AAWSAP DIRD Antigravity for Aerospace Applications March 30 2010 V2
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Defense
Intelligence
Reference
Document
Acquisition Threat Support
30 March 2010
!COD : 1 December 2009
DIA-08- 1003-018
Antigravity for Aerospace
Applications
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Antigravity for Aerospace Applications
Prepared by:
Acquisition Support Division (DW0- 3)
Defense Warning Office
Directorate for Analysis
Defense Intelligence Agency
Author:
AAP Person 58
Administrative Note
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This product is one in a series of advanced technology reports produced in FY 2009
under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace
Weapon System Applications (AAWSA) Program. Comments or questions pertaining to
this document should be addressed to lAAP Person 1 I AAWSA Prog ram
Manager, Defense Intelligence Agency, ATTN: CLAR/DWO-3, Bldg 6000, Washington,
DC 20340-5100.
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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 poss ibi lity of antigravity also appears in quantum gravity theories,
cosmological vacuum or dark energy, and quantum field theory. This report reviews all
of these top ics. 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 technologica l 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 {Fgrav) that mutually acts between two masses (Reference 1):
(1)
where the negative sign indicates t hat Fgrn" is a (mutual) force of attraction, G is
Newton's universal gravitation constant (6.673 x 10- 11 Nm 2/kg 2 ), m 1 and m 2 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) t o define the magnitude of the
gravitational acceleration a8 that acts on a small test mass m due to a larger
(gravitating) mass M (Reference 1):
GM
ag = - ? (2)
r-
If Earth is chosen to be the larger gravitating mass so that M = M@(5.972 x 10 24 kg),
then according to Equation (2) a small test mass m placed near the Earth's surface,
whereby r~ R @ (6.378 x 106 m), will experience a downward gravitational acceleration
of a g = g = 9.81 m/s 2 .
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 ( ~ 10 11
- 10 18 kg/m 3 ), 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 ba ll 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 ultra dense ball. Its gravitational acceleration is a g = 4Gp,, where p is the mass
density of the disk and t 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 th is 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
interesti ng 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
avai lable (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 E1ev for a
test mass m hovering at height It above the Earth's surface:
E
lcv
= GMffl
h
m (J / ko)
b
(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 ;::i Re and m = 1 kg in Equation (3), the result is E1ev = 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 t hat 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 E1cv ,
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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 t he 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 conta ined 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:
(4)
where µo is the vacuum electromagnetic permeability constant (4n x 10- 7 H/m), N is the
total number of turns of wire wound around the torus, i 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 Rt 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:
(5)
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where rio is the vacuum "gravitational permeability" constant(= 16nG/c2 = 3. 73 x 10- 26
m/kg), 1 N is the total number of turns of pipe wound around the torus, 'ft. 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, R1 is the radius of the torus, and c is the speed of light (3 x 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- 10a acc , where a acc is the acceleration of the
(dwarf star density) matter through the pipes. The tiny factor 10- 10 is composed of the
even smaller rio, 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 = 10 11 m/s 2 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, vis the mass velocity). This is also a feature of the
1 The vacuum "gravitational permittivity" constant is (Reference 12) : yo= (41tGJ-' = 1.19 x 109 kg -s2/m 3 •
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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 t he 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 cond ition) calculation showing that a mass, which radially
approaches or recedes from a payload at a relative velocity of Vcrir > c/3 112 ( vc,i1 = 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 Vent •
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The exact relativistic strong-field condition that establishes the lower limit criterion for
vcr;r 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):
(6)
In this expression, y = (I - p2)- 112 is the standard relativistic Lorentz transformation factor
which is a function of the normalized relativistic velocity parameter p = vie, 'I'= 'l'(r) = I -
(2GM/rc2) is the goo (or time-time) component of the static Schwarzschild spacetime
metric 2 of a source (or central) body of mass M, Lis 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
p (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 y2 > 3/2 or p > 3- 112 . 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 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 > vcri, , To delineate the propulsion benefit from this technique, Felber
determined that the maximum velocity ( vpmax-wf) that can be imparted to a payload
initially at rest by the weak (gravitationa l) field of a larger source mass moving toward
the payload at constant V > Vcrit is Vpmax-wf « c[P - (3Pt 1]. For the strong-field case, the
maximum velocity (vpma,-,r) that can be imparted to the payload (initially at rest) by the
larger source mass moving toward the payload at any constant v is vpma,-,r = pc. 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 ds 2 = g""dx 1'dx' , where g," is the metric tensor which is a 4 x4 matrix that encodes the geometry of
spacetime and dx" is the infinitesimal coordinate separation between two points. The Greek indices (µ,v = 0... 3)
denote spacetime coordinates, ><° ... x 3 , such that x 1... x 3 = space coordinates and ><°= time coordinate . The
Schwarzschild metric is: ds 2 = - (1 - 2GM/cir) c2dt 2 + (1 - 2GM/c2r )"1dr2 + r2 (d0 2 + sin 20dcp 2 ) . The corresponding
metric tensor is a diagonal matrix : g,,. = diag[- (1 - 2GM/c2r), (1 - 2GM/c2r)"1, r2, r2sin 2 0). (r, O,cp) 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 for 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 resu lts 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 flux (Reference 20). The energy density is "negative" in the
sense that a given (exotic) matter field must have an energy density, PE(= pc 2 , where
p is the rest-mass density), that is less than or equal to its pressures/tensions, p1
(Reference 21,22). 3 In many cases, these equations of state are also known to possess
an energy density that is algebra ically negative; that is, the energy density and flux are
less than zero. It is on t he 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 PE > pi and/or PE ~ 0. These conditions represent two
examples of what are variously ca lled the "standard" energy conditions: Weak Energy
Condition (WEC: PE ~ 0, PE +Pi~ 0), Null Energy Condition (NEC: PE+ p1 ~ 0), Dominant
Energy Cond iti on (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 thei r 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
11) (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
coord inates, x 1...x 3, indicati ng the spatial components of vector or tensor quantities.
