A Discussion of Subspace and Warp Fields
(Especially as they Apply to Momentum and Energy Conservation)

The following is a mixture of concepts mentioned in canon and simi-canon sources combined with a healthy dose of physical reasoning and a big spoonful of personal speculation to help it all go down.  It looks at the properties that subspace fields and warp fields are supposed to possess, and examines how these properties might live in harmony with certain physical laws (specifically, with conservation of energy and momentum).

The discussion is mainly written as if it were addressing a twenty-fourth century audience, and so the concepts I have developed for explaining various aspects of subspace physics are stated as facts.  In reality, even though the main properties of the subspace and warp fields come directly from canon sources, many of the other aspects of these fields are developed from physical reasoning with a  spattering of my own personal tastes.

For example, we know that to sustain a subspace or warp field, it is necessary to continually feed it energy.  So, where does this energy go?  Does the field continually build up energy, storing all of the energy being poured into it.  Even if this were the case, what happens to all that energy when the field is shut off.  The best answer to me seems to describe the field as "unstable" in that it doesn't stick around if you stop feeding it energy.  Instead, we might say that it continually "bleeds" this energy back into normal space in the form of heat in the field coils, electromagnetic radiation, and/or (perhaps) subspace radiation which can couple its energy back to normal space (like the shock wave in Star Trek VI).

So, I hope you enjoy this fairly lengthy discussion of subspace and warp fields.  Even if you disagree with the way some of the concepts are explained, at least understand that a lot of thought has gone into them in order to make the abilities of subspace and warp fields fit in with the concepts of momentum and energy conservation. Okay, prepare to take a little excursion.  As always, your thoughts and criticisms are welcome.

  1. Introduction:

In this discussion we will examine some of the basics of both simple subspace fields as well as warp fields.   In particular, we wish to look at how momentum and energy conservation come into play with the use of these fields.

Before discussing the subspace fields, we first want to talk in general about subspace and its frame of reference.  We will then see how the definition of the frame of reference of subspace allows for the creation of subspace fields (both simple subspace fields and warp fields).

After mentioning some general aspects that both types of fields possess, we will look individually at each type of field.  In each case, we will first go over some of the major characteristics of the particular field of interest.  We will then discuss how momentum and energy conservation come into play with that particular type of field. Finally, we will look at examples to further examine the conservation of momentum and energy with each type of field.

In addition to this, there are also a few technical notes at the end of some sections (specific to each section) which will be referred to at various times.  These will go into more technical detail concerning specific topics.

(A note to the 20th century reader:  The final section before the conclusion deals with the question of angular momentum conservation.   Throughout the other sections of the discussion, "momentum" is used to refer to linear momentum only.  This section will discuss for the 20th century reader why angular momentum has been left out everywhere else.)

2. Subspace and it's Frame of Reference:

Subspace is a continuum that exist in conjunction with our own space-time continuum.  Every point in our universe has a corresponding point in subspace.  Also, at every point in our universe, subspace has a particular frame of reference.  One could imagine subspace to be vaguely similar to a huge cloud-like field that pervades the known universe.  The particles in one area of such a cloud would be moving at some particular velocity, while the particles in another area may be moving at another particular velocity.  Similarly, at every point in our space, subspace has a particular "velocity" or frame of reference.

This fact is very important, because this feature of subspace is what allows us to travel faster than light without having to worry about such things as traveling back in time to meet ourselves every time we jump into warp.  The reason this is so will not be covered in this discussion, but there are texts available which explain why this is.

So, what is the frame of reference of subspace at a particular point in our universe?  Well, the frame of reference is defined by the local distribution of mass.  More specifically, it is defined by the distribution of mass and energy which is mathematically defined by what is known as the stress-energy tensor of local energy distributions.  However, for our purposes here, we will explain how the subspace frame of reference is approximately defined by using the less complicated concept of mass distribution. 

There is one other note that needs to be made before we get into defining the subspace frame of reference.  In subspace physics, there are three meanings to the word mass.  Classically, there are two "types" of mass theoretically believed to be equivalent.  They are gravitational mass and inertial mass.  With subspace physics, there is also the concept of subspace-equivalent mass.  This is the mass subspace "sees" which defines it's frame of reference.  Generally, this mass is equivalent to the gravitational and inertial mass; however, it can be different under certain circumstances.  Similarly, there is also a concept of the subspace-equivalent stress-energy tensor.

Now we will describe how someone can find the speed of the frame of reference of subspace with respect to their own frame of reference.  First, imagine dividing all the mass in the universe into sufficiently small chunks of mass "dm".  We then number each chunk so that the "i-th" chunk would have a mass of "dm_i".  We also note that for objects in the universe which are basically spherical and uniform, we can define the whole object as one of our chunks of mass (provided the object isn't a spherical shell which we might happen to be inside of).

So, we will be in one particular frame of reference (call it O). We want to find the speed of the frame of reference of subspace (in our frame of reference) at some point in the universe.  Well, in our frame of reference, the i-th chunk of mass (dm_i) has a particular velocity in the x direction (Vx_i).  It also has a particular distance away from the point of interested (R_i).  For each chunk, we then calculate the quantity:

   dm_i * Vx_i

  -------------.

     (R_i)^2

Once we calculate this quantity for every chunk of mass, we then sum up all the various quantities and call this sum "S":

     +---

      \    dm_i * Vx_i

  S = /   ------------ .

     +---    (R_i)^2

       i

Now, we want to consider another frame of reference which is moving with respect to our own.  We could figure out what velocities and distances would be measured for each chunk of mass in that frame, and we can calculate the sum, S, in that frame as well.  If we continue to do this for various frames of reference, then we will eventually find the frame of reference in which the absolute value of S is minimized.  The x velocity of that frame of reference will then be the x velocity of the of the subspace frame of reference as measured in our frame. We could then do similar calculations to find the y and z components of the velocity of the subspace frame of  reference.

(A note to the 20th century reader:  For now, this is only a) (preliminary way for determining the frame of reference of subspace. ) (There may be unforeseen problems in this definition, and I'll have  ) (to take some time to consider various aspects of this definition to ) (see if it is really what we want to use.)

So, what does all that mean?  Well, consider a bit of matter that is very close to the point of interest in the frame of reference you are considering (i.e. R_i for that bit of matter is quite small in that frame of reference).  That means that bit of matter provides a fairly large contribution to the sum, S, UNLESS the velocity of that bit of matter is very small in the frame of reference you are considering.  So, the speed of the subspace frame of reference will likely be close to the speed of that nearby bit of matter.  (Note: this is why we say that the subspace frame depends on the local distribution of mass.  For chunks of matter that are very far from you, their contribution to S is generally negligible.) 

However, also note that if there are many chunks of matter at some average distance from you which are all traveling at the same speed (like all the chunks of matter in a nearby star, for example) then all that mass provides a large contributes to the sum.  This means that the subspace frame of reference will be close to the frame of reference of those chunks (so that Vx in that frame of reference is small in order to canceling the large contribution created by the large mass).

Obviously, we could discuss the determination of the frame of reference of subspace for some time; however, for our purposes, it is only important to remember a couple of things about this determination: In the simplest idea, the subspace frame of reference is determined by the nearby distribution of mass.  However, in actuality, it is the distribution in the more complex structure known as the stress-energy tensor that determines the subspace frame of reference.

3. Creating Subspace Fields:

The creation of simple subspace fields as well as warp fields is closely related to the way in which the subspace frame of reference is defined (as described above).  Here we will look first at the creation of a simple subspace field and second at the creation of a warp field to show how these fields are produced.

3.1  Creating a Simple Subspace Field

Inside of a subspace field generator, generally a plasma stream is used to create a particular stress energy tensor within the generator.  Within the area of space where this stress-energy tensor is strongest, the frame of reference of subspace defined by the tensor is made to be radically different from the subspace frame of reference just outside of this area.  Thus, when produced correctly, the stress-energy tensor creates a large change in the frame of reference of subspace over a small area of space.

One might think that this could have the effect of "tearing" subspace in that area if it weren't for the fact that subspace has a natural mechanism for preventing this.  It creates what we call a subspace field which surrounds the offensive stress-energy tensor.  This field reduces the effect that the tensor has on the definition of the subspace frame of reference.  Basically, this reduces the effects of the subspace-equivalent stress-energy tensor.  However, at this point the subspace-equivalent stress energy tensor is still directly related to the real-space stress-energy tensor.  So, the field also lowers the effects of the stress-energy tensor as viewed in normal space (outside of the subspace field) as well.

By correctly producing the stress-energy tensor, one can create a subspace field which extends well beyond the localized area of the tensor (large enough, in fact, to surround a ship).  If we replace the concept of the stress-energy tensor for a moment with the simpler concept of mass, we see that this has the effect of lowering the apparent mass of anything within the subspace field.  In essence, the subspace field "submerges" a fraction of the mass into subspace so that it does not have to be considered as real-space mass when defining the subspace frame of reference.  Details on how momentum and energy remain conserved with this apparent mass reduction will be covered in a later section.

So, we see that by correctly manipulating the normal space effects which dictate the local frame of reference of subspace, we can create a simple subspace field.

3.2  Creating a Warp Field

The creation of the warp field isn't all that different in principle from the creation of a simple subspace field. The major differences are in the energy and configuration of the plasma stream and the exotic nature of the stress-energy tensor needed.

For the purposes of illustration, we will concentrate here on producing a warp field which is used for propulsion.  Other warp fields are produced in a similar manner by producing different stress-energy tensors.  Here we discuss the most basic components of warp field production; however, in section 6 we will mention a few more aspects that can come into play when producing warp fields.

