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DIA Defense Intelligence Reference Document DIA-08-1003-018, "Antigravity for Aerospace Applications," a 30 March 2010 AAWSAP technical review of theoretical paths to gravity control for propulsion.
DIA AATIP Defense Intelligence Reference Documents (38 DIRDs), FOIA Release, DIRD_19-DIRD_Antigravity_for_Aerospace_Applications.pdf
DIA Defense Intelligence Reference Document DIA-08-1003-018, "Antigravity for Aerospace Applications," a 30 March 2010 AAWSAP technical review of theoretical paths to gravity control for propulsion.
This is one of the 38 Defense Intelligence Reference Documents commissioned under the Defense Intelligence Agency's Advanced Aerospace Weapon System Applications (AAWSA) Program, dated 30 March 2010 with an information-cutoff of 1 December 2009. The paper surveys antigravity concepts drawn from Newtonian mechanics, Einstein's General Theory of Relativity (gravitomagnetic and Lense-Thirring effects, Forward's dipole gravitational field generator, Felber's relativistic repulsion above v > c/√3), and quantum field theory (Casimir effect, zero-point fluctuations, nonretarded interatomic dispersion forces). It quantifies the energy budget for sustained levitation at 62.5 MJ/kg, about 2.05 times the kinetic energy to reach low Earth orbit. The author and program manager are redacted under (b)(3):10 USC 424 and (b)(6); the document is marked UNCLASSIFIED//FOR OFFICIAL USE ONLY.
Antigravity effects can be implemented by manipulating spacetime. This paper reviews several different theoretical approaches for exploring the possibility of controlling gravity by generating forces that counteract, or otherwise modify, gravity for the purpose of aerospace propulsion. Einstein's General Theory of Relativity is the theoretical framework guiding this study.p.5
Gravity is the bane of aerospace transportation.p.6
For example, this limit implies that a single-stage rocket that accelerates to escape velocity must be composed of more than 93 percent fuel.p.6
To date, there is no technology that can achieve the active control of gravity.p.6
The reader should bear in mind that many of these concepts are nowhere near having any form of practicable engineering implementation. However, the report will provide theoretical estimates to guide the way toward technological implementation of antigravity.p.7
This is called the Lense-Thirring effect, or rotational frame dragging effect, in which rotating bodies literally drag spacetime around themselves.p.9
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UNCLASSIFIED//FOR OFFICIAL USE ONLY Defense Intelligence Reference Document [REDACTED] Acquisition Threat Support 30 March 2010 ICOD: 1 December 2009 DIA-08-1003-018 Antigravity for Aerospace Applications UNCLASSIFIED//FOR OFFICIAL USE ONLY
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UNCLASSIFIED//FOR OFFICIAL USE ONLY
Antigravity for Aerospace Applications
Prepared by:
(b)(3):10 USC 424
Defense Intelligence Agency
Author:
(b)(6)
Administrative Note
COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized.
This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, (b)(3):10 USC 424 Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining to
this document should be addressed to (b)(3):10 USC 424;(b)(6) , AAWSA Program
Manager, Defense Intelligence Agency, ATTN: (b)(3):10 USC 424 Bldg 6000, Washington,
DC 20340-5100.
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UNCLASSIFIED//FOR OFFICIAL USE ONLY [81] Robinson, A. L., "Now Four Laboratories Have Squeezed Light," Science, Vol. 233, 1986, pp. 280-281. [82] Saleh, B. E. A., and Teich, M. C., Fundamentals of Photonics, Wiley Series in Pure and Applied Optics, John Wiley & Sons, Inc., New York, 1991, pp. 414-416. [83] Caves, C. M., "Quantum-mechanical noise in an interferometer," Physical Review D, Vol. 23, 1981, pp. 1693-1708. [84] Pfenning, M. J., "Quantum Inequality Restrictions on Negative Energy Densities in Curved Spacetimes," Ph.D. Dissertation, Dept. of Physics and Astronomy, Tufts Univ., Medford, MA, 1998. [85] Casimir, H. B. G., "On the Attraction Between Two Perfectly Conducting Plates," Proc. Kon. Ned. Akad. Wetensch., Vol. 51, 1948, pp. 793-796. [86] Lamoreaux, S. K., "Demonstration of the Casimir Force in the 0.6 to 6 μm Range," Physical Review Letters, Vol. 78, 1997, pp. 5-8. [87] Mohideen, U., "Precision Measurement of the Casimir Force from 0.1 to 0.9 μm," Physical Review Letters, Vol. 81, 1998, pp. 4549-4552. [88] Chen, F., et al., "Theory confronts experiment in the Casimir force measurements: Quantification of errors and precision," Physical Review A, Vol. 69, 2004, 022117. [89] Brown, L. S., and Maclay, G. J., "Vacuum Stress between Conducting Plates: An Image Solution," Physical Review, Vol. 184, 1969, pp. 1272-1279. [90] Walker, W. R., "Negative energy fluxes and moving mirrors in curved space," Classical and Quantum Gravity, Vol. 2, 1985, pp. L37-L40. [91] Moore, G. T., "Quantum Theory of the Electromagnetic Field in a Variable-Length One-Dimensional Cavity," Journal of Mathematical Physics, Vol. 11, 1970, pp. 2679- 2691. 39 UNCLASSIFIED//FOR OFFICIAL USE ONLY
Concatenated page-by-page transcript. Born-digital pages came through pdf.js; scanned pages were transcribed by Claude vision OCR. Pages marked unreadable failed multiple OCR retries (heavy redaction, microfilm artifacts, or blank separators) and are kept in place for audit.
UNCLASSIFIED//FOR OFFICIAL USE ONLY Defense Intelligence Reference Document [REDACTED] Acquisition Threat Support 30 March 2010 ICOD: 1 December 2009 DIA-08-1003-018 Antigravity for Aerospace Applications UNCLASSIFIED//FOR OFFICIAL USE ONLY
UNCLASSIFIED//FOR OFFICIAL USE ONLY
Antigravity for Aerospace Applications
Prepared by:
(b)(3):10 USC 424
Defense Intelligence Agency
Author:
(b)(6)
Administrative Note
COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized.
