This figure however has one problem, the Minkowski space
is not plotted properly (this is because Mathcad does
not understand the concept of spacetime [to do so would
require a lot of programming =(
]), all that is shown is an arbritary spatial displacement,
but its good enough for all of us who think in three
dimensions.
The Alcubeirre Warp Drive:
The following graph is suppose to be given with the
following function z=q
(x,r
), in reference to the work of Alcubierre. However,
that would take much work for Mathcad to plot properly,
so the standard bivariate function z=f(x,y) is used
in its place, thus giving:
[note: this does not have to strictly mean the velocity
is given by one unit. When one defines G=c=1, the
velocity given can be equal to that of light (with
G being the gravitational constant and c the velocity
of light). This is also the same parameter defined
in Alcubierre's paper.]
Now for the really interesting
part, what does the Warp Drive look like, it can be
given as a graph of q
as a function x and
. Alcubierre defined the velocity of the warp bubble
to be
, such that theta could be set to q=vs(xs
/rs)(df/drs).
When the velocity of the warp drive is set to
then the Warp Drive Spacetime can be described by the
metric:
The Alcubierre top-hat function:
When this function is plotted is seen quite clearly
why the function chosen by Alcubierre is called the
top hat function:
One can now write the arbitrary function of the warp
drive spacetime as a function of the ship's positions
rs
. Since this is a numerical presentation done with
computer simulations I will now define the parameters
of the "top hat" function as defined in Alcubierre's
paper (Class Quant
Grav11
(1994) L73) as
for the radius of the function, and
for the energy-density:
With the ship's position
, the invariant lengths of this space is then given
by:
Imagine you are in space ship who's position is given
in Cartesian Coordinates by an arbitrary function xs
(0,0). Now imagine that your space ship is coasting
so that its future position is determined by an arbitrary
function of time xs
(t), this function yields the velocity of your space
ship by the relation vs(t)=dx
s
(t)/dt. One of the "artifacts" of the Special
Theory of Relativity (STR) is that nothing can travel
Faster Than Light (FTL), however this restriction only
applies to matter-energy and not space itself. During
the "Big Bang" epoch the expanding universe
would have appeared to exceed the speed of light as
seen from two spatially separated observers, although
locally the observers do not exceed the speed of light.
This is the same principle that the Alcubierre Warp
Drive is based on, since the General Theory of Relativity
(GTR) is a symmetrical theory it makes sense to also
include a contracting space (an ordinary gravitational
field) along with an expansion (as the case with the
Big Bang) to create an artificial velocity [this also
acts to smooth out the spacetime in the center of the
"warp drive" spacetime]. In the GTR the
contraction of spacetime can bring two distant points
to a common geodesic, thus greatly shorting the distance
between two points (a similar effect happens when one
compares a flat world projection to a globe). Therefore
the Alcubierre Warp Drive can be imagined as special
case geometry which brings a point in space to an observer
as well as pushing an observer from their point of
origin.
The purpose of the following work is to consider some
arbitrary parameters in order to form numerical representation
of the Warp Drive Spacetime. Thus the following does
not describe every detail of the Alcubierre spacetime,
but plots numerical sets in order to give graphical
depictions of the warp drive (i.e. just enough information
is inputted for Mathcad to understand the Alcubierre
Warp Drive to graph the appearance of the spacetime
theta function in three-dimensions) . In order to
do so one will need to construct an arbitrary function
f as to produce the contraction/expansion metric proposed
by the Alcubierre Spacetime. One would then have to
form a bubble around oneself, lets say that three spatial
coordinates are given by:
. Alcubierre defined the velocity of the warp bubble
to be 
, such that theta could be set to q=vs(xs
/rs)(df/drs).
then the Warp Drive Spacetime can be described by the
metric:
for the radius of the function, and 
for the energy-density:
, the invariant lengths of this space is then given
by:
