In this last part, I consider possible shortcuts in a "multiverse". One of the more controversial scenarios for superluminal travel entails a different sort of universe than the one we tend to think of. Most commonly it has been popularized by David Deutsch as the ‘multiverse’ in his book,

*The Fabric of Reality*. However, there are a number of variants of this. One hyper-dimensional form is based on a hyper-toroid such as depicted in the top figure.

Each longitude circle is a quotient space q1 and each latitude circle is a quotient space q2. More exactly, q2 denotes the fifth dimensional connector that links multiple parallel universes. Each of these can be represented by one single longitudinal loop marking off a coordinate on q1 (e.g. Θ). Since time can be broken down in fundamental units of tau (τ) then each q1 is separated by its neighbor universe by one tau. This neighbor is impossible to access from within the fixed coordinate quotient space (q1) but can be accessed if one can find a way to get to q2 and traverse it. Thus, in simplest terms, each “slice” q1 can be regarded as an alternate or parallel universe.

More explicitly, one may adopt absolute toroidal (Θ,φ) coordinates to locate events in the multiverse system. There are a number of fascinating aspects of this model. First, the toroidal hypersphere implies an endlessly repeating universe that is the same in each new cycle - instead of being different (i.e. with different physical constant as the reprocessed model demands). The key to this aspect is the fact that the radius R remains constant.

A second intriguing aspect is that the exact same point of space-time occurs for each "beginning" (Big Bang) and "end" ("Big Crunch"). This point is easily identified in the diagram as the most constricted part of the interior "hour-glass" shape defining the inner wall of the torus-hypersphere.

Third, the overall configuration is exactly the same as that defined for the high-energy "twistors" used in the 1980's by Roger Penrose and his colleagues at Oxford. The twistors are discrete points of space, each defined by 4 real numbers and 4 imaginary numbers. Without getting bogged down in the mathematics, I find it more than coincidental that the shape of the hypersphere model cosmos and the high-energy twistor are the same. Could it be that this relationship is holographic? That is, each twistor of space-time is the miniaturized "image" of the whole hypersphere?

If so, then a torsion acting in a localized space-time of the hypersphere could serve to twist it, permitting a shortcut. Under what conditions might such a torsion arise? One would require fiber bundles, which then have capacity for expansion to a sub-space large enough to provide a folded-space corridor. (A 'Klein bottle' is an example of a fiber bundle see, e.g. http://en.wikipedia.org/wiki/Klein_bottle).

The important geometric concept here, discerned from the situation shown, is that two distinct parallel universes (recall these are given by different longitudinal coordinates, Θ) are linked by the bundle’s vortex. Thus, such a linkage between parallel universes (within the defined multi-verse) is equivalent to an enormous short cut for the craft that seeks to exploit it.

What would the conditions have to be for linkage? Let the two parallel universes be distinguished by a 1-τ difference in fundamental time parameter, viz. [1 + 2τ] and [1 + 3τ], then we would require for connection, a mapping such that:

f:X -> X = f(Θ,φ) = (Θ, 2φ)

f:X -> X = f(Θ,φ) = (Θ, 3φ)

which means the absolute coordinate φ is mapped onto itself 2 times for [Universe A] and mapped onto itself 3 times for [Universe B]. Clearly, there’ll be coincidences for which: f(Θ,2φ) = f(Θ,3φ) wherein the two universes will 'interweave' a number of times. For example, such interweaving will occur when φ = π/2 in [A] and φ = π/3 in [B]. The total set or system of multiple points obtained in this way is called a Synchronicity Matrix. The distinguishing feature of this matrix is that once a single point is encountered, it is probable that others will as well. If one hyperspace transformation can occur linking parallel universes, A and B, then conceivably more such transformations can occur, linking A and C, D and E, or (exhausting the Latin alphabet), X1 and X oo(_1)