Generally, to create a warp field, the plasma is injected into warp field coals which are made of an appropriate material.  The material in the warp field coil is important because as the plasma is injected, the combination of the configuration of the plasma stream and the coil through which the plasma passes is what creates the exotic stress-energy tensor needed to produce the warp field.   The energizing of the field coil material with a properly configured plasma stream creates a stress-energy tensor that produces a much more violent change in the frame of reference of subspace over a much smaller area than is needed to produce a simple subspace field.  To counteract this violent change, subspace produces what we call a warp field, shifting the energy frequencies of the plasma deep into the subspace domain.  This shift has the effect of completely removing the significance of the stress energy tensor from the determination of the subspace frame of reference.

As with subspace fields, it is then possible to produce a warp field which extends far beyond the local area effected by the exotic stress-energy tensor.  When such a field surrounds an entire ship, everything within that ship can be removed from the determination of the subspace frame of reference.  This brings up two points to be discussed:

First we consider the frame of reference of the ship.  Because of the warp field, subspace and outside observers no longer consider the frame of reference of the ship when determining the subspace frame.  Instead, they considers all other "bits of matter" and determine the frame of reference from them.  Does the ship then NOT have a frame of reference from the point of view of subspace and outside observers? Not exactly.  The frame of reference of the ship instead becomes the frame of reference of subspace as it is defined without the ship's contribution.  Then, obviously subspace does not have to consider the ship when determining the subspace frame, because the ship's frame of reference perfectly matches the subspace frame of reference as it is determined from all other factors in the universe.  In other words, the ship's frame of reference is made to be such that it does not contribute to the sum, S, discussed earlier.  The only way this is possible is if the ship's frame of reference seems to be exactly the frame of reference of subspace defined as if the ship were not there.     Therefore, a warp field couples the frame of reference of everything inside the warp field to the frame of reference of subspace.  This becomes true regardless of what the frame of reference of the ship would be without the warp field there (i.e. it is true regardless of the actual speed of the ship with respect to subspace).  Thus, while the warp field is active, the ship's frame of reference remains the frame of reference of subspace and is NOT dependent on the ships speed.  This is what places the ship outside of the realm of relativity and allows it to travel faster than light without gross violations of causality.

Second, we note that this sounds like the warp field is completely removing the mass of the ship as viewed from outside of the warp field; however, this isn't the case.  Theory tells us that in order to completely remove the effects of a ship's mass from the universe, one would have to expend an infinite amount of energy.  What the warp field does is to de-couple the relationship between subspace-equivalent mass/stress-energy and normal space mass/stress-energy.  The subspace-equivalent mass becomes zero, while the normal space mass is reduced (in the eye of the outside observer) much like it is in the case of simple subspace fields.

So, this is how simple subspace fields and warp fields are formed by manipulating normal space material to produce desired effects on the frame of reference of subspace.  Next we will discuss certain aspects of these fields.

4. General Aspects of Subspace Fields:

All forms of subspace fields (be they simple subspace fields or warp fields) have certain general aspects.  For example, all subspace fields have effects in both space and subspace and form an interaction between the two.  We thus talk about such things as the shape of the field as it exists in the normal space domain or the subspace domain. The two shapes can be different, and a particular mapping will exist that maps one shape to the other.  The shape of the field in subspace will be mentioned later, but for other aspects of subspace fields, we will generally discuss only the effects they have in normal space.

All forms of subspace fields have three basic layers--the interior layer, the exterior layer, and the interaction layer. The interior layer is generally surrounded by the interaction layer.  Though the interior layer is usually normal space, there are some cases in which the field changes the characteristics of the space within the interior layer (such as the subspace fields used with today's faster than light computer cores which will be discussed later).  More often, the interior layer is basically a "bubble" of normal space surrounded by the interaction layer of the field.  The exterior layer is the part of the field which extends beyond the interaction layer.  This layer is generally filled with normal space with certain aspects of the interaction layer spilling over and mixing in with the normal space.

In the interaction layer, space and subspace combine.  The interaction of space and subspace within this layer is what gives subspace fields their unique capabilities.  For example, observers outside of the subspace field see various effects (such as a reduction of mass) when viewing objects within the subspace field.  The outside observers see these effects because they are viewing the objects through the influence of the interaction layer.  Also, the effects of the interaction layer are what causes subspace to ignore (to some extent) masses (or more appropriately, stress-energy tensors) which are inside of a subspace field, as mentioned earlier.  Subspace does this because it too is "viewing" those objects through the effects of the interaction layer.

With these common basics in mind, we can now discuss specific aspects of simple subspace fields and warp fields independently.

 5. Simple Subspace Fields:

A subspace field which is symmetric in the subspace domain causes subspace to (in essence) act as an energy reservoir.  Such a field is referred to as a simple subspace field (or just "a subspace field"). To outside observers, anything within such a field will appear to "loose" some of it's mass energy to subspace while the field is active (as discussed earlier.  (Equivalently, one could say that the field masks out part of the mass of objects inside the field as they are viewed from normal space.)  The amount of interior mass energy "placed" into subspace is dependent on the strength of the subspace field.  For all practical purposes, while the field is active, this mass energy disappears from normal space (see Technical Note 1 for this section).  However, it should be noted that when one compares the normal-space energy and momentum of a closed system before a subspace field is activated with that of the system after the field is deactivated, energy and momentum conservation must apply.  We will now look at momentum and energy conservation considerations with respect to simple subspace fields.

5.1  Momentum and Energy Conservation with Simple Subspace Fields

Here we will look separately at momentum conservation and energy conservation as they apply to subspace fields.  At the end of this section, examples will be considered to illustrate these conservation considerations.

5.1.1  Momentum Conservation

Consider a ship of mass M which surrounds itself in a simple subspace field.  To outside, normal space, the mass of the ship becomes m < M once the field is active.  This new, lower mass is called the apparent rest mass of the ship (or simply its "apparent mass").  If the normal space manifestation of the subspace field can be shaped so that the ship's fuel is kept outside of the field, the ratio of fuel mass to ship mass will be greatly increased.  In accordance with momentum conservation, fuel expelled with a given momentum will cause the ship to have an equivalent momentum in the opposite direction (thus conserving momentum).  However, with the subspace field activated, the speed this momentum gives to the ship would be calculated using the apparent (lower) rest mass of the ship. Thus, with the use of a subspace field one can achieve greatly improved acceleration rates as well as greatly lowered energy costs for reaching a given speed.

As long as the field is active, kinematic considerations of the ship will be calculated with the ship's apparent mass.  However, when the subspace field is deactivated, the masked mass of the ship returns.  The results of this returning mass as it applies to momentum conservation will be considered in the examples given after the energy conservation considerations have been discussed.

5.1.2  Energy Conservation

Once a subspace field is activated, energy conservation can be realized only if one includes the mass energy which is "submerged" into subspace.  This will be demonstrated in examples given at the end of this section.

There are, however, energy considerations other than kinematic ones.  Some of the energy that is internal to the ship must go into producing the subspace field.  Currently, subspace field generators produce unstable fields which continually "bleed" their energy back into normal space.  (This energy generally manifests itself as a combination of heat within the subspace generator, electromagnetic radiation, and/or subspace radiation which can couple it's energy into normal space.  Also, this energy bleeds off symmetrically so that momentum is conserved.)  Because of this bleed off, subspace field generators must continually supply energy to the subspace fields.  The same amount of energy supplied to the field is eventually bled back into regular space, thus conserving energy.

The final energy consideration involves internal ship energy which remains internal (producing life support, etc.).  Because the ship is within the interior of the subspace field, it appears to itself to be in a normal-space "bubble."  This means, for example, that to the ship's crew, the matter and antimatter on board do not loose any mass.  Objects on board the ship only seems to loose mass into subspace when the observer views the ship through the masking of the subspace field's interaction layer.  Inside the ship, the available energy does not change, and energy conservation goes on as it always did.

We can, however, show that even when viewed from normal space outside the subspace field, the energy released by the interaction of matter and anti-matter on board the ship is the same as if the matter and anti-matter hadn't "lost" mass to subspace.  It is true that once the field is activated, the matter and anti-matter aboard the ship will seem to "loose" some of it's mass energy to subspace in the point of view of the outside observer.  For the outside observer to realize that energy has been conserved, he must remember that this mass energy did not actually disappear from existence, but has simply been submerged into subspace.  However, as the matter and anti-matter interact, their mass is turned into other forms of energy.  Since this energy is no longer in the form of mass, the subspace field no longer masks part of that non-mass energy from the outside observer.  So, as the matter and anti-matter interact, the outside observer not only sees the reduced masses of the matter and anti-matter turn into other forms of energy, he also sees mass energy that had been masked by subspace being converted into normal, non-mass energy.  The result is that he sees as much normal, non-mass energy being produced as any inside observer would see, thus conserving energy from all points of view.

5.1.3  Some Examples

To analyze the conservation of energy and momentum involved with subspace fields, we will look at two examples.  In each example we will consider a ship which encloses itself within a subspace field and then expels fuel in order to take a trip.  At each step of the trip we will show that energy and momentum are conserved.

Example 1

In these examples, the ship of mass M begins in one particular frame of reference.  All energies and momentums will be calculated in this frame.  Initially, the ship's energy consists of its mass energy (M*c^2) and internal energy (E(int)--which will be used for various purposes).  During the trip, part of the internal energy will be used for on-ship purposes, and while this energy may change form (becoming heat and eventually being radiated into space, for example) we know that this energy is always present in some form.  Thus this part of the internal energy is preserved.  The rest of the energy involved will be considered at each step to show that it is also conserved along with momentum.