4 Planck's reduced constant, 11 = 1.055 x 10-34 J.s.
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Further investigation into this technica l issue showed t hat violations of the energy
cond itions 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
PE > Pi and/or PE;::,: 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 (Rµv) is
given by Roo z Gp/c2 z (7.41 x 10- 28 )p m -2 . This is the primary quantity inside the
general relativistic field equation 5 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 (p). For example, the Earth's mass
density is 5,500 kg/m 3 so Roo z 4.08 x 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, orb) 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:
(7)
where the definition
is used, PE*= rest-energy density+ compressional potential energy (a.k.a. pressure),
goo= goo(r) is the time-time component of the metric tensor 8r,v, and Tr(Tµv) = P\, is the
trace (sum of diagonal matrix elements) of the stress-energy-momentum tensor Tµv (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
v' 2 ,--::- _ 4rcG •
\I-goo - - 4 - P e (8)
C
5 The Einstein field equation is: G". = R". - (1/2)g". R = -(87[G/c:')°fi"'' where G". is the Einstein curvature tensor and
R = R"" (the matrix trace of R". ) is the Ricci scalar curvature. In simplest terms, this relation states that gravity is a
manifestation of the spacetime curvature (G".) induced by a source of matter (T". ).
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where V2 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
lv.J-g00 (r)I = G~ = g (acceleration, m/s 2 ) (9)
r
where the standard spherically symmetric spacetime (or Schwarzschild) coordinate
system (t,r,0,<p) in which time t, radial space coordinate r, and angular space coordinates
(0,<p) 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
,---- GM
✓-g oo (r) = 1 - - (10)
r
Since the definition is given that g = Iv
.J-g00 (r)I, 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 Min Equation
(9) can be replaced with the negative energy density - pE* = - p c2 = - Mc 2/V by using the
volume (V = 4nr28r) of a thin spherical shell of radius rand thickness 8r, and rearrange
quantities to solve for PE* to get the final result:
2
• - gc
PE=
41tG8r
(11)
-(1.05 X 10 27 )
=
or
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 8rwill
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 <<
PE cannot be assumed). The stress-energy tensor p v for this case is (Reference 16) :
(12)
where p is the fluid mass density, PE = pc2 is the fluid rest-energy density (or just
energy density), p is t he fluid pressure, u µ is the 4-velocity vector of the fluid, and gµ v is
the metric tensor. The Einstein general relativistic field equation using the identity gµµ =
4 to obtain R = (8nG!c4)T, which is the Ricci curvature scalar can be contracted. And so
Equation (12) becomes T = P E - 3p, which is just the trace of p•v. Since T = P E - 3p, a
modified Newtonian gravitational Poisson equation is produced:
(13)
where ~ 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 p(l + ii2/c2 ). Thus, for example, a
radiation-dominated fluid generates a gravitational attraction tw ice as strong as one
predicted by Newtonian gravity t heory according to Equation (13).
Vacuum Energy of Einstein's Cosmological Constant
A major consequence of the Einstein field equation is t hat pressure p becomes a source
of gravitational effects on an equal footing with the energy density P E• One consequence
of the gravitational effects of pressure is that a negative -pressure equation of state that
achieves PE + 3p < O in Equation (13) will produce gravitational repulsion (that is,
antigravity). The Einstein field equation that includes a cosmological constant A is:
(14)
where Gµv is the Einstein curvature tensor. The A 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 t imes,
then
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(15)
whereby this term now behaves like the stress-energy tensor of the vacuum, Tv~; ,
which acts as a gravitational source:
T r,v - Ac4 r,v (16)
vac - 87tG g
One should note that the absence of a preferred frame in special relativity means that
r;.; 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: riµv (the Minkowski flat
spacetime metric tensor in locally inertial frames). So in order for r;.; to remain
invariant under Lorentz transformations, the only requirement is that it must be
proportional to ri µv . But this generalizes in a straightforward way from inertial
coordinates to arbitrary coordinates by replacing ri µv with gµv , 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 P v•c •
Therefore, the vacuum must possess a negative-pressure equation of state (according
to the first law of thermodynamics):
P vac = -Pvac (17)
The vacuum energy density should be constant throughout spacetime, since a gradient
would not be Lorentz invariant. So by substituting Equation ( 17) into PE + 3p, the
following is produced
P vac + 3 P vac = P vac + 3 ( - P vac )
= -2Pvac (18)
< 0.
