Hopf tori warp fields

Two unit circles in , one in the x-y plane and the other in the z-w plane, form the basis of a curious structure on . The surface of a tubular neighborhood a fixed distance from either of these circles forms a torus in . The Clifford Tori are the family of tori generated by tubular neighborhoods around either of these centerlines. A tubular neighborhood at a distance of n from one centerline is the tubular neighborhood at a

distance of from the other, where . The family of tori generated by

varying are called the Clifford Tori. The tubular neighborhood at a distance of forms the flat torus in mentioned above.

This forms an equatorial torus.Not only does divide into two filled tori, but this division can be done in a geometrical way. In other words, if we cut the sphere into two pieces along this special torus, we could swap the two pieces (in ) of the and rearrange them back into a sphere. This means we might expect it to be possible to ``invert'' or swap these regions in

via isometries. In fact, this is possible. We can swap the two centerlines, by rotating

first by in the x-z plane, and then by in the y-w plane.

This operation also makes it possible to see another interesting fact about inverting in

. Since , we can distinguish two unique, perpendicular lines on the surface of (A and B) and call them the generators. If we choose the generators so that one is in the direction of one of the centerlines, and the other in the direction of the other centerline, and then watch these generators as we rotate the sphere, we can see that they are swapped as well.

Now that we have all of this machinery, we can consider another curious property of , the Hopf fibration. First we will consider a fibration on , and then examine what it looks like when this fibration is extended to the Clifford Tori.

We can think of the as a square, or , with edges identified. Each pair of parallel and identified edges becomes one of the generators. First, draw a diagonal from one corner to it's opposite. Now, draw a series of lines parallel to that diagonal.

Now consider what these lines become when the square is wrapped up into a torus. They become a series of circles, each one circling the torus exactly once in the direction of one generator, and once in the direction of the other generator. Notice that the circles are linked, each one passing through each other one.

In the Hopf torus chart ds^2 = cos(r)^2 dz^2 + dr^2 + sin(r)^2 du^2,

-Pi < z,u < Pi, 0 < r < Pi/2

the "obvious" frame field (ONB of vectors) is

e_1 = sec(r) d/dz

e_2 = d/dr

e_3 = csc(r) d/du

and the Clifford frame field is then

f_1 = cos(r) e_1 + sin(r) e_3 = d/dz + d/du

f_2 = e_2 = d/dr

f_3 = -sin(r) e_1 + cos(r) e_2 = -tan(r) d/dz + cot(r) d/duds^2 = cos(r)^2 dz^2 + dr^2 + sin(r)^2 du^2,

-Pi < z,u < Pi, 0 < r < Pi/2

the "obvious" frame field (ONB of vectors) is

e_1 = sec(r) d/dz

e_2 = d/dr

e_3 = csc(r) d/du

and the Clifford frame field is then

f_1 = cos(r) e_1 + sin(r) e_3 = d/dz + d/du

f_2 = e_2 = d/dr

f_3 = -sin(r) e_1 + cos(r) e_2 = -tan(r) d/dz + cot(r) d/duThis gives us a "extrinsic geometrical" descriptionof this frame for S^3 embedded as a round sphere in R^4. Here, the integral curves of theunit vector field f_1 are the Hopf circles, pairwise-linked great circleslying in Hopf tori r = r0. These circles are the fibers of the geometricHopf fibration

S^1 >>--> S^3 >-->> S^2

One also thinks of the Hopf tori as orbits under left multiplication bysome unit quaternion in S^3 = SU(2) = the group of unit norm quaternions.

Basically, you then have a bipolar ring type field around a central area of normal space-time. Since this could be formatted to be casually connected to the inner region(that of the craft). You’d have a bipolar region forming a wormhole throat attempting to pull the craft along. This would yield something akin to the sub-way to the stars method self generated by the craft itself. The interesting thing is the Hopf tori field itself would still form an embedded sub-space region. Yet, outside of transfer of velocity to the craft due to casual connection all of its Hawking radiation and stress effects are not focused at the craft itself, but out into space. Since the ship remains in normal space-time it can still detect only one lightcone system, however, highly Doppler shifted due to its abnormal Lorentz view. Even though its velocity would be higher than C, everything coming at it would be an at C motion with higher blueshifts than normal. Here some form of shielding would be required. Here is an interesting idea. If one could casually connect such a space-time field to a regular EM signal so that the Hopf tori becomes the carrier of that signal than one would have Warp Communication. This field method also lends itself with small

shielding to warp drive capable probes that could be lauched from a craft in warp. All the shields would have to do is deflect the oncoming photons in towards the Hopf tori region and thus away from the craft.Consider this method to reconcile the two sequence-approaches by using some ideas of Alastair Couper, who points out that "... as has been pointed out by mathematician Jason Sharples ... cross addition of any number n, in base b, is equivalent to the result of n mod(b-1) ... the column of cross added numbers [of the Fibonacci sequence]... will ... repeat every 24 elements of the Fibonacci sequence, ad infinitum. In addition the elements of this repeated pattern show an additional bipolar symmetry, whereby the sum of any element plus the element 12 places away always results in 9. These relations are shown by plotting the 24 repeating elements on a wheel, bringing out the symmetries clearly:

Combine this with the regular Hopf tori model inspected for Rodin's archetypal 'doubling circuits' and 3,6,9 'gap circuit', we find that they are indeed there:

engineer with rotating superconducting machines, perhaps using ideas from Gabriel Kron for example it might be possible to show that a lumped or distributed impedance network, surrounded by its own electromagnetic field, is actually the sum of four different types of multidimensional networks:

(1) the well-studied 1-network of branches in which the currents flow, (2) a 0-network formed by all the point generators, (3) a 2-network of equipotential surfaces that pass through the generators

perpendicularly to the branches, and (4) a 3-network composed of three-dimensional impedance blocks surrounding

the branches.

Thus the topological structure of a stationary or rotating, electric or electronic network is neither a graph nor a polyhedron, but a so-called fiber bundle over a non-Riemannian manifold. The fiber bundle structure used by Kron may be the Hopf fibration S3 / S1 = S2 where:

the 1-network corresponds to the S1 = U(1) Clifford-Hopf circles; the 2-network correponds to the S2 sphere of rotation axes; and the 3-network corresponds to a part of the 3-sphere S3 that is a bunch of layered

nested tori, such as used in Naudin's version of the Rodin coil. You can combine a bunch of 3-sphere S3 structures to make more complicated

structures, effectively using S3 structures as building blocks. We simply have to generate the 3-sphere S3 structures and combine them together to form the Hopf tori field. Just as

you can combine a bunch of S3 = SU(2) = Spin(3) = Sp(1) to make ANY semi-simple Lie algebra. You don't need huge radii to get meter-size non-holonomic torsion gaps, because if you think of the Lie bracket of rotations, which is what you have in S3 because of the S2 in the Hopf fibratioon, you can get a bracket product, or gap, that is on the same order of magnitude size as the two rotations whose product you are taking, so, with an S3 machine such as a Rodin coil, the gap can be roughly the same size as the Rodin coil machine itself, and you may be able to get a meter-size effect with a meter-size machine. The trick is projecting such engineered field gaps to line up a certain distance from the ship to achieve the effect you are after.

In cond-mat/9802259, Antonoyiannakis and Pendry has stated:  

“Our findings confirm that a body reacts to the EM field by minimising its energy, i.e. it is attracted (repelled) by regions of lower (higher) EM energy. When travelling waves (of real wavevector) are involved, forces can be additionally understood in terms of momentum exchange between the body and its environment. However when evanescent waves (of complex wavevector) dominate, the forces are complicated, often become attractive and cannot be explained by means of real momentum being exchanged. We have studied the EM forces induced by a laser beam ona crystal of dielectric spheres of GaP. We observe effects due to the lattice structure, as well as due to the single scattering from each sphere. In the former case the two main features are a maximum momentum exchange (and largest forces) when the frequency lies within a band gap; and a multitude of force orientations when the Bragg conditions for multiple outgoing waves are met. In the latter case the radiation couples to the EM eigenmodes of isolated spheres (Mie resonances) and very sharp attractive and repulsive forces occur. Depending on the intensity of the incident radiation these forces can overcome all other interactions present (gravitational, thermal and Van der Waals) and may provide the main mechanism for formation of stable structures in colloidal systems. ...".

So here again there is a possible EM method for generating a warp field. I still favor the EM approach, even if the devices to achieve this have not been built yet. We understand EM enough to mold it to our usage far better than we do gravity.

Recall that L2(R) means the Hilbert space of square-integrable complex function on the real line. If we define the unitary operators u and v on L2(R) given by

u = translation by the amount sv = multiplication by exp(ix)

we can see that they don't commute, but instead satisfy

uv = qvu

where q = exp(is).

The algebra of operators on L2(R) generated by u and v and their inverses is called the noncommutative torus T_q. (If you know how, it's better to take the C*-algebra generated by two unitaries u and v satisfying

uv = qvu.

This is actually quite a bit bigger for q = 1.) This is clearly a natural sort of thing because it's built up out of simple translation and multiplication operators, and all of Fourier theory is based on the interplay between translation and multiplication operators.

The shape of the field depends on the parameter q. If we take q = 1, T_q is the C*-algebra generated by two unitaries u and v that *commute*. This may identified the algebra of functions on a torus if we think of u as multiplication by exp(i theta) and v as multiplication by exp(i phi), where theta and phi are the two angles on the torus. So we've got a one-parameter family of algebras T_q such that when q = 1, it's just the algebra of (continuous) functions on a torus, but for q not equal to one we have some sort of noncommutative analog thereof. The parameter q measures noncommutativity or "quantum-ness", and one can relate it to Planck's constant (which also measures "quantum-ness") by

q = exp(i hbar).

