy that emanates from its surface and flows out into the space ahead
of the mirror (Reference 25, 56). This is essentially the simple case of an infinite plane
conductor undergoing acceleration perpendicular to its surface. If the acceleration
varies with time, the conductor will generally emit or absorb photons (i.e., exchange
energy with the vacuum), even though it is neutral. This is an example of the well
known quantum phenomenon of parametric excitation. The parameters of the
electromagnetic field oscillators (e.g., their frequency distribution function) change with
time owing to the acceleration of the mirror (Reference 57). However, this effect is
known to be exceedingly small, and it is not the most effective way to produce negative
energy. This scheme will not be considered any further.
6. Casimir Effect: Negative Energy for Traversable Wormholes
The electromagnetic Casimir effect can be used in principle to create a traversable
wormhole. The energy density pcE = -(n2nc/720)a4 within a Casimir cavity is negative
and manifests itself by producing a force of attraction between the cavity walls. But
cavity dimensions must be made exceedingly small in order to generate a significant
amount of negative energy. In order to use the Casimir effect to generate a spherically
symmetric traversable wormhole throat of radius r11zroar, there is need to design a cavity
made of perfectly conducting spherically concentric thin plates with a plate separation d
of (Reference 2):
d=
1t3)¼(
(30 ~ hroal
{he)½
\/7 (8)
= ( 4 .05 X 1o - JS) ✓ r lhroal (m)
To counteract the collapse of the cavity due to the Casimir Force acting between the
plates, the plates will have equal electric charges placed upon them to establish
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adequate Coulomb repulsion. 9 Equation (8) shows that a 1 km radius throat will require
a cavity plate separation of 1.28 x 10- 16 m (smaller than a nuclear diameter), which
gives pEc = - 1.62 x 10 36 J/m 3 for this configuration . In contrast, a wormhole with a
throat radius of 1 AU will require a plate separation of 1.57 x 10- 12 m (or 35% smaller
than the electron's Compton wavelength), which results in an energy density of -7.14 x
10 19 J/m 3 . 10 There is no technology known today t hat can engineer a cavity with such
minuscule plate separations. In addition, such minuscule plate separations are
unrealistic because the Casimir effect switches over to the non-retarded field behavior
a
( ~ 3 ) of van der Waals forces when plate separations go below the wavelength (::::: 10
nm) where they are no longer perfectly conducting (Reference 58). This scheme will not
be considered any further. However, future work will be necessary to elucidate whether
the various quantum field analogs of the Casimir effect can provide a more reaso nable
technical solution to this problem.
IV. Constructing a Traversable Wormhole is not Easy
A. NEGATIVE ENERGY REQUIREMENTS AND ENERGY CONDITION
VIOLATIONS
One knows how to make small quantities of negative energy in the lab. But one does
not know if it is possible to make large quantities of negative energy. It was pointed out
in Section III that one, some, or all of the classical energy conditions must be violated
in order to build a traversable wormhole. And it was also cautioned that this was not a
showstopper because the energy conditions have all been violated by nature or by lab
experiment prior to their formulation. However, the reader should be forewarned that
there are a number of published claims that the energy condition violations can be
avoided. These claims are just semantic games whereby investigators universally
invoke the following scenario: divide the total stress-energy into weird matter plus
normal matter, push all the energy condition violations into the weird matter so that
the normal matter does not violate the energy conditions. Given that the energy
conditions are not absolute, such rearranging approaches are not necessary.
Traversable wormhole throats violate the NEC (or ANEC). So how big a violation is
required? The answer is that there is only need to calculate the amount of negative
energy that will be needed to generate and hold open a wormhole throat. A simple
formula for short-throat wormholes using the thin shell formalism gives this quantity in
terms of the equivalent mass (note: the energy density derived from the general
relativistic field equation is too complex to use for this mass comparison) (Reference
3):
9 I n a detai led analysis the electrostatic energy required to support the Coulomb repulsion between the plates
would be considered separately.