Step 1:

The ship uses part of it's internal energy to create a subspace field.  As explained above, this energy is bled back into space, thus this energy is conserved.  As the field is turned on, part of the ship's mass is masked from outside observers, and the apparent mass of the ship becomes m.  To realize the conservation of energy, we must remember that this mass energy is still "present", but is submerged in subspace.  This submerged energy is the difference between the mass energy of the ship initially and its mass energy now--(M - m)*c^2. This makes it obvious that the energy is conserved (since the submerged energy of the ship plus its energy now is the same as it's initial mass energy).

Step 2:

The ship uses part of it's internal energy to produce a high energy photon (as fuel) with a certain momentum in a particular direction.  In accordance with conservation of momentum, the ship must gain an equivalent momentum in the opposite direction.  In accordance with conservation of energy, the internal energy used must be equal to the energy given to the photon plus the change in energy of the ship (which now has more energy since it is moving in the original frame of reference).  (See Technical Note 2 for this section.)  The change in energy of the ship is calculated with the ship's apparent mass (m), and the energy submerged in subspace is still equal to (M - m)*c^2.

So, part of the internal energy goes into the energy of the photon and increases the energy of the ship, while the energy submerged in subspace is still the same.  Meanwhile, the momentum of the photon is canceled out by the momentum of the ship.  Thus, energy  and momentum are conserved.

Step 3:

As the ship travels, it may experience "collisions" with other objects.  As long as these collisions do not collapse the subspace field, the ship's apparent rest mass will still be m as far as the collisions are concerned. This is no violation of energy or momentum, because for all intents and purposes, the missing mass of the ship has been "left" sitting still in the original frame of reference by keeping it submerged in subspace.  Thus the ship should interact with other objects as if it's mass is m.

 Step 4:

Part of the internal energy is used to produce another photon for fuel which brings the ship back to rest in the original frame of reference.  Energy and momentum are conserved here in the same way they were conserved in step 2.

Step 5:

The subspace field is disengaged, and the energy which had been submerged in subspace is returned to the mass energy of the ship. This is just the reverse of the first step, and energy is obviously conserved.

Example 2

This example is identical to the first example up to and including Step 3.  We will begin here with a new Step 4.

Step 4:

In the previous example, the ship "decelerated" to get back to the original frame of reference and then shut off its subspace field. Here we examine what happens if the subspace field is shut off (intentionally or accidentally) while the ship is still moving in the original frame of reference.

As the field is deactivated, the mass energy which was submerged in subspace will be added back to the ship.  This mass energy can be modeled as actual mass which is sitting at rest in the original frame of reference.  In this model, as the field dies, it is as if the ship runs into a portion of matter with a mass of (M - m).  This is not as harmful as it may seem.  A ship which actually runs into a chunk of matter with significant mass will be crushed because the force applied to the front of the ship will have to be transferred to the back of the ship before the back will stop moving.  This produces the crushing effect.  In our case, the mass is "added" throughout any objects within the subspace field at the same moment as the field is deactivated.  All particles throughout the interior of the subspace field are decelerated at the same time and at the same rate.

It is not that obvious what exactly takes place in this case to allow for the conservation of momentum and energy.  We can deduce what would happen by considering the model of the situation in which a ship runs into a mass of (M - m).  In this case, a ship of mass m and momentum p inelastically collides with an object of mass (M - m) which is at rest.  After the collision, the combined clump of ship plus object has a mass of M and a momentum p (to conserve momentum).  But, the energy of a mass m with momentum p plus the energy of a mass (M - m) does not generally equal to the energy of a mass M with a momentum p.  In order to conserve energy in this case, the final system must have internal energy in addition to it's mass energy and kinetic energy.  (See Technical Note 3 for this section.)  In our model, the collision will generally cause heating to produce this internal energy.  In the actual situation, the system after the subspace field has died will include electromagnetic radiation, and/or subspace radiation, and/or heat inside the ship to make up the extra energy needed for energy conservation.

In short, we have shown energy and momentum conservation in these examples with the following comparisons.  Turning on the subspace field is compared to a situation where the ship removes part of its mass, leaving it at rest in its original frame of reference.  The ship then continues along its trip, just as if it had a lower mass. Turning off the subspace field can then be compared to adding back on the previously removed mass which is still at rest in the original reference frame.  With these comparisons, one can see how energy and momentum are conserved in the use of simple subspace fields.

---------------------------------------------------------------------

5.2  Technical Notes for this Section (Simple Subspace Fields)

*Technical Note 1

We say that when a subspace field is activated, part of the mass energy of objects within the field disappears from normal space for all practical purposes (as seen by outside observers).  We should note, however, that other aspects of this matter (charge, baryon number, lepton number, etc.) are unaffected.

For example, an electron sitting within a subspace field will still seem to outside observers to have a charge of -1, a lepton number of 1, etc.  However, it will seems as if the normal rest mass of the electron has been reduced.

So, when a ship in a subspace field seems to loose part of its mass as seen by outside observers, it is not as if the ship has lost some of its particles.  Instead, it is as if all the particles individually became particles of lower rest mass.

*Technical Note 2

Here we examine the amount of energy needed to propel a ship with a reduced mass of m to a velocity v by expelling a photon.  We will be using regular relativistic equations for momentum and energy with the following notations:

   c    = the speed of light

   v    = the velocity of the ship

   beta = v/c

                 1

   gamma = ---------------.

             ____________

           \/ 1 - beta^2

Now, at some point the ship (whose reduced mass is m) uses part of its internal energy to expel a photon in a particular direction. If the photon is created correctly, afterwards the ship will be moving with the desired velocity v.  Its momentum and energy will thus be given by the following:

   p(ship) = gamma*m*v    (the relativistic momentum of the ship)

   E(ship) = gamma*m*c^2  (the relativistic energy of the ship)

 
Now, in order to conserve momentum, the photon's momentum will have to be equal and opposite to that of the ship.  The energy of the photon can then be calculated from its momentum.  We can thus write the following:

   p(photon) = p(ship) = gamma*m*v

   E(photon) = p(photon)*c = gamma*m*v*c

It is now possible for us to calculate how much of the internal energy of the ship would have to be used to expel this photon.  Before the photon was expelled, the energy of the system included the mass energy of the ship (m*c^2), the internal energy of the ship which would be used to expel the photon (E(fuel)), and some other internal energy which wouldn't be changed.  After the photon is expelled, the energy of the system includes the larger energy of the ship (gamma*m*c^2), the energy of the photon (gamma*m*v*c), and that part of the internal energy which wasn't changed.  The energy used to expel the photon must make up for the difference in energy between these two situations.  We can thus write the following:

   E(fuel) = (gamma*m*c^2 + gamma*m*v*c) - (m*c^2)

           = [gamma*(1 + beta) - 1]*m*c^2.

The interesting thing to note here is that if the subspace field hadn't been used to lower the apparent mass of the ship, this energy would be calculated with the same formula, except m would be replaced by M.  This means that the subspace field allows a savings of energy given by

   E(saved) = [gamma*(1 + beta) - 1]*(M - m)*c^2.

As long as the energy needed to produce and maintain the field is less than this energy, then there is an overall savings in energy for this particular example.

It should also be noted that for significantly high velocities, the E(fuel) could still be impractically high unless the apparent mass (m) is significantly small.  As it turns out, mass masking by subspace fields can provide the needed lowering in mass to make large changes in the velocity of the ship a practical ability.

*Technical Note 3

Here we examine the momentum and energy considerations of a collision between a mass m with momentum p and a mass (M - m) at rest. Consider the following diagrams of the situations before and after the collision:

Before:

            m                                  O   M - m

            O----------> p                    O O  P = 0

                                               O

(The total internal energy of these systems = E(int-before).)

After:

                       O  M

                      OOO---------->p

                       O

             Internal energy = E(int-after).

The momentum of the larger mass M (after the collision) will be equal to the momentum of the mass m (before the collision) in order to conserve momentum.  We are interested in the difference in the Energy between the two situations.  We will calculate this energy using the following notations:

   gamma = the relativistic gamma factor for the mass m

   GAMMA = the relativistic gamma factor for the mass M

We can then write the difference in energy as follows:

   E(After) - E(Before) =

    [E(int-after) + GAMMA*M*c^2] - [E(int-before) + gamma*m*c^2 +

                                                          (M-m)*c^2]

Conservation of energy requires this difference to be zero.  Using this, we will isolate the internal energies of the systems on one side of the equation.  This will be the difference in the internal energies before and after the collision (Delta(E-int)).  We thus write the following:

   Delta(E-int) = E(int-after) - E(int-before)

                = gamma*m*c^2 + (M-m)*c^2 - GAMMA*M*c^2

                = [(M-m) - (GAMMA*M - gamma*m)]*c^2

Now, we can rewrite the gammas by remembering that for any systemof mass m and momentum p, the energy can be written as

                       ___________________

   E = gamma*m*c^2 = \/p^2*c^2 + m^2*c^4

 We can thus write gamma for such a system as the following:

             ___________________

   gamma = \/ p^2/(m^2*c^2) + 1

Since the momentum of both m and M are the same in our example, we can rewrite the change in internal energy as the following:

                             _______________    _______________

  Delta(E-int) = [(M-m) - (\/ p^2/c^2 + M^2 - \/ p^2/c^2 + m^2  )]*c^2

Now, we know from the triangle inequality that

                _______________

   p/c + M >= \/ p^2/c^2 + M^2

and             _______________

   p/c + m >= \/ p^2/c^2 + m^2

(where ">=" denotes greater than or equal to).  Given this, we see that by subtracting one from the other we have

               _______________     _______________

   M-m  >=  (\/ p^2/c^2 + M^2  - \/ p^2/c^2 + m^2 ).