This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, (b)(3):10 USC 424 Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining to
this document should be addressed to (b)(3):10 USC 424;(b)(6) , AAWSA Program
Manager, Defense Intelligence Agency, ATTN: (b)(3):10 USC 424 Bldg 6000, Washington,
DC 20340-5100.
ii
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Contents
Foreword.........................................................................................................v
I. Introduction ................................................................................................. 1
II. Concepts for Antigravity Within Newtonian Physics ......................................... 2
Negating Newtonian Gravity .......................................................................... 2
Energy Estimate for Newtonian Levitation ....................................................... 3
III. Concepts for Antigravity Within General Relativity ......................................... 4
Antigravity via Gravitomagnetic Forces........................................................... 4
Historical Foundations .............................................................................. 4
Forward's Dipole Gravitational Field Generator............................................. 4
Felber's Relativistic Antigravity Effect............................................................ 7
Negative Energy-Induced Antigravity ............................................................. 9
Examples of Exotic or "Negative" Energy Found in Nature ........................... 10
Toy Model Estimate for Negative Energy-Induced Antigravity....................... 10
Cosmological Antigravity.............................................................................. 13
Pressure as a Source of Gravity................................................................ 13
Vacuum Energy of Einstein's Cosmological Constant................................... 13
Dark Energy ........................................................................................... 15
Antigravity Propulsion Application of Dark/Vacuum Energy ......................... 17
IV. Quantum Antigravity Propulsion Concepts ................................................... 17
Antigravity via Quantum Vacuum Zero-Point Fluctuation Force ....................... 19
Antigravity via Nonretarded Quantum Interatomic Dispersion Force ............... 21
V. Conclusion: The Way Forward ..................................................................... 24
Appendix A..................................................................................................... 29
Static Radial Electric & Magnetic Fields ........................................................ 29
Squeezed Quantum Vacuum......................................................................... 29
Gravitationally Squeezed Electromagnetic Zero-Point Fluctuations.................. 30
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Quantum Vacuum Field Stress: Negative Energy from the Casimir Effect......... 31
Dynamical Casimir Effect: Moving Mirrors ..................................................... 32
References .................................................................................................... 34
Figures
Figure 1. Dipole Electric Field Generator .......................................................... 5
Figure 2. Diople Gravitational Field Generator................................................... 6
Figure 3. Dipole Gravitational Field Generator: Inside-Out Whirling Dense
Matter Torus........................................................................................ 7
Figure 4. Illustration of the Casimir Effect. ....................................................... 31
Figure 5. Negative Energy Flux (Gold) Emanating From a Moving Mirror ............ 33
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Antigravity for Aerospace Applications
Foreword
Antigravity effects can be implemented by manipulating spacetime. This
paper reviews several different theoretical approaches for exploring the
possibility of controlling gravity by generating forces that counteract, or
otherwise modify, gravity for the purpose of aerospace propulsion. Einstein's
General Theory of Relativity is the theoretical framework guiding this study.
The paper also reviews other antigravity approaches via the interaction of
quantum theory with gravitation. And it explores the question of which
method or technique is best suited for aerospace applications and evaluates
the make-or-break issues that limit them.
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I. Introduction
Gravity is the bane of aerospace transportation. The force of the Earth's gravitational
field acts to pull all objects, whether in motion or at rest, downward towards the Earth's
surface. Because aerospace transportation involves the motion of vehicles through the
atmosphere and/or into space, propulsion engineers are always faced with the
requirement that aerospace vehicles will have to carry enough propellant and
associated tankage in order to provide enough propulsive thrust to overcome the
downward pull of gravity and achieve rectilinear motion. Energy has to be expended by
a propulsion system to overcome the force of gravity in addition to providing for
rectilinear motion, and the majority of propulsive energy is dedicated to overcome
gravity. The aerospace propulsion engineer is faced with two choices for the control of
gravity in this regard: passive control and active control. Modern aerospace propulsion
technology, which is based on accumulated scientific knowledge since recorded history,
can only achieve the passive control of gravity whereby a given propulsion device must
develop a thrust that will passively counteract the Earth's gravitational pull, lift a
vehicle off the surface, and propel it through the air or into space. Newton's laws of
motion and gravity require that the fuel fraction of any aerospace vehicle can never be
less than that given by a simple function of the ratio of the vehicle's maximum speed to
the speed of its rocket plume, jet, fan, or propeller wake. For example, this limit implies
that a single-stage rocket that accelerates to escape velocity must be composed of
more than 93 percent fuel. That is because a rocket must accelerate its working fluid
from rest (relative to the rocket) up to its exhaust speed. Thus, exhaust speeds for
aircraft and chemical rockets are limited by material science, chemical reaction rates,
and engineering factors to only a few thousand meters per second.
To date, there is no technology that can achieve the active control of gravity. If one
could eliminate or otherwise control the Earth's gravity field, then one has the ability to
dramatically reduce the amount of propellant, its tankage, and the overall structural
size and mass of an aircraft or rocket because there will no longer be any need for
these to overcome the pull of Earth's gravity while transporting a payload across the
globe or into space. Instead, aerospace vehicles will only need to have the propellant
mass and infrastructure necessary to change their kinetic energy from rest to a final
velocity necessary to achieve atmospheric flight or space orbit. The Earth's gravitational
well will no longer have any impact on aircraft, launch vehicle, or spaceflight dynamics
if one were to achieve active gravity control. Aerospace vehicles would merely "levitate"
in air and their propulsion systems would be optimized for change-in-velocity missions.
However, it is possible to envision a form of active gravity control propulsion that would
not require a change in kinetic energy.
One of the primary concepts for the goal of affecting gravity is "antigravity," which is a
colloquial expression that specifically means the negation or repulsion of the force of
gravity. A more general term that encompasses this notion and other possibilities is
"gravity control."
