UNCLASSIFIED//FOR OFFICIAL USE ONLY Defense Intelligence Reference Document ^^^^^^^^ Acquisition Threat Support 6 April 2010 ICO D : 1 D ecem ber 2009 D IA-08-1004-004 Traversable Wormholes, Stargates, and Negative Energy unclassifiED//ren official use unlf UNCLASSIFIED //TOR OFFICIAL USE ONL¥- Traversable Wormholes, Stargates, and Negative Energy Prepared by: Acquisition Support Division (DWO-3) Defense Warning Office Directorate for Analysis Defense Intelligence Agency AAP 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 to|AAP Person 1 | AAWSA Program Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington, DC 20340-5100. ii U N C LASSI FI E n//FHP nFrTrTi.1 urr pMHf UNCLASSI FIED//FOR OFFICIAL USE ONL¥- 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, fii and Qz, of Minkowski Space Used to Create the Wormhole Throat 611.5 Figure 4. The Same Diagram as in Figure 3 Except as Viewed by an Observer Sitting in Region Qi Who Looks Through the Wormhole Throat and Sees Remote Region Oa on the Other Side... 5 Figure 5. A Thin Shell of (Localized) Mass-Energy Possessing Two Principal Radii of Curvature, pi and pa. 6 iii UNCLASSI FIE n//FHP nrr^.. ..rrn>.Mr UNCLASSI FIED//FOR OFFICIAL USE ONL¥- 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 iv UNCLASSIFIEQ44EQB^EHGMM0MNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ 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 C on tact. 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 s 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. v UNCLASSIFIEQ44EQB^EHGMM0MNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONLY- 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, vi UNCLASSIFIEQX4EQB^EHGMMi0MNfe¥ UNCLASSiFiED//ron ornciAL use onl¥ 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 > 0.05c, 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 UNCI fl<^TFTFP//F^P AFH^Tftl UPC CNtY vii UNCLASSIFIED//BQR OKICIAh USB QNb¥ 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. viii UNCLASSIFIEQXXEQB^SHfiMMMMNfeY UNCLASSI FIED//FOR OFFICIAL USE ONLY- II. A Brief Review of Transversable Wormholes and the Stargate Solution How does one study the physics of FTL spacetim es within the fram ework of general relativity theory? When studying spacetim e physics, the norm al philosophy is to take the general relativistic field equation, add som e form of m atter, m ake sim plifying assum ptions, and then solve to deduce what the geom etry of spacetim e 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 spacetim e coordinates) and four constraints (stress-energy conservation). There is a trem endous body of research that takes exactly this approach, either analytically or num erically. However, this is not the best strategy for understanding worm hole spacetim es. The appropriate strategy is to decide beforehand on a definition of the traversable worm hole that you desire and decide what the spacetim e geom etry should look like. G iven the desired geom etry, use the general relativistic field equation to calculate the distribution of m atter required to set up this geom etry. Then one needs to assess whether the required distribution of m atter is physically reasonable and whether it violates any basic rules of physics, etc. The following sections briefly outline the key results for traversable worm holes. 'The Einstein field equation is: G„v^ R„r- [(1/2) gp.R] = -(BnG/dyr,,,., where G „.is the Einstein curvature tensor, R^iV is the Ricci curvature tensor, R = R^ (the trace of RPJ) is the Ricci scalar curvature, T^. is the stress-energy- m om entum tensor (a m atrix quantity that encodes the density and flux of a m atter source's energy and m om entum ), G is Newton's universal gravitation constant (6.673 x 10-11 Nm 2/kg2L and c is the speed of light. In sim plest term s, this relation states that gravity is a m anifestation of the spacetim e curvature (G J induced by a source of m atter (7^ ). The G reek indices (a v = 0...3) denote spacetim e coordinates, xo...xj, such that X1...X3 = space coordinates and xo = tim e coordinate. A. TRAVERSABLE WORMHOLES Traversable worm holes represent a class of exact m etric solutions of the general relativistic field equation. The solutions are "exact" in the sense that no approxim ations requiring a plethora of physical assum ptions have to be m ade to derive the appropriate spacetim e geom etry. To define a stable traversable worm hole one needs to define the desirable physical requirem ents it is to have in order to achieve the desired FTL travel benefit. The desired requirem ents are the following (Reference 1, 3): • Travel tim e through the worm hole tunnel or throat should be < 1 year as seen by both the travelers and outside static observers. • Proper tim e as m easured 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 /s) when going through the worm hole.2 • Travel speed through the tunnel/throat should be < c. • Travelers (m ade of ordinary m atter) m ust not couple strongly to the m aterial that generates the worm hole curvature; the worm hole m ust be threaded by a vacuum tube through which the travelers can m ove. • There is no event horizon at the worm hole throat. UNCLASSIFIEQX4EOR^SHGMM»E-eNfe¥ UNCLASSiFiED//ron official use onl¥ • There is no singularity of infinitely collapsed m atter residing at the worm hole throat. These requirem ents then lead us to define a spherically sym m etric Lorentzian spacetim e m etric, ds2,2 that prescribes the required traversable worm hole geom etry (Reference 1, 3): 2 A spacetim e m etric, ds2, is a Lorentz-invariant distance function between any two points in spacetim e that is defined by ds2 = g^dx^dx', where g,„. is the m etric tensor which is a 4x4 m atrix that encodes the geom etry of spacetim e and dx^ is the infinitesim al coordinate separation between two points. ds2 = -eWr)c2dt2 +[1 - b(r)/r]~' dr2 + r2d®2 (1) where standard spherical-polar coordinates are used (r: 2nr = circum ference; 0 < 0< jt; 0 < p< 2n), t is tim e (-co < t < co), d®2 = d^ + sm20d^r, ^ (r) is the freely specifiable redshift function that defines the proper tim e lapse through the worm hole throat, and b(r) is the freely specifiable shape function that defines the worm hole throat's spatial (hypersurface) geom etry. The throat is spherically shaped. There are a large num ber of variations of Equation (1), which define traversable worm holes 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 m ath, one can derive the density and flux of energy and m om entum (a.k.a. pressure) encoded by T^ for the source of m atter that is required to produce the traversable worm hole. The results show that the source of m atter m ust have zero or negative energy density and/or an outward radial tension (negative pressure) that is larger than the m agnitude of the