This example is actually called "deformation theory". One can read more about it Marc Rieffels' paper "Deformation quantization and operator algbras," Proc. Symp. Pure Math. 51 and in, "Deformation theory and quantization' by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer, in Ann. Physics 111, p. 61-151.

Now it shouldn't be too surprising that the noncommutative torus is an r-commutative algebra, since the commutation relations uv = qvu tell you exactly how to "switch" u and v. I've shown that there is a unique way to make the noncommutative torus into a strong r-commutative algebra such that

R(u x u) = u x uR(v x v) = v x v

and

R(u x v) = q v x u.

Strong simply means that R2 is the identity, so R(v x u) = q-1u x v.) One may thus go ahead and define "r-commutative differential forms" for the noncommutative torus, which satisfy

u(du) = (du)u, v(dv) = (dv)v, u(dv) = q(dv)u, (du)v = qv(du)

and more relations obtained by differentiating these.

One can then calculate the (r-commutative) de Rham cohomology of the noncommutative torus. We find it to be isomorphic to that of the usual torus. This would be like a negative energy solution. This fits into the philosophy that the noncommutative torus is obtained from the usual torus by a "continuous deformation" - no holes have been formed or gotten rid of. Thus, our needed negative energy state is self generated by the Torus itself.

Now let's set hbar = 1.

Suppose we have a particle on the plane - with no magnetic field. Then in quantum mechanics the momenta in the x and y directions are given by the operators,

p_x = -i d/dx and p_y = -i d/dy,

respectively. These commute, becuase mixed partials commute.

Now let's turn on the magnetic field pointing perpendicular to the plane. Let's say our particle has charge = 1, and the field strength is B. The curious fact about quantum theory is that (if we neglect the *spin* of the particle) the only effect of the magnetic field is to make us redefine the momentum operators to be

p_x = -i d/dx + A_x and p_y = -id/dy + A_y

where A, the vector potential, has curlA = B. Now p_x and p_y don't commute, and in fact the commutator

p_x p_y - p_y p_x

is just -iB

Now remember the transform from the prior articles.

This means that if you grab your charged particle and move it first along the x direction and then the y direction:

^||

------->

the particle winds up in a different state than if you first go in the y direction and then the x direction:

^------> | | |

Another way of thinking of it is as follows: take your particle, move it counterclockwise (say) around a rectangle:

v------< | | | | >------^

and it'll be back in the same place. Thus, the near-Kerr like frame dragging, but also the connection to the field generation idea.

If we call the particles wavefunction to start out with "psi," and when we're done "phi,"

phi = exp(is) psi

where s is just the line integral of A around the rectangle. routes. By Stoke's theorem, s is just the integral of B over the rectangle. We find that the magnetic field is really a curvature, and that the difference in phase obtained by taking two routes is called the "holonomy" of the connection. The idea then is to use that magnetic field curvature to produce our desired Hopf tori warp field. Anyway, now suppose that the magnetic field strength is a constant B. Let U denote the unitary operator corresponding to translation by a unit distance in the x direction, and let V be the unitary operator corresponding to a unit translation in the y direction. Then we have

UV = qVU

where q = exp(iB). Thus U and V satisfy the relations of a noncommutative torus. There are two different regions, the "inside" and the "outside" of the loop, which count as regions enclosed by the loop or a trapped surface embedded into normal 4D space-time. Together they form a bipolar region.

Imagine a Pendulum (a pivoted weightless rod with a mass on the end). Now imagine that you can turn gravity on and off. Turn on the gravity once a second. So the rotor spins freely except when "kicked" by gravity. Hence the name...

The classical kicked rotor can exhibit several types of motion depending on how strongly it is kicked. If the kicking is strong enough then almost all of the possible motions are stochastic, i.e., seemingly random. For weaker kicking, several types of motion are possible, depending on the speed at which the rotor is spinning when the kicking begins. For some initial speeds the rotor spins almost as it did without kicking (or with gravity on all of the time). For smaller speeds, the rotor vibrates around it's lowest point, exactly as it might if gravity were on all of the time.

The most interesting motion in the weakly kicked rotor happens around the so-called "separatrix" If gravity were on all of the time, there would be a certain initial speed and angle such that the rotor would almost stop entirely as it reached the top. The motion where the rotor does in fact stop and balance at the top is called the "separatrix motion"

In the kicked rotor, the separatrix motion becomes somewhat random. A "Poincare Surface of Section" is shown below.

Now suppose you have two or more rotors and, at the same time as you turn on gravity, you allow the rotors to influence each other. For instance, you could give rotor #1 a bigger kick if rotor #2 is swinging down rather than up.