If antigravity exists, it can be exploited to counteract or nullify the gravitational pull, or
attraction, of a planetary (or stellar) body that acts upon a much smaller body.
Einstein's General Theory of Relativity gives a prescription for a variety of different
antigravity generators. Even Newton's law of gravity offers several different classical
prescriptions. Newton's law of gravity can be used to simply nullify the gravity field of
one body acting on another body by using a clever arrangement of masses. The
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theoretical possibility of antigravity also appears in quantum gravity theories,
cosmological vacuum or dark energy, and quantum field theory. This report reviews all
of these topics. The report will also review the topics of gravity control that include the
production of antigravity (self-lifting) forces induced by quantum vacuum zero-point
energy and by nonretarded quantum interatomic dispersion forces in a curved
spacetime (that is, in a background gravitational field). The reader should bear in mind
that many of these concepts are nowhere near having any form of practicable
engineering implementation. However, the report will provide theoretical estimates to
guide the way toward technological implementation of antigravity.
II. Concepts for Antigravity Within Newtonian Physics
The basic form of Newton's law of gravity is given by the standard expression for the
gravitational force (F_grav) that mutually acts between two masses (Reference 1):
F_grav = -G m₁ m₂ / r² (1)
where the negative sign indicates that F_grav is a (mutual) force of attraction, G is
Newton's universal gravitation constant (6.673 × 10⁻¹¹ Nm²/kg²), m₁ and m₂ are two
interacting masses, and r is the radial distance between the two masses (note: MKS
units are used throughout). Observe in Equation (1) that the force of gravity acting on
a small test mass becomes stronger when the other (gravitating) mass is larger in
magnitude or when the distance between them is very small, or both. Also recall that
Equation (1) and Newton's second law of motion (F = ma) to define the magnitude of the
gravitational acceleration a_g that acts on a small test mass m due to a larger
(gravitating) mass M (Reference 1):
a_g = GM / r² (2)
If Earth is chosen to be the larger gravitating mass so that M = M_⊕ (5.972 × 10²⁴ kg),
then according to Equation (2) a small test mass m placed near the Earth's surface,
whereby r ≈ R_⊕ (6.378 × 10⁶ m), will experience a downward gravitational acceleration
of a_g ≡ g = 9.81 m/s².
NEGATING NEWTONIAN GRAVITY
It is possible to design an antigravity machine that can nullify Earth's gravity field using
Newton's law of gravity. One way to use Equation (1) to nullify the Earth's gravitational
pull at a particular location would be to locate another planet of equal mass above that
location (Reference 2,3). The forces from the two Earth masses will cancel each other
out over a broad region between them. Everything within this broad region will be in
free fall. However, this is not a practical solution for aerospace flight since there is no
way to manipulate and control another planetary sized body.
Along similar lines, Forward (Reference 2,3) suggested to consider using a ball of
ultradense compact matter, corresponding to dwarf star or neutron star matter (~ 10¹¹
- 10¹⁸ kg/m³), having a diameter of 32 cm and a mass of 4 million metric tons. This
ultradense ball will have a surface gravitational (attractive) force of 1-g. This small
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ultradense ball could be placed near the surface of the Earth and its 1-g gravity field will
cancel the Earth's 1-g gravity field. All test objects placed in the broad region between
the small ultradense ball and the Earth will thus be in free fall. Another option Forward
(Reference 2-5) suggested would be to shape the compact ultradense matter into a disk
that is 45 cm in diameter and 10 cm thick, and having the same mass and density as
the small ultradense ball. Its gravitational acceleration is a_g = 4Gρτ, where ρ is the mass
density of the disk and τ is its thickness. In this case, the disk will have a force of
gravitational attraction that is the same on both sides, and it will be uniform near the
center of the disk where the strength of the gravitational force will be 1-g. If this disk
were to be placed very close above the Earth's surface, then there will be a
gravitational force of 2-g above the disk (= 1-g due to the Earth's gravity field plus 1-g
due to the top-side gravity field of the disk) while underneath the disk near its center
there will be a gravity-free (or free fall) region because the Earth's gravity field
underneath is canceled by the gravity field of the disk's bottom-side. While these are
interesting antigravity machines, they are unfortunately not feasible from an
engineering standpoint since one does not yet have the technology or means to create
and handle ultradense compact matter.
ENERGY ESTIMATE FOR NEWTONIAN LEVITATION
An ideal propulsion breakthrough could take the form of the antigravity-based levitation
of an aerospace vehicle within the Earth's atmosphere. Rockets like the Air Force DC-XA
can hover above the ground for a time that is limited by the amount of rocket fuel
available (Reference 6). But an ideal antigravity propulsion device should allow for the
indefinite levitation of a vehicle above the Earth's surface. It is illustrative to estimate
the energy required to levitate a 1-kg test mass above the Earth's surface. This will
help quantify a potentially key engineering parameter for such a levitation system. A
generic estimate can be found by considering the amount of energy per unit mass
required to nullify the (magnitude of) the Earth's gravitational potential energy E_lev for a
test mass m hovering at height h above the Earth's surface:
E_lev = GM_⊕m / h (J / kg) (3)
Equation (3) can also be derived by calculating how much energy is required to
completely remove a test mass from the Earth's surface to infinity. This calculation is
more in line with the analogy to nullify the effect of gravitational energy. And Equation
(3) also represents the energy required to stop a test mass at the levitation distance h
if it were falling in from infinity with zero initial velocity.
Setting h ≈ R_⊕ and m = 1 kg in Equation (3), the result is E_lev = 62.5 MJ/kg. This is 2.05
times the kinetic energy required to put the test mass into low Earth orbit (LEO).
However, this estimate will require some adjustment that depends upon the type of
theory and its technological implementation. That is because the operational energetics
of a putative antigravity propulsion system must be considered in conjunction with E_lev.