energy density (Reference 1-3). Travelers m oving through the throat at very high speed will tend to m easure a negative energy density. These exotic properties are required to create and thread open the worm hole, and stabilize it against collapse (see Section III for m ore details). The technical description of a trip through a spherically sym m etric traversable worm hole is sim ply given by the proper tim e and/or the proper distance of travel through its throat as m easured by space travelers, while the (radial) travel velocity through the throat is v = v(r) < c. The proper tim e of travel as m easured by space travelers going through the worm hole is given by At = f(yv) ^ X, where y = [1 - (v/c)2] 1/2 and the integration (over the elem ent of proper distance, dX) is taken from the worm hole entrance to its exit. The proper distance of travel as m easured by the space travelers is AX = vAt. Rem ote static observers watching the space travelers go through the worm hole will m easure their travel tim e to be At = /(ve^ pt/X and their travel distance will be AX = vAt, where the integration is taken over the sam e lim its as before. 2 UNCLASSIFIEQXXEOR^SHCMb-WE-eNfeY UNCLASSI FIED//FOR OFFICIAL USE ONL¥ Figure 2 shows two diagram s representing the em bedded space (Flam m diagram ) representation of Equation (1), which depicts the geom etry of an equatorial (0= re/2) slice through space at a specific m om ent of tim e (t = const). The top of Figure 2 shows the em bedding diagram for a traversable worm hole that connects two different universes (i.e., an inter-universe worm hole). The bottom diagram in the figure is an intra­ universe worm hole with a throat that connects two distant regions of our own universe. These diagram s serve to aide in visualizing traversable worm hole geom etry and are m erely a geom etrical exaggeration. There was originally one other criterion for defining a traversable worm hole, which was that it m ust be em bedded Figure 2. Inter-Universe Wormhole (top) and Intra­ Universe Wormhole (bottom). within the surrounding (asym ptotically) flat spacetim e. However, Hochberg and Visser (Reference 4) proved that it is only the behavior near the worm hole throat that is critical to understanding the physics, and that a generic throat can be defined without having to m ake all the sym m etry assum ptions and without assum ing the existence of an asym ptotically flat spacetim e in which to em bed the worm hole. Therefore, one only needs to know the generic features of the geom etry 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 worm hole anywhere in spacetim e because it is only the geom etry and physics near the throat that m atters for any analysis. This fact led to the developm ent of a num ber of different traversable worm hole throat designs that are cubic shaped, polyhedral shaped, flat-face shaped, generic shaped, etc. The reader should consult (Reference 3) for a com plete technical review of the various types (and shapes) of traversable worm hole solutions found in general relativity theory. O ne knows that one needs exotic or negative energy to create and thread open a traversable worm hole. So in this regard, one asks what kind of worm hole one can m ake with less effort. To answer this question one can relate the local worm hole geom etry to the global topological invariant of the spacetim e via the G auss-Bonnet Theorem (Reference 5). In the G auss-Bonnet Theorem the local worm hole geom etry is quantified by the energy density, U (in geom etrodynam ic units, q = G = c = 1), threading the worm hole throat plus a spatial curvature constant (for the throat). The global topological invariant of spacetim e is quantified by the Euler Num ber, xe, which is itself defined in term s of the genus, g, representing the num ber of handles (or throats or tunnels) a worm hole can be assigned. These two topological quantities are related via Xe = 2(1 - g). Therefore, the (static) worm hole G auss-Bonnet relation is given by U < Xe/4 or 0 < (1 - g)/2 (Reference 5). (The case for dynam ic traversable worm holes has 3 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ results that are sim ilar to the static case.) This relation will help to decide if a traversable worm hole having one throat, or two or m ore throats should be built and at what energy cost this will incur. The following is the result of our analysis for traversable worm holes having: • 1-handle/throat (i.e., flat torus or spherical worm hole topology) giving g = 1, thus % e = 0, and so U< 0 • 2-handles/throats giving g = 2, thus % e = -2, and so U < -1/2 • 3-handles/throats giving g = 3, thus % e = -4, and so L/ < -1; and so on. It is clear from this that as the num ber of worm hole handles/throats increases the am ount of negative energy required to create the worm hole will grow larger in m agnitude. This is an undesirable dem and on any putative negative energy generator. It is clear then that item (a) defines the m ost desirable engineering solution one can hope for: a 1-handle/throat traversable worm hole that will require zero or (arbitrarily) little negative energy to create. The m agnitude of energy condition violations and the am ount of negative energy required to build a traversable worm hole will be addressed. B. THE "STARGATE" SOLUTION It is a straightforward exercise to design a real "stargate" from worm hole physics. A stargate is essentially a traversable worm hole with a flat-face shape for the throat as opposed to the spherical-shaped throat of the M orris and Thorne worm hole as discussed in the previous section. A traveler going through a stargate will sim ply be shunted into another rem ote spacetim e region within our universe or into another universe. The flat-face traversable worm hole solution is derived from the thin shell (a.k.a. junction condition or surface layer) form alism of the Einstein field equation (Reference 6, 7). The procedure is to take two copies of flat M inkowski space and rem ove from each identical regions of the form Q x w, where Q is a three-dim ensional com pact spacelike hypersurface and 'J? is a tim elike line (tim e axis). Then identify these two incom plete spacetim es along the tim elike boundaries cfl x 9 ?. The resulting spacetim e is geodesically com plete and possesses two asym ptotically flat regions connected by a traversable worm hole. The throat of the worm hole is just the junction SR, which is a two-dim ensional space-like hypersurface, at which the two original M inkowski spaces are identified (see Figures 3 and 4). 