AAWSAP DIRD, Traversable Wormholes, Stargates, and Negative Energy, April 2010

UNCLASSIFIED//1''111l '11'1'1@1JltL t:ISE 8Ptl¥ Defense Intelligence Reference Document Acquisition Threat Support 6 April 2010 !COD: 1 December 2009 DIA-08- 1004-004 Traversable Wormholes, Stargates, and Negative Energy UNCLASSIFIED/,'FQR 8FFlelAL tJ!I! OHLf UNCLASSIFIED//F8A &FFIOltlil YSE 8HL1f 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. UNCLASSIFIFQ { {FOR AfliiliCilil.L WSE 8HLY ii UNCLASSIFIED/fFOA OFFl&I.t.k Y&li &PtLY 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 4. The Same Diagram as in Figure 3 Except as Viewed by an Observer Figure 5. A Thin Shell of (Localized} Mass-Energy Possessing Two Principal Radii 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 Sitting in Region n1 Who Looks Through the Wormhole Throat and Sees Remote Region n2 on the Other Side....................................................... 5 of Curvature, p1 and p2 ............................................................................ 6 UNCLASSIFIEQ{ (FOR AFEICI.IJ.k Y&li 8HLY iii UNCLASSIFIED/fFOA OFFl&I.t.k Y&li &PtLY 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 UNCLASSIFIEQ{ {FOR AFEICIAk Y&li 8HLY iv UNCLASSIFIED/fFOA OFFl&I.t.k Y&li &PtLY 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. UNCLASSIFIEQ{ {FOR AFEICIAk Y&li 8HLY V UNCLASSIFIED/fFOA OFFl&I.t.k Y&& 8PtL\f Worm/Joie 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, UNCLASSIFIEQ{ {FOR AFEICIAk Y&& 8HLY vi UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 - 10100 (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 UNCLASSIFIEQ{ {FOR AFEICIAk Y&& 8HLY vii UNCLASSIFIED/}FOR OFFICIO Is lallii QI\II.¥ 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. UNCLASSIFIEQ{ {FOR AFFICIAk W&E 8HLY viii , UNCLASSIFIED/fFOA OFFl&I.t.k Y&& 8PtL\f II. A Brief Review of Transversable Wormholes and the Stargate Solution How does one study the physics of FTL spacetimes within the framework of general relativity theory? 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 that you desire and decide what the spacetime geometry should 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 the 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 should be ::; 1 go (go is the acceleration of gravity near the Earth's surface, 9.81 m/s2) when going through the wormhole. • Travel speed through the tunnel/throat should be < c. • Travelers (made of ordinary matter) must not couple strongly to the material that generates the wormhole curvature; the wormhole must be threaded by a vacuum tube through 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 Nm2/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. UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY 1 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f • 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!<rlc 2dt2 +[1-b(r)/rr' dr2 + r2d02 (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- + sin2Bdq}, r/1,.r) is the freely specifiable redshift function that defines the proper time 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, ds2, is a Lorentz-invariant distance function between any two points in spacetime that is defined by ds2 = 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. UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 2 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 Figure 2. Inter-Universe Wormhole (top) and Intra­ within the surrounding (asymptotically) 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 topological 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 UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY 3 UNCLASSIFIED/fFOA OFFl&I.t..k YS& 8Ptk\f 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-dimensional space-like hypersurface, at which the two original Minkowski spaces are identified (see Figures 3 and 4) . UNCLASSIFIEQ { {FOR OFFICIAL WS& 8HLY 4 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f _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 thin shell of matter (or mass­ energy) localized inside the wormhole throat an (Reference 8): UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 5 UNCLASSIFIED/ fFOA OFFI&il.t..k Y&& 8Ptk\f 4 S = - - c- (K' -8'K k) (2) J i J k 4rrG 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 the trace (sum of diagonal matrix elements) of KV K ;i is a diagonal matrix having the two principal radii of curvature, p1 and p2, of the thin 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 x1, x2"' space coordinates and><° = time coordinate. UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY 6 UNCLASSIFIED/ fFOA OFFl&I.t..k YS& 8Ptk\f (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 flat 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 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 UNCLASSIFIEQ { {FOR AEEiliCili.li.L YS& OHLY Figure 6. ASpherically Symmetric Traversable Wormhole Observed in Space 7 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 region 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) UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 8 UNCLASSIFIED/ fFOA OFFl&I.t.k Y&& 8PtL\f 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 this process as the occurrence of an unusually bright spot in the sky. Blue and red Doppler shifting of this bright spot would manifest when the 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 (= pc2, 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 4 From this point forward in the text, all Latin indices (e.g., i, j, k =1...3) that are affixed to physical quantities denote the usual 3-dimensional space coordinates, x1...x3, indicating the spatial components of vector or tensor quantities. UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY 9 UNCLASSIFIED/fFOA OFFl&I.t..k YS& 8Ptk\f 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 containing 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 reputation of alarming physicists. This is unfounded since all the energy condition hypotheses have been experimentally tested in the laboratory and experimentally shown to be false - 25 years before their