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III. Concepts for Antigravity Within General Relativity
In the Sections that follow the known types of antigravity that can be derived from
Einstein's General Theory of Relativity are described and summarized, which is the
modern relativistic theory of gravity.
ANTIGRAVITY VIA GRAVITOMAGNETIC FORCES
Historical Foundations
Heaviside (Reference 7) (in 1883), Einstein (prior to the 1916 publication of his General
Theory of Relativity), Thirring (Reference 8,9), and Thirring and Lense (Reference 10)
(see also, Reference 11) showed that general relativity theory provides a number of
ways to generate non-Newtonian gravitational forces via the splitting of gravitation into
electric and magnetic field type components. These forces can be used to counteract
the Earth's gravitational field, thus acting as a form of antigravity. General relativity
theory predicts that a moving source of mass-energy can create forces on a test body
which are similar to the usual centrifugal and Coriolis forces, although much smaller in
magnitude. These forces create accelerations on a test body that are independent of
the mass of the test body, and the forces are indistinguishable from the usual
Newtonian gravitational force. The Earth's gravitational field can be counteracted by
generating these forces in an upward direction at some spot on the Earth.
Forward (Reference 12) linearized Einstein's general relativistic field equation and
developed a set of dynamic gravitational field relations similar to Maxwell's
electromagnetic field relations. The resulting linearized gravitational field relations are a
version of Newton's law of gravitation that obeys special relativity. The linearized
gravitational field relations show that there is a unique correspondence between the
gravitational field and the electric field. For example, the Newtonian gravitational field
of an isolated mass is the gravitational analog to the electric field of an isolated electric
charge.
Likewise, there is an analogy to a magnetic field contained within the linearized
gravitational field relations. In Maxwellian electrodynamics, a magnetic field is due to
the flow of an electric charge or an electric current. In other words, the electric field
surrounding an electric charge in motion will appear as a magnetic field to stationary
observers. If the observers move along with the charge, they see no relative motion,
and so they will only observe the charge's electric field. Thus, the magnetic field is
simply an electric field that is looked at in a moving frame of reference. In an analogous
fashion, the linearized gravitational field relations show that if a (gravitational) mass is
set into motion and forms a mass current, then a new type of gravitational field is
created that has no source and no sink. This is called the Lense-Thirring effect, or
rotational frame dragging effect, in which rotating bodies literally drag spacetime
around themselves.
Forward's Dipole Gravitational Field Generator
Forward (Reference 13,14) used the linearized gravitational field relations plus aspects
of the Lense-Thirring effect to develop models for generating antigravity forces. One
example of an antigravity generator is based on a system of accelerated masses whose
mass flow can be approximated by the electrical current flow in a wire-wound torus.
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According to Maxwellian electrodynamics, an electric current flowing through a wire that
is wrapped around a torus (or ring) causes a magnetic field to form inside the torus. If
the current (I) in the wire increases with time, then the magnetic field B inside the
torus also increases with time. This time-varying magnetic field in turn creates a dipole
electric field E, as shown in Figure 1. The magnitude of the electric field at the center of
the torus is given by:
E = -μ₀Nr²İ / 4πR_t² (4)
where μ₀ is the vacuum electromagnetic permeability constant (4π × 10⁻⁷ H/m), N is the
total number of turns of wire wound around the torus, İ is the time rate-of-change of
the electric current flowing through the wire, r is the radius of one of the loops of wire,
and R_t is the radius of the torus.
[Figure 1. Dipole Electric Field Generator (Reference 14)]
In a similar fashion, Forward's antigravity device is a dipole gravitational field
generator. As shown in Figure 2, a mass flow T through a pipe wound around a torus
induces a Lense-Thirring field P to form inside the torus. If the mass flow is
accelerated, then the P-field increases with time, and thus a dipole gravitational field G
is created. The magnitude of the anti-gravitational field at the center of the torus is
given by:
G = η₀Nr²Ṫ / 4πR_t² (5)
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where η₀ is the vacuum "gravitational permeability" constant (≡ 16πG/c² = 3.73 × 10⁻²⁶
m/kg),¹ N is the total number of turns of pipe wound around the torus, Ṫ is the time
rate-of-change of the mass current flowing through the pipe, r is the radius of one of
the loops of pipe, R_t is the radius of the torus, and c is the speed of light (3 × 10⁸ m/s)
(Reference 14). One should note the striking similarity between Equations (4) and (5)
for the dipole electric and dipole gravitational fields.
[Figure 2. Diople Gravitational Field Generator (Reference 14)]
Using Equation (5), Forward (Reference 13,14) showed that there would be a need to
accelerate matter with the density of a dwarf star through pipes as wide as a football
field wound around a torus with kilometer dimensions in order to produce an antigravity
field (at the center of the torus) of G ≈ 10⁻¹⁰d_acc, where d_acc is the acceleration of the
(dwarf star density) matter through the pipes. The tiny factor 10⁻¹⁰ is composed of the
even smaller η₀, which is the reason why very large systems are required to obtain
even a measurable amount of acceleration. To counteract the Earth's gravitational field
of 1-g requires an antigravity field of 1-g (vectored upward), and thus the dwarf star
density material within the pipes must achieve a_acc = 10¹¹ m/s² in order to accomplish
this effect.
Forward (Reference 5) also identified a configuration comprised of a rotating torus of
dense matter that turns inside-out like a smoke ring as another type of dipole
gravitational field generator. As shown in Figure 3, an inside-out turning ring of very
dense mass (M) will create an upward force (of acceleration a) in the direction of the
(constant) mass motion (Mv, v is the mass velocity). This is also a feature of the
¹ The vacuum "gravitational permittivity" constant is (Reference 12): γ₀ = (4πG)⁻¹ = 1.19 × 10⁹ kg·s²/m³.