4 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ Figure 3. Diagram of a Simultaneous View of Two Remote Compact Regions, Hi and Qz, of Minkowski Space Used to Create the Wormhole Throat 60 (time is suppressed in this diagram) Figure 4. The Same Diagram as in Figure 3 Except as Viewed by an Observer Sitting in Region Qi Who Looks Through the Wormhole Throat 60 and Sees Remote Region Qi (dotted area inside the circle) on the Other Side It is a standard result of the thin shell form alism that the Einstein field equation m ay be cast in term s of the surface stress-energy tensor S) of a thin shell of m atter (or m ass­ energy) localized inside the worm hole throat 5Q (Reference 8): 5 UNCLASSlFIEaUEAB-AAUCMM&E-eNt^ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ .4 4kG (2) where the second fundam ental form K^ isa m atrix that represents the extrinsic curvature of 50 (telling how the worm hole throat is curved with respect to the enveloping four-dim ensional spacetim e), 3^ is the three-dim ensional unit m atrix, and A" kk is the trace (sum of diagonal m atrix elem ents) of K 'j.3 & 'j is a diagonal m atrix having the two principal radii of curvature, pi and p2, of the thin shell as its com ponents (see Figure 5). S ‘j m ay be interpreted in term s of the thin shell's surface energy density o and principal surface tensions, 9 i and 9 2, which are also diagonal m atrix com ponents. 3 The Latin indices (i, j, k = 0 -2) denote three-dim ensional hypersurface coordinates, x°...x2f such that x1, x2 * space coordinates and x° = tim e coordinate. Figure 5. A Thin Shell of (Localized) Mass-Energy Possessing Two Principal Radii of Curvature, pi and p2. thin shell of mass-energy Equation (2) is solved and the com ponents of S’ 'j are found to be (Reference 8): 4nG ^ Pi P2 > (3a) 6 UNCLASSIFIEQX^EOB^SHCMMSMfWr UNCLASSI FIED//FOR OFFICIAL USE ONL¥ 4nG p2 (3b) ^ =- ? 1 4 kG P] (3c) These are the Einstein field equations for a traversable worm hole that is produced by a thin shell of localized m atter. Equations (3a-c) im ply that (for oQ a convex hypersurface) one is dealing with negative surface energy density and negative surface tensions. This is exotic m atter! The negative surface tension (= positive outward pressure, a.k.a. gravitational repulsion) is required to keep the throat open and stable against collapse. To m ake this thin shell worm hole entirely flat requires that one chooses the throat 3Q to have at least one flat face (picture the thin shell in Figure 5 becom ing flat). O n that face the two principal radii of curvature becom e pi = p2 = ® as required by standard three-dim ensional geom etry; therefore, substituting this requirem ent into Equations (3a-c) gives: a = 9t = $ 2 = 0 which is a rem arkable result. This m eans that a traveler encountering and going through such a worm hole­ stargate will feel no tidal gravitational forces and see no exotic m atter threading the throat. A traveler stepping through the throat will sim ply be shunted into another rem ote spacetim e region or into another universe (note: the Einstein field equation does not fix the spacetim e topology, so it is possible that worm holes 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 m atter m uch like a thin film of soap stretched across a loop of wire. C. WHAT A WORMHOLE LOOKS LIKE IN THE REAL WORLD The exotic m atter threading a traversable worm hole throat produces repulsive gravity, which will then deflect light rays going through and around it. (4) Figure 6. A Spherically Symmetric Traversable Wormhole Observed in Space The entrance to the spherically sym m etric M orris & Thorne worm hole looks like a sphere that contains the m irror im age of a whole other universe or rem ote region within our own universe, incredibly shrunken and distorted (see Figure 6). This is an 7 IJNCI ASSIFIE Q XtfQ B O FFTfTffl U C F A N tY UNCLASSI FIED//FOR OFFICIAL USE ONL¥ exam ple of the topological inversion m anifested in worm hole geom etry. The spherical worm hole entrance/exit (a.k.a. the throat) is called a hypersphere because it is the hyperspace surface of our four-dim ensional spacetim e. If one were to travel through the worm hole and look back at it from the other side, then one would see a sphere (the entry way back hom e) that seem ed to contain the whole original universe or hom e region of space near Earth (within your universe). This would look just like a glass Christm as tree ornam ent, which is just a spherical m irror that reflects, in principle, the entire universe around it. A flat-faced worm hole, or stargate, which is also a hypersurface, would not distort the m irror im age of the rem ote space region or other universe seen through it because the negative surface energy density and negative surface tensions of the exotic m atter threading its throat is zero as seen and felt by light and m atter passing through it (recall Equation (4)). See Figures 7 and 8. Figure 7. A Stargate (adapted from Reference 9) UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONLY- Figure 8. A Stargate in Times Square If a sm all worm hole (three or m ore dim ensional) were to begin to appear or even bum p into our local space, one would perceive this process as the occurrence of an unusually bright spot in the sky. Blue and red D oppler shifting of this bright spot would m anifest when the intersection of the worm hole 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 m atter that is required to build traversable worm holes. What exactly is "exotic" m atter? In classical physics the energy density of all observed form s of m atter (fields) is non-negative. What is exotic about the type of m atter that m ust be used to generate traversable worm hole spacetim e is that it m ust have negative energy density and/or negative flux (Reference 10). The energy density is "negative" in the sense that the configuration of m atter fields one m ust deploy to generate and thread a traversable worm hole throat m ust have an energy density, pE (= pc2, where p is the rest-m ass density), that is less than or equal to its pressures/tensions, pi (Reference 1, 3).4 In m any 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-dim ensional space coordinates, xL.x3, indicating the spatial com ponents of vector or tensor quantities. 