
WARP DRIVE AND
GRAVITY CONTROL FOR SPACE TRAVEL
NEW RESEARCH
RESULTS
GO
RESULTS
TO DATE (POSITIVE RESULTS)
GO
ROTATION
SPEED OF THE MILKY WAY GALAXY
GO
UNIFYING GENERAL RELATIVITY AND QUANTUM
MECHANICS:
NEW RESEARCH
GO
NASA
CONCEPT OF AN ARTIFICIAL GRAVITY (1G) SPACESHIP
GO
GENERAL RELATIVITY THEORY AND APPLICATIONS
GO
GRAVITATIONAL WARP DRIVE FOR SPACE TRAVEL
GO
EFFECTS OF DARK ENERGY ON COSMOLOGY
GO
GRAVITY AND CURVATURE OF SPACETIME
GO
TEST FOR FLATNESS OF
SPACETIME
GO
HOW TO
DETERMINE, E = MC^{2}
GO
SPACETIME
CURVATURE
GO
 MAIN PAGE 
SOFTWARE LIST
 AEROTESTING 
MISSION
 RESUME 
Copyright © 19992018 John Cipolla/AeroRocket. All rights
reserved 
(1) WARP DRIVE FOR SPACE TRAVEL
TOP
John
Cipolla is performing unique research in the area of gravitational
warp drive
technology and gravity control for faster than light star travel. The
illustrations below show a spacecraft being accelerated
while enclosed within an artificially generated warp bubble.
The following results from
the theory of General Relativity illustrate how
a warp bubble uses opposing regions of expanding
and contracting spacetime for propelling a starship at velocities exceeding
the speed of light. This is a work in progress based on
a new method for warping spacetime to generate warp
bubble disturbances without the need for exotic matter
or negative energy. Experiments are being conducted to
evaluate the method's capability for generating the
theoretical warp metrics depicted in Figure2, Figure3, Figure4
and Figure5. 
Figure1: Warp bubble traveling
adjacent to the Earth (not to scale)
Figure2: Warp bubble geometry illustrating
how spacetime compression and expansion
propel a warp bubble and an enclosed starship through space to
distant stars
J. CIPOLLA
GENERAL RELATIVITY WARP METRIC RESULTS (10/15/2008)
Figure3: Warp bubble, Figure4: Spacetime interval (ds) and
Figure5 torsion of spacetime
where starship is located on the flat part of the disturbance
generated by torsion frame dragging
ALCUBIERRE WARP METRIC RESULTS (10/15/2008)
Figure6, Figure7 and Figure8: Theoretical Alcubierre warp metric derivation using MathCAD
MathCAD results for the
Relativistic analysis of the Alcubierre
faster than light
warp metric is illustrated in the above contour plots. Figure6 represents a light cone
where r_{s}(t) = [(xx_{s}(t))^{2} + y^{2}
+ z^{2}]^{1/2}. Figure7 represents the metricshape function, f(r_{s}) also called the "top hat"
function. Figure8 displays the resulting warp
metric for faster than light space travel. The complete MathCAD
analysis to determine the relativistic warp metric for faster than light star
travel is presented below. 
 End Warp Drive Analysis 
GENERAL RELATIVITY AND WARP DRIVE THEORY
This Relativistic Warp drive theory uses the
concept of a warp bubble to avoid violating the
universal speed limitation which is the speed of light, c.
Basic to the study of General Relativity is the concept of spacetime curvature embodied by the following
statement, "Matterenergy tells spacetime how to curve and spacetime tells
matterenergy how to move". The concept of spacetime curvature
is summarized in the Einstein equation which is a result of
the theory of General Relativity.
According
to the Einstein equation, matter and energy tell spacetime
how to curve and in turn spacetime tells matter and energy
how to move. Where, matter and energy are defined by the
stressenergy tensor (T) and spacetime curvature is defined
by the Riemann curvature tensor (R). In summation, the
Einstein equation relates spacetime curvature and
accelerated motion of a matterenergy system and the
implication that accelerated motion and the effects of
gravity are not distinguishable. Hence, artificial gravity
can be created by simply rotating a spacecraft to create the
effect of gravity on long journeys into space and a warp
bubble can be used to travel to distant places at many times
the speed of light without locally exceeding the speed of light
in the warp bubble.