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Lense-Thirring effect. Forward's linearization analysis generalizes all of these effects
into the following two key ingredients that are required to produce antigravity forces:
1) any mass with a velocity and an acceleration exerts many different general
relativistic forces on a test mass, and 2) these forces act in the direction of the velocity
and in the direction of the acceleration of the originating mass. In summary, these
forces are equivalent to gravitational forces, which can be used to cancel the Earth's
gravitational field.
[Figure 3. Dipole Gravitational Field Generator: Inside-Out Whirling Dense Matter Torus (Reference 5)]
One can also view this genre of devices as a gravity catapult machine in which the
machine pushes a body away using its general relativistic antigravity forces to impart a
change in velocity. A space launch operator on the ground wanting to send a payload
up into orbit would just ratchet up the strength of the (upward-directed) antigravity
field to some value above 1-g, and after pressing the release button the payload
accelerates up and away into orbit. These devices could also be placed in Earth orbit,
stationed anywhere within the solar system, or even distributed throughout the galaxy
in order to establish a network of gravity catapults. Space travelers could begin their
trip by being launched from the catapult on the Earth's surface, and when they reach
space they would jump through various catapults as needed to reach their destination.
FELBER'S RELATIVISTIC ANTIGRAVITY EFFECT
Felber (Reference 15) used the Schwarzschild solution of Einstein's general relativistic
field equation to find the exact relativistic motion of a payload in the gravitational field
of a mass moving with constant velocity. His analysis gives a relativistically exact
(strong gravitational field condition) calculation showing that a mass, which radially
approaches or recedes from a payload at a relative velocity of v_crit > c/3¹/² (v_crit ≡ critical
velocity), will gravitationally repel the payload as seen by distant inertial observers. In
other words, any source mass, no matter how large or small it is or how far away it is
from a test body (payload), will produce an antigravity field when moving at any
constant velocity above v_crit.
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The exact relativistic strong-field condition that establishes the lower limit criterion for
v_crit to induce antigravity repulsion of a payload (as measured by distant inertial
observers in the rest frame of the source or in the initial rest frame of the payload) is
given by (Reference 15):
γ² > (3/2)ψ[1 - L²/GMr(ψ/3 - GM/rc²)] (6)
In this expression, γ ≡ (1 - β²)⁻¹/² is the standard relativistic Lorentz transformation factor
which is a function of the normalized relativistic velocity parameter β = v/c, ψ = ψ(r) = 1 -
(2GM/rc²) is the g₀₀ (or time-time) component of the static Schwarzschild spacetime
metric² of a source (or central) body of mass M, L is the constant specific angular
momentum of a ballistic payload of mass m, and r is the radial distance of the
approaching/receding payload from M. One can solve the inequality in Equation (6) for
β (or v) under the condition that a payload far from M, such that r >> b (b is the periapsis
distance of the payload from M) and r >> GM/c², and find that the payload will become
gravitationally repelled by M whenever γ² > 3/2 or β > 3⁻¹/². In order to derive an exact
solution, Felber considered the case for which M >> m so that the energy and momentum
delivered to the payload has a negligible back-reaction on the source body's motion.
And he found that a strong gravitational field is not required for antigravity propulsion
because a weak-field solution achieves the same results.
Felber discovered another interesting facet about this new relativistic antigravity effect.
He found that there is also an antigravity field that repels bodies in the backward
direction with a strength that is one-half of the strength of the antigravity field in the
forward direction. Thus a stationary body will repel a test body that is radially receding
from it at any v > v_crit. To delineate the propulsion benefit from this technique, Felber
determined that the maximum velocity (v_pmax-wf) that can be imparted to a payload
initially at rest by the weak (gravitational) field of a larger source mass moving toward
the payload at constant v > v_crit is v_pmax-wf << c[β - (3β)⁻¹]. For the strong-field case, the
maximum velocity (v_pmax-sf) that can be imparted to the payload (initially at rest) by the
larger source mass moving toward the payload at any constant v is v_pmax-sf = βc. Felber's
analysis includes examples where he uses black holes for the large source mass.
This form of antigravity propulsion is not too surprising because Misner et al.
(Reference 16), Ohanian and Ruffini (Reference 17), and Ciufolini and Wheeler
(Reference 18) report that general relativistic calculations show that the time-
independent Kerr (spinning black hole) gravitational field exhibits an inertial frame
dragging effect similar to gravitational repulsive forces in the direction of a moving
mass at relativistic velocities. This and Felber's exact solution are among the genre of
Lense-Thirring type effects that produce antigravity forces. It is interesting to note that
even though general relativity theory admits the generation of antigravity forces at
relativistic velocities (Reference 19), they have not been seen in laboratory experiments
² A spacetime metric (ds²) is a Lorentz-invariant distance function between any two points in spacetime that is
defined by ds² = g_μν dx^μ dx^ν, where g_μν is the metric tensor which is a 4×4 matrix that encodes the geometry of
spacetime and dx^μ is the infinitesimal coordinate separation between two points. The Greek indices (μ,ν = 0...3)
denote spacetime coordinates, x⁰...x³, such that x¹...x³ = space coordinates and x⁰ = time coordinate. The
Schwarzschild metric is: ds² = -(1 - 2GM/c²r)c²dt² + (1 - 2GM/c²r)⁻¹dr² + r²(dθ² + sin²θdφ²). The corresponding
metric tensor is a diagonal matrix: g_μν = diag[-(1 - 2GM/c²r), (1 - 2GM/c²r)⁻¹, r², r²sin²θ]. (r,θ,φ) are the usual
spherical polar coordinates in 3-dimensional space.
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because repulsive force terms are second and higher-order in the source mass velocity.
To invent a relativistic driver for a captured astronomical body in order to use it to
launch payloads into relativistic motion presents a large technical challenge to future
experimenters. For this reason, this paper will not consider this concept any further.
However, it does serve the useful purpose of illustrating the unusual antigravity forces
that can appear in Einstein's general relativity theory.
NEGATIVE ENERGY-INDUCED ANTIGRAVITY
Negative energy density and negative pressure are acceptable results both
mathematically and physically in general relativity and quantum field theories, and
negative energy/pressure manifests as gravitational repulsion (that is, antigravity).