Thus,

             _______________     _______________

   [M-m - (\/ p^2/c^2 + M^2  - \/ p^2/c^2 + m^2 )]*c^2

is always greater than or equal to zero.

So, we see that the change in internal energy is always positive. That means that in order for energy and momentum to be conserved in this type of collision regardless of the masses and momentum involved, the overall system must increase in internal energy.  Generally, this would mean that the collision would cause heating and this additional heat would allow for energy to be conserved.

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6. Warp Fields:

There is one major difference between simple subspace fields and warp fields.  A field is labeled as a warp field when it produces a reference-frame coupling.  The reference frame of objects within the real-space manifestation of the warp field must be coupled in some way to the reference frame of subspace, as discussed in section 3.

In section 3 we mentioned that we would discuss other aspects of warp field production in this section.  What we want to consider is the difference in the "exotic" nature of the stress-energy tensors needed to produce simple subspace fields and those needed to produce warp fields.  There are essentially two ways in which one could imagine changing a subspace-field-producing stress-energy tensor so that it becomes a warp-field-producing stress-energy tensors.  As it turns out, the easiest way to do this is to change the exotic nature of the tensor so as to skew the subspace manifestation of the subspace field until it is no longer symmetric in that domain. Interestingly, manipulating a subspace-field-producing tensor in this way creates an exotic enough effect to produce a reference-frame coupling at the interaction layer of the field.  Observers in the interior of such a field will measure space and time outside of the field as if they were viewing it from within the subspace frame of reference--regardless of the velocity of these observers.  This  feature is what allows for the faster than light travel on which we so depend.

Another useful features of skewed subspace fields is that the depositing of mass energy into subspace which occurs is not symmetric. This asymmetric placing of energy into subspace manifests itself as momentum transfer, and this causes subspace to act as a momentum reservoir as well as an energy reservoir.  Momentum is essentially deposited within subspace, and to conserve overall momentum, the combination of all objects within the warp field will gain an equivalent momentum in the opposite direction.  Only when the momentum transferred into subspace is taken into account can momentum conservation be realized.  At the time the ship's momentum is changed, no actual fuel is expelled to produce this momentum, and normal-space-only momentum conservation is essentially ignored as long as subspace is masking the momentum.  Therefore, this method of warp travel is labeled as non-Newtonian propulsion.  The use of warp propulsion will be discussed in a later subsection.

It is also possible to change the exotic nature of the stress-energy tensor in order to produce warp fields which are non-propulsive.  This is generally done simply by intensifying the exotic nature of the tensor by increasing its strength alone, and without skewing the subspace field.  Such tensors are generally called subspace-symmetric warp tensors, and they produces a field which provides a reference frame coupling while the subspace manifestation of the field is still symmetric.  By changing the characteristics of such tensors, one can produce many different varieties of these fields, and even though they are technically warp fields (because they produce a reference frame coupling) certain varieties are sometimes still referred to simply as subspace fields (because of they are in fact symmetric within the subspace domain.)

Perhaps the most useful non-propulsive warp fields in use today are ones which provide a subspace reference frame coupling to every point within the interior of the field as views by every other point within the interior of the field. Unlike the warp propulsion field, this field allows objects within its interior to travel faster than light with respect to one another. These fields are the ones in which modern shipboard computer cores are placed so that signals can be sent faster than light between various computer components.

Another type of symmetric, non-propulsive field which has been studied with interest are known as static warp bubbles.  These have been known to have the odd effect of coupling people inside the field not back to real space-time, but to a virtual space-time created within the bubble.

There are, as mentioned, many different types of non-propulsive warp fields, and we will not consider them all here.  What we wish to stress here is that the one major component which all warp fields share (propulsive/asymmetric or non-propulsive/symmetric) is a reference frame coupling of one type or another.

6.1  Warp Propulsion

Producing a warp propulsion field causes subspace to act as both an energy and a momentum reservoir.  The ship within the warp field will have a lower apparent mass, and it will gain a momentum equivalent to and in the opposite direction of the momentum placed into subspace.  Because there is also a reference frame coupling, the relationship between the momentum and the velocity of the ship is not calculated using Einsteinian physics.  This allows the ship to have a real (non-imaginary) momentum and energy even though its apparent speed is greater than the speed of light.  Energy and momentum conservation will be discussed in a later subsection.

6.1.1  Single-Layered Warp Fields

First we will consider warp propulsion produced with a single-layer warp field.  As such a field is activated, the momentum of the ship (and thus it's speed) will increase.  At first, the ship will be traveling at slower than light speeds, and the energy of the ship increases dramatically as its speed approaches that of light.  Only after the jump to faster than light speeds occurs will the reference frame coupling take full effect, and the energy of the ship be completely outside of the realm of Einsteinian physics.

Once the reference frame coupling takes effect, all measurements with respects to the ship are done as if the ship is in the frame of reference of subspace.  That means that at any particular moment, properties such as distances, times, etc. are measured just as if the ship were sitting still for that moment in the frame of reference of subspace.  As in illustration, one could imagine taking a snapshot of a ship in warp and finding that it is indistinguishable for that one moment from a ship who is not moving with respect to the subspace frame of reference.  Yet, we attribute kinetic energy (energy of motion) to such a ship, even if we view it from the subspace frame of reference.  This is because the kinetic energy of the ship is actually held within the warp field itself.

Thus, to keep the ship at a certain speed, one must keep the warp field at a constant energy level which is seen as the energy of the ship itself.  But, today's warp field generators produce unstable fields (similar to subspace field generators.)  Thus, warp fields also bleed off there energy back to the normal universe (in the form of heat in the field coils, electromagnetic energy given off nearby the ship, etc.).  Therefore, the warp field must be given a constant supply of energy from the ship.  (This, too, will be discussed in a later subsection. The important thing to understand here is that the warp field does need a constant supply of energy).

To increase the speed of the ship, one must increase the energy level of the warp field.  However, at higher energy levels, a warp field becomes much less efficient (bleeding off it's energy at much larger rates).  Therefore, the power output of the ship must increase dramatically to hold the warp field at a higher energy level (thus holding the ship at a large velocity).

For our examples, we will use a model which approximates warp field energy levels in certain geometries.  The power (the amount of energy given to the field per unit time) given to a field layer depends on the energy of that layer, and in our model that dependence is as follows:

    Power = P_0*(E/E_0)^3

Where E is the energy of the layer (and thus the energy of the ship) and P_0 & E_0 are a power level and an energy level intrinsic to the model.

For example, a ship traveling at a particular warp velocity may have an energy of 2*E_0 associated with its motion.  In order to keep the warp field up, the ship would have to output energy at a particular rate, providing a power of 8*P_0.  If the ship increases its speed so that its energy is now 4*E_0 (twice as much as before), the ship will have to provide a power of 64*P_0 (8 times as much as before) in order to keep the warp field up.  The energy of the ship itself (associated with it's velocity) has only increased by a factor of 2, while the warp engines are now having to output eight times as much power into the warp field because the higher energy warp field is much less efficient (quickly bleeding it's energy back into normal space).

6.1.2  Multi-Layered Warp Fields

As the ship's speed increases, the correlation between space and  subspace at the interaction layer becomes greater and greater.  More and more of the ship's mass energy is masked by (or submerged into) subspace, and more and more momentum is placed into subspace.  We thus say that the ship is submerged to a deeper subspace level as its speed increases.  (We should, however, remember that the interior of the warp field is essentially still normal space. It is only the relationship between the interior and exterior of the field that becomes deeper interlaced with subspace.)  We can use this analogy to understand why multi-layer warp fields are used today for warp propulsion.

By correctly setting up the geometry of the stress-energy tensor within a warp field generator, one could produce a double layered warp field which conceptually divides subspace into two levels (an "upper" level and a "lower" level).  This is basically done by creating a two stage stress energy tensor which when both stages are active looks like the usual warp-field-creating stress energy tensor.  However, when only one stage of the tensor is active, its effects would not be able to "submerge" a ship deeper than the "bottom" of the upper level of subspace, regardless of how much energy was provided to the tensor.  For our purposes we will say that if the ship were "submerged" as deep as this first stage could take it, it would have an energy of E_th (the threshold energy between the two subspace levels).  Then, when the first layer of the warp field was active, the ship's energy would be between zero and E_th. For illustration, we can assume that E_th is a particular value, say 4*E_0 (where E_0 comes from our model mentioned above).

     With the first warp layer active, one can supply it with more and more power up to the point where the energy in that layer is 4*E_0. At that point, one would be supplying a power of 64*P_0 (as seen earlier).  This is no different from having a single-layer geometry to the warp field rather than a double-layer warp field.  The difference will be evident if one attempts to supply even more power to the first layer of the double-layer warp field.  With only the first layer active, the energy of the warp field can be no higher than 4*E_0 (the energy associated with being "half way deep" into subspace).  Any power supplied to the layer above 64*P_0 will be instantly bleed back into normal space rather than pushing the warp field to a higher energy.