10 Mean Earth-Sun distance, 1 AU = 1.50 x 10 11 m .
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2
} throat c
M wh =
G
= -(1. 35 X 1027 kg) J throat (9)
1 meter
= - (0.71MJ) ] throat
1 meter
where Mwh is the (equivalent) mass required to build the wormhole, A.t hroat is a suitable
measure of the linear dimension (width or diameter) of the throat, and M1 is the mass
of the planet Jupiter. One can also obtain the required energy, E..,1i, by multiplying both
sides of Equation (9) by c2 . Equation (9) shows that a mass of -0. 71 M1 will be required
to build a wormhole 1-m in size. As the wormhole size increases, the mass requirement
grows negative-large. Table 2 presents a tabulation of the required negative
( equivalent) mass as a function of sample wormhole throat sizes. After being alarmed
by the magnitude of the results, one should note that M wh is not the total mass of the
wormhole as seen by remote observers. The non-linearity of the general relativistic field
equation dictates that the total mass is zero (actually, the total net mass being positive,
negative or zero in the Newtonian approximation depending on the details of the
negative energy configuration constituting the wormhole system). Finally, Visser et al.
(Reference 59) demonstrated the existence of spacetime geometries containing
traversable wormholes that are supported by arbitrarily small quantities of negative
energy, and this was proved to be a general resu lt. The next section will expand on this
further.
Table 2. Negative Equivalent Mass Required for
Traversable Wormhole
A.throat (m} Mwh
1000 - 709.9 MJ
100 - 71 MJ
10 -7.1 MJ
1 - 0. 71 MJ
0.1 -22.6 M©
0.01 - 2.3 M©
M1 = 1.90 x 1027 kg, Mo= 5.98 x 1024 kg
B. PHYSICAL CONSTRAINTS ON NEGATIVE ENERGY
The Quantum Inequalities (QI) conjecture is an extension of the Heisenberg Uncertainty
Principle to curved spacetimes. Much research has been conducted around this one
topic alone. The literature is too numerous to cite here but the reader should consult
(Reference 10) and (Reference 46) for detailed information. The QI conjecture relates
(via model dependent time integrals of the energy density along geodesics) the energy
density of a free quantum field and the t ime during which this energy density is
observed. This conjecture was devised as an attempt to quantify the amount of
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negative energy or energy condition violations required to build a traversable wormhole
spacetime. Investigators have invoked the QI to rule out many of the macroscopic
wormhole spacetimes. When generating negative energy the QI postulate that: a) the
longer the pulse of negative energy lasts, the weaker it must be; b) a pulse of positive
energy must follow and the magnitude of the positive pulse must exceed that of the
initial negative pulse; and c) the longer the time interval between the two pu lses, the
larger the positive pulse must be. This actually sounds quite reasonable on energy
conservation grounds until one discovers that the Casimir effect and its non-Maxwellian
quantum field analogs violate all three conditions. There are also a number of squeezed
vacuum sources and Dirac field states that manifestly violate all three conditions.
Cosmological inflation, cosmological particle production, classical scalar fields, the
conformal anomaly, and gravitational vacuum polarization are among the many other
examples that also violate the QI. Visser (Reference 60) also points out that
observational data indicate that large amounts of "exotic matter" are required to exist
in the universe in order to account for the observed cosmological evolution parameters.
The QI have also not been verified by laboratory experiments. The assumptions used to
derive the QI and the efficacy of their derivation for various cases has been called into
question by numerous investigators. Krasnikov (Reference 61) constructed an explicit
counterexample for generalized FTL spacetimes showing that the relevant QI breaks
down even in the simplest FTL cases. And he also addressed Fewster's (Reference 62)
technical arguments on this issue. It is important to point out that the Qis have been
mainly proven for free massless scalar fields in flat two-dimensional Minkowski
spacetime, so there remains the unanswered questions of extending the QI into a four
dimensional curved spacetime model (with or without boundaries) and how much
negative energy density can arise for interacting fields.