Negative energy is also known as a form of "exotic matter."
In classical physics the energy density of all observed forms of matter (fields) is non-
negative. What is exotic about negative energy is that it must have negative energy
density and/or negative energy flux (Reference 20). The energy density is "negative" in
the sense that a given (exotic) matter field must have an energy density, ρ_E (= ρc², where
ρ is the rest-mass density), that is less than or equal to its pressures/tensions, ρ_i
(Reference 21,22).³ In many cases, these equations of state are also known to possess
an energy density that is algebraically negative; that is, the energy density and flux are
less than zero. It is on the basis of these conditions that this material property is called
"exotic." The condition for ordinary, classical (non-exotic) forms of matter that one is
familiar with in nature is that ρ_E > ρ_i and/or ρ_E ≥ 0. These conditions represent two
examples of what are variously called the "standard" energy conditions: Weak Energy
Condition (WEC: ρ_E ≥ 0, ρ_E + ρ_i ≥ 0), Null Energy Condition (NEC: ρ_E + ρ_i ≥ 0), Dominant
Energy Condition (DEC), and Strong Energy Condition (SEC). These energy conditions
forbid negative energy density between material objects to occur in nature, but they
are mere hypotheses. Hawking and Ellis (Reference 23) formulated the energy
conditions in order to establish a series of mathematical hypotheses governing the
behavior of collapsed-matter singularities in their study of cosmology and black hole
physics. More specifically, classical general relativity allows one to prove lots of general
theorems about the behavior of matter in gravitational fields.
The bad news is that real physical matter is not "reasonable" because the energy
conditions are in general violated by semiclassical quantum effects (occurring at order
ℏ) (Reference 22).⁴ More specifically, quantum effects generically violate the average
NEC (ANEC). Furthermore, it was discovered in 1965 that quantum field theory has the
remarkable property of allowing states of matter containing local regions of negative
energy density or negative fluxes (Reference 24). This violates the WEC, which
postulates that the local energy density is non-negative for all observers. "Negative
energy" has the unfortunate reputation of alarming physicists. This is unfounded since
all the energy condition hypotheses have been experimentally tested in the laboratory
and experimentally shown to be false – 25 years before their formulation (Reference
25).
³ Latin indices (e.g., i, j, k = 1...3) that are affixed to physical quantities denote the usual 3-dimensional space
coordinates, x¹...x³, indicating the spatial components of vector or tensor quantities.
⁴ Planck's reduced constant, ℏ = 1.055 × 10⁻³⁴ J·s.
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Further investigation into this technical issue showed that violations of the energy
conditions are widespread for all forms of both "reasonable" classical and quantum
matter (Reference 26-30). Furthermore, Visser (Reference 22) showed that all
(generic) spacetime geometries violate all the energy conditions. So the condition that
ρ_E > ρ_i and/or ρ_E ≥ 0 must be obeyed by all forms of matter in nature is spurious.
Violating the energy conditions commits no offense against nature. Negative energy has
been produced in the laboratory and this will be discussed in the following sections.
Examples of Exotic or "Negative" Energy Found in Nature
The exotic (energy condition-violating) fields that are known to occur in nature are:
• Static, radially-dependent electric or magnetic fields. These are borderline exotic, if
their tension were infinitesimally larger, for a given energy density (Reference
23,31).
• Squeezed quantum vacuum states: electromagnetic and other (non-Maxwellian)
quantum fields (Reference 21,32).
• Gravitationally squeezed vacuum electromagnetic (or other field) zero-point
fluctuations (Reference 33).
• Casimir effect; that is, the Casimir vacuum in flat, curved, and topological spaces
(Reference 34-40).
• Other quantum fields/states/effects. In general, the local energy density in quantum
field theory can be negative due to quantum coherence effects (Reference 24).
Other examples that have been studied are Dirac field states: the superposition of
two single particle electron states and the superposition of two multi-electron-
positron states (Reference 41,42). In the former (latter), the energy densities can
be negative when two single (multi-) particle states have the same number of
electrons (electrons and positrons) or when one state has one more electron
(electron-positron pair) than the other.
Cosmological inflation (Reference 22), cosmological particle production (Reference 22),
classical scalar fields (Reference 22), the conformal anomaly (Reference 22), and
gravitational vacuum polarization (Reference 26-29) are among many other examples
that also violate the energy conditions. Since the laws of quantum field theory place no
strong restrictions on negative energies and fluxes, then it might be possible to produce
exotic phenomena such as faster-than-light travel (Reference 43-45), traversable
wormholes (Reference 21,22,46), violations of the second law of thermodynamics
(Reference 47,48), and time machines (Reference 22,46,49). There are several other
exotic phenomena made possible by the effects of negative energy, but they lie outside
the scope of this report. See Appendix A for more technical details on items 1 through
4.
Toy Model Estimate for Negative Energy-Induced Antigravity
For the purpose of this report, the discussion will be confined to how negative energy
can be used to produce antigravity for the simplest case of counteracting the Earth's
gravitational field. To counteract or otherwise reduce gravity merely requires the
deployment of a thin spherical shell (bubble) of negative energy around an aerospace
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vehicle. This particular case study will serve as a useful illustrative comparison with the
Newtonian antigravity case discussed in Section II-A.
Interest is only in the slow (non-relativistic) motion, weak (gravity) field regime that
characterizes the physics of the Earth, Sun, other forms of solar system matter, most
interstellar matter (excluding compact dense stars and black holes), and small test
masses. In this case the time-time component of the Ricci curvature tensor (R_μν) is
given by R₀₀ ≈ Gρ/c² ≈ (7.41 × 10⁻²⁸)ρ m⁻². This is the primary quantity inside the
general relativistic field equation⁵ that encodes and measures the curvature of
spacetime around a source of matter and characterizes the weak or strong gravity field
regime for all forms of astronomical mass density (ρ). For example, the Earth's mass
density is 5,500 kg/m³ so R₀₀ ≈ 4.08 × 10⁻²⁴ m⁻², which indicates that an extremely flat
space surrounds the Earth and thus the system is within the weak field regime.