When two weakly kicked rotors are made to "interact" the stochastic regions in one rotor can make the motion in the other rotor "diffuse" in ways it could not if the interaction were not present. This diffusion is called "Arnold Diffusion"(1-5) because it was V.I. Arnold who first realized that such diffusion was possible.

Below is a "Husimi" plot of a quantum wavefunction corresponding to the separatrix region of the kicked rotor surface of section (gray). Superimposed in blue and red are the classical stable and unstable manifolds respectively.

Now, following this line of thought applied to our warp field if we consider each kicked rotor as a separate Hopf Tori Warp Field Generator and move the separation between the field till they overlap we’d have a canceled region in the center with to rotor frame dragging the outer region much a a Kerr solution frame drags space-time. When Classically diffusive systems are quantized, diffusive motions can be localized. For instance, when the strongly kicked rotor is quantized the stochastic diffusion of momentum which is seen in the classical rotor is localized in the quantum rotor. That is, in the classical rotor, after enough kicks the rotor's speed is essentially random (inside some range which is controlled by how many kicks have occured. More kicks give you a greater range of possible speeds) Not so for the quantum rotor where only a certain range of speeds is possible no matter how many kicks are allowed. This is often called "Dynamical Localization" because it is a localization of the speed not the mass itself. Two or more interacting Hopf Tori Warp Fields can exhibit a huge range of possible speeds and angles even if the kicking and the interaction is weak. Thus Arnold diffusion is a classical type of phase space transport. We could in theory achieve the effects of a non-minimual coupling to gravity of an EM scalar with actual minimual coupling. As the classical stable and unstable manifolds, from the diagram above, are brought together their lines begin to overlap and fold together yielding a direct inline to the plane of motion if that motion is viewed as towards the right of the graph. This pulls the jet trails directly inline with this plane too. This yields us a space jet effect. The two outer

darkened regions would then form our space-time matter sync region and the space-time restoration region via the combined effect. These translate to the compressed region in the forward area and expanding region in the rear region of a normal warp metric. This type of metric would have actual flow of energy in and out exactly like a wormhole with no rearward trapping surface.

THE HOPF TORI WARP DRIVE

Starting with the metricds^2 = dt^2/sqrt(cos(2 theta)) - (d theta)^2 - cos^2 theta dx^2 - sin^2 theta dy^2We set G^{theta theta} = 0So that the pressure between adjacent tori to vanish and, equally obviously, G^tt has to be a constant.We derive no flux terms, so that matter elements are co-moving with the coordinates. (That means {theta}, x, and y are constant.)In the central region theta=0 cos0=1 sin0 =0

ds^2 = dt^2/sqrt(cos(2 theta)) - (d theta)^2 - cos^2 theta dx^2 - sin^2 theta dy^2

ds^2 = dt^2/sqrt(cos(0)) - (d 0)^2 - cos^2 (0) dx^2 - sin^(0) dy^2

ds^2 = dt^2 - dy^2We find then that:ds^2 = dt^2 - [dx-vs*f(rs)]^2 - dy^2 - dz^2

if x=xs we have

ds^2 = dt^2 - dy^2 - dz^2to achieve warp drive behaviour into x-axis only we make

ds^2 = dt^2 - [dx-vs*f(rs)]^2

giving

ds^2 = dt^2

for the Hopf Tori warp drive we set

ds^2 = dt^2/sqrt(cos(2 theta)) - (d theta)^2 - cos^2 theta dx^2 - sin^2 theta dy^2

supressing dy

ds^2 = dt^2/sqrt(cos(2 theta)) - (d theta)^2 - cos^2 theta dx^2

"central region" (emanating around theta = 0)

ds^2 = dt^2 - dy^2You now have a basic metric that displays warp behavior. We will now add two such fields side by side.

This gives two such fields on edge and in line side by side. Now imagine sliding the fields inward towards each other till the fields overlap with just the outer walls of each remaining. This is the setup of a dual field setup. What ever effects from the field exist in the original centers are now nulled. The craft sits in a space normal region. Yet the field provides warp motion by frame dragging of the local space-time.Imagine a Pendulum (a pivoted weightless rod with a mass on the end). Now imagine that you can turn gravity on and off. Turn on the gravity once a second. So the rotor spins freely except when "kicked" by gravity. Hence the name...

The classical kicked rotor can exhibit several types of motion depending on how strongly it is kicked. If the kicking is strong enough then almost all of the possible motions are stochastic, i.e., seemingly random. For weaker kicking, several types of motion are possible, depending on the speed at which the rotor is spinning when the kicking begins. For some initial speeds the rotor spins almost as it did without kicking (or with gravity on all of the time). For smaller speeds, the rotor vibrates around it's lowest point, exactly as it might if gravity were on all of the time.

The most interesting motion in the weakly kicked rotor happens around the so-called "separatrix" If gravity were on all of the time, there would be a certain initial speed and angle such that the rotor would almost stop entirely as it reached the top. The motion where the rotor does in fact stop and balance at the top is called the "separatrix motion" In the kicked rotor, the separatrix motion becomes somewhat random. A "Poincare Surface of Section" is shown below.