It turns out that Visser and coworkers (Reference 59, 63, 64) developed a superior way
to properly quantify the amount of negative energy or energy condition violations
required to build a traversable wormhole spacetime. They propose a quantifier in terms
of a spatial volume integral, wh ich amounts to calculating the following definite
integrals (Reference 59, 63, 64):
(10)
with an appropriate choice of the integration measure dV ( = 4n?dr or glf 2 drd(:k:J<p, where
g = det(gpv) is the matrix determinant of gw), The amount of energy condition violation
is defined as the extent to which Equation (10) can become negative. The value of
Equation (10) provides information about the total amount of energy condition violating
matter that must exist for any given FTL spacetime under study (e.g., warp drives and
traversable wormholes). It was further shown that Equation (10) can be adjusted to
become vanishingly small by appropriate choice of parameters; therefore, examples
can be constructed whereby the energy condition violation can be made arbitrarily
small. But the violation cannot be made to vanish entirely.
Equation (10) also gives the result that traversable wormholes require arbitrarily small
amounts of negative energy to build (whereby Equation (9) serves only as a gross
upper limit) such that within a wormhole spacetime (Reference 59):
PE= O; fp ,dV ➔ 0 (11)
C
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where p,- is the outward radial pressure required to hold a wormhole throat open. The
Gauss-Bonnet Theorem (discussed in Section II-A) predicted this result beforehand.
Equation (11) is a result that is also due to the intrinsic nonlinearity of the general
relativistic field equation. This nonlinearity also impacts the coupling of a finite
spaceship mass with each side of a wormhole's throat (or the mouth on each side of the
throat) leading to a specialized mass conservation law for the combined system of
spacecraft and wormhole: when finite mass spaceships traverse a wormhole they alter
the (equivalent) mass of the wormhole mouths they pass through (Reference 3). The
entrance mouth absorbing the spacecraft gains ( equivalent) mass while the exit mouth
emitting it loses (equivalent) mass.11 (This mass coupling and conservation law takes
into account the possibility that spaceships traversing the wormhole may lose or gain
some momentum and kinetic energy in the process, and it is assumed that the two
mouths are sufficiently far apart that their mutual gravitational interaction is
negligible.) This unusual result suggests, but does not prove, the possibility of a
fundamental limit on the total mass that can traverse a wormhole. The coupled mass
conservation law shows that for a sufficiently large net transfer of mass the final
(equivalent) mass of the exit mouth becomes negative. This is actually a beneficial
result because ANEC violations are required just to hold the wormhole throat open in
the first place. If it appears that a runaway reaction might occur, then it would be
prudent for wormhole engineers to simply "turn off" the wormhole for a brief moment
and then "turn it back on" (i.e., "reset" the wormhole) to restart space transportation
operations.
It is on the basis of the foregoing discussion that traversable wormholes appear to be
the most viable form of FTL transport. However, one still does not know how to
construct a traversable wormhole because general relativity theory only provides a
recipe for the essential geometric and material ingredients required to open and
maintain one, but not the required assembly instructions. Will one need to pull a
traversable wormhole out of the quantum spacetime foam and enlarge it to
macroscopic scale or will there be need to use extremely large spacetime curvatures to
"punch a hole" through space? Or are there construction techniques yet to be
identified? The author is convinced that the answer can only be found through empirical
studies designed to decide whether the present general relativistic recipe is enough to
work with or an additional construction mechanism will be required.
On physical grounds Equation (10) appears to be the correct negative energy/energy
condition violation quantifier. However, further work is needed to establish whether
Equation (10) is the correct quantifier to use overall and whether all (averaged) energy
condition theorems can be extended to include it.