The vacuum equation of state is therefore manifestly negative. Last, when incorporating
pvac into the Einstein field equation as a gravitational source term, and comparing its
corresponding (Lorentz invariant) stress-energy tensor P vacg•" ' with Equation (16), then
the usual identification (or definition) is made that:
A c4
P vac = S1tG (19)
Thus the terms "cosmological constant" and "vacuum energy" are essentially
interchangeable in this perspective and mean the same th ing (whereupon P vac = p.,. ),
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 A will act
to cause a large-scale repulsion of space (because this gives a negative vacuum
pressure), whereas a negative A (giving a positive vacuum pressure) will cause a large
scale contraction of space. Because A is a constant, the vacuum energy is a constant
(that is, time independent) . Th is then implies a problem with energy conservation in an
expand ing 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 t hi s apparent energy conservation problem is the vacuum equation of state
given by Equation (17) . A negative pressure is something like a tensio n in a rubber
band. It takes work to expand the volume rather than work to compress it. The proof of
this is as fo llows (Reference 51): the energy created in the vacuum by increasing
(expand ing) space by a volume element dV is PvacdV, wh ich must be supplied by t he
work done by the vacuum pressure -pvac dV dur ing the expansion of space, t herefore /Jvac
= - p vac , In other words, t he work done by t he vacuum pressure maintains t he constant
vacuum energy density as space expands. Therefore, t he vacuum act s as a reservoir of
unlimited energy that provides as much energy as needed to inflat e any region of space
to any given size at const ant energy densit y.
Dark Energy
Dark energy is an easily misunderstood fo rm of energy in cosmology. There are two
sets of evidence pointing toward the existence of something else beyond the ra diati on
and ( ordinary and dar k) matt er itemized in t he overall cosmic energy budget. 6 The first
comes from a simple budgetary shortfall. The t ot al energy density of t he universe is
very close to critica l. This is expected theoretica lly and it is observed in the anisotropy
patt ern of the cosmic microwave backgroun d (CMB). Yet , t he tota l matter density
inferred from observations is 26 percent of crit ica l. 7 The remaining 74 percent of t he
energy densit y in the un iverse must be in some smooth, uncl ust ered form that is
dubbed " dark energy ." The second set of evidence is more direct. Given the ene rgy
composition of t he universe, one can compute a t heoretica l distance vs. redshift
diagram . This re lation can t hen 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 t hey have nearly identical absolute magn itudes at any cosmological redsh ift-
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 v ia gravitational and weak nuclear forces.
Dark matter has a positive rest-energy density and a nearly negl igible positive pressu re. Thus, it has no beneficial
appl ication for breakthrough propulsion physics.
7 26% total matter density = 4% ordinary (baryonic) matter + 22% dark matter.
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distance). 8 The supernovae data strongly disfavored (with high confidence) the flat
matter-dominated (nm = 1, n /\ = 0) universe and the pure open universe (nm = 0.3, n /\
= 0) models. 9 After this discovery, a lot of attention was paid to choosing an
appropriate name for this new energy. "Quintessence" was one good choice because it
expresses the fact that, after cosmological photons, baryons, neutrinos, and dark
matter, there is a fifth essence in the universe. More recently, "dark energy" is used
more often, with quintessence referring to the subset of models in which the energy
density can be associated with a time-dependent scalar field or a time-dependent
cosmological vacuum energy.
In analyzing the cosmological modeling results suggested by the Type Ia supernovae
data, it becomes apparent that the only form of dark energy budgeted for in the models
is the cosmological constant. To consider other possibilities one evaluates the time
evolution of the general relativistic conservation law for energy, VJ:= V,l ; = O, where
v = 0 to signify time evolution and V r, is the covariant derivative (or spacetime
curvature gradient), in an expanding universe as applied to the cosmological constant
(Reference 16):
8PE +~3PE + 3p] = O (20)
at a
where a is the scale factor of the universe and t& is the time derivative of a. Equation
(20) is derived using Equation (12) in the case of a perfect isotropic fluid where there is
no gravity and velocities are negligible such that u µ = (1, 0, 0, 0), and the energy
density and pressure evolve according to the continuity and Euler equations. The only
way Equation (20) can be satisfied with constant energy density is if the pressure is
defined by Equation (17). One might imagine energy with a slightly different pressure
and therefore energy evolution. Define the equation of state w:
(21)
A cosmological constant corresponds to w"' = wvac = -1, matter (ordinary and dark) to
w mauer ::::: 0, and radiation to W ract = 1/3. 10 The earlier Riess and Perlmutter supernovae
data (fixing the universe to be flat) showed that values of Wcte > - 0.52 for dark energy
are strongly disfavored. In fact, Riess and a team of collaborators (a.k.a. the "Higher-Z
team") recently published new observational data and analysis that includes a much
larger survey of Type Ia supernovae that are at much higher cosmological redshift
(Reference 54). The measured spectra of ancient (z ~ 1, or up to 10 billion light-years
distance or a look-back time of up to 10 billion years ago) and recent (z ~ 0.1, or ~ 1
billion light-years distance or a look-back time of ~ 1 billion years ago) were compared
and showed that there was no evolutionary change in the physics that drives Type Ia
supernovae explosions and their subsequent spectral luminosity output. This establishes
8 In cosmology, the redshi~ z serves as a surrogate for distance (in light-years) or look-back time.
9 Om = ratio of energy density contained in matter (as measured today) to the critical energy density; n A = Ovac =
ratio of energy density in a cosmological constant to the critical energy density; Per= 3Hl /8rcG is the critical energy
density, where Ho is the present-day Hubble expansion rate .