What if both absolute toroidal coordinates (Θ,φ) map into themselves the same number of times? Say, something like:

f:X -> = f(Θ, φ) = (2Θ, 2φ): Universe A

f:X -> = f(Θ, φ) = (3Θ, 3φ): Universe B

For example, given the previous conditions for coordinate φ, now let 2Θ = 3Θ for discrete values of Θ (e.g. 2π). For all multiples of 2π, the same toroidal cosmos will be experienced - if the absolute time coordinates are equal (e.g. π/2 = φ in A, and π/3 = φ in B) then we will have: Universe A = Universe B. The exact same physical state-space prevails in both A and B. For all intents and purposes there is no difference between them. In this case, we say that there exists an interpenetration of different parallel universes. Note that though the physical state spaces (e.g. with constants h, G, e/m, etc.)may be alike, they can still differ in dimensionality. And we cannot disregard fractal dimensionality.

If this is so, then it may account for a frequently cited phenomenon connected to many UFO observations, that of seemingly ‘instant’ appearance and disappearance. To fix ideas, assume you are a denizen of a two-dimensional world, confined to the plane of that world. How would you be aware of contact with a hyper-dimensional entity? Imagine a 3-D ellipsoidal disk (like a UFO) as an example, which is 'hyperdimensional' compared to a 2-D plane (like a sheet of paper). To enable detection, the disk must progressively intersect the plane. The observer sees a mystifying object suddenly grow to vast size, then contract and finally disappear! (See the diagram below - where the horizontal line is a 1D infinite object, intersected by an elliptical 2D object, which might be the 'cross-section' of the UFO).

( U F O )

----------------------------A

--------( U F O )--------B

----------------------------C

( U F O)

In the changing situation or transition from A to B to C, the interpenetration is such that the dimensions of the intruder appear to contract or shrink. But this is only a consequence of its departure from our own (linear, in this case) dimensionality. Say, in transit to a parallel universe with (slightly) different dimensionality. This is precisely analogous to the case of UFOs regularly observed to 'shrink' or contract.

A similar manifestation is called ‘skipping’. The UFO appears to be at one location, say over a building, and in the next instant far displaced, without having traversed the intervening distance! It appears to have simply jumped or 'skipped' the space from A to B! This sort of quantized travel is another example of hyperdimensional passage, since no normal macroscopic object exhibits quantum jumps in transport. In our (‘Universe A’) frame of reference, all discrete objects must transit from point X say to point Y. However, if our ‘Universe A’ is really interpenetrated by an adjoint ‘Universe B’ of slightly different dimension, then non-local jumps are possible. In fact, it’s not really a ‘jump’ per se, merely the craft or object exploiting the inherent geometric features of two parallel universes that are intermeshed. This can also account for the apparent FTL displacements observed.

Brane space folding:

What if space is not itself structured for shortcuts, or folding? Might it be feasible for a device to be designed that can change the structure of space and facilitate FTL transport? This is, incidentally, of a totally distinct nature from Alcubierre’s warp drive (Section 1). While the warp drive mechanism produced local effects in the form of a wake in space-time, D-brane craft would generate non-local effects to enable it to instantly connect to a distant region of space.

Branes are surface representations of strings, and include p-, and D-branes. The latter are most useful for our purposes since they comprise a five dimensional hyper-surface in which propagates in time t, thus sweeping out a six dimensional space-time.

How to apply to a transport device? I suggest here that an internal brane is manipulated so that its physical parameters comport to those of a distant region of space. (Given the fact that branes are manifestations of charges, fields, etc.).

The principle at work is so profoundly radical that it requires further explication. We are not talking so much of a “propulsion” system as a resonance mechanism capable of instantaneously connecting distant regions. Thus, the ‘craft’ depicted uses no “fuel” as such but rather a form of local engineering on a distinct and malleable mem-brane that alters its (electrical, magnetic, etc.) properties to adjust to some brane-space, X(B). When a suitable adjustment is made between two branes (one for UFO, the other for a distant target) an instantaneous linkage is forged as a brane-hole expands and detaches in the distant space – emerging as the UFO. Let’s look at it in stages, from the diagram of Fig. 2 (above).