Gravitational physics in the weak field regime is completely described by the standard
Schwarzschild spacetime metric, which leads to the usual Newtonian and post-
Newtonian gravitational physics.
Two simple approaches can be used to determine the negative energy density required
to counteract the Earth's gravitational field: a) integrate the Einstein general relativistic
field equation, or b) use an already derived result from general relativity that gives the
repulsive force acceleration in terms of the spacetime metric components. For the first
case, the generalized gravitational Poisson equation from the Einstein field equation is:
-R₀₀√(-g₀₀) = (4πG/c⁴) Tr(T_μν)√(-g₀₀)
= (4πG/c⁴) T^μ_μ √(-g₀₀) (7)
= (4πG/c⁴) ρ*_E,
where the definition
ρ*_E ≡ T^μ_μ √(-g₀₀)
is used, ρ*_E ≡ rest-energy density + compressional potential energy (a.k.a. pressure),
g₀₀ ≡ g₀₀(r) is the time-time component of the metric tensor g_μν, and Tr(T_μν) ≡ T^μ_μ is the
trace (sum of diagonal matrix elements) of the stress-energy-momentum tensor T_μν (a
matrix quantity that encodes the density and flux of a matter source's energy and
momentum). Using tensor identities and grinding the algebra, Equation (7) can be re-
written as
∇²√(-g₀₀) = (4πG/c⁴) ρ*_E (8)
⁵ The Einstein field equation is: G_μν = R_μν - (1/2)g_μν R = -(8πG/c⁴)T_μν, where G_μν is the Einstein curvature tensor and
R = R^μ_μ (the matrix trace of R_μν) is the Ricci scalar curvature. In simplest terms, this relation states that gravity is a
manifestation of the spacetime curvature (G_μν) induced by a source of matter (T_μν).
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where ∇² is the standard Laplace differential operator. The left-hand-side of Equation
(8) is the gravitational potential. Integrating Equation (8) once over a region of space
exterior to a ball (or thin spherical shell) of rest-energy density to obtain
|∇√(-g₀₀(r))| = GM/r² = g (acceleration, m/s²) (9)
where the standard spherically symmetric spacetime (or Schwarzschild) coordinate
system (t,r,θ,φ) in which time t, radial space coordinate r, and angular space coordinates
(θ,φ) have their usual meaning is used.
The second approach (case b) can be derived by recalling that in the exterior
Schwarzschild spacetime around a central mass M (a ball or thin spherical shell) is
√(-g₀₀(r)) = 1 - GM/r (10)
Since the definition is given that g ≡ |∇√(-g₀₀(r))|, then perform the radial derivative of
Equation (10) and again arrive at Equation (9).
Since from special relativity M = E/c² (for a given rest-energy E), a negative energy
state is identical to a negative mass state (Reference 50). Thus the mass M in Equation
(9) can be replaced with the negative energy density -ρ_E* = -ρc² = -Mc²/V by using the
volume (V = 4πr²δr) of a thin spherical shell of radius r and thickness δr, and rearrange
quantities to solve for ρ_E* to get the final result:
ρ*_E = -gc² / 4πGδr
= -(1.05×10³⁷) / δr (J/m³), (11)
where g is now the acceleration due to gravity near the Earth's surface. If one desires
to use other geometries (for example, torus, cylinder, prism, cone, and pyramid)
instead of a thin spherical shell, then Equation (11) will admit minor numerical
adjustments to accommodate the relevant geometrical factors associated with different
geometrical volumes. Equation (11) gives the negative energy density required to
generate a repulsive gravitational force that counteracts the Earth's gravity field from
the surface all the way up to LEO (since g in LEO is only a few percent smaller than on
the surface). Any realistic value that one chooses for the bubble wall thickness δr will
give a negative energy density that will always be on the order of the equivalent
negative energy density of a dwarf star or neutron star. The technical challenge to
implement this kind of antigravity, however, is daunting.
In the next section the case of a cosmological antigravity that is generated by a form of
matter having a positive energy density and negative pressure is discussed.
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COSMOLOGICAL ANTIGRAVITY
It turns out that there is already a naturally occurring antigravity force that acts
throughout the universe. Actually, this force acts upon the entire spacetime structure of
the universe, and it is called cosmological inflation. Cosmological inflation causes the
universe to expand at an ever accelerating rate. In what follows, the nature of this
cosmological antigravity force and its potential aerospace propulsion application is
examined.
Pressure as a Source of Gravity
Newtonian gravitation is modified in the case of a relativistic perfect-fluid (where p <<
ρ_E cannot be assumed). The stress-energy tensor T^μν for this case is (Reference 16):
T^μν = (ρ_E + p)U^μU^ν - pg^μν (12)
where ρ is the fluid mass density, ρ_E ≡ ρc² is the fluid rest-energy density (or just
energy density), p is the fluid pressure, U^μ is the 4-velocity vector of the fluid, and g^μν is
the metric tensor. The Einstein general relativistic field equation using the identity g^μ_μ =
4 to obtain R = (8πG/c⁴)T, which is the Ricci curvature scalar can be contracted. And so
Equation (12) becomes T = ρ_E - 3p, which is just the trace of T^μν. Since T = ρ_E - 3p, a
modified Newtonian gravitational Poisson equation is produced:
∇²φ = 4πG(ρ_E + 3p) (13)
where φ is the gravitational potential. It should be noted that the energy density and
pressure are kept as separate terms as opposed to Equations (7) and (8) in the
previous section. Equation (13) means that a gas of particles all moving at the same
speed u has an effective gravitational mass density of ρ(1 + u²/c²). Thus, for example, a
radiation-dominated fluid generates a gravitational attraction twice as strong as one
predicted by Newtonian gravity theory according to Equation (13).
