Subject: Subspace Physics
From: [email-masked] (Jason W. Hinson)
Date: Thu, 11 Jan 2001 20:56:49 +0000 (UTC)
Newsgroups: rec.arts.startrek.tech

This is the version 2.0 of this post.  It has changed from version 1.0 
in that I have added three new sections and change a few things in 
some of the other sections accordingly.  Two of these new sections 
deal with subspace itself as well as how one might fictionally produce 
subspace fields.  The other added section (Section 7) deals with the 
question of angular momentum conservation, which is ignored elsewhere 
in the post.  I hope you enjoy it.

So, what's all this then?
 
     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).
     Please let me know if you find any problems with the explanations
I give (either problems related to the physics or problems with
contradicting canon sources).  I tried to keep in mind all the various
aspects given by canon sources, but I could have trampled over a few.
For example, one part of the Technical Manual (page 49) mentions
symmetric fields as non-propulsive (indicating that a propulsive field
could be created by producing an asymmetric field).  However, in
another section (page 54) it indicates that the key to "non-Newtonian"
propulsion (which would seem to me to be a propulsive warp field) "lay
in the concept of nesting many layers of warp field energy."  I
therefore tried to incorporate both ideas into my explanation of warp
propulsion.
     Oh, one other thing before we get to the discussion.  I have
assumed in writing this that the reader has read my monthly
"Relativity and FTL Travel" post.  In the discussion below, I assume
concepts discussed in the Relativity post (specifically, the idea of
subspace providing a special frame of reference for the purpose of
faster than light travel such that warp fields "plug into" that frame
of reference).  The Relativity post should be posted at approximately
the same time as this discussion, so if you haven't read it, you may
want to give it a look.
     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.

 
 
               A Discussion of Subspace and Warp Fields
    (Especially as they Apply to Momentum and Energy Conservation)
 
Contents:
1. Introduction:
2. Subspace and its Frame of Reference:
3. Creating Subspace Fields:
3.1  Creating a Simple Subspace Field
3.2  Creating a Warp Field
4. General Aspects of Subspace Fields:
5. Simple Subspace Fields:
5.1  Momentum and Energy Conservation with Simple Subspace Fields
5.1.1  Momentum conservation
5.1.2  Energy Conservation
5.1.3  Some Examples
5.2  Technical Notes for this Section (Simple Subspace Fields)
6. Warp Fields:
6.1  Warp Propulsion
6.1.1  Single-Layered Warp Fields
6.1.2  Multi-Layered Warp Fields
6.1.3  Development of Modern Warp Propulsion Fields
6.1.4  Modern Warp Propulsion Field Generation
6.2  Momentum and Energy Conservation with Warp Propulsion
6.2.1  Some Examples
6.3  Technical Notes for this Section (Warp Fields)
(7. Angular Momentum Conservation--for the 20th Century reader:)
8. Conclusions

 

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 its 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 its 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 its 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 its 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 its 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 its 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 its
initial mass energy).
 
Step 2:
     The ship uses part of its 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 its 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 its 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 system
of 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, since M > m, and both momenta (p) above are the same, we can draw
the following diagram representing the relationships between M, m, and
p as follows:

             +     -                       +
            /|     |                      /|
           / |     |                     / | 
          /  |     |                    /  | M-m
         /   |     |M                  /   |
        H    + -   |        =>        H    +
       /   / | |   |                 /   /
      /  h   | |m  |                /  h
     / /     | |   |               / /
    //       | |   |              //
   o---p/c---+ -   -             o

               _______________               _______________
  Note:  H = \/ p^2/c^2 + M^2    and   h = \/ p^2/c^2 + m^2

Looking at the right hand diagram, it's simple to show that

   H <= h + (M-m)
so
   (M-m) >= H - h
or              _______________     _______________
   (M-m) >=  (\/ p^2/c^2 + M^2  - \/ p^2/c^2 + m^2 ).
 
(where ">=" denotes greater than or equal to).  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.
---------------------------------------------------------------------

 
 
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 its 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 its 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 its 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 its 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 its 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 its
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 its 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 its 
warp field to adjust its 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 its 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 its destination and shuts off its warp field.  
What happens here will depend on which of the two cases (mentioned 
above) was chosen.  If the ship changed its 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
outmatched, 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 subspace 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.
---------------------------------------------------------------------

(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)
   Py' = Py
   Pz' = Pz.

     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 its 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.

                                                  Jason Hinson

{Author's Note: Some Final Words:
     Well, there it is.  Please let me know what you think of the
ideas and the reasoning (as they relate to canon sources as well as to
real physics).  I'll try to refine the concepts as much as possible
and I'll make this little excursion into the physics of Star Trek's
science fiction a regular monthly post.}

-Jay