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 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 = MC2 GO SPACETIME CURVATURE GO | MAIN PAGE | SOFTWARE LIST | AEROTESTING | MISSION | RESUME | Copyright © 1999-2015 John Cipolla/AeroRocket. All rights reserved

Figure-1: Warp bubble traveling adjacent to the Earth (not to scale)

Figure-2: 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)
       
Figure-3: Warp bubble, Figure-4: Spacetime interval (ds) and Figure-5 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)
       
Figure-6, Figure-7 and Figure-8: 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. Figure-6 represents a light cone where rs(t) = [(x-xs(t))2 + y2 + z2]1/2. Figure-7 represents the metric-shape function, f(rs) also called the "top hat" function. Figure-8 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, "Matter-energy tells spacetime how to curve and spacetime tells matter-energy 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 stress-energy 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 matter-energy 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 mass-energy 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 Figure-11 of the RESULTS TO DATE section. SPECIAL REFERENCES: Note-1: 2-D warp bubble from John Cipolla's Warp Drive Notes, 1974. Note-2: 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. Note-3: Sci Fi Science video, warp theory: Traveling at Warp Speed Note-4: Sci Fi Science video, starship design: Exploring the Universe using the Warp Drive Figure-9: 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: Hyper-fast 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

       
Figure-10, Figure-11 and Figure-12: Experimental and theoretical warp metrics
 

These results compare the warp signature of spacetime generated using frame dragging as displayed in Figure-3 and Figure-4 to the warp signature of the Alcubierre warp metric displayed in Figure-11 and Figure-12. 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)

Figure-13: 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)

Figure-14: 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 (Figure-3) that superficially  matches the signature warp bubble (Figure-1) 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 hyper-fast interstellar travel."

INERTIAL MASS REDUCTION EXPERIMENT (7/21/2011)   Figure-15: 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 Figure-14 is composed of a ring of six ceramic magnets that rotate at 1720 RPM. Experiments conducted using the force balance system illustrated in Figure-14 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 in-turn 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 electro-magnetic 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 Figure-15 or Figure-18 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 mass-energy warps spacetime has been proven by Gravity Probe-B (GP-B) a NASA spacecraft that validated Eienstein's 4-dimensional Theory of General Relativity. The NASA probe proved that higher dimensional gravitational theories, M-Theory (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 4-dimensional spacetime exists. Therefore, M-Theory 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 electro-magnetic field has on the metrics of space-time. 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 space-time. 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 space-time 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, GP-B researcher statements, free-surface shape extracted from Schwarzschild’s metric, a unit analysis of the curvature and energy-momentum 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 space-time 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 energy-momentum rotates in the gravitational field of another massive object a Magnus effect based on the superfluid of space-time 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, space-time will be compressed on one side of the object and expanded on the other side of the object generating an imbalance in space-time. The deformed space-time surrounding the spinning object could be called a warp bubble that uses the imbalance within space-time 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 space-time around a circular cylinder with circulation is defined as a uniform flow plus a doublet plus a vortex.   Rotating mass-energy and resulting warped space-time (3)   Superfluid warp drive operating in the Solar System (3) Superfluid vortex experiment (1, 2)   4-d space-time interpreted by GP-B as the surface of a superfluid. See Gravity Probe-B (GP-B) for information Related Publications by John Cipolla 1) “Potential Vortex Transient Analysis and Experiment”, viXra e-print archive, (2014) 2) "Hydrodynamic Analogue for Curved Space-Time and General Relativity", viXra e-print archive, (2014) 3) "Rudimentary Warp Drive Propulsion", Warp-Drive.pdf, (2015) 4) "Does Time Exist", Does-Time-Exist.pdf, (2015) TESTS AND ANALYSES TO BETTER UNDERSTAND GRAVITY GRAVITATIONAL WAVES (8/9/2011) Figure-16: 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. Figure-17: Test results (red dots) verses an inverse square law curve fit for magnetic force verses distance. Figure-18: 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 Figure-18 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, Figure-16 demonstrates how force verses distance were experimentally determined to generate the magnetic force verses distance data presented in Figure-17. 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 Figure-16 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 Figure-18 is a reasonable analogy for the gravity wave generator presented in Figure-19 where several masses possessing mass and energy are rotated at high speed. During operation the cylindrical mass in Figure-18 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 Figure-18 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. Figure-18 illustrates how a massive rotating system analogous to a binary star generate gravitational disturbances in spacetime. Gravity waves are generated by a rotating mass-energy 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. Figure-22 presents a simple gravitational-wave analysis of a binary star. This example is similar to the example displayed in GRAVITATION on pages 979 and 980 where the gravitational-wave 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 2.27E-22 ergs/sec and the force imparted to an area 500 meters away is only 1.89E-42 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 Figure-20 and Figure-21 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 gravitational-wave 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 mass-energy 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 Figure-21 for the basic methodology required for carrying out this analysis. However, achieving the mass-energy 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 Figure-3 was created using frame dragging and not negative energy as required by Alcubierre's warp bubble. While the theoretical warp bubble illustrated in Figure-3 looks similar to the negative energy warp bubble illustrated in Figure-1 and Figure-2 the frame dragging warp bubble needs to be more clearly understood to determine its true physical characteristics. Figure-19: Reduced quadrupole moment generation of gravitational waves through spacetime. Figure-20: Methodology to approximate quadrupole gravitational-wave power Figure-21: Order of magnitude gravitational-wave power analysis Figure-22: 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. Figure-2 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 space-extension or distortion of the volume is caused by the curvature of spacetime.