9 UNCLASSIFIEQXXEOR^SHGMb-WE-eNfeY UNCLASSiFiED//ron ornciAL use onl¥ one can call this m aterial property "exotic." The condition for ordinary, classical (non­ exotic) form s of m atter that all are fam iliar with in nature is that pc > pi and/or pe > 0. These conditions represent two exam ples 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), D om inant Energy Condition (D EC), and Strong Energy Condition (SEC). These energy conditions forbid negative energy density between m aterial objects to occur in nature, but they are m ere hypotheses. Hawking and Ellis (Reference 11) form ulated the energy conditions in order to establish a series of m athem atical hypotheses governing the behavior of collapsed-m atter singularities in their study of cosm ology and black hole physics. M ore specifically, classical general relativity allows one to prove lots of general theorem s about the behavior of m atter In gravitational fields. The im pact or im plications of the D EC or SEC will not be considered because they add no new inform ation beyond the WEC and NEC. The bad news is that real physical m atter is not "reasonable" because the energy conditions are in general violated by sem iclassical quantum effects (occurring at order q) (Reference 3).5 M ore specifically, quantum effects generically violate the average NEC (ANEC). Furtherm ore, it was discovered in 19 65 that quantum field theory has the rem arkable property of allowing states of m atter 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 theorem s of differential geom etry that guarantee that there m ust be a violation of one, som e, or all of the energy conditions (m eaning exotic m atter is present) for all traversable worm hole spacetim es. With respect to creating traversable worm hole spacetim es, "negative energy" has the unfortunate reputation of alarm ing physicists. This is unfounded since all the energy condition hypotheses have been experim entally tested in the laboratory and experim entally shown to be false - 25 years before their form ulation (Reference 13). 5 Planck's reduced constant, q = 1.055 x 10-34 I s. Further investigation into this technical issue showed that violations of the energy conditions are widespread for all form s of both "reasonable" classical and quantum m atter (Reference 14-18). Furtherm ore, Visser (Reference 3) showed that all (generic) spacetim e geom etries violate all the energy conditions. So the condition that pe > pi and/or pe > 0 m ust be obeyed by all form s of m atter in nature is spurious. Violating the energy conditions com m its 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 m agnetic fields. These are borderline exotic, if their tension were infinitesim ally larger, for a given energy density (Reference 11, 19 ). • Squeezed quantum vacuum states: electrom agnetic and other (non-M axwellian) quantum fields (Reference 1, 20). 10 U N CLASSI FI E QXXEOR^BHGMb-WE-eNfeY UNCLASSI FIED//FOR OFFICIAL USE ONLY- • G ravitationally squeezed vacuum electrom agnetic zero-point fluctuations (Reference 21). • Casim ir effect, i.e., the Casim ir vacuum in flat, curved, and topological spaces (Reference 22-28). • O ther quantum fields/states/effects. In general, the local energy density in quantum field theory can be negative due to quantum coherence effects (Reference 12). O ther exam ples that have been studied are D irac field states: the superposition of two single particle electron states and the superposition of two m ulti-electron­ positron states (Reference 29 , 30). In the form er (latter), the energy densities can be negative when two single (m ulti-) particle states have the sam e num ber of electrons (electrons and positrons) or when one state has one m ore electron (electron-positron pair) than the other. Cosm ological inflation (Reference 3), cosm ological particle production (Reference 3), classical scalar fields (Reference 3), the conform al anom aly (Reference 3), and gravitational vacuum polarization (Reference 14-17) are am ong m any other exam ples that also violate the energy conditions. Since the laws of quantum field theory place no strong restrictions on negative energies and fluxes, then it m ight be possible to produce exotic phenom ena such as faster-than-light travel (Reference 31-33), traversable worm holes (Reference 1-3), violations of the second law of therm odynam ics (Reference 34, 35), and tim e m achines (Reference 2, 3, 36). There are several other exotic phenom ena m ade possible by the effects of negative energy, but they lie outside the scope of the present study. This section will review the previously listed item s 1 thru 4 and exam ine their applicability and technical m aturity. D irac 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 m ore desirable option for inducing traversable worm holes 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 m agnetic fields. Suffice it to say that ultrahigh-intensity tabletop lasers have been used to generate extrem e electric and m agnetic field strengths in the lab. Ultrahigh-intensity lasers use the chirped-pulse am plification (CPA) technique to boost the total output beam power. All laser system s sim ply repackage energy as a coherent package of optical power, but CPA lasers repackage the laser pulse itself during the am plification process. In typical high-power short-pulse laser system s, it is the peak intensity, not the energy or the fluence, which causes pulse distortion or laser dam age. However, the CPA laser dissects a laser pulse according to its frequency com ponents, and reorders it into a tim e-stretched lower- peak-intensity pulse of the sam e energy (Reference 37-39 ). This benign pulse can then be am plified safely to high energy, and then only afterwards reconstituted as a very short pulse of enorm ous peak power - a pulse which could never itself have passed safely through the laser system . M ade m ore tractable in this way, the pulse can be 11 UNCLASSIFIEQX^EfiR^SHGMMGE-eNfeY UNCLASSiFiED//ron ornciAL use onl¥ am plified to substantial energies (with orders of m agnitude greater peak power) without encountering intensity-related problem s, The extrem e 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.m 2 (1034 W/m 2 using SLAC as a booster). • Peak Power Pulse < 103 fs. • Electric field, E « 1014 to 1018 V/m [note: com pare this with the critical quantum electrodynam ic (QED ) vacuum breakdown E-field intensity, Ec = Zm^/^e a 1018 V/m , defined by the total rest-energy of an electron-positron pair created from the vacuum divided by the electron's Com pton wavelength] .6 • M agnetic field, B » several x 10 Tesla (note: the critical QED vacuum breakdown B- field intensity is & = E(/c ~ 1010 Tesla). 6 • Ponderom otive 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. • Plasm a Tem peratures > 1010 K. 6 Electron m ass, m e = 9 .11 x 10 31 kg; electron charge, e = 1.602 x 10 ” C. The vigilant reader m ight assert that the electric and m agnetic 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 ~ 1O 1S to 1028 J/m 3 and a m agnetic field energy density ~ 1019 J/m 3. However, there rem ains the problem of engineering this type of experim ent because classical electrom agnetic theory states that every observer associated with the experim ent will see a non­ negative energy density that is x E2 + B2, where E and Bare m easured in an observer's reference fram e. It is not known how to increase the tension in these fields using current physics, but som e new physics m ay provide an answer. This technical problem m ust be left for future investigation. 