formulation (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. UNCLASSIFIEQ { (FOR OFFICIAL WS& 8HLY 10 UNCLASSIFIED/ fFOA OFFl&I.t..k YS& 8Ptk\f • Gravitationally squeezed vacuum electromagnetic zero-point fluctuations (Reference 21). • Casimir effect, i.e., the Casimir 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-pulse 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 UNCLASSIFIEQ { {FOR AEEiliCili.li.L WS& OHLY 11 UNCLASSIFIED/fFOA OFFl&I.t..k YS& 8Ptk\f 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 "' 1019 to 1030 W/m2 (1034 W/m2 using SLAC as a booster). • Peak Power Pulse :-::: 103 fs. • Electric field, E"' 1014 to 1018 V/m [note: compare this with the critical quantum electrodynamic (QED) vacuum breakdown E-field intensity, Ee = 2me2c3/rie"' 1018 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"' 1010 Tesla). • Ponderomotive Acceleration of Electrons "' 1017 to 1030 go (go is the acceleration of gravity near the Earth's surface, 9.81 m/s2). • Light Pressure"' 109 to 1015 bars. • Plasma Temperatures > 1010 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 ~1016 to 1028 J/m 3 and a magnetic field energy density ~ 1019 J/m3 . 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. This 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 10-3i kg; electron charge, e = 1.602 x 10-19 C. UNCLASSIFIEQ { (FOR OFFICIAL WS& 8HLY 12 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 semiminor 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 cos2[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 = ( 21/"} inh s[sinh s + cosh scos ( 2w(t- z I c) + 8)] (J / m3) (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. UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 13 • • UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f through an optical cavity resonator made of a lithium niobate (LiNbQ3) crystal that is shaped like a cylinder with rounded silvered ends to reflect light. The resonator will act to produce a secondary lower frequency light beam in which the pattern of photons is rearranged into pairs. The squeezed light beam emerging from the resonator will contain pulses 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 would ensure that the negative energy pulses would be reflected at a slightly different angle 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. + + ). + Positive Energy + r, + Pulses Rotating Redirector + Mirror Sy&.em Laser & -+-+-+­ LiNl:DJ •IIIIJ Resonator 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 pulses by careful design (Reference 41). An illustration of this is shown in Figure 10. UNCLASSIFIEQ { {FOR OFFICIAL Y&& 8HLY 14 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 15 UNCLASSIFIED/fFOA OFFl&I.t.k Y&li &PtLY Finally, Ries et al. (Reference SO) experimentally demonstrated the very first simple, scalable squeezed vacuum source in the laboratory that consisted of a continuous-wave diode laser and an atomic rubidium vapor cell. The experimental tools one needs to begin exploring the generation of negative energy for the purpose of creating traversable wormholes are just now becoming available. 3. Gravitationally Squeezed Electromagnetic ZPF A natural source of negative energy comes from the effect that gravitational fields (of astronomical bodies) in space have upon the surrounding quantum vacuum. For example, the gravitational field of the Earth produces a zone of negative energy around it by dragging some of the virtual 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 this 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 1024 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 8 r, = 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. 16 UNCLASSIFIEQ { {FOR OFFICIAL Y&li 8HLY UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f gravitational field is squeezing the vacuum). The corresponding local vacuum state energy density is: pE-gsvac = -211h1c!A4. 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 (;::: 1011 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): 2n2hc PE-gsvac = - ~ 2n2hc == - -l- 4- (7) c 2n2hG2M 2 (JI m3) UNCLASSIFIEQ { {FOR OFFICIAL W&& 8HLY 17 UNCLASSIFIED/fFOA OFFl&I.t..k YS& 8Ptk\f Table 1. Substantial Gravitational Squeezing Occurs for Vacuum ZPF When )., ~ 8 1trs Mass of body (kg} rs (m} )., (m} PE-gsvac ( J / m3 } Sun = 2.00 x 1030 2.95 X 103 2': 78.0 X 103 - 1.69 X lQ-44 Jupiter = 1.90 x 1027 2.82 2': 74 - 2. 08 X lQ­32 Earth = 5.98 x 1024 8.87 X 10-3 2': 0.23 - 2.23 X 10-22 Typical mountain:::::: 1011 :::::: 10-16 2': 10-15 - 6,25 X 1035 Pla nck mass= 2.18 x 10-8 3.23 X 10-35 2': 8 .50 X 10-34 -1. 20 X l O lOB Proton = 1.67 x 10-27 2.48 X 10-54 2': 6.50 X 10-5) - 3,50 X 10184 For example, near the surface of the Earth (r ~ Ro, M = Mo), 11.c:::::: 2.42 x 1011 m and hence, Equation (7) gives PE-gsvac:::::: - 1.82 x 10-70 J/m3. Compare these values with 11. 2': 0.23 m and PE-gsvac:::::: - 2.23 x 10-22 J/m3 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 astronomical matter. Naturally occurring traversable wormholes in the vicinity of astronomical matter would therefore become possible. 4. Vacuum Field Stress: Negative Energy from the Casimir Effect The Casimir effect is by far the easiest and most well known way to generate negative energy in the lab. The Casimir effect that is familiar to most people is the force that is associated with the electromagnetic quantum vacuum (Reference 51). This is an attractive force that must exist between any two neutral (uncharged), parallel, flat, conducting surfaces (e.g., metallic plates) in a vacuum. This force has been well measured and it can be attributed to a minute imbalance in the vacuum electromagnetic zero-point energy density inside the cavity between the conducting surfaces versus the vacuum electromagnetic zero-point energy density in the free-space region outside of the cavity (Reference 52-54). See Figure 12 for an illustration of this effect. UNCLASSIFIEQ { {FOR AEEICili.li.L WS& 8HLY Figure 12. Schematic of the Casimir Effect 18 UNCLASSIFIED/fFOA OFFI&il.t..k Y&& 8Ptk\f 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 giv