WARP BUBBLE PHYSICS
According to General Relativity gravity and acceleration are not
distinguishable and are caused by the curvature or warp metric of
spacetime.
A warp bubble is a specific warp metric solution of General
Relativity and is a combination of positive and negative energy
fields that pushes and pulls our starship forward to bring our
destination to us just like a conveyer belt. The exotic ingredient required to make a warp
bubble is negative energy which has the unusual property of
being able to make ordinary matter fall up in a gravitational
field. According to General
Relativity the spacetime in front of a warp bubble is
compressed pulling our destination to us. At the same time the spacetime behind a warp bubble is expanding pushing us to our
destination. The compression and expansion process happens in an instant and at many times
the speed of light making faster than light travel possible. The
combination of positive and negative energy produces an
expansion of space behind the bubble and a contraction of space
in front of the bubble. in other words, creating space behind the bubble pushes us to our destination
and destroying space in front of the bubble pulls us to our destination.
This mechanism allows us to travel many times faster than the
speed of light (see Starship Warp Velocity) relative to the
Earth without exceeding the speed of light in our local frame of
reference, the warp bubble.
The warp bubble itself is made of fields of positive energy at
either end and a band of negative energy around the middle.
These energy fields create huge gravitational effects so powerful the
warp bubble can distort spacetime without having to accelerate
the traveler to achieve faster than light velocity. The main requirement,
negative energy also called vacuum energy is a property of a vacuum where subatomic
particles smaller than an atom dart into and out of existence
almost instantaneously. According to the rules of quantum mechanics
negative energy creates a negative quantum pressure that
propels the warp bubble and therefore our starship forward. An
interesting observation is that we may already see the effects
of negative energy because astronomers have observed that our
universe is expanding due to the presence of dark energy. It is
theorized that dark energy fills the vacuum of space between the
galaxies and is the cause for the expansion and increasing
acceleration of the universe.
Therefore, dark energy and negative energy are probably the same
"stuff" required to make a warp bubble possible.
General Relativity states the equivalent massenergy of a planet
the size of Jupiter is required to create a warp bubble. Because
producing negative energy is beyond our capability the objective
of this research is to find an alternate way to create a
relativistic warp bubble without the need for exotic matter or negative
energy. It is proposed that a replacement for negative energy
may be possible by using positive energy in unique ways to generate an energy signature
equivalent to the Alcubierre warp metric displayed in
Figure11 of the
RESULTS TO DATE
section.
SPECIAL REFERENCES:
Note1: 2D warp bubble from John
Cipolla's Warp Drive Notes, 1974.
Note2: Negative energy composite view based on Sci Fi
Science, How to Explore the Universe: Where Dr Michio
Kaku reveals how we could one day build a warp drive.
Note3: Sci Fi Science video, warp theory:
Traveling at Warp Speed
Note4: Sci Fi Science video, starship design:
Exploring the Universe using the Warp Drive
Figure9: MathCAD
warp bubble analysis of a hypothetical flight to a star 4.3 light years away
REFERENCES FOR
GENERAL RELATIVITY
Gravitation, Charles W. Misner, Kip S. Thorne and John
A. Wheeler
SPACETIME and GEOMETRY An Introduction to General
Relativity, Sean M. Carroll
Relativity Demystified, David McMahon
WARP DRIVE
REFERENCES
The
Warp Drive: Hyperfast Travel Within General Relativity,
Miguel Alcubierre
Breakthrough Propulsion Physics (NASA)
Warp Drive, When? (NASA)
Back to TOP 
WARP BUBBLE TEST RESULTS
TOP
See the new "breakthrough" results
Figure10, Figure11 and Figure12: Experimental and theoretical warp metrics
These results compare
the warp signature of spacetime generated using frame dragging
as displayed in Figure3 and Figure4 to the warp signature
of the Alcubierre warp metric
displayed in Figure11 and Figure12. Work continues using a laser to map
spacetime around the experimental warp metric, clocks
to measure time dilation within and around the warp
bubble and force measurements to determine inertia effects. 
LASER SPACETIME WARP EXPERIMENT
(2/12/2009)
Figure13: Laser spacetime warp
experiment
Laser
experiment used to map spacetime around the
proposed warp drive demonstration device. This laser
measurement system is mechanically and acoustically isolated
from the system used to warp spacetime. Similar experiments
are being designed using synchronized clocks to measure time
dilation effects. No positive results to date. 