Figure-1, 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.



Figure-2, 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 Figure-3. Several observations for the deflection of light by the Sun during solar eclipses are in agreement with this simple light ray deflection equation.

Figure-3, 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 light-travel time (T ~ 3 seconds). Using the Riemann curvature tensor determine the uncertainty of measurement for the volume as it traverses the space around Earth.

Figure-4, Example from Gravitation, page 30
 

(4) SPACETIME CURVATURE TOP The following is a general method or procedure to determine the non-relativistic 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 non-Euclidian.

Figure-5, 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.

Figure-6, 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 VH = H*(t/t0)*d, where t0 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.

Figure-7, 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

 

7) UNIFYING GENERAL RELATIVITY AND QUANTUM MECHANICS TOP First Published by John Cipolla April 25, 2012 This discussion is an attempt to unify Einstein's theory of general relativity which governs gravitational fields around massive objects and quantum mechanics or the wave nature of matter and energy. String theory postulates that electrons and quarks in atoms are one-dimenional oscillating line segments. The first elementary particle string models were called bosonic strings because only bosons or force carriers like photons, gluons  and the Higgs were modeled. Later, superstring theory was developed that predicted a connection or a "supersymmetry" between bosons and fermions where fermions are elementary particles like electrons, protons and quarks that compose all ordinary matter. This analysis will assume that massive objects like planets, people and spacecraft are heterotic (closed) superstrings. In this case, we are attempting to relate the gravitational forces predicted by string theory and Einstein's theory of general relativity. To establish the relationship between general relativity and quantum mechanics requires a relatively simple "model equation" that represents the basic features of the physics but at a very simplistic level. A "model equation" for this analysis is required because the full string theory equations are far too complex for this author at the present time. This same technique is used in computational fluid dynamics where solution methods are used on one-dimensional "model equations" and later extended to the three dimensional nature of the flow. For this analysis the one-dimensional string illustrated in Figure-8 represents the wave equation in Figure-9 and its solution. This is a notional string theory solution not a "real" solution. But, the physics of any problem can sometimes be revealed using this basic technique. The solution for the wave equation in Figure-9 reveals the traveling wave velocity of a disturbance on a string equals the square root of the tension on a string divided by the linear mass density of the string. Figure-8, String line segment Figure-9, Wave equation and the wave velocity solution for the wave equation The MathCAD analysis illustrated in Figure-12 was performed to determine the gravitational force at the surface of the Earth using the wave velocity displayed in Figure-9 of the wave equation. For this analysis the length of the string is equal to the circumference of the Earth at the equator and the wave velocity or traveling velocity of the disturbance in the string is equal to the velocity of the Earth around the Sun. According to one interpretation of superstring theory the heterotic strings of multiple particles merge to eventually form the massive heterotic string notionally located at the surface of the Earth. Please see the image on the left which presents this concept. Following the analysis illustrated in Figure-12 the tension in the string is analogous to the gravitational force at the surface of the Earth where the string is notionally located. The gravitational force predicted by general relativity and Newtonian mechanics at the surface of the Earth uses the equation for force (F) and acceleration of gravity (g) expressed in Figure-10. Comparison using the two methods to determine gravitational force at the surface of the Earth shows the force predicted by general relativity and Newtonian mechanics is 43% of the value determined by the simplified string theory. This difference can be explained by various factors such as dark matter and dark energy. A difference of only 43% indicates something fundamental has appeared which deserves more research. Could missing dark matter be causing the difference between simplified string theory and general relativity results?

Figure-10, Gravitational force predicted by general relativity and Newtonian mechanics

Figure-11, Schrödinger's equation for a free particle which describes the traveling wave on a string

Analysis Relating General Relativity and Simplified String Theory

Figure-12, MathCAD solution of the wave velocity solution of the
wave equation to estimate gravitational force at the surface of Earth
 

Solar System Analysis: The following graphic plots the ratio of gravitational potential computed using general relativity (F) to the gravitational potential computed by one-dimensional string theory (t). Notice the wide variation of F/t from planet to planet in the solar system but average F/t for the solar system is approximately 1.0 indicating the over-all correlation between general relativity and string theory is good. These results indicate gravity can be modeled as a heterotic string for a first order approximation of a grand unifying field theory of gravitation and quantum mechanics.

REFERENCES FOR STRING THEORY ANALYSIS
Halliday and Resnick, The Wave Equation, page 425 to page 426, 7th Edition
Halliday and Resnick, Schrödinger's Equation, page 1071 to page 1072, 7th Edition
Calculus Early Transcendentals, Partial Differential Equations, page 917, Edition 5
Introduction to Superstrings and M-Theory, Michio Kaku, 2nd Edition

EINSTEIN'S HYPOTENUSE AND E = mc² TOP
Einstein's hypotenuse is derived from Minkowski's flat space-time metric, below.


The following light cone plot displays space-time for S = 0.5, Xmax = 2 and c = 1.
Note that ct verses x (blue) approaches the light cone (red) as S approaches zero.


Figure-13, Plot of the space-time interval, s² verses distance, x in Minkowski space-time

NASA CONCEPT OF AN ARTIFICIAL GRAVITY (1G) SPACESHIP TOP


Figure-13, NASA concept for using artificial-gravity (AG) for Mars exploration. R = 56m and 4 rpm.

Figure-14, Rotation radius of 56 meters and rotation rate of 4 rpm generates 1.0 g artificial gravity.


Figure-15, Free-body 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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