2. Squeezed Quantum Vacuum Substantial theoretical and experim ental work has shown that in m any quantum system s the lim its to m easurem ent precision im posed by the quantum vacuum zero­ point fluctuations (ZPF) can be breached by decreasing the noise in one observable (or m easurable quantity) at the expense of increasing the noise in the conjugate observable; at the sam e tim e the variations in the first observable, say the energy, are reduced below the ZPF such that the energy becom es "negative." "Squeezing" is thus the control of quantum fluctuations and corresponding uncertainties, whereby one can squeeze/reduce the variance of one (physically im portant) observable quantity provided the variance in the (physically unim portant) conjugate variable is stretched/increased. The squeezed quantity possesses an unusually low variance, m eaning less variance than would be expected on the basis of the equipartition theorem . O ne can in principle 12 UNCLASSI FIEny/FOP QMJCWW ..rr »»^r UNCLASSI FIED//FOR OFFICIAL USE ONL¥- exploit quantum squeezing to extract energy from one place in the ordinary vacuum at the expense of accum ulating excess energy elsewhere (Reference 1). The squeezed state of the electrom agnetic field is a prim ary exam ple of a quantum field that has negative energy density and negative energy flux. Such a state becam e a physical reality in the laboratory as a result of the nonlinear-optics technique of "squeezing," i.e., of m oving som e of the quantum -fluctuations of laser light out of the cos[«)(t - z/c)] part of the beam and into the sin[a(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 transform s the phase space circular noise profile characteristic of the vacuum into an ellipse, whose sem im ajor and sem im inor axes are given by unequal quadrature uncertainties (of the quantized electrom agnetic field harm onic 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 co, these unequal quadrature uncertainties m anifest them selves in the electrom agnetic field oscillator energy by periodic occurrences, which are separated by one quarter cycle, of both sm aller and larger fluctuations com pared to the unsqueezed vacuum . 7 < o is the angular frequency of light, t is tim e, and z denotes the z-axis direction of beam propagation. M orris 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 electrom agnetic field with weaker fluctuations and thus less energy density than the vacuum at locations where cos2[o(t- z/c)] = 1 and sin2[w(t - z/c)] < < 1; but with greater fluctuations and thus greater energy density than the vacuum at locations where cos2[®(t - z/c)] < < 1 and sin2[®(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 (renorm alized) expectation value for the energy density. Therefore, a squeezed vacuum state consists of a traveling electrom agnetic wave that oscillates back and forth between negative energy density and positive energy density, but has positive tim e-averaged energy density. For the squeezed electrom agnetic vacuum state, the energy density pE-sqvac is given by (Reference 46): PE-sqvac =[-p-jsinh^[sinH + cosh^c°s(2tt(7^^ (J/m3) (5) where L3 is the volum e of a large box with sides of length L (i.e., the quantum field is placed in a box with periodic boundary conditions), £ is the squeezed state am plitude (giving a m easure of the m ean photon num ber 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 cosh £, > sinh ^ is m et. It turns out that this is always true for every nonzero value of ^ , so pE-sqvac becom es negative at som e point in the cycle for a general squeezed vacuum state. O n another note, when a quantum state is close to a squeezed vacuum state, there will alm ost always be som e negative energy densities present. Negative energy can be generated by an array of ultrahigh-intensity lasers using an ultra-fast rotating m irror system (Reference 47). In this schem e a laser beam is passed 13 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSiFiED//ron ornciAL use onl¥ through an optical cavity resonator m ade of a lithium niobate (LiNbO s) 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 em erging 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 ~ 1015 second duration. In principle a set of rapidly rotating m irrors could be arranged to separate the positive and negative energy pulses from each other. The light beam would be set to strike each m irror 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 sm all spatial separation of the two different energy pulses would occur at som e distance from the rotating m irror. Another system of m irrors 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. Rotating Redirector Mirror System Positive Energy Pulses Laser & LiNbOj »||i|i Resonator Alternating Pulses of Negative& Positive Energy Negative Energy Pulses Concentrated Negative Energy Figure 9. Conceptual Squeezed Light Negative Energy Generator The rotating m irror system can actually be im plem ented via non-m echanical m eans. A cham ber 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 m irror to form a standing wave within the sodium cham ber. 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. 14 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONL* Another way to generate negative energy via squeezed light would be to m anufacture extrem ely reliable light pulses containing precisely one, two, three, etc., photons apiece and com bine them together to create squeezed states to order (Reference 47). Superim posing m any 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 dem onstrated the feasibility of using photonic crystal waveguides (m ixing together the classical and quantum properties of optical m aterials) to engineer light sources that produce beam s containing precisely one, two, three, etc., photons. For exam ple, researchers at M elbourne University used a m icrowave oven to fuse a tiny diam ond, just l/1000th of a m illim eter long, onto an optical fiber, which could be used to create a single photon beam of light (Reference 48, 49 ). The com bining of different beam s containing different (finite integer) num bers of photons is already state-of-the- art practice via num erous optical beam com bining m ethods that can readily be extended to our application. Figure 11 Alternative Conceptual Squeezed Light Negative Energy Generator 15 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ Finally, Ries et al. (Reference 50) experim entally dem onstrated the very first sim ple, scalable squeezed vacuum source in the laboratory that consisted of a continuous-wave diode laser and an atom ic rubidium vapor cell. The experim ental tools one needs to begin exploring the generation of negative energy for the purpose of creating traversable worm holes are just now becom ing available. 