LASER SPACETIME WARP EXPERIMENT (8/12/2009)
Figure14: Laser spacetime warp
experiment using fog
The same experiment
using fog to trace a laser beam around the warp
bubble disturbance. The laser beam was not deflected
indicating spacetime is not being warped to any
measurable degree by the proposed warp generator. Several
configurations of the warp generator resulted in the same
null result. The experiments are continuing using
conventional energy sources to affect spacetime.
The aim of these experiments
is an attempt to
show that a warp bubble may be generated using
electromagnetic energy instead of negative energy as required by Alcubierre's
faster than light warp metric. The method proposed here to warp
spacetime
theoretically generates a warp bubble (Figure3) that
superficially matches the signature warp bubble (Figure1) predicted using negative energy.
General Relativity, Dark Matter and Dark
Energy cosmology indicate that a warp bubble is analogous to
expanding spacetime observed to occur between distant
galaxies. In
effect, expanding spacetime between galaxies is like a
conveyer belt pushing galaxies apart at a rate of approximately
71 km/sec/mega parsec. While measuring the warp effect
predicted by this method has not proven successful the paper
by Alcubierre states the following, "The need for exotic
matter therefore doesn't necessarily eliminate the
possibility of using spacetime distortion like the one
described here for hyperfast interstellar travel."

INERTIAL MASS REDUCTION EXPERIMENT (7/21/2011)
Figure15: Warp chamber shown
rotating at 1720 RPM and the theoretical warp bubble
This
experiment is an attempt to determine if the inertial
mass
of a 15.5 gram cylindrical aluminum object is reduced within
a proposed gravitational warp bubble. The warp
chamber illustrated in Figure14 is composed of a ring of
six ceramic magnets that rotate at 1720 RPM. Experiments
conducted using the force balance system illustrated in
Figure14 has so far not detected any measurable reduction
in the weight or inertial mass of the cylindrical aluminum object. These
experiments will continue using improved force measurement
devices and various orientations as these valuable resources become available ...
The theoretical basis for the
operation of this experiment is that a massive object causes
spacetime to curve and inturn spacetime tells a
massive object how to move and accelerate.
It is postulated
here that spacetime curvature can be modified
using powerful electromagnetic fields
to reduce the inertial mass of a starship. In the
weakest implementation of this theory a starship can be made
to accelerate as if the inertial mass of the starship were
reduced making near light speed possible using simple
electric thrusters. However, in the most advanced
implementation of this experiment when the energy of the
electromagnetic fields cause the inertial mass of a starship
to become imaginary the starship in the warp
bubble will become a tachyon capable of moving
faster than the speed of light. In its advanced form the
object in the warp bubble is isolated from the rest of the
universe allowing the warp bubble to become a local frame of
reference where Faster Than Light (FTL) travel does
not violate the local speed of light (c).
Tachyon: A particle postulated to move at a velocity
greater than the speed of electromagnetic radiation, such
that as the particle accelerates it loses energy. Of the two
properties rest mass and energy, one must be
real and the other imaginary. If a tachyon
exists it may be detected through the emission of
Cerenkov radiation (a kind of electromagnetic shock wave) or by cosmic
ray collisions. 
INERTIAL MASS REDUCTION EXPERIMENT
FIRST POSITIVE RESULTS (11/26/2012 to 12/6/2012)
Positive test
results have been achieved in recent tests "indicating" that
a warp bubble based on free vortex theory can be generated and that propulsion using
these methods may be possible. However, these results have not
been confirmed to be purely a consequence of spacetime
warping or an electromagnetic effect so further
experimentation is necessary. A new load cell and data
acquisition system made it possible to "tease out" very
small reductions of test item gravitational force. A totally new
configuration not represented by Figure15 or Figure18 and
a slight change in methodology made these results possible.