On another note, Borde et al. (Reference 65) have recast the QI conjecture into a new
program which seeks to study the allowed spatial distributions of negative energy
density in quantum field theory. Their study models free massless scalar fields in flat
two-dimensional Minkowski spacetime. Several explicit examples of spacetime averaged
QI were studied to allow or rule out some particular model (spatial) distributions of
negative energy. Their analysis showed that some geometric configurations of negative
energy can either be ruled out or else constrained by the QI restrictions placed upon
11 Similar coupling and conservation results hold for the case of electrically charged matter that traverse a (charged
or uncharged) wormhole.
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the allowable spatial distributions of negative energy. And there were found to be
allowable negative energy distributions in which observers would never encou nter the
accompanying positive energy distribution so long as the QI restrictions and
correspond ing energy conditions are violated . The extent to which the results of Borde
et al. 's analysis can be generalized to a four-d imensional curved spacetime (with or
without boundaries) and interacting fields remain unsolved.
C. OBSERVING NEGATIVE ENERGY IN THE LAB
Negative energy should be observable in lab experiments . The presence of naturally
occurring negative energy regions in space is pred icted to produce a unique signature
correspond ing to lensing, chromaticity and intensity effects in micro- and macro-l ensing
events on galactic and extragalactic/cosmological scales (Reference 66-71) . It has been
shown that these effects provide a specific signature that allows for discrimination
between ordinary (positive energy) and negative energy lenses via the spectra l analysis
of astronomical lensing events. Theoretical modeling of negative energy lensing effects
has led to intense astronomical searches for naturally occurring traversable wormholes
in the universe. Computer model simulations and comparison of their results with
recent satellite observations of gamma ray bursts (GRBs) has shown that putative
negative energy (i.e., traversable wormhole) lensing events very closely resemble the
main features of some GRBs. Other research has found that current observational data
suggests that large amounts of naturally occurring "exotic matter" must have existed
sometime between the epoch of galaxy formation and the present in order to (properly)
quantitatively account for the "age-of-the-oldest-stars-in-the-galactic halo" problem
and the cosmological evolution parameters (Reference 60).
When background light rays strike a negative energy lensing region, they are swept out
of the central region thus creating an umbra region of zero intensity. At the edges of
the umbra the rays accumulate and create a rainbow-like caustic with enhanced light
intensity. The lensing of a negative energy region is not analogous to a diverging lens
because in certa in circumstances it can produce more light enhancement than does the
lensing of an equivalent positive energy region. Real background sources in lensing
events can have non-uniform brightness distributions on their surfaces and a
dependency of their emission with the observing frequency. These complications can
result in chromaticity effects, i.e., in spectral changes induced by differential lensing
during the event. The quantification of such effects is quite lengthy, somewhat model
dependent, and with recent application only to astronomical lensing events. Suffice it to
say that future work is necessary to scale down the predicted lensing parameters and
characterize their effects for lab experiments in which the negative energy will not be of
astronomical magnitude. Present ultrahigh-speed optics and optical cavities, lasers,
photonic crystal (and related switching) technology, sensitive nano-sensor technology,
and other techniques are very likely capable of detecting the very small magnitude
lensing effects expected in lab experiments.
A non-optical scheme for detecting negative energy in experiments was recently
reported by Davies and Ottewill (Reference 72) who studied the response of switched
particle detectors to static negative energy densities and negative energy fluxes. Their
model is based on a free (massless) scalar field in flat four-dimensional Minkowski
spacetime and utilized a simple generalization of the standard monopole detector,
which is switched on and off to concentrate the measurements on periods of isolated
negative energy density (or negative energy flux). The detector model includes an
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explicit switching factor whereby five different switching functions (based on data
windowing theory) are defined and evaluated. In order to isolate the effects of negative
energy a comparison is made for the response of a detector switched on and off during
a period of negative energy density (or negative energy flux) and that switched on and
off in the vacuum. The results shed light on t he response of matter (detectors) to
pulses of negative energy of finite duration, and they showed that negative energy
should have the effect of enhancing deexcitation (i.e., induce cooling) of the detector.
This is the opposite of our experience with detectors that undergo excitation when
encountering "normal" matt
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