At the far left we see the craft’s internal brane, labeled ‘B’ in the previous diagram. This D-brane must be manipulated so that it’s ambient physical properties (charge q, B-, E-field strengths, polarizations, etc.) are altered to mimic those of the distant ‘target’ space. As the manipulation proceeds, the D-brane fluctuates and its manifolds separate as depicted in the center frame. The arc deforming from the brane’s top edge represents tachyon’s localizing. These are hypothetical faster-than-light particles, not yet observed but with a mathematical basis. The closed oval (actually called a “hole”) near the top of the fluctuating D-Brane manifolds is the crude ‘mirror’ image of the craft. It may be thought of roughly as its “echo” in imaginary time. In the last phase of the sequence (far right) the “hole” detaches entirely from the embedding brane-space and the craft re-materializes. Now, many light years from where it originally had been located.

Underpinning the above portrayal are several key principles:

i)D-branes, by definition, are dynamical objects that can fluctuate. This means they are perfectly adapted for manipulation – given the necessary energy requirements- of UFOs.

ii)The dynamical aspect is a result of the ability of D-branes to couple to closed strings. Recall here that the latter embody a description of gravity. (See also Fig. 3c and the manner in which a closed string can append to a D-brane manifold. Thereby weakening it, leading to fluctuation).

iii)D-branes act as fundamental sources of the various physical fields (e.g. E-, B-, G-)

iv)D-branes as sources of fields carry the smallest charge(-s) possible. For example, q E for E.

The quantitative description of brane phenomena is enormously complex and far beyond the scope of this Appendix, or book. However, some simple insight and appreciation for the complexity can be gained by looking at the simplest brane of all, the D-0 brane consistent with the point particle. Without going into the details of obtaining the (fiber) bundle map, we require for completeness the tachyon field described by:

T(x) = SIGMA_n=1..9 GAMMA_n X^n

where the summation is taken from n = 1 to n = 9, with x^n the coordinates for the rotation space (group) R(9), and GAMMA_n , the matrices needed. Then the tachyon field consistent with the above would be:

T =

( x^9 GAMMA*x)

( GAMMA^T*x -x^9)

Here, the Gamma matrices (GAMMA) and their transposes (GAMMA^ T ) are predicated on the basis for SO(9) spinors, viz. S = [S+ , S - ], such that the mappings: GAMMA : S - -> S +, and GAMMA^ T : S + -> S - apply, respectively. For a system of D-0 branes parameterized by such matrices and defined by the given tachyon field (T(x)), we have a system of 9 x 9 matrices, defined by [x_t9]. This will feature a diagonal specified by: x_N (τ), τ = 1…..9, and zeros everywhere else.

Note that in the special case of D-branes widely separate, say analogous to the UFO example described at the beginning of the section, the matrix can be rendered as the much simpler diagonal case above.

All of the preceding is intended to show how the very basis for even the simplest (D-0) brane rests on a dynamical (fluctuating) vacuum, strings and in particular the faster-than –light tachyon. It remains to be ascertained, however, how a UFO- brane can be manipulated in terms of the energy required. At the very least, we’d like to set some kind of threshold on the magnitude of energy.

Since the fields, and dynamics (as well as the tachyon field) are occurring at the level of the vacuum, that is the first place to start. Thus, to manipulate the brane B, in Fig. 3a, it seems clear that energy consonant with a vacuum energy – or more exactly, a vacuum energy density- is needed.

The conclusion from all this is that, assuming we take the brane-driven UFO seriously as an FTL device, it must somehow make use of dark energy (repulsive gravity) to manipulate distant branes to achieve nonlocal (FTL) transitions. This is also the only venue in which the enormous energy density makes any sense.