DISCUSSION:
According to the well tested and validated Theory of
General Relativity, gravity
is a direct result of the curvature of spacetime caused by
the presence of mass and energy. It has been proven by observation
of gravitational lensing around
massive objects that mass
and energy warp the fabric of spacetime. The warping of
spacetime by massive objects is a scientific fact where
General Relativity accurately predicts the deflection of
visible light around massive objects like stars, clusters of
stars and galaxies. The fact that massenergy warps
spacetime has been proven by
Gravity ProbeB (GPB) a NASA spacecraft that validated Eienstein's 4dimensional Theory of General
Relativity. The NASA probe proved that higher dimensional
gravitational theories, MTheory (superstrings) and quantum
mechanics
are
not required for warping spacetime. In addition, very
complex theories are probably not correct because as
Occam's razor states the theory that makes the fewest
assumptions is probably correct. Because General Relativity
is the "simplest" explanation for gravity and because
experiments have proven that gravity is a physical
effect caused by the presence of mass and energy it seems
reasonable that gravity can be controlled and manipulated by
mass and energy in a laboratory experiment. It is not a
theory that 4dimensional spacetime exists. Therefore,
MTheory which
has not been proven by a single experiment and quantum mechanics
should not be
required to manipulate the physical large scale
aspects of spacetime and gravity. For example, it is well
known that rotating masses like binary stars and rotating
beams generate gravitational waves in spacetime. 
PLASMA WARP CORE
EXPERIMENT
RESULTS (12/17/2014
to PRESENT)
A rotating
plasma analogous to the inertial mass reduction
experiment that used a rotating magnet has been fabricated to
document the influence a rotating plasma generated
electromagnetic field has on the
metrics of spacetime. The beam of a Helium Neon laser
has been used to penetrate the warp core region of a rotating plasma
core with no deflection observed to date. The challenge has
been to accurately measure laser beam deflection. This experiment is important
because a laser beam is capable of penetrating the
interior of the warp core region without outside
influence. The experiment is considered successful if the laser beam
deflects by even the smallest amount indicating warped
spacetime. More images of the
experiment will be posted as the experiments proceed. 
ESTABLISHING THE ANALOGY
BETWEEN
GENERAL RELATIVITY AND POTENTIAL VORTEX FLOW
AND RUDIMENTARY WARP DRIVE PROPULSION
BY JOHN CIPOLLA (1990 to 2015)
NEW PAPERS AVAILABLE
By John R. Cipolla,
Copyright 2015
Research has
shown that an analogy exists between potential vortex flow
and the generation of spacetime curvature around massive
objects as predicted by Einstein’s theory of General
Relativity (GR). The analogy between GR and potential vortex
flow is based on results from potential vortex
experimentation, GPB researcher statements, freesurface shape extracted from Schwarzschild’s metric, a
unit analysis of the curvature and energymomentum
components of potential vortex flow and the analogous
components from Einstein’s Field Equations and black hole
dynamics compared to potential vortex dynamics. Predictions
based on this research
are made that indicate gravity control and rudimentary warp
drive is possible.
An implication for the existence of a superfluid potential vortex substratum is that interesting
fluid mechanical characteristics of spacetime can be
revealed. Specifically, an interesting by product of a
superfluid substratum is the Magnus effect. The Magnus
effect is the force exerted on a rapidly spinning cylinder
or sphere moving through air or another fluid in a direction
at an angle to the axis of spin. The sideways force is
responsible for the swerving of balls when hit or thrown
with spin. For example, if an object composed of
energymomentum rotates in the gravitational field of
another massive object a Magnus effect based on
the superfluid of spacetime will impart a sideways force on
the object and an associated acceleration in the substratum.
In exactly the same way the surrounding fluid is deformed by
a spinning object, spacetime will be compressed on one side
of the object and expanded on the other side of the object
generating an imbalance in spacetime. The deformed
spacetime surrounding the spinning object could be called a
warp bubble that uses the imbalance within spacetime to
propel an object perpendicular to the field lines of the
surrounding superfluid. Speeds approaching the speed of
light are not practical but exotic materials are not
required for a device based on this technology. The
analogous Magnus effect in General Relativity that uses the
principals of fluid mechanics to model
spacetime around a circular cylinder with circulation is
defined as a uniform flow plus a doublet plus a
vortex.
Rotating
massenergy and resulting warped spacetime (3)

Superfluid warp drive operating
in the Solar System (3) 
Superfluid vortex
experiment (1, 2)

4d spacetime
interpreted by
GPB as the surface of a
superfluid.