Now suppose you have two or more rotors and, at the same time as you turn on gravity, you allow the rotors to influence each other. For instance, you could give rotor #1 a bigger kick if rotor #2 is swinging down rather than up.

When two weakly kicked rotors are made to "interact" the stochastic regions in one rotor can make the motion in the other rotor "diffuse" in ways it could not if the interaction were not present. This diffusion is called "Arnold Diffusion"(1-5) because it was V.I. Arnold who first realized that such diffusion was possible.

Below is a "Husimi" plot of a quantum wavefunction corresponding to the separatrix region of the kicked rotor surface of section (gray). Superimposed in blue and red are the classical stable and unstable manifolds respectively.

Now, following this line of thought applied to our warp field if we consider each kicked rotor as a separate Hopf Tori Warp Field Generator and move the separation between the field till they overlap we’d have a canceled region in the center with to rotor frame dragging the outer region much a a Kerr solution frame drags space-time. When Classically diffusive systems are quantized, diffusive motions can be localized. For instance, when the strongly kicked rotor is quantized the stochastic diffusion of momentum which is seen in the classical rotor is localized in the quantum rotor. That is, in the classical rotor, after enough kicks the rotor's speed is essentially random (inside some range which is controlled by how many kicks have occured. More kicks give you a greater range of possible speeds) Not so for the quantum rotor where only a certain range of speeds is possible no matter how many kicks are allowed. This is often called "Dynamical Localization" because it is a localization of the speed not the mass itself. Two or more interacting Hopf Tori Warp Fields can exhibit a huge range of possible speeds and angles even if the kicking and the interaction is weak. Thus Arnold diffusion is a classical type of phase space transport. We could in theory achieve the effects of a non-minimual coupling to gravity of an EM scalar with actual minimual coupling. As the classical stable and unstable manifolds, from the diagram above, are brought together their lines begin to overlap and fold together yielding a direct inline to the plane of motion if that motion is viewed as towards the right of the graph. This pulls the quadrapole trails directly inline with this plane too. This yields us a space jet effect.

The two outer darkened regions would then form our space-time matter sync region and the space-time restoration region via the combined effect. These translate to the compressed region in the forward area and expanding region in the rear region of a normal warp metric. This type of metric would have actual flow of energy in and out exactly like a wormhole with no rearward trapping surface.

The twin fields behave as if they were each a kicked rotor obeying the Arnold Solution. They can exhibit a huge range of possible speeds and angles even if the kicking and the interaction is weak. But the key is to keep the fields balanced and aligned exactly right. This will be an area of concern for Engineers that develop this system.

The quadrapole effect the field shows in twin format till aligned comes from the least radiation produced by gravity. Below is a figure comparing EM to GR.

What we are doing as we shape the field by the overlap of the two field generator patterns is reshaping gravity into the dipole which can be produced by electromagnetic sources. It is a dipole gravity field that is required to utilize warp drive in the first place. If you look at figure A and compare it to the below diagram you will see the similarity in the two field setups.

This second field shape example is derived through the following.

Two unit circles in , one in the x-y plane and the other in the z-w plane, form the basis of a curious structure on . The surface of a tubular neighborhood a fixed distance from either of these circles forms a torus in . The Clifford Tori are the family of tori generated by tubular neighborhoods around either of these centerlines. A tubular neighborhood at a distance of n from one centerline is the tubular neighborhood at a

distance of from the other, where . The family of tori generated by

varying are called the Clifford Tori. The tubular neighborhood at a distance of forms the flat torus in mentioned above.

This forms an equatorial torus.Not only does divide into two filled tori, but this division can be done in a geometrical way. In other words, if we cut the sphere into two pieces along this special torus, we could swap the two pieces (in ) of the and rearrange them back into a sphere. This means we might expect it to be possible to ``invert'' or swap these regions in

via isometries. In fact, this is possible. We can swap the two centerlines, by rotating

first by in the x-z plane, and then by in the y-w plane.

This operation also makes it possible to see another interesting fact about inverting in

. Since , we can distinguish two unique, perpendicular lines on the surface of (A and B) and call them the generators. If we choose the generators so that one is in the direction of one of the centerlines, and the other in the direction of the other centerline, and then watch these generators as we rotate the sphere, we can see that they are swapped as well.

Now that we have all of this machinery, we can consider another curious property of , the Hopf fibration. First we will consider a fibration on , and then examine what it looks like when this fibration is extended to the Clifford Tori.

We can think of the as a square, or , with edges identified. Each pair of parallel and identified edges becomes one of the generators. First, draw a diagonal from one corner to it's opposite. Now, draw a series of lines parallel to that diagonal.

Now consider what these lines become when the square is wrapped up into a torus. They become a series of circles, each one circling the torus exactly once in the direction of one generator, and once in the direction of the other generator. Notice that the circles are linked, each one passing through each other one.

In the Hopf torus chart ds^2 = cos(r)^2 dz^2 + dr^2 + sin(r)^2 du^2,

-Pi < z,u < Pi, 0 < r < Pi/2

the "obvious" frame field (ONB of vectors) is

e_1 = sec(r) d/dz

e_2 = d/dr

e_3 = csc(r) d/du

and the Clifford frame field is then

f_1 = cos(r) e_1 + sin(r) e_3 = d/dz + d/du

f_2 = e_2 = d/dr

f_3 = -sin(r) e_1 + cos(r) e_2 = -tan(r) d/dz + cot(r) d/duds^2 = cos(r)^2 dz^2 + dr^2 + sin(r)^2 du^2,

-Pi < z,u < Pi, 0 < r < Pi/2

the "obvious" frame field (ONB of vectors) is

e_1 = sec(r) d/dz

e_2 = d/dr

e_3 = csc(r) d/du

and the Clifford frame field is then

f_1 = cos(r) e_1 + sin(r) e_3 = d/dz + d/du

f_2 = e_2 = d/dr

f_3 = -sin(r) e_1 + cos(r) e_2 = -tan(r) d/dz + cot(r) d/duThis gives us a "extrinsic geometrical" descriptionof this frame for S^3 embedded as a round sphere in R^4. Here, the integral curves of theunit vector field f_1 are the Hopf circles, pairwise-linked great circleslying in Hopf tori r = r0. These circles are the fibers of the geometricHopf fibration

S^1 >>--> S^3 >-->> S^2

One also thinks of the Hopf tori as orbits under left multiplication bysome unit quaternion in S^3 = SU(2) = the group of unit norm quaternions.

Basically, you then have a bipolar ring type field around a central area of normal space-time. Since this could be formatted to be casually connected to the inner region(that of the craft). You’d have a bipolar region forming a wormhole throat attempting to pull the craft along. This would yield something akin to the sub-way to the stars method self generated by the craft itself. The interesting thing is the Hopf tori field itself would still form an embedded sub-space region. Yet, outside of transfer of velocity to the craft due to casual connection all of its Hawking radiation and stress effects are not focused at the craft itself, but out into space.

Recall that L2(R) means the Hilbert space of square-integrable complex function on the real line. If we define the unitary operators u and v on L2(R) given by

u = translation by the amount sv = multiplication by exp(ix)

we can see that they don't commute, but instead satisfy

uv = qvu

where q = exp(is).

The algebra of operators on L2(R) generated by u and v and their inverses is called the noncommutative torus T_q. (If you know how, it's better to take the C*-algebra generated by two unitaries u and v satisfying

uv = qvu.

This is actually quite a bit bigger for q = 1.) This is clearly a natural sort of thing because it's built up out of simple translation and multiplication operators, and all of Fourier theory is based on the interplay between translation and multiplication operators.

The shape of the field depends on the parameter q. If we take q = 1, T_q is the C*-algebra generated by two unitaries u and v that *commute*. This may identified the algebra of functions on a torus if we think of u as multiplication by exp(i theta) and v as multiplication by exp(i phi), where theta and phi are the two angles on the torus. So we've got a one-parameter family of algebras T_q such that when q = 1, it's just the algebra of (continuous) functions on a torus, but for q not equal to one we have some sort of noncommutative analog thereof. The parameter q measures noncommutativity or "quantum-ness", and one can relate it to Planck's constant (which also measures "quantum-ness") by

q = exp(i hbar).

This example is actually called "deformation theory". One can read more about it Marc Rieffels' paper "Deformation quantization and operator algbras," Proc. Symp. Pure Math. 51 and in, "Deformation theory and quantization' by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer, in Ann. Physics 111, p. 61-151.

Now it shouldn't be too surprising that the noncommutative torus is an r-commutative algebra, since the commutation relations uv = qvu tell you exactly how to "switch" u and v. I've shown that there is a unique way to make the noncommutative torus into a strong r-commutative algebra such that

R(u x u) = u x uR(v x v) = v x v

and

R(u x v) = q v x u.

Strong simply means that R2 is the identity, so R(v x u) = q-1u x v.) One may thus go ahead and define "r-commutative differential forms" for the noncommutative torus, which satisfy

u(du) = (du)u, v(dv) = (dv)v, u(dv) = q(dv)u, (du)v = qv(du)

and more relations obtained by differentiating these.

One can then calculate the (r-commutative) de Rham cohomology of the noncommutative torus. We find it to be isomorphic to that of the usual torus. This would be like a negative energy solution. This fits into the philosophy that the noncommutative torus is obtained from the usual torus by a "continuous deformation" - no holes have been formed or gotten rid of. Thus, our needed negative energy state is self generated by the Torus itself.

Now let's set hbar = 1.