3. Gravitationally Squeezed Electromagnetic ZPF A natural source of negative energy com es from the effect that gravitational fields (of astronom ical bodies) in space have upon the surrounding quantum vacuum . For exam ple, the gravitational field of the Earth produces a zone of negative energy around it by dragging som e of the virtual quanta (a.k.a. vacuum ZPF) downward. This concept was initially developed in the 19 70s as a byproduct of studies on quantum field theory in curved space (Reference 25). However, Hochberg and Kephart (Reference 21) derived an im portant application of this concept to the problem of creating and stabilizing traversable worm holes. 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 m atter having negative energy density. The analysis, via quantum optics, showed that gravitation itself provides the m echanism for generating the squeezed vacuum states needed to support stable traversable worm holes. The production of negative energy densities via a squeezed vacuum is a necessary and unavoidable consequence of the interaction or coupling between ordinary m atter and gravity, and this defines what is m eant by gravitationally squeezed vacuum states. The m agnitude of the gravitational squeezing of the vacuum can be estim ated from the quantum optics squeezing condition for given transverse m om entum and (equivalent) energy eigenvalues, j, of two electrom agnetic ZPF field m odes, such that this condition is subject to j ^ 0, and it is defined as (Reference 21): te2 Mo _ 8r, A?0 I ^0 J M 1 (6) where 2 is the ZPF m ode wavelength, r is the radial distance from the center of the astronom ical body in question, Ro is the radius of the Earth (6.378 x 106 m ), Mo is the m ass of the Earth (5.9 72 x 1024 kg), M is the m ass of the astronom ical body, and rs is the Schwarzschild radius of the astronom ical body.8 Note that rs is only a convenient radial distance param eter for any object under exam ination 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 m akes the equation sim pler in form . Also note that Equation (6) contains an extra factor of two (com pared to the j derived in Reference 21) in order to account for the photon spin. The squeezing condition plus Equation (6) sim ply states that substantial gravitational squeezing of the vacuum occurs for those ZPF field m odes with X > 8^ of the m ass in question (whose 8 r, = ZGM/c2. According to general relativity theory, this is the critical radius at which a spherically sym m etric m assive body becom es a black hole, i.e., at which light is unable to escape from the body's surface. UNCLASSIFIEaUEOB-AAUCMM&E-eNt^ UNCLASSiFiED//ron ornciAL use onl¥ gravitational field is squeezing the vacuum ). The corresponding local vacuum state energy density is: pe-gsvac = -27rqc/X4 . 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 becom es im portant for sam ple bodies and their associated pE-gsvac. The table shows that in the case of the Earth, Jupiter and the Sun, the squeezing effect is extrem ely feeble because only ZPF m ode wavelengths above 0.2 m to 78 km are affected, each having very m inute pe-gsvac. For a solar m ass black hole (radius of 2.9 5 km ), the effect is still feeble because only ZPF m ode wavelengths above 78 km are affected. But note that Planck m ass bodies will have an enorm ously strong negative energy zone surrounding them because all ZPF m ode 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 com parison because the squeezing effect includes all ZPF m ode wavelengths above 6.50 x 10 53 m . Furtherm ore, a body sm aller than a nuclear diam eter (= IO -16 m ) and containing the m ass of a m ountain (= 1011 kg) has a fairly strong negative energy zone because all ZPF m ode wavelengths above 10 15 m will be squeezed. In each of these cases, the m agnitude of the corresponding pe-gsvac is very large. However, the estim ates for the wavelengths in Table 1 m ight be too sm all. Ford (private com m unication, 2007) argues that Reference 21 is in error because spacetim e is flat on scales sm aller than the local radius of curvature, which is defined by the inverse square root of the typical Riem ann curvature tensor com ponent in a local orthonorm al fram e, or Xc ~ (P^/GM)1^. According to Ford, only ZPF m odes with 1 > Xc will be squeezed by the gravitational field. This leads to a different local vacuum state energy density (for r» rs) (Reference 15): (7) 17 UNCLASSIFIEaUEOB-AAUCMM&E-eNt^ UNCLASSI FIED//FOR OFFICIAL USE ONL¥ Table 1. Substantial Gravitational Squeezing Occurs for Vacuum ZPF When X > Siers Mass of body (kg) rs (m) X (m) PE-gsvac (J/m 3) Sun = 2.00 x 1030 2.9 5 x 103 > 78.0 x 103 -1.69 x 10 44 Jupiter = 1.9 0 x 1027 2.82 > 74 -2.08 x IO’32 Earth = 5.9 8 x 1024 8.87 x 10 3 > 0.23 -2.23 x 10 22 Typical m ountain « 1011 »io-16 > IO 15 -6.25 x 1035 Planck m ass = 2.18 x 10‘8 3.23 x 10 35 > 8.50 x 10 34 -1.20 x 10108 Proton = 1.67 x IO -27 2.48 x 10 s4 > 6.50 x 10 s3 -3.50 x 10184 For exam ple, near the surface of the Earth (r« Ao. M = Mo), Xc * 2.42 x 1011 m and hence, Equation (7) gives pE-gsvac = -1.82 x IO -70 J/m 3. Com pare these values with X > 0.23 m and pE-gsvac « -2.23 x 10-22 J/m 3 in Table 1. The resolution of this disagreem ent rem ains an open question. O ne 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 m atter. Naturally occurring traversable worm holes in the vicinity of astronom ical m atter would therefore becom e possible. 4. Vacuum Field Stress: Negative Energy from the Casimir Effect The Casim ir effect is by far the easiest and m ost well known way to generate negative energy in the lab. The Casim ir effect that is fam iliar to m ost people is the force that is associated with the electrom agnetic quantum vacuum (Reference 51). This is an attractive force that m ust exist between any two neutral (uncharged), parallel, flat, conducting surfaces (e.g., m etallic plates) in a vacuum . This force has been well m easured and it can be attributed to a m inute im balance in the vacuum electrom agnetic zero-point energy density inside the cavity between the conducting surfaces versus the vacuum electrom agnetic 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. Figure 12. Schematic of the Casimir Effect 18 UNCLASSIFIEQXtffiR^SHGMMSE-eNfeY UNCLASSiFiED//ron ornciAL use onl¥ It turns out that there are m any different types of Casim ir effects found in quantum field theory (Reference 22-24, 28, 55). For exam ple, if one introduces a single infinite plane conductor into the M inkowski (flat spacetim e) vacuum by bringing it adiabatically from infinity so that whatever quantum fields are present suffer no excitation but rem ain in their ground states, then the vacuum (electrom agnetic) stresses induced by the presence