See
Gravity ProbeB (GPB) for information 
Related Publications by John Cipolla
1) “Potential Vortex
Transient Analysis and Experiment”, viXra
eprint archive, (2014)
2)
"Hydrodynamic Analogue for Curved SpaceTime and General
Relativity", viXra eprint
archive, (2014)
3) "Rudimentary Warp
Drive Propulsion",
WarpDrive.pdf,
(2015)
4) "Does Time Exist",
DoesTimeExist.pdf, (2015) 
TESTS AND
ANALYSES TO BETTER UNDERSTAND GRAVITY
GRAVITATIONAL WAVES (8/9/2011)
Figure16: Experiment to determine
magnetic force (F) verses
distance (r) separating a magnet from a small
cylindrical steel mass
and to prove magnetic forces obey the inverse square law
relationship.
Figure17: Test results (red
dots) verses an inverse square law curve fit for magnetic
force verses distance.
Figure18: Magnetic field analogy for a
gravity wave generator to determine distant particle motion.
Vector,
V illustrates the motion and velocity of a cylindrical steel mass
exposed to a rotating pair of magnets.
The steel mass is exposed to the quadrupole moment generated by the
rotating pair of ceramic magnets.
The mass follows an elliptical orbit that is perpendicular to the axis of
the rotating pair of magnets.
GRAVITATIONAL WAVES: The law of
gravitation is an inverse square law relationship as
are the laws relating the forces associated with monopole
static charges and dipole magnetism. In general the inverse
square law relates the intensity of a field effect to the
reciprocal of the square of the distance from the source of
the effect. The experiment illustrated in Figure18 uses a
magnetic field analogy of a gravity wave generator to
demonstrate the effect quadrupole gravitational waves
have on spacetime and particle motion.
To demonstrate that dipole magnetic fields obey an inverse
square law relationship and therefore are a useful mechanism
to visualize quadrupole gravitational radiation for rotating
systems, Figure16 demonstrates how force verses distance were
experimentally determined to generate the magnetic force
verses distance data presented in Figure17. As expected
from field theory, dipole magnetism obeys the inverse square
law relationship. The following equation fits the force
verses distance data measured using the method illustrated
in Figure16 where the relationship is F = C/r^2 and
C = 1.786E5 dyne*mm^2. Because dipole magnetism
obeys the inverse square law it can be assumed the experiment
illustrated in Figure18 is a reasonable analogy for the
gravity wave generator presented in Figure19 where several
masses possessing mass and energy are rotated at
high speed. During operation the cylindrical mass in Figure18 follows a
highly elliptical orbit indicating the presence of an
external magnetic quadrupole field. Therefore, to understand
how gravitational quadrupole radiation affects particle
motion the rotating magnetic field experiment in Figure18
is useful.
It is well known and documented in GRAVITATION
and other books
about general relativity that rotating systems like binary stars, black holes
and all rotating massive objects generate
gravitational waves due to the reduced quadrupole moment
of the rotating disturbance. Figure18 illustrates how a
massive rotating system analogous
to a binary star generate
gravitational disturbances in spacetime. Gravity waves are
generated by a rotating massenergy system because the
differential arrival time from opposite sides of the system
cause a phase angle between gravitational vectors. Gravitational vectors from opposite sides of a
rotating system that initially oppose each other when the
system is stationary are drawn inclined at phase angle,
dq
during rotation. The amplitude of the resulting
gravitational wave generates a reduced quadrupole moment
that when squared is proportional to the generated
gravitational power. Further, it can be
shown that like electromagnetic waves, gravitational waves
have energy, U that delivers
momentum, p to a point in spacetime causing a small net force,
F to act at that point. The force, F is the net
gravitational wave force this research is attempting to
generate, enhance and measure.
Figure22 presents a simple gravitationalwave analysis of a binary star. This example is similar to the example
displayed in GRAVITATION on pages 979 and 980 where the gravitationalwave power output of a massive rotating beam
is computed when the beam rotation frequency is determined
by balancing centrifugal force and beam material tensile
strength. The power radiated in the form of gravitational
waves by the rotating beam is only 2.27E22 ergs/sec and the force
imparted to an area 500 meters away is only 1.89E42
newtons. However, if the mass or the rotation rate of
the beam are greatly increased possibly to speeds
approaching the speed of light then a form of gravity
propulsion may be possible. In ways similar to Alcubierre's
warp metric, gravity waves
produce repeated regions of compressed spacetime followed
immediately by regions of expanded spacetime.