Suppose we have a particle on the plane - with no magnetic field. Then in quantum mechanics the momenta in the x and y directions are given by the operators,

p_x = -i d/dx and p_y = -i d/dy,

respectively. These commute, becuase mixed partials commute.

Now let's turn on the magnetic field pointing perpendicular to the plane. Let's say our particle has charge = 1, and the field strength is B. The curious fact about quantum theory is that (if we neglect the *spin* of the particle) the only effect of the magnetic field is to make us redefine the momentum operators to be

p_x = -i d/dx + A_x and p_y = -id/dy + A_y

where A, the vector potential, has curlA = B. Now p_x and p_y don't commute, and in fact the commutator

p_x p_y - p_y p_x

is just -iB

Now remember the transform from the prior articles.

This means that if you grab your charged particle and move it first along the x direction and then the y direction:

^||

------->

the particle winds up in a different state than if you first go in the y direction and then the x direction:

^------> | | |

Another way of thinking of it is as follows: take your particle, move it counterclockwise (say) around a rectangle:

v------< | | | | >------^

and it'll be back in the same place. Thus, the near-Kerr like frame dragging, but also the connection to the field generation idea.

If we call the particles wavefunction to start out with "psi," and when we're done "phi,"

phi = exp(is) psi

where s is just the line integral of A around the rectangle. routes. By Stoke's theorem, s is just the integral of B over the rectangle. We find that the magnetic field is really a curvature, and that the difference in phase obtained by taking two routes is called the "holonomy" of the connection. The idea then is to use that magnetic field curvature to produce our desired Hopf tori warp field. Anyway, now suppose that the magnetic field strength is a constant B. Let U denote the unitary operator corresponding to translation by a unit distance in the x direction, and let V be the unitary operator corresponding to a unit translation in the y direction. Then we have

UV = qVU

where q = exp(iB). Thus U and V satisfy the relations of a noncommutative torus. There are two different regions, the "inside" and the "outside" of the loop, which count as regions enclosed by the loop or a trapped surface embedded into normal 4D space-time. Together they form a bipolar region.

If we consider this:

ds^2 = dt^2/sqrt(cos(2 theta)) - sec^2 (2 theta) (d theta)^2 - cos^2 theta dx^2 - sin^2 theta dy^2

The matter tensor is diagonal and for low values of theta, we do not violate the energy conditions. So with this we have density decreasing from the central region and emanating from theta = 0) and tension along the x-direction of each Hopf tori along with an increasing compression around the y-direction. For small values of theta, where the energy conditions are not violated, the Riemann tensor takes the form:

R^t_(theta t theta) = -1 + 3 theta^2R^t_(x t x) = theta^2 R^t_(y t y) = -1 + 3 theta^2R^theta_(x theta x) = 1 - 8 theta^2R^theta_(y theta y) = 5 - 16 theta^2R^x_(yxy) = 1 - 4 theta^2

The obvious result: There is a lot of tidal compression along theta and the y-coordinate and a minimal tidal expansion along the x-direction - all without violating the energy conditions (again, for small values of theta).

Also consider:

ds^2 = dt^2 - h(t)^2[g(theta)^2 (d theta)^2 + cos^2 theta dx^2 + sin^2 theta dy^2]

Diagonal matter tensor again and if pressures are all equal we have h as a constant value. From that we can try to find electromagnetic solutions by using:

ds^2 = exp(2 theta) [dt^2 - (d theta)^2] - cos^2(t) [ cos^2 theta dx^2 + sin^2 theta dy^2 ]

So here our matter tensor is:

G^(tt) = 1/theta + 1 -16/3 theta + t^2 + 22/3 theta^2 - 2 theta t^2 + 2 theta^2 t^2G^(theta theta) = 1/theta - 1 -4/3 theta - t^2 + 10/3 theta^2 + 2 theta t^2 - 2 theta^2 t^2G^(t theta) = t/theta - 4 t + 14/3 theta t -8/3 theta t^2

So it seems that theta satisfies the energy conditions here as well. We do not really have a co-moving coordinate system from this at this point. This seems at least a promising way to (a) show positive matter formualations of "warp metrics", and (b) provide falsification parameters within the larger concept of those "warp metrics".

Consider the following.

Showing a a singular solution using the Hopf Tori model. As compared to the same type of model with a regular negative energy wormhole.

The effect is the same. But the means of generating the solution is different. The object in the first is to create a compression in front of the craft that goes by the craft and extends to the rear behind the craft at some desired distance. At this point the field ends and the compressed space-time rapidly inflates causing a pressure difference from front to back without the field. This lower pressure region creates the same effect negative energy does in a regular Warp Metric solution. It moves space-time by the craft. The velocity in this simply concept depends upon these factors:

1.) Amount of compression.2.) Amount of inflation difference in pressure.3.) Distance from front to back of the field.4.) Rate of the field’s pulsing.

