Rueda - Gravity and the Quantum Vacuum Inertia Hypothesis (2001), Energy from the Vacuum
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Gravity and the Quantum Vacuum Inertia Hypothesis
I. Formalized Groundwork for Extension to Gravity
Alfonso Rueda
Department of Electrical Engineering & Department of Physics, ECS Building
California State University, 1250 Bellower Blvd., Long Beach, CA 90840
arueda@csulb.edu
Bernard Haisch and Roh Tung
California Institute for Physics & Astrophysics
366 Cambridge Ave., Palo Alto, CA 94306
haisch@calphysics.org, tung@calphysics.org
Abstract
It has been shown [1,2] that the electromagnetic quantum vacuum makes a contribution to the inertial
mass, m
i
, in the sense that at least part of the inertial force of opposition to acceleration, or inertia reaction
force, springs from the electromagnetic quantum vacuum (see also [3] for an earlier attempt). Specically, in
the previously cited work, the properties of the electromagnetic quantum vacuum as experienced in a Rindler
constant acceleration frame were investigated, and the existence of an energy-momentum ux was discovered
which, for convenience, we call the Rindler ux (RF). The RF, and its relative, Unruh-Davies radiation, both
stem from event-horizon eects in accelerating reference frames. The force of radiation pressure produced by
the RF proves to be proportional to the acceleration of the reference frame, which leads to the hypothesis that
at least part of the inertia of an object should be due to the individual and collective interaction of its quarks
and electrons with the RF. We call this the quantum vacuum inertia hypothesis. We demonstrate that this
approach to inertia is consistent with general relativity (GR) and that it answers a fundamental question left
open within GR, viz. is there a physical mechanism that generates the reaction force known as weight when a
specic non-geodesic motion is imposed on an object? Or put another way, while geometrodynamics dictates
the spacetime metric and thus species geodesics, is there an identiable mechanism for enforcing the motion
of freely-falling bodies along geodesic trajectories? The quantum vacuum inertia hypothesis provides such a
mechanism, since by assuming the Einstein principle of local Lorentz-invariance (LLI), we can immediately
show that the same RF arises due to curved spacetime geometry as for acceleration in at spacetime. Thus the
previously derived expression for the inertial mass contribution from the electromagnetic quantum vacuum
eld is exactly equal to the corresponding contribution to the gravitational mass, m
g
. Therefore, within the
electromagnetic quantum vacuum viewpoint proposed in [1,2], the Newtonian weak equivalence principle,
m
i
= m
g
, ensues in a straightforward manner. In the weak eld limit it can then also be shown, by means of
a simple argument from potential theory, that because of geometrical reasons the Newtonian gravitational
force law must exactly follow. This elementary analysis however does not pin down the exact form of the
gravitational theory that is required but only that it should be a theory of the metric type, i.e., a theory like
Einstein's GR that can be interpreted as curvature of spacetime. While the present analysis shows that our
previous quantum vacuum inertial mass analysis is consistent with GR, the extension of these two analyses
to components of the quantum vacuum other than the electromagnetic component, i.e. the strong and weak
vacua, remains to be done.
1. INTRODUCTION
Using the semiclassical representation of the electromagnetic quantum vacuum embodied in Stochastic
Electrodynamics (SED), it has been shown that a contribution to the inertial mass, m
i
, of an object must
result from the interactions of the quantum vacuum with the electromagnetically-interacting particles (quarks
and electrons) comprising that object [1,2]. Specically, the properties of the electromagnetic quantum
vacuum as measured in a Rindler constant acceleration frame were investigated, and the existence of an
energy-momentum ux was discovered which, for convenience, we now call the Rindler ux (RF). The
RF, and its relative, Unruh-Davies radiation, both stem from event-horizon eects in accelerating reference
frames. Event horizons create an asymmetry in the quantum vacuum radiation pattern.
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The force of radiation pressure on a massive object produced by the RF proves to be proportional to
the acceleration of the reference frame attached to the object. This leads naturally to the hypothesis that
at least part of the inertia of an object is due to the acceleration-dependent drag force that results from
individual and collective interactions of the quarks and electrons in the object with the RF. For simplicity of
reference, we refer to this concept, and an earlier derivation of a similar result using a completely dierent
approach (involving a perturbation technique due to Einstein and Hopf on an accelerating Planck oscillator
[3]), as the quantum vacuum inertia hypothesis.
SED is a theory that includes the eects of the electromagnetic quantum vacuum in physics by adding to
ordinary Lorentzian classical electrodynamics a random uctuating electromagnetic background constrained
to be homogeneous and isotropic and to look exactly the same in every Lorentz inertial frame of reference
[4,5]. This replaces the zero homogeneous background of ordinary classical electrodynamics. It is essential
that this background not change the laws of physics when exchanging one inertial reference system for
another. This translates into the requirement that this random electromagnetic background must have a
Lorentz invariant energy density spectrum. The only random electromagnetic background with this property
is one whose spectral energy density, (!), is proportional to the cube of the frequency, (!)d! !
3
d!. This
is the case if the energy per mode is h!=2 where ! is the angular frequency. (The h!=2 energy per mode is of
course also the minimum energy of the analog of an electromagnetic eld mode: a harmonic oscillator.) The
spectral energy density required for Lorentz invariance is thus identical to the spectral energy density of the
zero-point eld of ordinary quantum theory. For most purposes, including the present one, the zero-point
eld of SED may be identied with the electromagnetic quantum vacuum. However SED is essentially a
classical theory since it presupposes only ordinary classical electrodynamics and hence SED also presupposes
special relativity (SR).