In order to push the ship deeper into subspace and further increase its energy, the second warp field layer needs to be activated.  One therefore turns on the second stage of the stress-energy tensor, creating the second warp field layer.  This can only happen once the first layer has taken the ship deep enough into the first level of subspace to "jump" into the second level as the second layer is activated.  This is due to the fact that if one tries to energize the second stage of a two-stage stress-energy tensor before the first stage is sufficiently energized, the overall tensor will not have the geometry needed to sustain a warp field.  However, once the first stage is sufficiently energized, the second stage will complement the overall geometry of the tensor, producing the second field layer.  Once the second layer is activated, the total energy of the warp field is _divided_ among the two stages of the tensor, and thus among the two layers of the subspace field.

In our example, one could hold the ship just above 4*E_0 (close enough for us to estimate it with 4*E_0) with each layer holding 2*E_0 of energy apiece.  This means that the power needed by each of the two layers is only 8*P_0 apiece (as calculated in our model) for a total of 16*P_0 rather than 64*P_0.  This is a substantial savings in power consumption.

To sum up... As one pushes one layer of a warp field to higher and higher energies, the efficiency of that layer drops dramatically. However, one can use multi-layer warp fields to divide subspace into many levels.  By adding enough energy to the warp field while N layers are active, one can go deeper and deeper into level N of subspace. Once one is close enough to level N + 1, one can activate the next warp field layer and "jump" into the next subspace level. This divides the energy of the warp field among more layers, lowering the energy level of each individual layer.  This in turn increases the efficiency of each individual layer (thus increasing the overall efficiency of the warp field as a whole).

The actual calculation of the power requirements for a warp field is more complicated than in our simple model.  However, the principle is the same, and multi-layer warp fields do increase power efficiency. When this discovery was made, it had a profound effects on the future of Warp Propulsion.

6.1.3  Development of Modern Warp Propulsion Fields

Just after the discovery of increased efficiency with the use of multi-layer warp fields, many research teams started working to produce various multi-layer strategies and maximizing there efficiencies.  One particular team jumped ahead of the rest and fairly easily developed a 9 layer warp field design (the first layer beginning at the speed of light).  While work started on maximizing the efficiency of this new 9 layer design, still other teams moved on to try and produce strategies with even higher numbers of layers. However, no such attempts were successful.

Work done to maximize the 9 layer design soon lead to theories which suggested that the success of the 9 layer strategy wasn't simply luck or coincidence.  These theories suggested that subspace actually possessed an intrinsic 9 level nature--that there really were 9 preexisting subspace levels.  Such theories correctly predicted the proper method for maximizing the 9 layer warp field design, and they suggested that it was impossible to produce warp fields with more than 9 levels.

Today, many aspects of these theories are widely accepted, and the 9 layer warp field is the standard by which warp factors are defined.  The full development of the first warp field layer (Warp 1) in today's warp systems constitutes the entrance into the first level of subspace.  Each consecutive warp factor constitutes the entrance into the next consecutive subspace level.  As one approaches warp 10, one presses deeper towards the "bottom" of the ninth subspace level, and warp 10 corresponds to being fully submerged into subspace.  Thus, fully submerging a vessel into subspace theoretically gives the vessel infinite velocity, requires an infinite amount of energy to get the vessel there, and requires an infinite output of power to hold the ship there.

Unfortunately, the 9 levels of subspace (which is theoretically natural and cannot be bypassed) is the limiting factor of the speeds maintainable by today's warp vessels.  Past warp 9 the power requirements for higher warp speeds continues to increase without another power threshold like those found at the integer warp factors. The fact that current theory rules out the possibility of producing a tenth highly efficient warp factor is generally referred to as the "warp 10 barrier." (Note: Sometimes this phrase is used to refer to the infinite speed one would theoretically obtain at warp 10. However, this is a less proper use of the phrase.  Thus, the statement "perhaps one day we will break the warp 10 barrier" would more likely refer to the possibility of finding an efficient means for traveling much faster than warp 9 rather than referring to the possibility of traveling faster than infinite speed.)

Though our current technology still supports the theories behind the warp 10 barrier, certain brushes with advanced non-federation technology suggests that some linking of warp field production and strong gravimetric distortion may hold the key to producing fantastic speeds through energy and power outputs easily attainable by today's starships.  Still, skepticism abounds, and only time will tell whether we will every be able to "break" the warp 10 barrier.

6.1.4  Modern Warp Propulsion Field Generation

There was one very important problem with multi-layered warp fields that we have yet to mention.  The geometry of a multi-stage stress-energy tensor inherently produces a warp field which is symmetric in the subspace domain.  That means that the multi-layered warp field produced by such a tensor cannot be propulsive.

In order for propulsive fields to gain the benefits which multi-layered warp fields possess, a new way to produce multi-layered fields needed to be found.  As it turns out, the key to regaining the non-Newtonian drive came in nesting many layers of warp field energy within one another.  In today's warp engines, a series of single-stage tensors are activated in a particular way to produce a warp field which has the desired effects.  We will now examine how the "trick" of producing multi-layered, propulsive warp fields is performed by considering an example using 3 single layer field generators.

The 3 field generators are placed in a row with a particular distance between each of them.  The generators are then activated in sequence, one after the other, at a particular frequency.  This means that plasma is ejected for a moment into each field coil, and then it is quickly shut off.  Each coil then produces its own warp field layer which dissipates energy as it expands and eventually disappears once it has lost all its energy.  Before the field layer produced by the first generator dies, the second field generator is activated, and so on.

Because the tensors used to create the 3 fields are each single-stage tensors, the three fields themselves do not form a three layer warp field like we have previously described.  Instead, they act as three separate, nested layers of warp field energy.  However, when the  frequency at which the three fields are produced is just right (the actual value depends on the precise geometry of the situation) the nested field layers form at just the right spacing so that they interact to produce a single warp field.  At that point, the three nested field layers appear to subspace to be one warp field which consists of the first layer of a multi-layer design.  If the tensors used to produce the fields have the correct geometry (which in part depends on the number and placements of the field coils), then this multi-layer design seen by subspace will be the natural 9 layer design which we want.  Also, because the nested layers that make up this field are produced at different points in space (and thus at different corresponding points in subspace) the overall warp field appears to be asymmetric in the subspace domain.  Thus, this "first-level" warp field will be a propulsive field.

At this point, we could increase the energy input to each of the field coils in order to make the field press deeper into subspace.  However, when we do this we increase the energy of the overall warp field being created, thus lowering the efficiency of the overall field.  This means that each nested field layer will dissipate its energy more rapidly, thus expanding and dying more rapidly.  Remember that the key to having the 3 nested layers act as a single warp field was that they were created with just the right spacing to interact properly.  Thus, because the higher energy field layers are expanding more rapidly, we must produce the layers at a higher frequency if we still want them to interact properly and form a single warp field.

At some point, the energy in the overall warp field will be enough to press the ship into the second level of subspace.  When this happens, we will have a second-level warp field--subspace will see the three nested field layers as a single warp field consisting of 2 un-nested field layers.  Conceptually we can then think of the total energy of the field being divided among these two "virtual" un-nested field layers, thus increasing the total efficiency of the warp field as discussed earlier.  With the efficiency increased, each layer now dissipates and expands more slowly.  However, in the second level of subspace, each field layer needs to interact more strongly with the next, and thus they must be created closer together.  The combination of slower expansion and the need to create the fields closer together exactly cancel each other out such that the frequency just before interring the second level is approximately equal to the frequency just after interring the second level.

This process can be continued--increasing the energy of the warp field and increasing the frequency at which the nested layers are created--in order to press deeper into subspace and pass through the higher efficiency points at the integer warp values.  And there we have it--the effects of a multi-layered warp field design which is produced with some number of nested layers of warp field energy, each created at a different point in space and subspace such that the field is asymmetric (and thus propulsive).  We should note that this means that the asymmetry of the field (and thus the direction of propulsion) is not controlled by changing the complex geometry of the tensor used to create the field, but rather by sequencing the field coils in a particular way.  With modern ship design, an optimal number of field coils are placed within two warp nacelles on either side of the ship.  This means that by properly sequencing the coils in the two nacelles, the ship will be able to maneuver in various directions during warp.  We could also produce maneuverability in a single nacelle design by changing the geometry of the tensors used such that they give a left-right asymmetry.  However, this has been found to be much less efficient and much more difficult than simply using two nacelles and sequencing the field coils properly to produce the desired effects.

Finally, we should note that in this modern design, the momentum coupling (the placement of momentum into subspace as mentioned earlier) manifests itself as a force coupling between the various layers of warp field energy.  During the coupling, part of the mass energy of the ship becomes masked by (or submerged into) subspace in an asymmetric way (because of the geometry of the field) to produce the momentum masking which creates the non-Newtonian propulsion.

6.2  Momentum and Energy Conservation with Warp Propulsion

In this section we will consider the conservation of momentum and energy as it applies to warp propulsion.  When we did this with normal subspace fields, we looked separately at each issue (energy and momentum), however, here they are so integrated that it will be easier to consider them both at once.

Again, we look at two types of energy separately--the internal energy of the ship, and the energy associated with the mass of the ship and its motion.  The momentum is, of course, closely related to the energy of the ship and its motion, so we will look at the two together.  For the internal energy of the ship, the conservation of energy takes place much the same way it did with subspace fields.  The mass of any matter/anti-matter is lowered, but energy is seen to be conserved by all observers, just as it is with subspace fields.  Part of the internal energy will go to produce the warp field, and this will eventually be bleed back into real space.

(Note: Since the warp field produces the motion of the ship in real space, and this bleeding off of energy makes it necessary to output energy at a constant rate in order to keep moving, one can also explain this as "continuum drag."  This is done by associated the motion of the ship to the motion of a classic vessel moving through the use of friction.  In this model, subspace is said to provides a constant force against the ship while the ship provides a constant force in order to keep moving at a constant velocity.  (See Technical Note 1 for this section.))