WHAT RADIATES GRAVITATIONAL WAVES: In applying the
equations that appear in Figure20 and Figure21 one must be
careful to ignore internal power flows that cannot radiate
gravitationally, that is internal motions that do not
accompany a time changing quadrupole moment. For example, a
normal star does not radiate gravitational waves because the
internal power flows associated with spherical pulsation and
axially symmetric rotation are not unbalanced motions.
However, dynamic astrophysical systems that do radiate
gravitational waves include stars that pulsate and rotate
wildly, collapsing stars, exploding stars, feeding black
holes and chaotic systems of stars.
GRAVITY WAVE PROPULSION  HYPOTHESIS:
The power output by a laboratory sized gravitationalwave
generator is very small unless the rate of rotation or the
mass of the beam is greatly increased. However, it is hypothesized
that if the ordinary massenergy of a rotating beam
is increased to that of the planet Jupiter and if the
rate of rotation is kept the same at 4.456 revolutions per
second it may be possible to impart a force of 28.5 newtons
to an object 500 meters away. Please see Figure21 for the
basic methodology required for carrying out this analysis.
However, achieving the massenergy density for
successfully conducting this experiment does not yet exist
on the planet Earth. But, it is encouraging that negative
energy of the same density is not be required.
FURTHER INVESTIGATION: Using the reduced quadrupole
moment of rotating systems deserves further investigation.
For example, the theoretical warp bubble illustrated in
Figure3 was created using frame dragging and
not negative energy as required by Alcubierre's
warp bubble. While the
theoretical warp bubble illustrated in Figure3 looks
similar to the negative energy warp bubble illustrated in
Figure1 and Figure2 the frame dragging warp bubble needs
to be more clearly understood to determine its true
physical characteristics. 
Figure19: Reduced quadrupole moment
generation of gravitational waves through spacetime.
Figure20: Methodology to approximate quadrupole
gravitationalwave power
Figure21: Order of magnitude
gravitationalwave power analysis
Figure22: More precise method to
determine gravitational power radiated by a binary star from
GRAVITATION
(2) GRAVITY AND CURVATURE OF SPACETIME
TOP
According to Einstein's
General Theory of Relativity gravitation is a
manifestation of the curvature of spacetime. Light and
particles of matter travel along geodesics while the geometry
in
which travel occurs takes place in spacetime not just
space. A geodesic is the shortest line between two
points that lies in a given surface. In curved space two
separate geodesics that start off parallel will
eventually cross or intersect. Because gravity is a
manifestation of geometry this behavior will occur in
the motion of particles on geodesics in spacetime. The
intersection of initially parallel geodesics is an
expression of gravitational tidal effects while
traveling within a gravitational field. For example, two
particles in free fall in a gravitational field will
initially move parallel to each other as they approach
the ground. However, because the particles are moving on
radial paths to the center of the massive object they
will seem to move toward each other if the distance
traveled is great enough. This is a description of the
tidal effects of gravity and the spacetime effect on
particles moving in spacetime. This phenomenon is also
called geodesic deviation.
Figure2 represents the gravitational field determined
using the Schwarzschild metric solution for the
curvature of spacetime outside any spherically
symmetric mass like the Earth, Sun or a black hole. The
tidal effects of gravity on a volume of space as the
volume approaches a massive object is displayed. Changes
of spaceextension or distortion of the volume is caused by the
curvature of spacetime. 
Figure1, Schwarzschild metric or line element for static,
spherically symmetric fields outside spherically
symmetric bodies. This equation describes the metric
structure of empty spacetime surrounding a massive body.
Figure2, Volume entering the gravitational field of an object
modeled by the Schwarzschild solution
Furthermore,
the curvature of spacetime causes the path of a light ray
to bend in the region around a massive object. A ray of
light as it approaches the gravitational charge of a massive
object undergoes a deflection through the angle,
F
when the separation distance, D
is small enough. Using the Schwarzschild metric
solution given by the principle of equivalence the equation
for the deflection angle, F
of a ray of light is illustrated in Figure3.
Several observations for the deflection of light by the Sun
during solar eclipses are in agreement with this simple
light ray deflection equation. 
Figure3, Deflection of light determined by the Schwarzschild
metric
(3) TEST FOR FLATNESS
OF SPACETIME
TOP
Sometimes it's necessary to
determine the degree to which spacetime is curved. The
following test for spacetime flatness is useful to
determine if the influence of a nearby massive object can be
ignored when trying to determine the relative position of
two particles or two space ships in orbit. The following example
is from page 30 of Gravitation by Misner and Thorne.