10 Non-relativistic (ordinary and dark) matter has a very tiny positive pressure, p oc Temp/m (Temp is absolute
temperature, m is mass), while a relativistic gas (of radiation) hasp = pE/3 > 0.
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the efficacy of using Type Ia supernovae as a standard distance candle for cosmological
dark energy surveys. The Higher-Z team's results also concluded, with 98 percent
confidence, that wde = - 1.0, and that th is is a perpetual constant (over at least 10 billion
years time) (Reference 54). This result falsifies all quintessence models for cosmology.
Therefore, a cosmological constant is consistent with the dark energy data to a high
degree of precision and statistical confidence whereby one can now state that dark
energy is the vacuum energy of Einstein's cosmological constant because wdc = wA = -1
(Reference 55,56).
Equation (20) can be integrated to find the evolution of the dark energy density, Pde =
PA,as a function of the cosmological scale factor a :
(22)
where a' is the dummy integration variab le for the scale factor. Since wde = - 1 (= wA ) is
a constant in Equation (22), then Pc,•. oc aexp[-3(1 + ~ 1,)] or Pde = P A oc a0 . This is exactly
what is expected on the basis of previous analysis in Section III-D-2. For a comparison
with this result, one should note that pc2 oc a- 3 for (ordinary and dark) matter and P rn<1 oc
a-4 for radiation such that pc2 ➔ 0 and p,,d ➔ 0 as a ➔ oo while Pde = PA remains constant.
Antigravity Propulsion Application of Dark/Vacuum Energy
If one could somehow harness a local amount of dark/vacuum energy, then use can be
made of its negative pressure property to produce an antigravity propulsion effect? To
answer this question one can use the estimated value for Pde = P A:=:; 2.4poc2 ~ 10-9 J/m 3 ,
where po is the present-day value of the total cosmologica l mass density of (ordinary
and dark) matter (Reference 54,57). Using this number one can work through the math
and estimate that the total amount of dark/vacuum energy contained within our solar
system amounts to the mass equivalent of a small asteroid. This means that its
repulsive gravitational influence upon planetary orbital dynamics inside the solar
system is completely inconsequential. Only on the extragalactic-to-cosmological scale
will its repulsive gravitational property achieve strong enough influence over matter and
spacetime. On this basis, one can conclude that it is highly unlikely, if not impossible,
that one will be able to invent a technology in the near future that can acquire and
exploit a near-cosmological amount of dark/vacuum energy to implement a useful
antigravity propulsion system.
IV. Quantum Antigravity Propulsion Concepts
Quantum antigravity can be found within the very large genre of quantum gravity
theories in which repulsive gravity terms appear as quantum corrections to the classical
Newtonian gravitational force law. Generally, one can derive such correction terms by
quantizing the Einstein general relativistic field equation or by starting with a particular
type of quantum field theory (for example, supersymmetric field theory, quantized 5-
dimensional Kaluza-Klein unified field theories, quantum superstrings/D-Brane theory,
quantum loops or knots, and Yang-Mills theories) and work backwards to find the
corresponding gravity theory. The particular mathematical form and quantitative
magnitude that quantum correction terms can have totally depends upon the
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quantization procedure and order of approximation used in a given quantum gravity
theory. However, the linearized semi-classical quantum gravity theory is related to
Einstein's classical nonlinear General Relativity Theory whereby the former uniquely
implies the latter provided that the graviton, which exchanges the gravitational force
between two massive particles or photons, is a pure spin-2 particle. In this theory, the
stress-energy tensor of the source matter fields is quantized while gravitation (via the
Einstein curvature tensor) is still treated classically. Semi-classical quantum gravity is a
quantum field theory in curved spacetime that has been successful in reproducing a few
of the predictions and many of the foundational precepts of General Relativity Theory.
A particular example of what a quantum antigravity correction term looks like was
derived in 1984 by R. L. Forward and the author, with instruction provided by R. P.
Feynman and M. Scadron, during a summer quantum gravity seminar sponsored by the
Hughes Research Labs in Malibu, CA. One began by studying the Feynman quantization
procedure for the case of single-photon exchange between two charged particles, which
tells us about the underlying nature and quantum corrections to the static Coulomb
force. From this study discovered that the same is also true for the case of single
graviton exchange between two massive spin-0 particles in connection with the static
Newtonian force. By applying Feynman's quantization procedure (Reference 58-60) to
the linearized Einstein field equation in the nonrelativistic limit, the following static
graviton -exchange potential, V gr.,v (r), for two spin - 0 particles undergoing a gravitational
interaction can be derived:
(23)
where m1 and m 2 are the masses of the interacting particles, r is their radial separation,
and 83(r) is the 3-dimensional Dirac 8-function with r the position vector of some
reference point in space. The first term in Equation (23) is immediately recognized as
the attractive Newtonian gravitational potential while the second quantum correction
term is repulsive. Also, the second term is independent of the interacting particle
masses and can only be measured for bound quantum s-states because the product of
the coefficient 4n(Gli2/c2 ) ~ 10-94 with the 8-function gives only a minute physical effect
at the atomic scale. The second term happens to be analogous to the usual quantum
correction to the Coulomb or nuclear force. If the two particles were to have non-zero
quantum spin, then Vgrav(r) will be modified by additional spin-orbit and spin-spin
correction terms. Furthermore, there are additional velocity-dependent corrections to
Vgrav(r) that generate the general relativistic post-Newtonian modifications of the
classical equation of motion of a particle in a gravitational field.