Just as it was with simple subspace fields, a warp field masks part of the mass of the enclosed ship from outside observers.  This leaves a ship of mass M with a new "apparent mass" of m.  Again, overall energy conservation can be realized only when one takes into account the mass energy submerged into subspace.

Now, it is the kinematic energy of the ship that is associated with its momentum.  They both increase as the actual velocity of the ship increases.  However, the velocity increases as the warp field increases, and this reduces the ship's apparent mass.  All of this can be accounted for with a simple association.  We associate the actual, faster than light velocity of the ship (v) with a slower-than-light, "energy-equivalent" velocity (v').  We then use the actual mass of the ship (M) and the energy-equivalent velocity (v') in conjunction with normal, relativistic equations to calculate the momentum and energy of the ship.  (Note: the relationship between v and v' is discussed in Technical Note 2.)  This association allows us to easily calculate the momentum and energy of the ship, and all the complexity of increasing the actual velocity while decreasing the apparent momentum are all rolled into the association.  So, where does this energy and momentum of the ship come from, and how are they conserved?  Well, remember that part of the internal energy goes into maintaining the warp field at a constant energy level.  That means that part of the internal energy must go into the warp field to raise it to that constant energy level in the first  place.  As mentioned earlier, this constant energy level of the warp field IS the energy of the ship's motion.  They are one and the same.

The momentum comes directly from the fact that a propulsive warp field causes subspace to act as a momentum reservoir.  There is a momentum being masked by subspace which is equal but opposite to the momentum of the ship.  Only when this masked momentum is taken into account can conservation of momentum be realized.  One could think of this situation as equivalent to a Newtonian drive situation by equating the momentum masked by subspace to the momentum of the expelled fuel in a Newtonian drive situation.  However, there is a major difference--anything in normal space which has momentum also has energy, and the energy of the expelled fuel in the Newtonian drive situation must come from the ship's internal energy. However, the momentum masked by subspace has no energy associated with it, and so it doesn't take away from the ship's internal energy.

The fact that subspace takes up for the momentum of the ship (momentum which seems to come from nowhere in the eyes of outside observers who only consider normal-space momentum) has some rather interesting effects, as we will see in examples below.

6.2.1  Some Examples

To analyze the conservation of energy and momentum involved with warp propulsion fields, we will look at two examples (similar to what we did when considering simple subspace fields).  In each example we will consider a ship which takes a trip using warp.  At each step of the trip we will show that energy and momentum are conserved.

Example 1

In these examples, the ship of mass M begins in one particular  frame of reference.  All energies and momentums will be calculated in this frame.  Initially, the ship's energy consists of its mass energy (M*c^2) and internal energy (E(int)--which will be used for various purposes).  During the trip, part of the internal energy will be used for on-ship purposes, and while this energy may change form (becoming heat and eventually being radiated into space, for example) we know that this energy is always present in some form.  Thus this part of the internal energy is preserved.  The rest of the energy involved will be considered at each step to show that it is also conserved.

Step 1:

The ship uses part of it's internal energy to create a warp field.  As discussed above, part of this energy is bled back into space, while the rest accounts for the kinematic energy of the ship, thus this energy is conserved.  As the field is turned on, part of the ship's mass is masked from outside observers, and the apparent mass of the ship becomes m.  To realize the conservation of energy, we must remember that this mass energy is still "present", but is submerged in subspace.  This submerged energy is the difference between the mass energy of the ship initially and its mass energy now--(M - m)*c^2. This makes it obvious that this energy is conserved (since the submerged energy of the ship plus its energy now is the same as it's initial mass energy).

The warp field also causes subspace to act as a momentum reservoir, and so a certain momentum becomes masked by subspace.  As mentioned above, this momentum has no energy associated with it.  To conserve overall momentum, the ship gains an equivalent momentum in an opposite direction.  The motion of the ship gives the ship kinematic energy.  Again, this energy is part of the energy contained in the warp field, and thus it comes from part of the internal energy.  We have thus shown overall conservation of momentum and energy in this step.

Step 2:

As the ship travels, it may experience "collisions" with other objects.  Though these collisions may not collapse the warp field, they would have interesting effects.  We will wait to consider these effects in example 2.  Collisions which do collapse the warp field can have very damaging effects.  (See Technical Note 3 for this section.)

 Step 3:

As the ship comes to its destination, it shuts down its warp field.  As this is done, the momentum masked by subspace becomes unmasked, and the ship in turn looses its momentum. The energy contained in the warp field is bleed back into normal space as the warp field collapses.  Remember that this energy also accounts for the energy of the ships motion, thus as the ship looses momentum, it also loses it's kinematic energy which is bleed back into normal space. Finally, the mass energy that was masked by subspace returns to the ship, bringing its mass back to the original M.  So, here we again see that the overall energy and momentum are conserved.

Example 2

The first step in this example is identical to the previous  example.  We will thus start with the second step and more closely examine the collisions mentioned in step 2 of example 1.

Step 2:

During the travel, the ship encounters a large object.  For convenience, we will assume that the object is at rest in the original rest frame of the ship so that it must be deflected away from the path of the ship.  As the object is deflected, the ship's momentum is effected as if it were a ship with a momentum calculated by using its energy-equivalent velocity (v').  That is, the ship acts no different (kinematically speaking) from a ship of mass M and velocity v'.

Deflected the object will give it energy and momentum.  The energy can come in part from the kinematic energy of the ship and in part from the internal energy of the ship (if a tractor beam is used to deflect the object, for example).  But, in addition, internal energy must be transferred to the warp field in order to keep it from collapsing during the interaction with the object.  How much internal energy needs to be expended and why will be explained as we look at momentum conservation.

To conserve momentum, the total change of the ship's momentum will be equal and opposite to the change in the momentum of the object.  The deflection of the object will cause the warp field to become imbalanced in the direction of the ship's change in momentum. 

This happens as the additional energy is feed to the warp field to keep it from collapsing.  After the interaction, the ship can do one of two things.  First, it could continue on its changed course, coming out of warp at some later point in time; or, second, it could use it's warp field to adjust it's momentum (and its course) to get to its original destination.

In the first case, the ship will continue its journey along its changed course until step 3.  In the second case, the ship will use the warp field to readjust its course.  As this readjustment is made, the imbalanced warp field deposits actual momentum into space (generally in the form of photons) rather than "putting" the momentum into subspace.  This means that the real change in momentum of the object will be counteracted by the real momentum of the expelled photons--thus conserving normal space momentum. 

The energy needed to produce these photons comes from the energy placed into the warp field (to keep it from collapsing) as the interaction with the object took place.  Also note that as the photons are emitted, the ship gains back the momentum it lost during the collision.  That means that it must also gain back the kinetic energy that it lost.  This energy must also be supplied by the energy stored in the warp field while the interaction took place.  Since this energy is exactly the energy lost to the object during the interaction, the object's energy eventually comes from the internal energy of the ship.  Therefore, as the object is deflected, the energy feed into the warp field is just enough to produce photons (whose momentum will be equal and opposite to the change in the object's momentum) and to restore the kinematic energy lost by the ship.

(Note: The ship could continually adjust its warp field during the collision so that it's momentum and velocity don't change.  In this case, energy is still feed to the warp field during the interaction, but the continually adjusting warp field will continually use that energy to immediately create the photons necessary to conserve momentum. The end result is the same--the ship has changed the momentum of the object, a momentum equal and opposite to that of the change becomes real in the form of photons, and the ship's momentum remains unchanged.  Meanwhile, the internal energy of the ship has been used to produce the photons and to give the object its energy.)

So, energy and momentum in real space are conserved during and after a "collision" with an object.

Step 3:

The ship reaches it's destination and shuts off it's warp field.  What happens here will depend on which of the two cases (mentioned above) was chosen.  If the ship changed it's course after the collision (thus completely making up for the collision), then as the ship comes out of warp it will come back to rest in its original frame of reference (just as it did in example 1).  However, if the ship did not change its course, then it will have to make up for the collision as it comes out of warp.  As the imbalanced warp field collapses, the energy that was placed in the warp field during the interaction will produce the photons necessary to make up for the real momentum given to the object.  As the momentum of these photons gives momentum back to the ship, the ship will gain energy which must also come from the energy stored in the warp field during the interaction.  Then the re-balanced warp field can completely collapse, bringing the ship to rest in its original frame (just as it did in example 1).

Note, that if the energy needed to create the photons and restore the ships lost kinetic energy were not stored in the warp field during the collision, then they would have to be supplied by the internal energy of the ship as the warp field collapses.  That means that one would actually have to expend energy just to shut off the warp field (which makes no sense because the warp field must collapse when you stop feeding energy to it, even if you have no more energy left to create photons, etc.).  This is why it is important that all the energy needed to make up for the collision is stored in the warp field during the collision.

So, we see conservation of energy and momentum in all the stages of this example as well.

6.3  Technical Notes for this Section (Warp Fields)

*Technical Note 1

     Here we consider the model of warp travel which involves the concept of continuum drag.  In this model, the constant power supplied to the warp field to keep the ship at a constant speed is required because a constant force (continuum drag) is said to be applied to the ship.  To examine this, we consider a classical case of supplying a constant force against a friction force in order to maintain a constant velocity.

In this situation, a vehicle which has already reached a particular velocity (v) continues to supply a constant force equal and opposite to an opposing frictional force to maintain its velocity.  So we write

   F(vehicle) = -F(friction) = constant (in, say, the x direction).

