
A Discussion of Subspace and Warp Fields
(Especially as they Apply to Momentum and Energy Conservation)
The following is a mixture of concepts
mentioned in canon and simicanon 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 twentyfourth century audience, and so the concepts I have
developed for explaining various aspects of subspace physics are stated as
facts. In reality, even though the main properties of the subspace and warp fields come directly from canon
sources, many of the other aspects of these fields are developed from physical
reasoning with a spattering of my own personal tastes.
For example, we know that to sustain a
subspace or warp field, it is necessary to continually feed it energy. So,
where does this energy go? Does the field continually build up energy, storing
all of the energy being poured into it. Even if this were the case, what
happens to all that energy when the field is shut off. The best answer to me
seems to describe the field as "unstable" in that it doesn't stick around if you
stop feeding it energy. Instead, we might say that it continually "bleeds" this
energy back into normal space in the form of heat in the field coils,
electromagnetic radiation, and/or (perhaps) subspace radiation which can couple
its energy back to normal space (like the shock wave in Star Trek VI).
So, I hope you enjoy this fairly lengthy
discussion of subspace and warp fields. Even if you disagree with the way some
of the concepts are explained, at least understand that a lot of thought has
gone into them in order to make the abilities of subspace and warp fields fit in with the concepts of momentum and
energy conservation. Okay, prepare to take a little excursion. As always, your
thoughts and criticisms are welcome.
1. Introduction:
In this discussion we will examine some of
the basics of both simple subspace fields as well as warp fields. In
particular, we wish to look at how momentum and energy conservation come into
play with the use of these fields.
Before discussing the subspace fields, we
first want to talk in general about subspace and its frame of reference. We
will then see how the definition of the frame of reference of subspace allows
for the creation of subspace fields (both simple subspace fields and warp
fields).
After mentioning some general aspects that
both types of fields possess, we will look individually at each type of field.
In each case, we will first go over some of the major characteristics of the
particular field of interest. We will then discuss how momentum and energy conservation come into play with that
particular type of field. Finally, we will look at examples to further examine
the conservation of momentum and energy with each type of field.
In addition to this, there are also a few
technical notes at the end of some sections (specific to each section) which
will be referred to at various times. These will go into more technical detail
concerning specific topics.
(A note to the 20th century reader: The
final section before the conclusion deals with the question of angular momentum
conservation. Throughout the other sections of the discussion, "momentum" is
used to refer to linear momentum only. This section will discuss for the 20th
century reader why angular momentum has been left out everywhere
else.)
2. Subspace and it's Frame of Reference:
Subspace is a continuum that exist in
conjunction with our own spacetime 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 cloudlike 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 stressenergy 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 subspaceequivalent mass. This
is the mass subspace "sees" which defines it's frame of reference. Generally,
this mass is equivalent to the gravitational and inertial mass; however, it can be different under certain
circumstances. Similarly, there is also a concept of the subspaceequivalent
stressenergy 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 "ith"
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 ith 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 stressenergy 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 stressenergy 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 stressenergy 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 stressenergy 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 subspaceequivalent stressenergy tensor. However,
at this point the subspaceequivalent stress energy tensor is still directly related to the realspace stressenergy tensor.
So, the field also lowers the effects of the stressenergy tensor as viewed in
normal space (outside of the subspace field) as well.
By correctly producing the stressenergy
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 stressenergy 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 realspace 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 stressenergy 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
stressenergy 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 stressenergy tensor needed to produce the warp field.
The energizing of the field coil material with a properly configured plasma
stream creates a stressenergy 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 stressenergy 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
decouple the relationship between subspaceequivalent mass/stressenergy and
normal space mass/stressenergy. The subspaceequivalent 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 layersthe 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,
stressenergy tensors) which are inside of a subspace field, as mentioned
earlier. Subspace does this because it too is "viewing" those objects through
the effects of the interaction layer.
With these common basics in mind, we can
now discuss specific aspects of simple subspace fields and warp fields
independently.
5.
Simple Subspace Fields:
A subspace field which is symmetric in the
subspace domain causes subspace to (in essence) act as an energy reservoir.