Statement of the Problem: A region just above the
surface of the Earth, 100 m x 100 m x 100 m (space
extension) is followed for 10^6 m of lighttravel time (T ~
3 seconds). Using the Riemann curvature tensor determine the
uncertainty of measurement for the volume as it traverses
the space around Earth. 
Figure4, Example from Gravitation, page 30
(4) SPACETIME
CURVATURE
TOP
The following is a general
method or procedure to determine the nonrelativistic
change in the space extension of a volume, region or object
in the vicinity of a massive object caused by tidal effects
of gravity and spacetime curvature. This example is useful
to determine the dimensions of an object as it approaches a
black hole or to determine when spacetime can be considered
Euclidian (flat) or nonEuclidian. 
Figure5, Simple application of the Riemann curvature tensor
(5) GENERAL RELATIVITY THEORY AND APPLICATIONS
TOP
The following series of simple
analyses are applications of General Relativity to
the study of Cosmology. Gravity dominates on large scales
making it possible to neglect nuclear and electromagnetic
forces for cosmological approximations. In addition, the
universe is to a very high degree "homogeneous" (the same at
every point) and "isotropic" (the same in every direction) making the spacetime metric nearly the same from one
point to another over large distances. For more information
please see the references especially
Relativity Demystified. 
Figure6, General Relativity theory and applications
6) EFFECTS OF DARK ENERGY ON COSMOLOGY
TOP
The cosmology presented here is based on the
concept of dark energy and the resulting negative
pressure required for an expanding universe. These
concepts are important because designing a warp drive
depends on dark energy or something similar to generate
the signature warp bubble required for faster than light
star
travel.
The following are plots of the scale factor (a), Hubble
parameter (H), energy density (r)
and expansion velocity (VH/c) of the universe as a function
of time from the Cosmic Microwave Background which occurred 380,000 years after the
Big Bang. Generating these Cosmology results require the
following equations from Sean M. Carroll's text book, SPACETIME and GEOMETRY
an Introduction to General
Relativity. The equations required for this analysis are: Scale
factor (a), equation 8.183 on page 367, Hubble parameter
(H), equation 8.184 on page 367 and average mass density (r) of the
universe, equation 8.67 on page 336. Equation 8.67 is the Friedmann equation
which relates spacetime curvature (K), mass density and the expansion rate
(H) of the universe. Using the Friedmann equation average mass
density of the universe is determined by substituting K = 0 because the universe is observationally flat
over great distances. Finally, the expansion velocity of the universe is
V_{H} = H*(t/t_{0})*d,
where t_{0} is the present time from the CMB and d
is our present distance from the CMB. The CMB
is defined as the Cosmic Microwave Background which occurred
380,000 years after the Big Bang. 
Figure7, Results for scale factor, Hubble
parameter, energy density and expansion velocity of the
universe as
a function of time from when the Cosmic
Microwave Background (CMB) occurred 380,000 years after the
Big Bang
EINSTEIN'S HYPOTENUSE AND E = mc^{2}
TOP
Einstein's hypotenuse is derived from Minkowski's flat
spacetime metric, below.
The following light cone
plot displays spacetime for S = 0.5, Xmax = 2 and c = 1.
Note that ct verses x (blue)
approaches the light cone (red)
as S approaches zero.
Figure13,
Plot of the spacetime interval, s^{2}
verses
distance, x
in Minkowski spacetime
NASA CONCEPT OF AN ARTIFICIAL GRAVITY (1G) SPACESHIP
TOP
Figure13, NASA concept for using artificialgravity (AG) for
Mars exploration. R = 56m and 4 rpm.
Figure14, Rotation radius of 56 meters and rotation rate of
4 rpm generates 1.0 g artificial gravity.
Figure15, Freebody diagram illustrating how rotation
radius and rotation rate create gravity.
This concept illustrates equivalence between gravity
and normal acceleration.
REFERENCES FOR
GENERAL RELATIVITY
Gravitation, Charles W. Misner, Kip S. Thorne and John
A. Wheeler
SPACETIME and GEOMETRY An Introduction to General
Relativity, Sean M. Carroll
Relativity Demystified, David McMahon
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