of the infinite plane conductor produces a Casim ir effect. This result holds equally well when two parallel plane conductors (with separation distance d) are present, which gives rise to the fam iliar Casim ir effect inside a cavity. Note that in both cases, the spacetim e m anifold is m ade incom plete 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 Casim ir vacuum . The generic expression for the energy density of the Casim ir effect is pee = -A^c)^, where A = ^ (D )/8n2 in spacetim es of arbitrary dim ension D (Reference 22-24). The appearance of the zeta-function Q(D) is characteristic of expressions for vacuum stress-energy tensors,T^. In our fam iliar four-dim ensional spacetim e (D = 4), A = n2/720. To calculate ^ vfor a given quantum field is to calculate its associated Casim ir effect. Analogs of the Casim ir effect also exist for fields other than the electrom agnetic field. When considering the vacuum state of other fields, one m ust consider boundary conditions that are analogous to the perfect-conductor boundary conditions for the electrom agnetic field at the surfaces of the plates (Reference 22-24, 28). O ther fields are not electrom agnetic in nature, that is to say they are non-M axwellian, and so the perfect-conductor boundary conditions do not apply to them . It turns out that com plete m anifolds exhibit what is called the topological Casim ir effect for any non-M axwellian fields. In order to define boundary conditions for other fields the conductor boundary conditions are replaced and M inkowski spacetim e by a m anifold of the form 9 1 x S (i.e., a product space), where 9 1 is the real line defining the tim e dim ension for this particular product space and S is a flat three-dim ensional m anifold having any one of the following topologies: 9 12 x S1, 9 1 x T2, T3, 9 1 x K2, etc., 9 1 being the real line that defines any linear space dim ension (e.g., 9 1 = line, SR2 = two-dim ensional plane, etc.), Tn being the n-torus, K2 the two-dim ensional Klein bottle, S1 the circle, etc. The case S = 9 12 x S' has the closest resem blance to the electrom agnetic Casim ir effect, the difference being that instead of im posing conductor boundary conditions, one im poses periodic boundary conditions on som e of the space coordinates in the three­ dim ensional m anifold. When im posing this topological constraint on the field theoretic calculation of the topological Casim ir effect (for linear m assless fields), one finds that the generic expression for the energy density is also pce = -A(qc)^ , where A = ±df (tt/9 0), df is the num ber 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 ferm ion fields (giving a positive energy density). If one were to adm it spin structure in the m anifolds described above and the field is spinorial, then there is another im portant subtlety that m ust be taken into account when evaluating T^, However, this introduces an additional com plexity 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 19 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSiFiED//ron ornciAL use onl¥ m anifold, but of the fibre bundle itself has an effect on T^. In addition to this, there are (com pactified) extra-space dim ensional quantum field (i.e., D -Brane or "brane world") analogs of the Casim ir effect yet to be explored. But a detailed consideration of these for producing traversable worm holes is beyond the scope of this report and will be left for future investigation. As a final note, the m ethods used to obtain the electrom agnetic 7^ "'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 sim ilar to the case of intersecting conductors. These geom etries have also been evaluated for the case of dielectric m edia. These particular cases will not be considered further since there are technical subtleties involved that com plicate 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 m oving reflecting (conducting) surface (a.k.a. a m oving m irror). A m irror m oving with increasing acceleration generates a flux of negative energy that em anates from its surface and flows out into the space ahead of the m irror (Reference 25, 56). This is essentially the sim ple case of an infinite plane conductor undergoing acceleration perpendicular to its surface. If the acceleration varies with tim e, the conductor will generally em it or absorb photons (i.e., exchange energy with the vacuum ), even though it is neutral. This is an exam ple of the well- known quantum phenom enon of param etric excitation. The param eters of the electrom agnetic field oscillators (e.g., their frequency distribution function) change with tim e owing to the acceleration of the m irror (Reference 57). However, this effect is known to be exceedingly sm all, and it is not the m ost effective way to produce negative energy. This schem e will not be considered any further. 6. Casimir Effect: Negative Energy for Traversable Wormholes The electrom agnetic Casim ir effect can be used in principle to create a traversable worm hole. The energy density pee = -(7r2^ c/720)r^ 4 within a Casim ir cavity is negative and m anifests itself by producing a force of attraction between the cavity walls. But cavity dim ensions m ust be m ade exceedingly sm all in order to generate a significant am ount of negative energy. In order to use the Casim ir effect to generate a spherically sym m etric traversable worm hole throat of radius nitmai, there is need to design a cavity m ade of perfectly conducting spherically concentric thin plates with a plate separation d of (Reference 2): ’^ oJ^ Vj (8) = (4.05x10'")^ (m) To counteract the collapse of the cavity due to the Casim ir Force acting between the plates, the plates will have equal electric charges placed upon them to establish 20 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfe¥ UNCLASSI FIED//FOR OFFICIAL USE ONLY- adequate Coulom b repulsion.9 Equation (8) shows that a 1 km radius throat will require a cavity plate separation of 1.28 x io46 m (sm aller than a nuclear diam eter), which gives pec = -1.62 x 1036 J/m 3 for this configuration. In contrast, a worm hole with a throat radius of 1 AU will require a plate separation of 1.57 x IO -12 m (or 35% sm aller than the electron's Com pton wavelength), which results in an energy density of-7.14 x 1019 J/m 3.10 There is no technology known today that can engineer a cavity with such m inuscule plate separations. In addition, such m inuscule plate separations are unrealistic because the Casim ir effect switches over to the non-retarded field behavior (~ d~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 schem e will not be considered any further. However, future work will be necessary to elucidate whether the various quantum field analogs of the Casim ir effect can provide a m ore reasonable technical solution to this problem . 