Such a field is referred to as a simple subspace field (or just "a subspace
field"). To outside observers, anything within such a
field will appear to "loose" some of it's mass energy to subspace while the
field is active (as discussed earlier. (Equivalently, one could say that the
field masks out part of the mass of objects inside the field as they are viewed
from normal space.) The amount of interior mass energy "placed" into subspace
is dependent on the strength of the subspace field. For all practical purposes,
while the field is active, this mass energy disappears from normal space (see
Technical Note 1 for this section). However, it should be noted that when one
compares the normalspace energy and momentum of a closed system before a
subspace field is activated with that of the system after the field is
deactivated, energy and momentum conservation must apply. We will now look at
momentum and energy conservation considerations with respect to simple subspace
fields.
5.1 Momentum and Energy Conservation with
Simple Subspace Fields
Here we will look separately at momentum
conservation and energy conservation as they apply to subspace fields. At the
end of this section, examples will be considered to illustrate these
conservation considerations.
5.1.1 Momentum Conservation
Consider a ship of mass M which surrounds
itself in a simple subspace field. To outside, normal space, the mass of the
ship becomes m < M once the field is active. This new, lower mass is called the
apparent rest mass of the ship (or simply its "apparent mass"). If the normal space manifestation of
the subspace field can be shaped so that the ship's fuel is kept outside of the
field, the ratio of fuel mass to ship mass will be greatly increased. In
accordance with momentum conservation, fuel expelled with a given momentum will cause the ship to have an
equivalent momentum in the opposite direction (thus conserving momentum).
However, with the subspace field activated, the speed this momentum gives to the
ship would be calculated using the apparent (lower) rest mass of the ship. Thus,
with the use of a subspace field one can achieve greatly improved acceleration
rates as well as greatly lowered energy costs for reaching a given speed.
As long as the field is active, kinematic
considerations of the ship will be calculated with the ship's apparent mass.
However, when the subspace field is deactivated, the masked mass of the ship
returns. The results of this returning mass as it applies to momentum
conservation will be considered in the examples given after the energy conservation considerations have been discussed.
5.1.2 Energy Conservation
Once a subspace field is activated, energy
conservation can be realized only if one includes the mass energy which is
"submerged" into subspace. This will be demonstrated in examples given at the
end of this section.
There are, however, energy considerations
other than kinematic ones. Some of the energy that is internal to the ship must
go into producing the subspace field. Currently, subspace field generators
produce unstable fields which continually "bleed" their energy back into normal
space. (This energy generally manifests itself as a combination of heat within
the subspace generator, electromagnetic radiation, and/or subspace radiation
which can couple it's energy into normal space. Also, this energy bleeds off
symmetrically so that momentum is conserved.) Because of this bleed off,
subspace field generators must continually supply energy to the subspace
fields. The same amount of energy supplied to the field is eventually bled back
into regular space, thus conserving energy.
The final energy consideration involves
internal ship energy which remains internal (producing life support, etc.).
Because the ship is within the interior of the subspace field, it appears to
itself to be in a normalspace "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 antimatter on board the ship is the same as if the
matter and antimatter hadn't "lost" mass to subspace. It is true that once the
field is activated, the matter and antimatter aboard the ship will seem to
"loose" some of it's mass energy to subspace in the point of view of the outside
observer. For the outside observer to realize that energy has been conserved,
he must remember that this mass energy did not actually disappear from
existence, but has simply been submerged into subspace. However, as the matter
and antimatter 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 nonmass energy from the outside observer. So, as the matter and antimatter interact, the outside
observer not only sees the reduced masses of the matter and antimatter turn
into other forms of energy, he also sees mass energy that had been masked by
subspace being converted into normal, nonmass energy. The result is that he
sees as much normal, nonmass 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
onship purposes, and while this energy may change form (becoming heat and
eventually being radiated into space, for example) we know that this energy is
always present in some form. Thus this part of the internal energy is
preserved. The rest of the energy involved will be considered at each step to
show that it is also conserved along with momentum.
Step 1:
The ship uses part of it's internal energy
to create a subspace field. As explained above, this energy is bled back into
space, thus this energy is conserved. As the field is turned on, part of the
ship's mass is masked from outside observers, and the apparent mass of the ship
becomes m. To realize the conservation of energy, we must remember that this
mass energy is still "present", but is submerged in subspace. This submerged
energy is the difference between the mass energy of the ship initially and its
mass energy now(M  m)*c^2. This makes it obvious that the energy is conserved
(since the submerged energy of the ship plus its energy now is the same as it's
initial mass energy).