If field is pulsed at 100000 cycles per second and the field length was 1 mile you have 186300 Miles effected per second. Which translates, depending upon the pressure difference to a moving frames system of 1C velocity. The reason is the compression and inflation will always produce a 1 to 1 ratio unless we modify the field a bit. So we have one unit movement per cycle.The Lorentz system displays sensitive dependence in which neighboring trajectories separate exponentially with time. Neighboring trajectories emanating from a are visibly separate by b and diverge into distinct spirals by c1 and c2, so that their subsequent dynamics is unrelated. The Hopf bifurcation forms a periodic closed orbit attractor. For v < 0 there is only a sink (attractor). As v crosses 0 an attracting periodic orbit and source (repellor) are created.

The Hopf bifurcation results in the formation of a closed orbit attractor (oscillation) from a point attractor (sink) at the origin as v crosses 0. Because it uses non-linear differential equations it cannot in general be integrated directly. However, we can resort to techniques of numerical integration in which a discrete transfer function is constructed which approximates a stroboscopic representation of the flow at discrete time intervals

by using numerical methods such as the Runge-Kutta (Butcher 1987) :

 

Repeated iteration of the corresponding chaotic map similarly causes the separation of two adjacent points to become exponentially increased. This provides a means of calculating the exponent of growth, called the Liapunov exponent.

Consider the repeated action of the discrete map increasing separation by a

factor :                

Hence we can write .

If the separation varies along the path, we can take limits as we have

This formula makes it easy to calculate the Liapunov exponent for any iteration. In a chaotic system in one or more variables, sensitive dependence requires at least one of the Liapunov exponents to be greater than 1, thus resulting in exponential separation of trajectories.

Note that in the case of a continuous flow, the role of the constant a  is slightly different.

In the flow ,

whereas with the map,

The formula also naturally represents the loss of information, or entropy :

The Shannon informational entropy is  

where  is the probability of being in state i and .

Consider a single iteration in which [0,1] maps to [0,1] under separation a.. At the initial

stage, we have n states each with probability , so :

where  is the probability of being in state i and .

After one iteration, the resolution is reduced by factor , following the same

reasoning as in the above example for one step, giving states each with probability ,

so we have :

Thus there is thus a difference .

Averaging this over many iterations, we have

which is obviously the same, except for a factor of log 2.

The above is an example of repeated Hopf bifurcation which results in Tori. Creation of two oscillations results in a flow on the 2-torus. Periodic flow on 2-torus results in closed orbits which meets themselves exactly. Anything beyond this results in chaos and disrupts the periodic relationships.

This also sets some limits on the type of field designed around a Hopf Tori metric we could use. The attarctor or sink and the repeller points I would say are actually the quadrapole being compressed into a bipolar solution. They tend to focus ahead and behind the Tori itself once engaged and thus, increase the actual range over which the field is working. Something in our favor here. In some graphs I have looked at they actually double the effect range of the field. So, if our field reached a 1 mile limit it would actually be effecting 2 miles. If the field pulsed at 100000 times a second you get a 2C motion when the field is bipolar given a 1/1 front to back ratio on pressure differences. If you could add negative energy into this approach you could get an even better ratio. Say 1 to 2 which could even double this. The odd thing is this range on the attarctor or sink and the repeller points varies the higher the compression goes following that Clifford scale. This is the tie in to velocity. If you tailored a field just right in its static natural case you could derive a velocity scale exactly equal to Startrek’s. With each full compression jump you’d get velocities that are x^3. However, considering the range of variables on this any velocity is possible as long as you have a field capable of doing the required compression.A Blackhole, no matter the type, can be considered a compression of space-time. Now if that compression is tied to a region that is reversed so that it reexpands, even just back to its natural state you form a bipolar region. The Event Horizon for such a compression is described by:

2GM/(c^2) Where: G = Newton's Constant of gravity M = Mass of star c = Velocity of light

The event horizon for the reexpansion is simply the original starting horizon at the entry to our original compression event horizon. If the compression is prevented from hitting

a singularity point simply by the field’s structure itself then you have a warp drive metric field being generated by positive energy alone since you do create a bipolar flow.

Consider

Versus

Same Geometry altered to eliminate singularity and tied to an exist region. You have compression region in front and an expansion region in rear. The amount of

compression now controls velocity, not the amount of negative energy. You achieve the same effect negative energy does, but with no negative energy needed to generate the field. The effect is warp drive without negative energy. This same approach could be utilized with any metric. Just the field generator shaping would have to be altered.

References

1.) Rasband, S. N. "Arnold Diffusion." §8.6 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 179-181, 1990.

2.) Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion, 2nd ed. New York: Springer-Verlag, 1994.

3.) Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 74, 1989.

4.) http://atlas-conferences.com/c/a/d/y/20.htm5.) Amadeu Delshams Tere M. Seara, Rafael de la Llave, Universitat Politècnica de Catalunya Geometric methods for instability in Hamiltonian systems