But the most important characteristic to observe about the quantum antigravity
correction term in Equation (23) is that its magnitude is incredibly minute, only
affecting bound quantum s-states. In general, quantum gravity correction terms at any
level of approximation, whether gravitationally repulsive or attractive, will have
coefficients ~ G(li0/cK) (for 8, K > 1), and therefore will not have a measurable impact on
any macroscopic system that embodies any form of propulsion. Because these quantum
corrections are so minute, and because there is no single universally accepted quantum
gravity theory to work with, investigators have had little reason to look into the
potential application of quantum gravity correction terms to antigravity propulsion
physics.
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However, this isn't the entire story because there are many interesting quantum field
theoretic phenomenon that exist outside of that which arises in quantum corrections to
Newtonian gravity. In what follows, the recent discovery of antigravity forces that arise
within both QED vacuum fluctuation and nonretarded quantum interatomic dispersion
force theories in curved spacetime are reviewed.
ANTIGRAVITY VIA QUANTUM VACUUM ZERO-POINT FLUCTUATION
FORCE
Calloni et al. (Reference 61,62) explored the possibility of verifying the equivalence
principle for the zero-point energy of quantum electrodynamics (QED). They used semi
classical quantum gravity theory to evaluate the net force produced by quantum
vacuum zero-point fluctuations (ZPF) acting on a rigid Casimir cavity in a weak
gravitational field. Their analysis assumed the rigid Casimir cavity to be a non-isolated
system at rest in the Earth's gravitational field, which is modeled using the standard
Schwarzschild spacetime metric geometry, so that they could evaluate the regularized
(or renormalized) stress-energy tensor, ('r,,:: I..,.,
11 of the quantized vacuum
electromagnetic field between two plane-parallel ideal metallic plates lying in a
horizontal plane. (r.:::J .. encodes the Casimir Effect which has a negative energy
density and a negative pressure along the vertical (acceleration) axis between the
plates. {See Appendix A for more information about the Casimir Effect.) Their results
agreed with the equivalence principle because they showed that quantum vacuum ZPF
(that is, virtual quanta) do gravitate because the energy of each ZPF mode is redshifted
by the factor (-g 00 )"' =[1-(2cM tc'r)J'" even though the modes remain unchanged. In
other words, the electromagnetic vacuum state in a weak gravitational field is
redshifted. This effect remains true for strong gravitational fields.
The resulting antigravity force (Fc asGrav ) derived by Calloni et al. is (Reference 62):
F. _ 11:2A hg
CasGrav - 180cd3
(24)
:::; ( 1.89x 10-43) ; I
in Newtons (N), where A is the area of the plates and dis their separation. Equation
(24) states that a Casimir device in a weak gravitational field will experience a tiny
push in the upwards direction (that is, the opposite direction with respect to the Earth's
gravitational acceleration). This is consistent with the interpretation that the negative
Casimir energy in a gravitational field behaves like a negative mass (Reference 63) .
F ca,arnv is actually the sum of two separate force terms: the first term arises from the
Casimir energy encoded in (r.,: ),,. which is interpreted as the Newtonian repulsive force
on an object with negative energy, and the second term arises from the pressure along
the vertical (acceleration) axis which is interpreted as the mass contribution of the
spatial part of the stress-energy tensor. To evaluate F casGrav for the case of any
gravitating body of interest, one must replace gin Equation (24) with Equation (2).
11The angular brackets denote the quantum (vacuum state) expectation value of the stress-energy tensor P ". Also
note that stress-energy is synonymous with energy-momentum.
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Calloni et al. further point out that a real Casimir cavity is an isolated system in wh ich
the actual (total) resulting force is the Newtonian force on the sum of the rest-Casimir
energy and rest-mechanical mass whereby the contribution of the vacuum ZPF leads to
a gravitational repulsion (FcGexp ) on the Casimir device that is given by (Reference 62):
1
FCGexp =
4 FcasGrav
(25)
which is the force that should be experimentally tested. Equation (25) takes into
consideration that the contribution to the total force on a real cavity resulting from the
spatial part of the stress-energy tensor is balanced by the contribution from the
mechanical stress-energy tensor. Given that the typical dimensions of a Casimir device
are very small, it appears that FcGexp will be very difficult, if not impossible, to measure
using present-day lab technology.
However, Calloni et al. propose an experimental device that could significantly magnify
the repulsive force up to a measurable scale. Their proposed device is a multilayered
series of rigid Casimir cavities with each cavity consisting of two thin metallic disks that
are separated by a dielectric material which is inserted to maintain rigidity. They
suggest SiO2 for the dielectric material because it is an efficient dielectric with low
absorption over a wide range of frequencies, and it is an inexpensive material that is
easy to fabricate into layers. The introduction of the dielectric material is equivalent to
enlarging the optical path length by the refractive index n so that the cavity plate
separation d and. The Casimir Effect has been tested down to plate separations ~ 60
nm while separations~ 10 nm is possible with present technology. But at :::; 10 nm
distances dielectric absorption and finite conductivity are expected to decrease the
effective Casimir pressure compared to a cavity comprised of perfect mirrors. For
example, a plate separation of 6.5 nm corresponds to a decreasing factor (½) of 0.07 for
plates made of aluminum. Finite temperature and plate surface roughness could also
introduce additional corrections to the Casimir pressure. Calloni et al. propose to
magnify the total force by using Ne = 10 6 layers of rigid cavities with each cavity having
a diameter of 35 cm and thickness of 100 nm, for a total device thickness of 10 cm.