Step 2:
The ship uses part of it's internal energy
to produce a high energy photon (as fuel) with a certain momentum in a
particular direction. In accordance with conservation of momentum, the ship
must gain an equivalent momentum in the opposite direction. In accordance with
conservation of energy, the internal energy used must be equal to the energy
given to the photon plus the change in energy of the ship (which now has more
energy since it is moving in the original frame of reference). (See Technical
Note 2 for this section.) The change in energy of the ship is calculated with
the ship's apparent mass (m), and the energy submerged in subspace is still
equal to (M  m)*c^2.
So, part of the internal energy goes into
the energy of the photon and increases the energy of the ship, while the energy
submerged in subspace is still the same. Meanwhile, the momentum of the photon
is canceled out by the momentum of the ship. Thus, energy
and momentum are conserved.
Step 3:
As the ship travels, it may experience
"collisions" with other objects. As long as these collisions do not collapse
the subspace field, the ship's apparent rest mass will still be m as far as the
collisions are concerned. This is no violation of energy or momentum,
because for all intents and purposes, the missing mass of the ship has been
"left" sitting still in the original frame of reference by keeping it submerged
in subspace. Thus the ship should interact with other objects as if it's mass
is m.
Step 4:
Part of the internal energy is used to
produce another photon for fuel which brings the ship back to rest in the
original frame of reference. Energy and momentum are conserved here in the same
way they were conserved in step 2.
Step 5:
The subspace field is disengaged, and the
energy which had been submerged in subspace is returned to the mass energy of
the ship. This is just the reverse of the first step, and energy is obviously
conserved.
Example 2
This example is identical to the first
example up to and including Step 3. We will begin here with a new Step 4.
Step 4:
In the previous example, the ship
"decelerated" to get back to the original frame of reference and then shut off
its subspace field. Here we examine what happens if the subspace field is shut
off (intentionally or accidentally) while the ship is still moving in the original frame of reference.
As the field is deactivated, the mass
energy which was submerged in subspace will be added back to the ship. This
mass energy can be modeled as actual mass which is sitting at rest in the
original frame of reference. In this model, as the field dies, it is as if the
ship runs into a portion of matter with a mass of (M
 m). This is not as harmful as it may seem. A ship which actually runs into a
chunk of matter with significant mass will be crushed because the force applied
to the front of the ship will have to be transferred to the back of the ship
before the back will stop moving. This produces the crushing effect. In our
case, the mass is "added" throughout any objects within the subspace field at
the same moment as the field is deactivated. All particles throughout the
interior of the subspace field are decelerated at the same time and at
the same rate.
It is not that obvious what exactly takes
place in this case to allow for the conservation of momentum and energy. We can
deduce what would happen by considering the model of the situation in which a
ship runs into a mass of (M  m). In this case, a ship of mass m and momentum p inelastically collides with an object of mass (M  m) which is at rest. After
the collision, the combined clump of ship plus object has a mass of M and a
momentum p (to conserve momentum). But, the energy of a mass m with momentum p
plus the energy of a mass (M  m) does not generally equal to the energy of a
mass M with a momentum p. In order to conserve energy in this case, the final
system must have internal energy in addition to it's mass energy and kinetic
energy. (See Technical Note 3 for this section.) In our model, the collision
will generally cause heating to produce this internal energy. In the actual
situation, the system after the subspace field has died will include
electromagnetic radiation, and/or subspace radiation, and/or heat inside the
ship to make up the extra energy needed for energy conservation.
In short, we have shown energy and momentum
conservation in these examples with the following comparisons. Turning on the
subspace field is compared to a situation where the ship removes part of its
mass, leaving it at rest in its original frame of reference. The ship then
continues along its trip, just as if it had a lower mass. Turning off the
subspace field can then be compared to adding back on the previously removed
mass which is still at rest in the original reference frame. With these
comparisons, one can see how energy and momentum are conserved in the use of
simple subspace fields.

5.2 Technical Notes for this Section (Simple
Subspace Fields)
*Technical Note 1
We say that when a subspace field is
activated, part of the mass energy of objects within the field disappears from
normal space for all practical purposes (as seen by outside observers). We
should note, however, that other aspects of this matter (charge, baryon number, lepton number, etc.) are unaffected.
For example, an electron sitting within a
subspace field will still seem to outside observers to have a charge of 1, a
lepton number of 1, etc. However, it will seems as if the normal rest mass of
the electron has been reduced.
So, when a ship in a subspace field seems
to loose part of its mass as seen by outside observers, it is not as if the ship
has lost some of its particles. Instead, it is as if all the particles
individually became particles of lower rest mass.