Vacuum Energy of Einstein's Cosmological Constant
A major consequence of the Einstein field equation is that pressure p becomes a source
of gravitational effects on an equal footing with the energy density ρ_E. One consequence
of the gravitational effects of pressure is that a negative-pressure equation of state that
achieves ρ_E + 3p < 0 in Equation (13) will produce gravitational repulsion (that is,
antigravity). The Einstein field equation that includes a cosmological constant Λ is:
G^μν + Λg^μν = -(8πG/c⁴) T^μν (14)
where G^μν is the Einstein curvature tensor. The Λ term, as it appears in Equation (14),
represents the curvature of empty space. Now if one moves this term over to the right-
hand-side of Equation (14), which has become widespread practice in modern times,
then
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G^μν = -(8πG/c⁴ T^μν + Λg^μν) (15)
whereby this term now behaves like the stress-energy tensor of the vacuum, T^μν_vac,
which acts as a gravitational source:
T^μν_vac = Λc⁴/8πG g^μν (16)
One should note that the absence of a preferred frame in special relativity means that
T^μν_vac must be the same (that is, isotropic or invariant) for all observers. There is only
one isotropic tensor of rank 2 that meets this requirement: η^μν (the Minkowski flat
spacetime metric tensor in locally inertial frames). So in order for T^μν_vac to remain
invariant under Lorentz transformations, the only requirement is that it must be
proportional to η^μν. But this generalizes in a straightforward way from inertial
coordinates to arbitrary coordinates by replacing η^μν with g^μν, thus justifying the curved
spacetime metric tensor in Equation (16). By comparing Equation (16) with the perfect-
fluid stress-energy tensor in Equation (12), one finds that the vacuum looks like a
perfect fluid with an isotropic pressure p_vac opposite in sign to the energy density ρ_vac.
Therefore, the vacuum must possess a negative-pressure equation of state (according
to the first law of thermodynamics):
p_vac = -ρ_vac (17)
The vacuum energy density should be constant throughout spacetime, since a gradient
would not be Lorentz invariant. So by substituting Equation (17) into ρ_E + 3p, the
following is produced
ρ_vac + 3p_vac = ρ_vac + 3(-ρ_vac)
= -2ρ_vac (18)
< 0.
The vacuum equation of state is therefore manifestly negative. Last, when incorporating
ρ_vac into the Einstein field equation as a gravitational source term, and comparing its
corresponding (Lorentz invariant) stress-energy tensor ρ_vac g^μν with Equation (16), then
the usual identification (or definition) is made that:
ρ_vac ≡ Λc⁴/8πG (19)
Thus the terms "cosmological constant" and "vacuum energy" are essentially
interchangeable in this perspective and mean the same thing (whereupon ρ_vac = ρ_Λ),
which is seen in the present-day cosmological literature.
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By substituting Equation (19) into Equation (17), one observes that a positive Λ will act
to cause a large-scale repulsion of space (because this gives a negative vacuum
pressure), whereas a negative Λ (giving a positive vacuum pressure) will cause a large-
scale contraction of space. Because Λ is a constant, the vacuum energy is a constant
(that is, time independent). This then implies a problem with energy conservation in an
expanding universe since one expects that energy density decreases as a given volume
of space increases, which is the case for the ordinary matter and cosmic microwave
background that is observed in extragalactic space. In other words, the matter and
radiation energy densities decay away as the universe expands while the vacuum
energy density remains constant.
The cure for this apparent energy conservation problem is the vacuum equation of state
given by Equation (17). A negative pressure is something like a tension in a rubber
band. It takes work to expand the volume rather than work to compress it. The proof of
this is as follows (Reference 51): the energy created in the vacuum by increasing
(expanding) space by a volume element dV is p_vac dV, which must be supplied by the
work done by the vacuum pressure -p_vac dV during the expansion of space, therefore p_vac
= -p_vac. In other words, the work done by the vacuum pressure maintains the constant
vacuum energy density as space expands. Therefore, the vacuum acts as a reservoir of
unlimited energy that provides as much energy as needed to inflate any region of space
to any given size at constant energy density.
Dark Energy
Dark energy is an easily misunderstood form of energy in cosmology. There are two
sets of evidence pointing toward the existence of something else beyond the radiation
and (ordinary and dark) matter itemized in the overall cosmic energy budget.⁶ The first
comes from a simple budgetary shortfall. The total energy density of the universe is
very close to critical. This is expected theoretically and it is observed in the anisotropy
pattern of the cosmic microwave background (CMB). Yet, the total matter density
inferred from observations is 26 percent of critical.⁷ The remaining 74 percent of the
energy density in the universe must be in some smooth, unclustered form that is
dubbed "dark energy." The second set of evidence is more direct. Given the energy
composition of the universe, one can compute a theoretical distance vs. redshift
diagram. This relation can then be tested observationally.
Riess et al. (Reference 52) and Perlmutter et al. (Reference 53) reported direct
evidence for dark energy from their supernovae observations. Their evidence is based
on the difference between the luminosity distance in a universe dominated by dark
matter and one dominated by dark energy. They showed that the luminosity distance is
larger for objects at high redshifts in a dark energy-dominated universe. Therefore,
objects of fixed intrinsic brightness will appear fainter if the universe is composed of
dark energy. The two groups measured the apparent magnitudes of a few dozen Type
Ia supernovae at redshifts z ≤ 0.9, which are known to be standard distance candles
(meaning they have nearly identical absolute magnitudes at any cosmological redshift-
⁶ Dark matter and dark energy are not to be confused. Dark matter is a non-luminous, non-absorbing, non-
baryonic form of matter that only interacts with all other forms of matter via gravitational and weak nuclear forces.
Dark matter has a positive rest-energy density and a nearly negligible positive pressure. Thus, it has no beneficial
application for breakthrough propulsion physics.
⁷ 26% total matter density = 4% ordinary (baryonic) matter + 22% dark matter.
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