Now, if the vehicle starts at a position x = 0 and at some point the vehicle has traveled to the position x, then we can calculate the amount of work done by (and thus the amount of energy supplied by) the vehicle during the trip:

       x

       /

   E = | F(x') dx'   (the integral from 0 to x of F(x'), dx').

       /

       0

But since the force is constant over time (and thus over distance), this reduces to the following:

   E = F*x

Finally, we can calculate the amount of power output one would need to keep supplying this force during the whole trip:

       dE     dx

   P = -- = F*-- = F*v

       dt     dt

Under normal circumstances, the vehicle would not be able to get to a velocity grater than c, and so this formula (though it itself doesn't indicate a problem at v = c) would never be used for such a velocity.  In our case, however, this formula works for our continuum drag model.

For a particular warp factor, the ship travels at a particular velocity v, and there is an associated continuum drag "force" F. Given those, one can calculate the power output needed to keep the ship at that warp factor.  For modern multi-layered warp fields, the force of the continuum drag is lowest at the integer warp values. Thus, this model gives alternate explanations for the concepts discussed in this section.

*Technical Note 2

Consider a ship of mass M traveling in warp with a faster than  light velocity v.  The apparent mass of the ship will be m < M, and the momentum and energy of the ship depends directly on its apparent mass m and velocity v in a non-trivial way.  Also note that since the apparent mass m depends on the strength of the warp field (and thus on the warp factor), it can then be seen as dependent on the ship's velocity v.

The easiest way to incorporate all the velocity dependence and calculate the momentum and energy of the ship is to make an  association between the actual, faster than light velocity (v) and an "energy-equivalent" velocity (v').  Using this velocity and the actual mass of the ship (M), one can calculate the momentum and energy of the ship.

We could calculate the momentum and energy using the apparent mass of the ship and the actual, faster than light velocity.  However,  the equations would look much different from those we are used to seeing in relativistic physics.  When the ship exchanges momentum and energy with an outside object, the exchange will be governed by these non-relativistic equations. 

In the end, the ship does not act like a relativistic ship with a mass equal to the reduced apparent mass of the ship.  So, though the  ship does have a lower apparent mass which facilitates the slippage of the ship through subspace, from the kinematics point of view, the ship's mass is M and its velocity is v'.  Of course, this is only the case with propulsive warp fields (where the momentum and energy calculations are outside of the realm of relativistic physics).  With non-propulsive warp fields and with simple subspace fields, the mass reduction carries over into the kinematics of the situation.

So, how is this energy-equivalent velocity (v') calculated?  As an example, we consider a simple model that is actually useful with certain warp field geometries.  In this model the relationship between v' and v is given as follows:

   v' = (1 - exp(-A*v/c))*c

where A is a constant intrinsic to the model and c is the speed of light.  Notice that as the actual velocity of the ship approaches  infinity, the energy-equivalent velocity will approach the speed of light.  Thus, as the velocity of the ship approaches infinity, so does its energy and momentum.

To use this formula in an example, consider this.  A poorly designed warp field geometry might yield an A value of 1.  In that  case, at a speed of only 2.01c (less than warp 2), the energy- equivalent velocity will be 0.866c.  At this velocity the energy of the ship would be

   E = gamma*M*c^2 = 2*M*c^2.

Before the warp field was active, the energy of the ship was M*c^2. This means that the ship now has an additional energy equivalent to the mass energy of the entire ship, and this incredible amount of energy would have to come from the energy reserves of the ship itself.

A more desirable geometry might yield an A value of 0.0001.  In that case, at a speed of 1000c the energy-equivalent velocity would be 0.95c.  In such a case, the energy of the ship is only 1.005 times the mass energy of the ship.  Still, an additional 0.005*M*c^2 of energy can be a phenomenal amount of energy for a large ship.  Half of a percent of the entire mass of the ship would need to be matter and anti-matter just to have enough energy to get the ship to this velocity (not counting the additional energy needed to sustain the warp field during the acceleration).

Today's warp fields (if modeled in this simplistic way) would yield extremely small A values so that a typical ship would easily be able to produce the energy needed to travel at high warp velocities.

 *Technical Note 3

Fortunately, the energy contained in the motion of a ship in warp is not very great (as discussed in the previous technical note).  If this were not the case, the ship would have to supply an extreme amount of energy in order to accelerate to a given warp speed.

The small energy of the ship translates into a small momentum as well.  That is, ships in warp do not carry a large amount of momentum. However, we should not discount the amount of damage that can be done by a warp collision.  To examine the damage potential of a warp collision, we will consider the following example.

During a battle with a hostile ship, our ship finds itself out matched, and it is decide to ram the hostile ship in the hopes of crippling its ability to cause more harm.  In addition to a warp core breach and the associated explosion to follow, we also want the actual collision to cause as much damage as possible.

In the time one has to accelerate before the collision, one could use the impulse engines to accelerate to a significant velocity.  However, the quick acceleration is only possible because a subspace field is used to greatly reduce the apparent mass of the ship.  The lower mass means that the momentum, energy, and damage potential are not necessarily that great.

On the other hand, one could jump into maximum warp to ram the  hostile ship.  Again, a quick acceleration (this time, to a faster than light velocity) is possible.  However, the velocity v translates to a fairly small energy-equivalent velocity v', and (as we have discussed) the momentum and energy of the ship's motion are again fairly small.

However, we have left out one part of the collision.  As the subspace field or warp field interacts with the hostile ship, it will deposit energy into the ship and collapse.  In the case of the subspace field, the collapse of the field will cause the mass energy of our ship to be returned (however momentum will be conserved) and will produce an increase in internal energy or radiated energy (which can have some damaging effects on the hostile ship).  In addition, the energy held in the field itself can be partially transferred to the hostile ship.

In the case of the warp field, as the field collapses, the mass energy and the momentum held in the field will return to the ship. Here, there has been no fuel expelled, and so there is no real momentum held in the ship's motion.  The momentum is completely held within sub space while the warp field is active.  However, as the warp field interacts with the hostile ship, the momentum that is held within the field can be coupled onto the hostile ship.  As the field collapses, rather than slow the motion of the ramming ship, the momentum in the field can be imparted to part of the hostile ship, causing more damage.  In addition, the energy held within the warp field (which is generally larger than the energy held in a subspace field) is imparted onto the hostile ship.

As it turns out, with everything taken into consideration, the damage potential is significantly greater when one chooses to use warp drive to ram the hostile ship.

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(7. Angular Momentum Conservation--for the 20th Century reader:

Throughout the other sections of this discussion, the term "momentum" was used to mean linear momentum only.  The reason why we haven't discussed angular momentum conservation as well is that it doesn't seem it can exist if we want to get the effects we desire.  Here I will point out why this is, and I will try to explain why it might not be so bad.

I will look at one specific example where angular momentum cannot be conserved in all frames of reference if we want to get a desired effect.  This is an example where a subspace field is used to lower the apparent mass of the ship in order to make it easier to get from place to place.  What I will do is look at the situation in one frame of reference where we can have angular momentum conservation.  Then I will transform into another frame of reference and show that angular momentum conservation in this frame requires that we use just as much energy to move the ship as if its mass during the trip were the total mass that it begins and ends with (before and after the subspace field is activated).  Thus I will show that we cannot gain any advantage by using subspace fields if we want to have angular momentum conservation.

Before I can do this, however, I must give the equations that are  used to relativistically transform positions, times, momentums, and energies.  In relativity, transformations generally concern four related properties.  If four particular properties can be transformed in a particular way into another frame of reference, then each of the four properties is a component of a "four-vector"--one component in the "t" direction, one in the x direction, one in the y direction and one in the z direction.  The transformations which relate to these four properties are usually written to transform from one frame into another frame which is moving in the x direction with respect to the first.  For example, consider some four-vector that might be denoted (Ft, Fx, Fy, Fz) in one frame of reference.  Consider a second frame of reference moving with respect to the first at a velocity v in the x direction.  Then, the four components of this arbitrary four-vector in this second frame of reference can be found with the following formulas:

  Ft' = gamma*(Ft - beta*Fx)

  Fx' = gamma*(Fx - beta*Ft)

  Fy' = Fy

  Fz' = Fz

where

  beta  = v/c

  gamma = 1/SQRT(1-beta^2)

  c     = the speed of light.

A note here--these transformations assume that the space-time involved is "flat" (meaning that it is not very curved by gravitational effects). 

Now, it turns out that if an event occurs in one frame of reference at a time t and at a position (x,y,z), then we can use these four properties to form a proper four-vector in the following way:   "position" four-vector = (c*t, x, y, z). That means that if we transform the occurrence of this event into another frame of reference moving with velocity v in the x direction (with respect to the first frame), then the occurrence of the event in this second frame is given by

  c*t' = gamma*(c*t - beta*x)

    x' = gamma*( x  - beta*c*t)

    y' = y

    z' = z.

One can also form a proper four-vector using the energy and momentum of an object in the following way:  "momentum" four-vector = (E/c, Px, Py, Pz), where Px, Py, and Pz are the three spatial components of the momentum. So, to find the energy and momentum of the object in another frame of reference moving with velocity v in the x direction (with respect to the first frame), we use the formulas

  E'/c = gamma*(E/c - beta*Px)

   Px' = gamma*(Px  - beta*E/c)

     With these transformations understood, we can now look at our example.  In this example. we will first consider a frame of reference in which a ship is initially at rest.  At some point in time, the ship activates it's subspace field and emits a photon in the -y direction (thus giving the ship some momentum in the +y direction).  After some time, the ship will emit a second photon in the +y direction to bring the ship to a halt.  Then, the ship will shut off its subspace field.  What we will do is to write down the time for each occurrence of these events.  We will also note the positions, energies, and momentums of each of the objects involved.  Next we will compute the angular momentums at the beginning and end of this sequence of events, and see what is necessary for them to be the same.  Finally, we will transform all of the information to another frame of reference and see what is necessary for the angular momentum to be conserved in that second frame of reference as well.