*Technical Note 2
Here we examine the amount of energy needed
to propel a ship with a reduced mass of m to a velocity v by expelling a
photon. We will be using regular relativistic equations for momentum and energy
with the following notations:
c = the speed of light
v = the velocity of the ship
beta = v/c
1
gamma = .
____________
\/ 1  beta^2
Now, at some point the ship (whose reduced
mass is m) uses part of its internal energy to expel a photon in a particular
direction. If the photon is created correctly, afterwards the ship will be
moving with the desired velocity v. Its momentum and energy will thus be
given by the following:
p(ship) = gamma*m*v (the relativistic
momentum of the ship)
E(ship) = gamma*m*c^2 (the relativistic
energy of the ship)
Now, in order to conserve momentum, the photon's momentum will have to be
equal and opposite to that of the ship. The energy of the photon can then be
calculated from its momentum. We can thus write the following:
p(photon) = p(ship) = gamma*m*v
E(photon) = p(photon)*c = gamma*m*v*c
It is now possible for us to calculate how
much of the internal energy of the ship would have to be used to expel this
photon. Before the photon was expelled, the energy of the system included the
mass energy of the ship (m*c^2), the internal energy of the ship which would be used to expel the photon (E(fuel)), and
some other internal energy which wouldn't be changed. After the photon is
expelled, the energy of the system includes the larger energy of the ship
(gamma*m*c^2), the energy of the photon (gamma*m*v*c), and that part of the
internal energy which wasn't changed. The energy used to expel the photon must
make up for the difference in energy between these two situations. We can thus
write the following:
E(fuel) = (gamma*m*c^2 + gamma*m*v*c) 
(m*c^2)
= [gamma*(1 + beta)  1]*m*c^2.
The interesting thing to note here is that
if the subspace field hadn't been used to lower the apparent mass of the ship,
this energy would be calculated with the same formula, except m would be
replaced by M. This means that the subspace field allows a savings of energy
given by
E(saved) = [gamma*(1 + beta)  1]*(M 
m)*c^2.
As long as the energy needed to produce and
maintain the field is less than this energy, then there is an overall savings in
energy for this particular example.
It should also be noted that for
significantly high velocities, the E(fuel) could still be impractically high
unless the apparent mass (m) is significantly small. As it turns out, mass
masking by subspace fields can provide the needed lowering in mass to make large
changes in the velocity of the ship a practical ability.
*Technical Note 3
Here we examine the momentum and energy
considerations of a collision between a mass m with momentum p and a mass (M 
m) at rest. Consider the following diagrams of the situations before and after
the collision:
Before:
m
O M  m
O> p O
O P = 0
O
(The total internal energy of these systems =
E(intbefore).)
After:
O M
OOO>p
O
Internal energy = E(intafter).
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(intafter) + GAMMA*M*c^2]  [E(intbefore)
+ gamma*m*c^2 +
(Mm)*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(Eint)). We thus write the following:
Delta(Eint) = E(intafter)  E(intbefore)
= gamma*m*c^2 + (Mm)*c^2 
GAMMA*M*c^2
= [(Mm)  (GAMMA*M 
gamma*m)]*c^2
Now, we can rewrite the gammas by
remembering that for any systemof mass m and momentum p, the energy can be
written as
___________________
E = gamma*m*c^2 = \/p^2*c^2 + m^2*c^4
We can thus write gamma for such a system as
the following:
___________________
gamma = \/ p^2/(m^2*c^2) + 1
Since the momentum of both m and M are the same
in our example, we can rewrite the change in internal energy as the following:
_______________
_______________
Delta(Eint) = [(Mm)  (\/ p^2/c^2 + M^2  \/
p^2/c^2 + m^2 )]*c^2
Now, we know from the triangle inequality that
_______________
p/c + M >= \/ p^2/c^2 + M^2
and _______________
p/c + m >= \/ p^2/c^2 + m^2
(where ">=" denotes greater than or equal to).
Given this, we see that by subtracting one from the other we have
_______________
_______________
Mm >= (\/ p^2/c^2 + M^2  \/ p^2/c^2 +
m^2 ).