All these engineering factors taken together led Calloni et al. to recast FcGexp into the
following new form (Reference 62):
1t 2 Ahg
F r::;rN---
cc.e,r "' 1 720 c( nd)3
(26)
r:,;(4.73x10-44 )sN, ~
(nd)
in Newtons. Calloni et al. also suggest that a feasible experiment will require
modulating F cGexp in order to obtain a measurable force. They are investigating the
possibil ity of modulating ½by varying the temperature in order to induce a periodic
transition from conducting state to superconducting state. They estimate that doing this
could achieve ½max z 0.5, and thus produce a force F cGexp ~ 10- 14 N at a modulation
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frequency on the order of tens of mHz ford = 5 nm and n = 1.46 (for SiO2 dielectric).
This result is more than two orders of magnitude larger than the force which the VIRGO
gravitational wave antenna is expected to detect at severa l tens of Hz. If one could
fabricate a device consisting of 109 layers, then F coexp 10- 11 N. This suggests that
IV
cavities made from thin-film deposited surfaces or photonic band-gap materials would
be the best approach for fabricating a multilayer Casimir device.
Bimonte et al. (Reference 63-65) also derived Equation (24) for th is very same problem
by using Green -function techniques in the Schwinger-DeWitt quantum ether
prescription for (T..:
~),., in a curved spacetime. They also computed the weak
gravitational field-induced correction terms for the Casimir pressure on the plates,
(r:: ).., , and t he total energy ( EcasGrav) stored in the Casimir device which is given by
(Reference 63,64):
2
_ _ 1t Ahc ( l+~ gd)
G:asGrav - 720 d3 2 C2 (27)
in Joules (J). The correction terms for the different (measurable) physical quantities of
interest are generally g!c?- .
IV
Finally, Calloni et al. point out that the overriding concern with performing an
experiment to test Fcocxr is whether cavities can be made sufficiently rigid, if the effect
of surface roughness and defects can be quantified to improve the force estimate, and if
the necessary signal modulation can be achieved in the lab. However, micro- and nano
manufacturing is maturing to the point where rigidity, surface roughness, and close
plate separations are becoming routinely controllable. While the numerical estimate for
F cGexp is quite feeble, it is still significant since it is at the very low end of the
macroscopic scale, and it might be possible to devise advanced methods to magnify the
force to a magnitude that benefits a propulsion application. However, the upward force
will have to be larger than the weight of the propulsion system in order to achieve
levitation. This could be very difficult to do, but th is is a concept that is ripe for further
exploration .
ANTIGRAVITY VIA NONRETARDED QUANTUM INTERATOMIC
DISPERSION FORCE
Pinto (Reference 66) evaluated the net lifting force produced by nonretarded
electrostatic dipole-dipole interactions (that is, nonretarded van der Waals dispersion
forces) acting on a quantum system of polarizable particles in a curved spacetime. The
foundation of Pinto's study was the original discovery made by Fermi (Reference 67)
that classical electrostatic theory must be reformulated in a curved spacetime in order
to properly evaluate the effects of gravitation upon the Coulomb electric field of a single
charged particle. In this case, the Laplace equation of electrostatics for a single charged
particle can be generalized in the presence of a gravitational field and then extended to
show that a system of classical charged particles undergoes a gravity-induced self
lifting force. Fermi and other investigators arrived at this counterintuitive result by
computing the gravity-induced self-force acting on an isolated electric dipole in a weak
gravitational field and showing that the self-force (times dipole size) is exactly equal to
the gravitational equivalent of the electrostatic internal energy of the dipole.
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The net gravity-induced (electrostatic levitation) self-force (Fo;pGrav ) is given by
(Reference 66):
(28)
where qe is the electric charge on a particle and r is the radial distance between two
charged particles in the dipole. There is an additional term of order g2/c4 in Fo;porav that is
neglected because it is negligible in magnitude. Equation (28) states that an electric
dipole will experience a push in the upwards direction (opposite direction with respect
to the Earth's gravitational acceleration); that is, the dipole undergoes self-acceleration
in which one charged particle in the dipole appears to be chasing the other charged
particle. As an example, for a dipole comprised of two charges (for example, an
electron-proton system) held at fixed r to levitate in the Earth's gravitational field, r
would have to be ~ 10- 15 m (the size of an atomic nucleus). An experiment to test this
prediction on such a small scale is too difficult to control or measure.
An energy analysis done by Pinto showed that there is a distance r between two
charges (each of rest-mass mo) in a dipole (of mass Mdip = 2mo) such that their
electrostatic potential energy, U d;p = - qe2/4rcEor, becomes equal to the unrenormalized
mass of the system as r ➔ oo . At this distance, the effective total gravitational mass Mct;p
+ Ud;p/c2 = O and the self-force alone can support the dipole at rest against its own weight.
The self-acceleration of the dipole is such that the acceleration process can continue
indefinitely, which poses a problem for energy conservation because the dipole can be
left to self-accelerate for an arbitrary period of time and then stopped to harness the
resulting kinetic energy. This process could be used to extract unlimited energy from
the system. Pinto claims that there is no conflict with energy conservation because the
renormalized inertial mass of the accelerating system is Mdip-ren =M dir + U ct;p/c2 = 0 and the
total energy of the system is zero at all times regardless of speed. This claim requires
reevaluation because there are subtle boundary conditions involved that might have
been overlooked in the analysis.
Fermi's discovery led to a new subfield of research devoted to the study of
electrodynamics and dipole and interatomic dispersion forces in a curved spacetime.
Pinto's theoretical program extended the result of these studies by considering a
system of polarizable atoms and adopting an approach in which the effect of a
gravitational field in general relativity is modeled as an effective optical medium. In
other words, the spacetime vacuum is treated as a non-uniform optical medium with a
varying index of refraction that defines the components of a flat spacetime metric
geometry (Reference 68). There is no spacetime curvature due to sources of matter in
this model, instead its equivalent general relativistic effects (that is, gravitation) are
produced by varying the vacuum index of refraction, comprised of the vacuum
electromagnetic permittivity and permeability constants, in response to the presence of
matter sources. Pinto's lengthy analysis gives the van der Waals dispersion self-force
for two polarizable atoms in a curved spacetime (that is, a weak gravitational field) as
(Reference 66):
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6q; ai )( cg)
= - ( - 4nc:0 r 6 2
(29)
2
~ ( 2.44 X 10-57 ) q:
r
in Newtons, where ao is the Bohr radius (5.292 x 10- 11 m), r is the radial distance
between two atoms, and u:;,~
is the flat spacetime van der Waa ls (interatom ic potential)
interaction energy to second-order in quantum perturbation theory. Pinto used Equation
(29) to estimate the gravity-induced self-acceleration ( a1;fl ) for the case of two hydrogen
atoms in their ground state at r = 20ao, and found that a lir1. H = F vdworavl2mtt z 4 x 10- 15 m/s2
( mH = mass of hydrogen atom). For the case of two positronium (Ps) atoms, he found
that GJift,Ps z 8 X 10- 12 m/s2 .
Pinto's strategy is to dramatically magnify F vdWGrav to a large enough magnitude that it
becomes viable for propulsion applications. He cla ims that this can be done by
manipulating u:;:v , which depends on the atomic polarizability and is strongly affected
by the quantum state in which the atoms are prepared. Interatomic forces can also be
manipulated by means of external electromagnetic fields that can transform van der
Waals forces into a first-order interaction. He evaluated a number of schemes and
settled on the following techniques for manipulating dispersion forces: 1) excitation of
polarizable atoms to Rydberg states in external time-dependent electric fields, 2)
polarizability resonant enhancement by laser radiation, and 3) laser-induced near-zone
orientational average of the dispersion force. Also, in order to generate a macroscopic
self-lifting force, it wi ll be necessary to apply these techniques to a cluster of trapped
atoms because the total self-lifting force acting on the center-of-mass of a trapped gas
composed of N. identical polarizable atoms is N.2 times the self-lifting force acting on a
single pa ir of interacting atomic dipoles. Item 1 has a two-part contribution to the
magnification of the self-l ifting force: 1) one part from a2(ro)E2 due to the effect of
external time-dependent electric fields on atomic polarization, where a(ro) is the atomic
polarizability as a function of the electric field frequency ro and Eis the electric field
intensity; 2) another part from using highly-excited Rydberg atoms (with principal
quantum number nr >> 1 and Bohr radius an= np2ao ) whose polarizability scales as n p7 ,
Item 2 leads to a magnification by factors of a(ro)/ao z 10 3 - 10 5 (ao is the static value of
the polarizability) via detuning of the (laser) excitation radiation frequency from the
nearest atomic transition resonance of the atoms in the trapped cluster. Item 3 leads to
a further magnification due to the effect of the incident laser radiation on the dispersion
force being averaged over all directions, which changes the interatomic potentia l (cx:
l/r6) into a gravity-like 1/r potential.
Pinto's study suggests that the combined effect of items 1 - 3 will magnify the self
lifting force to the point where a cluster of trapped atoms will not only hover
unsupported in the Earth's gravitational field, but will also generate an additional
upward thrust. On the basis of extensive theoretical and empirical studies, along with
the typical parameters for laboratory laser and optical atomic matter trap technologies,
he estimates that a1ir1 ~ 1.5-g (in the upward direction). Trapped atom gravimeters can
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be used to observe this effect in the lab. Pinto also points out that other polarizable
systems such as nanoparticles, microspheres, and quantum dots can be used in place
of atoms. The trapping of latex spheres into a form of optical matter by means of
intense laser radiation has already been demonstrated in the lab. In addition, an
analogy to the item 1 - 3 manipulations that produce dramatically enlarged
polarizabilities in trapped interacting nanoparticles and microspheres have also been
demonstrated in the lab.
Pinto proposes a levitation propulsion thruster in which the combined system of trapped
interacting polarizable particles and external confining fields forms a single thruster
element comprising a fraction of the mass of the entire vehicle. The reaction of the self
lifting force exerted by this element aga inst the external confining fields results in the
transfer of force (thrust) to the entire vehicle. In order to achieve levitation, this
requires that the upward thrust per polarizable particle be larger than its own weight if
the fraction of the thrusting mass is smaller than the mass of the rest of the vehicle.