Frame 1:

Time: t0 = 0

The ship is at a position x0 = 0, y0 = 0, z0 = 0; its momentum is also zero; and its energy is a combination of mass energy and internal energy which together give it an energy of E0.

  Four-vectors:

   Ship's position: (c*t, x, y, z) = (0, 0, 0, 0)

   Ship's momentum: (E/c, Px, Py, Pz) = (E0/c, 0, 0, 0)

Time: t1

The ship has turned on its subspace field.  This means that we will not be able to look at normal-space-only momentum and energy conservation between this time and time t0.  Once the subspace field is off (at t3), then we can look at the momentum and energy and compare it to time t0.

At time t1, the ship emits a photon (labeled A) from its position with momentum -Py in the y direction.  At that split second, the ship is still at its original position, but it has just gained a momentum equal to Py in the y direction.  We also note that the energy of the  photon can be given by the magnitude of its momentum times the speed of light so that E(A) = c*Py (or E(A)/c = Py).  So we have the following four-vectors at this moment.

  Four-vectors:

   Ship's position: (c*t1, 0, 0, 0)

   Ship's momentum: (E1/c, 0, Py, 0)

   Photon A's position: (c*t1, 0, 0, 0)

   Photon A's momentum: (Py, 0, -Py, 0)

Time: t2

The ship has traveled to a new position, y2, at which point it emits a photon (labeled B) with a momentum of Py.  (again, we can calculate E(B)/c for this photon to be the magnitude of its momentum, Py) This brings the ship to rest in frame 1.  Meanwhile, photon A has been traveling in the negative x direction at speed c since it was created at time t1.  That means that its position in y is now given by -c*(t2-t1).

  Four-vectors:

   Ship's position: (c*t2, 0, y2, 0)

   Ship's momentum: (E2/c, 0, 0, 0)

   Photon A's position: (c*t2, 0, -c*(t2-t1), 0)

   Photon A's momentum: (Py, 0, -Py, 0)

   Photon B's position: (c*t2, 0, y2, 0)

   Photon B's momentum: (Py, 0, Py, 0)

Time: t3

Finally, the ship turns off its subspace field, bringing its mass energy back to what it was earlier.  It has not changed its position or momentum, but the positions of the photons have changed as they kept moving between t2 and now, t3.  Photon A's position can be found by realizing that it has been moving in the negative y direction at speed c from its starting point of y = 0 for a time (t3-t1).  Photon B started at the position y2 and has been moving in the +y direction for a time (t3-t2).

  Four-vectors:

   Ship's position: (c*t2, 0, y2, 0)

   Ship's momentum: (E3/c, 0, 0, 0)

   Photon A's position: (c*t3, 0, -c*(t3-t1), 0)

   Photon A's momentum: (Py, 0, -Py, 0)

   Photon B's position: (c*t3, 0, y2 + c*(t2-t1), 0)

   Photon B's momentum: (Py, 0, Py, 0)

Now, we can look at the original situation (t0) and this final situation (t3) to look at conservation of energy and momentum.  First we can sum together the energies and momentums in the momentum four-vectors of situation t0 and then we can do the same with t3.

Sum of four-momentums:

 t0:  Sum = (E0/c,        0, 0, 0)

 t3:  Sum = (E3/c + 2*Py, 0, 0, 0)

The momentum conservation is obvious, and the energy conservation requires that

  E0/c = E3/c + 2*Py.

We can rewrite this as

  E0 - E3 = 2*Py*c

which says that difference in the energy associated with the ship between the two times must be made up by the energy that produced the two photons.

Next we can look at the angular momentum (about the origin) between the two situations.  Since all motions are in the x, y plane, the angular momentum of each object will either be in the plus or minus z direction.  To calculate the angular momentum of an object at position x, y and with momentum Px, Py we would perform a vector operation known as the cross product:

Angular momentum in the z direction = Lz = x*Py - y*Px.

We therefore find that the angular momentums in situations t0 and t3:

Sum of Lz's:

 t0:  Lz(total) = Lz(Ship) = 0*Py - 0*0 = 0

 t3:  Lz(total) = Lz(Ship) + Lz(A) + Lz(B)

                = 0*0 - y2*0 + (-0*Py - -c*(t2-t1)*0) +

                                        (0+Py - (y3 + (c*(t3-t2))*0)

                = 0

So, obviously we have angular momentum conservation for this frame of reference.

Now let's transform all the four-vectors from t0 and t3 into another frame of reference which is moving with velocity Vx in the x direction.  Doing so we find the following:

Frame 2:

 t0 Four-vectors:

   Ship's position: (0, 0, 0, 0)

   Ship's momentum: (gamma*E0/c, -gamma*beta*E0/c, 0, 0)

 t3 Four-vectors:

   Ship's position: (gamma*c*t3, -gamma*beta*c*t3, y2, 0)

   Ship's momentum: (gamma*E3/c, -gamma*beta*E3/c, 0, 0)

   Photon A's position: (gamma*c*t3, -gamma*beta*c*t3, -c*(t3-t1), 0)

   Photon A's momentum: (gamma*Py, -gamma*beta*Py, -Py, 0)

   Photon B's position: (gamma*c*t3, -gamma*beta*c*t3, y2+c*(t3-t2),0)

   Photon B's momentum: (gamma*Py, -gamma*beta*Py, Py, 0)

Again, let's compare the sums of the four-momenta for each situation:

Sum of four-momentums:

 t0:  Sum = (gamma*E0/c,         -gamma*beta*E0/c,        0, 0)

 t3:  Sum = (gamma*(E3/c + 2*Py),-gamma*beta*(E3 + 2*Py), 0, 0)

Note that this says that if E0/c = E3/c + 2*Py (which was what we said was true to conserve energy in frame 1) then both linear momentum and energy will also be conserved in this frame.  It turns out that if we have energy and linear momentum conservation in one frame, then we have it in all frames.  But this is not so with angular momentum, as

we will now see.

We will now calculate the total Lz for t0 and t3 in this second frame:

Sum of Lz's:

 t0:  Lz(total) = Lz(Ship) = 0

 t3:  Lz(total) = Lz(Ship) + Lz(A) + Lz(B)

                = gamma*beta*[(y2*E3/c) + (Py*c*t3 - Py*c*(t3-t1))

                                      + (-Py*c*t3 + Py*(y2+c*(t3-t2))]

                = gamma*beta*[y2*E3/c - Py*c*t3 + Py*c*t1 + Py*y3

                                      + Py*c*t3 - Py*c*t2]

                = gamma*beta*[y2*E3/c - Py*c*(t2 - t1) + Py*y2]

If these two total angular momentums are to be equal, then we must set the t3 angular momentum to zero.  We then divide by gamma*beta and find that

  y2*E3/c - Py*c*(t2 - t1) + Py*y2 = 0

so

  y2*(E3/c + Py) = Py*c*(t2 - t1)

But to conserve linear momentum and energy we have shown that E3/c + 2*Py  = E0/c.  So we can say that E3/c + Py = E0/c - Py.  Applying this above we find

  y2*E0/c - y2*Py = Py*c*(t2 - t1)

Again we rewrite this to get

  y2*E0 = Py*c*(y2 + c*(t2-t1))

or

  Py*c = energy of each photon = E0/[1 + c*(t2-t1)/y2]

So, what does all this mean?  Well, this says that if we are going to have conservation of angular momentum in this second frame of reference, then the energy we must use to produce each photon must be related to the ORIGINAL energy of the ship, the distance the ship travels during its motion (y2), and the time it takes for the trip to travel that distance (t2-t1) in the first frame of reference.

But that means that if angular momentum is to be conserved in all frames of reference, then the amount of energy we expend to get the ship from place to place cannot be dependent on the mass energy the ship has with its subspace field active, but rather on the energy it has before it activates its field.  And there you have it--we cannot gain anything with the use of subspace fields and also have angular momentum in all frames of reference.

The only thing left to note here is that the subspace field might somehow change the way we transform momentums and energies.  However, we were transforming at two situations (t0 and t3) which could be a long time before and a long time after the local subspace field is active.  Therefore, the transformations we have performed should hold.

One could perform similar sorts of transformations to show that angular momentum also poses problems with any type of FTL travel and with any type of non-Newtonian based travel as well.  It would thus seem that in the future depicted on Star Trek, real-space angular momentum conservation simply doesn't occur when using subspace and warp fields.

Is this such a bad thing, though?  For the purposes of the science fiction, perhaps not.  You see, I don't see how non-conservation of angular momentum would allow for any fantastic things such as infinite energy supplies which would make the science fiction future to "easy" a place to live.  All and all, we may just have to live with the idea that angular momentum is not conserved with the use of subspace and warp fields.  If I find the time (yeah, right) I might try to look further into the consequences of that.

8. Conclusion:

In this discussion, we have considered the basics of simple subspace fields and warp fields.  We have discussed at length how energy and momentum are conserved with the use of these fields.  In the end, we find that with simple comparisons to normal space situations, one can understand how momentum and energy conservation occurs with the various uses of these fields.