Thus,
_______________ _______________
[Mm  (\/ 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 referenceframe coupling. The reference frame of objects within
the realspace 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 stressenergy 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 subspacefieldproducing stressenergy tensor so that
it becomes a warpfieldproducing stressenergy 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 subspacefieldproducing tensor in
this way creates an exotic enough effect to produce a referenceframe 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 referenceregardless 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 normalspaceonly momentum
conservation is essentially ignored as long as subspace is masking the
momentum. Therefore, this method of warp travel is labeled as nonNewtonian
propulsion. The use of warp propulsion will be discussed in a later subsection.
It is also possible to change the exotic
nature of the stressenergy tensor in order to produce warp fields which are
nonpropulsive. 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 subspacesymmetric 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
nonpropulsive 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, nonpropulsive
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 spacetime, but to a virtual spacetime created within the
bubble.
There are, as mentioned, many different
types of nonpropulsive 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 nonpropulsive/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 (nonimaginary)
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
SingleLayered Warp Fields
First we will consider warp propulsion
produced with a singlelayer warp field. As such a field is activated, the
momentum of the ship (and thus it's speed) will increase. At first, the ship
will be traveling at slower than light speeds, and the energy of the ship increases dramatically as its speed approaches
that of light. Only after the jump to faster than light speeds occurs will the
reference frame coupling take full effect, and the energy of the ship be
completely outside of the realm of Einsteinian physics.
Once the reference frame coupling takes
effect, all measurements with respects to the ship are done as if the ship is in
the frame of reference of subspace. That means that at any particular moment,
properties such as distances, times, etc. are measured just as if the ship were sitting still for that moment in the
frame of reference of subspace. As in illustration, one could imagine taking a
snapshot of a ship in warp and finding that it is indistinguishable for that one
moment from a ship who is not moving with respect to the subspace frame of
reference. Yet, we attribute kinetic energy (energy of motion) to such a ship,
even if we view it from the subspace frame of reference. This is because the
kinetic energy of the ship is actually held within the warp field itself.
Thus, to keep the ship at a certain speed,
one must keep the warp field at a constant energy level which is seen as the
energy of the ship itself. But, today's warp field generators produce unstable
fields (similar to subspace field generators.) Thus, warp fields also bleed off
there energy back to the normal universe (in the form of heat in the field coils,
electromagnetic energy given off nearby the ship, etc.). Therefore, the warp
field must be given a constant supply of energy from the ship. (This, too, will
be discussed in a later subsection. The important thing to understand here is
that the warp field does need a constant supply of energy).
To increase the speed of the ship, one must
increase the energy level of the warp field. However, at higher energy levels,
a warp field becomes much less efficient (bleeding off it's energy at much
larger rates). Therefore, the power output of the ship must increase dramatically to hold the warp field at a higher
energy level (thus holding the ship at a large velocity).
For our examples, we will use a model which
approximates warp field energy levels in certain geometries. The power (the
amount of energy given to the field per unit time) given to a field layer
depends on the energy of that layer, and in our model that dependence is as
follows:
Power = P_0*(E/E_0)^3
Where E is the energy of the layer (and thus the
energy of the ship) and P_0 & E_0 are a power level and an energy level
intrinsic to the model.
For example, a ship traveling at a
particular warp velocity may have an energy of 2*E_0 associated with its
motion. In order to keep the warp field up, the ship would have to output
energy at a particular rate, providing a power of 8*P_0. If the ship increases its speed so that its energy is now 4*E_0 (twice
as much as before), the ship will have to provide a power of 64*P_0 (8 times as
much as before) in order to keep the warp field up. The energy of the ship
itself (associated with it's velocity) has only increased by a factor of 2, while the warp engines are now having to
output eight times as much power into the warp field because the higher energy
warp field is much less efficient (quickly bleeding it's energy back into normal
space).
6.1.2 MultiLayered 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
multilayer warp fields are used today for warp propulsion. By correctly setting up the geometry
of the stressenergy 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 warpfieldcreating 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 singlelayer geometry to the warp
field rather than a doublelayer warp field. The difference will be evident if
one attempts to supply even more power to the first layer of the doublelayer
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 stressenergy
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 twostage
stressenergy 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 multilayer 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 multilayer 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 multilayer warp fields, many research teams started
working to produce various multilayer 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 naturethat 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
nonfederation 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
multilayered warp fields that we have yet to mention. The geometry of a
multistage stressenergy tensor inherently produces a warp field which is
symmetric in the subspace domain. That means that the multilayered warp field produced by such a tensor cannot be
propulsive.