9 In a detailed analysis the electrostatic energy required to support the Coulom b repulsion between the plates would be considered separately. 10 M ean Earth-Sun distance, 1 AU = 1.50 x 1011 m . IV. Constructing a Traversable Wormhole is not Easy A. NEGATIVE ENERGY REQUIREMENTS AND ENERGY CONDITION VIOLATIONS O ne knows how to m ake sm all quantities of negative energy in the lab. But one does not know if it is possible to m ake large quantities of negative energy. It was pointed out in Section III that one, som e, or all of the classical energy conditions m ust be violated in order to build a traversable worm hole. And it was also cautioned that this was not a showstopper because the energy conditions have all been violated by nature or by lab experim ent prior to their form ulation. However, the reader should be forewarned that there are a num ber of published claim s that the energy condition violations can be avoided. These claim s are just sem antic gam es whereby investigators universally invoke the following scenario: divide the total stress-energy into weird m atter plus norm al m atter, push all the energy condition violations into the weird m atter so that the norm al m atter does not violate the energy conditions. G iven that the energy conditions are not absolute, such rearranging approaches are not necessary. Traversable worm hole throats violate the NEC (or ANEC). So how big a violation is required? The answer is that there is only need to calculate the am ount of negative energy that will be needed to generate and hold open a worm hole throat. A sim ple form ula for short-throat worm holes using the thin shell form alism gives this quantity in term s of the equivalent m ass (note: the energy density derived from the general relativistic field equation is too com plex to use for this m ass com parison) (Reference 3): 21 UNCLASSIFIEQX4EfiR^SHGMM»E-eNfeY UNCLASSI FIED//FOR OFFICIAL USE ONL¥ C2 M.= ---S> «_ wh G = -(1.35xl027 ^) 1,hM 1 meter = -(0.71 A/,) ljte& ■ J 1 meter (9 ) where M Wh is the (equivalent) m ass required to build the worm hole, /.throat is a suitable m easure of the linear dim ension (width or diam eter) of the throat, and M j is the m ass of the planet Jupiter. O ne can also obtain the required energy, EWh, by m ultiplying both sides of Equation (9 ) by c2 . Equation (9 ) shows that a m ass of -0.71 M j will be required to build a worm hole 1-m in size. As the worm hole size increases, the m ass requirem ent grows negative-large. Table 2 presents a tabulation of the required negative (equivalent) m ass as a function of sam ple worm hole throat sizes. After being alarm ed by the m agnitude of the results, one should note that M Wh is not the total m ass of the worm hole as seen by rem ote observers. The non-linearity of the general relativistic field equation dictates that the total m ass is zero (actually, the total net m ass being positive, negative or zero in the Newtonian approxim ation depending on the details of the negative energy configuration constituting the worm hole system ). Finally, Visser et al. (Reference 59 ) dem onstrated the existence of spacetim e geom etries containing traversable worm holes that are supported by arbitrarily sm all quantities of negative energy, and this was proved to be a general result. The next section will expand on this further. Table 2. Negative Equivalent Mass Required for Traversable Wormhole Xthroat (iTl) Mwh 1000 -709 .9 M j 100 -71 M j 10 -7.1 M j 1 -0.71 M j 0.1 -22.6 M ® 0.01 -2.3 M ® M j = 1.9 0 x 1027 kg, M o = 5.9 8 x 1024 kg B. PHYSICAL CONSTRAINTS ON NEGATIVE ENERGY The Quantum Inequalities (QI) conjecture is an extension of the Heisenberg Uncertainty Principle to curved spacetim es. M uch research has been conducted around this one topic alone. The literature is too num erous to cite here but the reader should consult (Reference 10) and (Reference 46) for detailed inform ation. The QI conjecture relates (via m odel dependent tim e integrals of the energy density along geodesics) the energy density of a free quantum field and the tim e during which this energy density is observed. This conjecture was devised as an attem pt to quantify the am ount of 22 UNCLASSI FIEny/FOP nmcTn. ..rr m.Mr UNCLASSiFiED//ron ornciAL use onl¥ negative energy or energy condition violations required to build a traversable worm hole spacetim e. Investigators have invoked the QI to rule out m any of the m acroscopic worm hole spacetim es. When generating negative energy the QI postulate that: a) the longer the pulse of negative energy lasts, the weaker it m ust be; b) a pulse of positive energy m ust follow and the m agnitude of the positive pulse m ust exceed that of the initial negative pulse; and c) the longer the tim e interval between the two pulses, the larger the positive pulse m ust be. This actually sounds quite reasonable on energy conservation grounds until one discovers that the Casim ir effect and its non-M axwellian quantum field analogs violate all three conditions. There are also a num ber of squeezed vacuum sources and D irac field states that m anifestly violate all three conditions. Cosm ological inflation, cosm ological particle production, classical scalar fields, the conform al anom aly, and gravitational vacuum polarization are am ong the m any other exam ples that also violate the QI. Visser (Reference 60) also points out that observational data indicate that large am ounts of "exotic m atter" are required to exist in the universe in order to account for the observed cosm ological evolution param eters. The QI have also not been verified by laboratory experim ents. The assum ptions used to derive the QI and the efficacy of their derivation for various cases has been called into question by num erous investigators. Krasnikov (Reference 61) constructed an explicit counterexam ple for generalized FTL spacetim es showing that the relevant QI breaks down even in the sim plest FTL cases. And he also addressed Fewster's (Reference 62) technical argum ents on this issue. It is im portant to point out that the Qis have been m ainly proven for free m assless scalar fields in flat two-dim ensional M inkowski spacetim e, so there rem ains the unanswered questions of extending the QI into a four­ dim ensional curved spacetim e m odel (with or without boundaries) and how m uch 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 am ount of negative energy or energy condition violations required to build a traversable worm hole spacetim e. They propose a quantifier in term s of a spatial volum e integral, which am ounts to calculating the following definite integrals (Reference 59 , 63, 64): fft^ O ; J(fe + A)dV< 0 (10) with an appropriate choice of the integration m easure dV (= ^r^dr or g1/2drd0d