The propulsive levitation cond ition is expressed as (Reference 66): F1hrus1 = (Mveh + mAN)g or
F1hrus1lmANg ~ 1, where F1hrus1 is the total gravity-induced thrust, M vch is the vehicle mass,
mA is the mass of individual polarizable particles, and N is the total number of trapped
polarizable particles.
Pinto identified numerous technical challenges that will have to be overcome before this
concept can be put to practice. One challenge is that polarizability resonant
enhancement also leads to atomic transitions and decay which result in the recoil and
evaporation of atoms from inside the trap. Another is the difficulty of maintaining
continued confinement of a trapped cluster of polarizable particles in a specific 3-
dimensional array while the cluster is simultaneously opposing the amplified interatomic
forces and producing thrust. The confinement lifetime of trapped polarizable particles is
finite and there is the possibility that these particles might be evaporated away or
destroyed in a time that is too short to deliver the required thrust to the vehicle.
Therefore, a scheme for active repopulation of the trapped cluster will have to be
developed. The design of particle cluster traps and associated external confinement
fields are of primary importance to determine the effective thrusting time of every
polarizable particle. In addition, Rydberg atoms suffer from finite radiative lifetimes and
are sensitive to external perturbations, so dispersion force manipulation might lead to
the ionization of atoms. Tradeoffs will have to be made between all of the relevant
system parameters in order to discover the "sweet spot" that achieves levitation and
upward acceleration. These and other yet to be identified technical challenges need to
be addressed via further empirical and theoretical studies.
V. Conclusion: The Way Forward
This report has reviewed and analyzed a number of antigravity concepts that are found
within Newtonian gravity theory, General Relativity Theory, sem i-classical quantum
gravity theory, quantum field theory, and nonretarded quantum interatomic dispersion
force theory. One found that plausible mechanisms exist within Newtonian and general
relativistic theories whereby one could embody a realistic device that produces a
significant antigravity force. However, one discovered that there are daunting technical
challenges that arise in each of the proposed embodiments. Mechanical embodiments
that produce antigravity forces require kilometer-sized apparatus, astronomical-sized
masses and densities, or extreme mass velocities and accelerations. There are other
subtleties involved, such as the possibility of different forms of matter having a highly
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nonlinear gravitational permeability, which could dramatically mitigate such large-scale
requirements (see item 1 below for further discussion).
Negative energy has been produced in the lab in very small quantities. The technologies
used for producing negative energy are nascent, and so it will be some time before it
can be ascertained whether they are capable of producing the astronomical amounts of
negative energy required to generate significant antigravity forces as discussed in
Section III-C-2 (see item 2 below for further discussion).
Antigravity forces produced by quantum electromagnetic vacuum ZPF or by
nonretarded quantum interatomic dispersion forces in a curved spacetime (that is,
gravitational field) are very feeble, but there are proposals based on other theoretical
and empirical studies wh ich suggest that these forces can be amplified to macroscopic
level. However, there are a number of difficult technical challenges to overcome in
order to achieve success.
Going forward toward the demonstration of an antigravity generator will require the
following steps to be taken:
• Antigravity via Dipole Gravitational Field Generators: Presently, the technology does
not exist to achieve the astronomical mass densities, extreme velocities or
accelerations of mass motion, and the large device dimensions required to produce
large enough antigravity forces for useful propulsion. The issues are: 1) dense
materials, and 2) gravitational properties of matter. Forward (Reference 14)
suggests investigating neutron-neutron interactions. One could cool a gas of thermal
neutrons from a nuclear reactor to extremely low temperatures using magnetic
confinement or magneto-gravitational traps, and concentrate them into a small
region through the interaction of the trap's magnetic field with the magnetic
moment of the neutrons. The Fermi energy 12 of the bound neutrons limits the
neutron density to ~
10- 3 kg/m 3 . However, the formation of putative
13
tetraneutrons or the existence of a superconductive-type phase space
condensation will create bosons that do not have this limitation. It turns out that
exotic quantum states of matter such as Bose-Einstein (BE) and Fermionic
condensates 14 transcend the Fermi energy limit and thus possess highly unusual
material properties. BE condensates were first created in 1995 and Fermionic
condensates were first created in 2003, but both are still undergoing laboratory
exploration. As for the gravitational properties of matter, one knows from
electromagnetism that the permeability (µ) of magnetic materials such as iron is
anomalously large and nonlinear, which allows for the construction of highly efficient
electromagnetic field generators. The gravitational equivalent to the magnetic
permeability is a property of matter that is still largely unexplored. A material
possessing an anomalously large, very nonlinear gravitational permeability (ri)
would be useful in the construction of highly efficient, very small scale gravitational
field generators. One would expect all materials to have an ri that is different from
rio because the atoms comprising any material have quantum spin. Forward
12 In condensed matter physics, this is the energy of the highest occupied quantum state in a system of fermions
(that is, spin-½ particles such as electrons, nucleons, or atoms) at zero absolute temperature.
13 A hypothetical stable cluster of four neutrons whereby recent empirical evidence suggests it exists. Readers
should consult the technical literature for more information by using "tetraneutrons" as a search term .
14 BE or Fermionic condensates are a macroscopic collection of bosons (spin-1 particles such as nucleons or atoms)
or fermions that collapse into the same quantum state when they fo
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