In order for propulsive fields to gain the
benefits which multilayered warp fields possess, a new way to produce
multilayered fields needed to be found. As it turns out, the key to regaining
the nonNewtonian drive came in nesting many layers of warp field energy within
one another. In today's warp engines, a series of singlestage 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 multilayered,
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 singlestage 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 multilayer 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 multilayer 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 "firstlevel" 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 secondlevel warp fieldsubspace
will see the three nested field layers as a single warp field consisting of 2
unnested field layers. Conceptually we can then
think of the total energy of the field being divided among these two "virtual"
unnested 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 continuedincreasing
the energy of the warp field and increasing the frequency at which the nested
layers are createdin order to press deeper into subspace and pass through the
higher efficiency points at the integer warp values. And there we have itthe effects of a
multilayered 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 leftright 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 nonNewtonian 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
separatelythe 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/antimatter 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
slowerthanlight, "energyequivalent" velocity (v'). We then use the actual
mass of the ship (M) and the energyequivalent 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 differenceanything 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 normalspace 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
onship purposes, and while this energy may change form (becoming heat and
eventually being radiated into space, for example) we know that this energy is
always present in some form. Thus this part of the internal energy is
preserved. The rest of the energy involved will be considered at each step to
show that it is also conserved.
Step 1:
The ship uses part of it's internal energy
to create a warp field. As discussed above, part of this energy is bled back
into space, while the rest accounts for the kinematic energy of the ship, thus
this energy is conserved. As the field is turned on, part of the ship's mass is
masked from outside observers, and the apparent mass of the ship becomes m. To
realize the conservation of energy, we must remember that this mass energy is
still "present", but is submerged in subspace. This submerged energy is the
difference between the mass energy of the ship initially and its mass energy
now(M  m)*c^2. This makes it obvious that this energy is conserved (since the
submerged energy of the ship plus its energy now is the same as it's initial
mass energy).
The warp field also causes subspace to act
as a momentum reservoir, and so a certain momentum becomes masked by subspace.
As mentioned above, this momentum has no energy associated with it. To conserve
overall momentum, the ship gains an equivalent momentum in an opposite
direction. The motion of the ship gives the ship kinematic energy. Again, this energy is part of the
energy contained in the warp field, and thus it comes from part of the internal
energy. We have thus shown overall conservation of momentum and energy in this
step.
Step 2:
As the ship travels, it may experience
"collisions" with other objects. Though these collisions may not collapse the
warp field, they would have interesting effects. We will wait to consider these
effects in example 2. Collisions which do collapse the warp field
can have very damaging effects. (See Technical Note 3 for this section.)
Step 3:
As the ship comes to its destination, it
shuts down its warp field. As this is done, the momentum masked by subspace
becomes unmasked, and the ship in turn looses its momentum. The energy contained
in the warp field is bleed back into normal space as the warp field collapses. Remember that this energy
also accounts for the energy of the ships motion, thus as the ship looses
momentum, it also loses it's kinematic energy which is bleed back into normal
space. Finally, the mass energy that was masked by subspace returns to the ship,
bringing its mass back to the original M. So, here we again see that the overall
energy and momentum are conserved.
Example 2
The first step in this example is identical
to the previous example. We will thus start with the second step and more
closely examine the collisions mentioned in step 2 of example 1.
Step 2:
During the travel, the ship encounters a
large object. For convenience, we will assume that the object is at rest in the
original rest frame of the ship so that it must be deflected away from the path
of the ship. As the object is deflected, the ship's momentum is effected as if it were a ship with a momentum
calculated by using its energyequivalent velocity (v'). That is, the ship acts
no different (kinematically speaking) from a ship of mass M and velocity v'.
Deflected the object will give it energy
and momentum. The energy can come in part from the kinematic energy of the ship
and in part from the internal energy of the ship (if a tractor beam is used to
deflect the object, for example). But, in addition, internal energy must be
transferred to the warp field in order to keep it from collapsing during the
interaction with the object. How much internal energy needs to be expended and
why will be explained as we look at momentum conservation.
To conserve momentum, the total change of
the ship's momentum will be equal and opposite to the change in the momentum of
the object. The deflection of the object will cause the warp field to become
imbalanced in the direction of the ship's change in momentum.
This happens as the additional energy is feed to
the warp field to keep it from collapsing. After the interaction, the ship can
do one of two things. First, it could continue on its changed course, coming
out of warp at some later point in time; or, second, it could use it's warp
field to adjust it's momentum (and its course) to get to its original
destination.
In the first case, the ship will continue
its journey along its changed course until step 3. In the second case, the ship
will use the warp field to readjust its course. As this readjustment is made,
the imbalanced warp field deposits actual momentum into space (generally in the form of photons) rather than
"putting" the momentum into subspace. This means that the real change in
momentum of the object will be counteracted by the real momentum of the expelled
photonsthus conserving normal space momentum.
The energy needed to produce these photons
comes from the energy placed into the warp field (to keep it from collapsing) as
the interaction with the object took place. Also note that as the photons are
emitted, the ship gains back the momentum it lost during the collision. That means that it must also gain
back the kinetic energy that it lost. This energy must also be supplied by the
energy stored in the warp field while the interaction took place. Since this
energy is exactly the energy lost to the object during the interaction, the
object's energy eventually comes from the internal energy of the ship.
Therefore, as the object is deflected, the energy feed into the warp field is
just enough to produce photons (whose momentum will be equal and opposite to the
change in the object's momentum) and to restore the kinematic energy lost by the
ship.
(Note: The ship could continually adjust
its warp field during the collision so that it's momentum and velocity don't
change. In this case, energy is still feed to the warp field during the
interaction, but the continually adjusting warp field will continually use that
energy to immediately create the photons necessary to conserve momentum. The end
result is the samethe ship has changed the momentum of the object, a momentum
equal and opposite to that of the change becomes real in the form of photons,
and the ship's momentum remains unchanged. Meanwhile, the
internal energy of the ship has been used to produce the photons and to give the
object its energy.)
So, energy and momentum in real space are
conserved during and after a "collision" with an object.
Step 3:
The ship reaches it's destination and shuts
off it's warp field. What happens here will depend on which of the two cases
(mentioned above) was chosen. If the ship changed it's course after the
collision (thus completely making up for the collision), then as the ship comes
out of warp it will come back to rest in its original frame of reference (just
as it did in example 1). However, if the ship did not change its course, then
it will have to make up for the collision as it comes out of warp. As the
imbalanced warp field collapses, the energy that was placed in the warp field
during the interaction will produce the photons necessary to make up for the
real momentum given to the object. As the momentum of these photons gives
momentum back to the ship, the ship will gain energy which must also come from
the energy stored in the warp field during the interaction. Then the
rebalanced 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 multilayered 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 nontrivial 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
"energyequivalent" 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 nonrelativistic 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
nonpropulsive warp fields and with simple subspace fields, the mass reduction carries over into the kinematics of
the situation.
So, how is this energyequivalent 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 energyequivalent 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 energyequivalent
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 antimatter just to have
enough energy to get the ship to this velocity (not counting the additional
energy needed to sustain the warp field during the acceleration).
Today's warp fields (if modeled in this
simplistic way) would yield extremely small A values so that a typical ship
would easily be able to produce the energy needed to travel at high warp
velocities.
*Technical Note 3
Fortunately, the energy contained in the
motion of a ship in warp is not very great (as discussed in the previous
technical note). If this were not the case, the ship would have to supply an
extreme amount of energy in order to accelerate to a given warp speed.
The small energy of the ship translates
into a small momentum as well. That is, ships in warp do not carry a large
amount of momentum. However, we should not discount the amount of damage that
can be done by a warp collision. To examine the damage potential of a warp
collision, we will consider the following example.
During a battle with a hostile ship, our
ship finds itself out matched, and it is decide to ram the hostile ship in the
hopes of crippling its ability to cause more harm. In addition to a warp core
breach and the associated explosion to follow, we also want the actual collision to cause as much damage as possible.
In the time one has to accelerate before
the collision, one could use the impulse engines to accelerate to a significant
velocity. However, the quick acceleration is only possible because a subspace
field is used to greatly reduce the apparent mass of the ship. The lower mass means that the momentum, energy, and
damage potential are not necessarily that great.
On the other hand, one could jump into
maximum warp to ram the hostile ship. Again, a quick acceleration (this time,
to a faster than light velocity) is possible. However, the velocity v
translates to a fairly small energyequivalent velocity v', and (as we have discussed) the momentum and energy of the ship's
motion are again fairly small.
However, we have left out one part of the
collision. As the subspace field or warp field interacts with the hostile ship,
it will deposit energy into the ship and collapse. In the case of the subspace
field, the collapse of the field will cause the mass energy of our ship to be
returned (however momentum will be conserved) and will produce an increase in
internal energy or radiated energy (which can have some damaging effects on the
hostile ship). In addition, the energy held in the field itself can be
partially transferred to the hostile ship.
In the case of the warp field, as
the field collapses, the mass energy and the momentum held in the field will
return to the ship. Here, there has been no fuel expelled, and so there is no
real momentum held in the ship's motion. The momentum is completely held within sub space while the warp field is
active. However, as the warp field interacts with the hostile ship, the
momentum that is held within the field can be coupled onto the hostile ship. As
the field collapses, rather than slow the motion of the ramming ship, the momentum in the field can be imparted to part of
the hostile ship, causing more damage. In addition, the energy held within the
warp field (which is generally larger than the energy held in a subspace field)
is imparted onto the hostile ship.
As it turns out, with everything taken into
consideration, the damage potential is significantly greater when one chooses to
use warp drive to ram the hostile ship.

(7. Angular Momentum Conservationfor 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
"fourvector"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 fourvector 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 fourvector 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(1beta^2)
c = the speed of light.
A note herethese transformations assume that
the spacetime 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 fourvector in the following way:
"position" fourvector = (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 fourvector
using the energy and momentum of an object in the following way: "momentum"
fourvector = (E/c, Px, Py, Pz),
where Px, Py, and Pz are the three spatial
components of the momentum. So, to find the energy and momentum of the object in
another frame of reference moving with velocity v in the x direction (with
respect to the first frame), we use the formulas
E'/c = gamma*(E/c  beta*Px)
Px' = gamma*(Px  beta*E/c)
With these transformations understood, we
can now look at our example. In this example. we will first consider a frame of
reference in which a ship is initially at rest. At some point in time, the ship
activates it's subspace field and emits a photon in the y direction (thus
giving the ship some momentum in the +y direction). After some time, the ship
will emit a second photon in the +y direction to bring the ship to a halt.
Then, the ship will shut off its subspace field. What we will do is to write
down the time for each occurrence of these events. We will also note the
positions, energies, and momentums of each of the objects involved. Next we
will compute the angular momentums at the beginning and end of this sequence of
events, and see what is necessary for them to be the same. Finally, we will
transform all of the information to another frame of reference and see what is
necessary for the angular momentum to be conserved in that second frame of
reference as well.
Frame 1:
Time: t0 = 0
The ship is at a position x0 = 0, y0 = 0,
z0 = 0; its momentum is also zero; and its energy is a combination of mass
energy and internal energy which together give it an energy of E0.
Fourvectors:
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 normalspaceonly 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 fourvectors at this moment.
Fourvectors:
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*(t2t1).
Fourvectors:
Ship's position: (c*t2, 0, y2, 0)
Ship's momentum: (E2/c, 0, 0, 0)
Photon A's position: (c*t2, 0, c*(t2t1), 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 (t3t1). Photon B started at the
position y2 and has been moving in the +y direction for a time (t3t2).
Fourvectors:
Ship's position: (c*t2, 0, y2, 0)
Ship's momentum: (E3/c, 0, 0, 0)
Photon A's position: (c*t3, 0, c*(t3t1), 0)
Photon A's momentum: (Py, 0, Py, 0)
Photon B's position: (c*t3, 0, y2 +
c*(t2t1), 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
fourvectors of situation t0 and then we can do the same with t3.
Sum of fourmomentums:
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*(t2t1)*0) +
(0+Py 
(y3 + (c*(t3t2))*0)
= 0
So, obviously we have angular momentum
conservation for this frame of reference.
Now let's transform all the fourvectors 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 Fourvectors:
Ship's position: (0, 0, 0, 0)
Ship's momentum: (gamma*E0/c,
gamma*beta*E0/c, 0, 0)
t3 Fourvectors:
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*(t3t1), 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*(t3t2),0)
Photon B's momentum: (gamma*Py, gamma*beta*Py,
Py, 0)
Again, let's compare the sums of the fourmomenta
for each situation:
Sum of fourmomentums:
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*(t3t1))
+ (Py*c*t3
+ Py*(y2+c*(t3t2))]
= 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*(t2t1))
or
Py*c = energy of each photon = E0/[1 +
c*(t2t1)/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 (t2t1) 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 itwe 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 nonNewtonian based travel as well. It would
thus seem that in the future depicted on Star Trek, realspace 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
nonconservation 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.