According to the weak equivalence principle (WEP) of Newton and Galileo, inertial mass is equal to
gravitational mass, m
i
= m
g
. If the quantum vacuum inertia hypothesis is correct, a very similar mechanism
involving the quantum vacuum should also account for gravitational mass. This novel result, restricted for
the time being to the electromagnetic vacuum component, is precisely what we show in x2 by means of
formal but simple and straightforward arguments requiring physical assumptions that are uncontroversial
and widely accepted in theoretical physics. In x3 the consistency of this argument with so-called metric
theories of gravity (i.e. those theories characterized by spacetime curvature) is exhibited. In addition to the
metric theory, par excellance, Einstein's GR, there is the Brans-Dicke theory and other less well known ones,
briey discussed by Will [6]. x4 briey discusses a non-metric theory.
Nothing in our approach points to any new discriminants among the various metric theories. Never-
theless, our quantum vacuum approach to gravitational mass will be shown to be entirely consistent with
the standard version of GR. Next, in x5, we take advantage of geometrical symmetries and present a short
argument from standard potential theory to show that in the weak eld limit a Newtonian inverse square
force must result from our approach.
A new perspective on the origin of weight is presented in x6. In x7 we discuss an energetics aspect,
related to the derivation presented herein, and resolve an apparent paradox. A brief discussion on the nature
of the gravitational eld follows in x8, but we infer that the present development of our approach does not
provide any deeper or more fundamental insight than GR itself into the ability of matter to bend spacetime.
We present conclusions in x9. A full development within GR is left for the accompanying article [7].
2. ON WHY THE ELECTROMAGNETIC VACUUM CONTRIBUTION TO GRAVITATIONAL MASS IS
EXACTLY THE SAME AS FOR INERTIAL MASS
Table 1 compares the quantum vacuum inertia hypothesis with the standard view on mass. We intend
to show { in this and a companion paper [7] { not only that the quantum vacuum inertia hypothesis is
consistent with GR, but that it answers an outstanding question regarding a possible physical origin of the
force manifesting as weight. We also intend to show that just as it becomes possible to identify a physical
process underlying the f=ma postulate of Newtonian mechanics (as well as its extension to SR [1,2]), it
is possible to identify a parallel physical process underlying the weak equivalence principle, m
i
= m
g
, viz.
interaction of matter with the RF.
Within the standard theoretical framework of GR and related theories, the equality (or proportionality)
of inertial mass to gravitational mass has to be assumed. It remains unexplained. As correctly stated by
2
Rindler [8], \the proportionality of inertial and gravitational mass for dierent materials is really a very
mysterious fact." However here we show that { at least within the present restriction to electromagnetism
{ the quantum vacuum inertia hypothesis leads naturally and inevitably to this equality. The interaction
between the electromagnetic quantum vacuum and the electromagnetically-interacting particles constituting
any physical object (quarks and electrons) is identical for the two situations of acceleration with respect to
constant velocity inertial frames or remaining xed above some gravitating body with respect to freely-falling
local inertial frames.
A related theoretical lucuna involves the origin of the force which manifests itself as weight. Within GR
theory one can only state that deviation from geodesic motion results in a force which must be an inertia
reaction force. We propose that it is possible in principle to identify a mechanism which generates such
an inertia reaction force, and that in curved spacetime it acts in the same way as acceleration does in at
spacetime.
Begin by considering a macroscopic, massive, gravitating object, W, which is xed in space and for
simplicity we assume to be solid, of constant density, and spherical with a radius R, e.g. a planet-like object.
At a distance r >> R from the center of W there is a small object, w, that for our purposes we may regard
as a point-like test particle. A constant force f is exerted by an external agent that prevents the small body
w from falling into the gravitational potential of W and thereby maintains w at a xed point in space above
the surface of W. Experience tells us that when the force f is removed, w will instantaneously start to move
toward W with an acceleration g and then continue freely falling toward W.
Next we consider a freely falling local inertial frame I
(in the customary sense given to such a local
frame [6]) that is instantaneously at rest with respect to w. At w proper time , that we select to be = 0,
object w is instantaneously at rest at the point (c
2
=g; 0; 0) of the I
frame. The x-axis of that frame goes
in the direction from W to w and, since the frame is freely falling toward W, at = 0 object w appears
accelerated in I
in the x-direction and with an acceleration g
w
= xg. As argued below, w is performing
a uniformly-accelerated motion, i.e. a motion with a constant proper acceleration g
w
as observed from
any neighboring instantaneously comoving (local) inertial frame. In this respect we introduce an innite
collection of local inertial frames I
, with axes parallel to those of I
and with a common x-axis which is
that of I
. Let w be instantaneously at rest and co-moving with the frame I
at w proper time . So the
parameter representing the w proper time also serves to parametrize this innite collection of (local) inertial
frames. Clearly then, I
is the member of the collection with = 0, so that I
= I
=0
. At the point in time of
coincidence with a given I
, w is found momentarly at rest at the (c
2
=g; 0; 0) point of the I
frame. We select
also the times in the (local) inertial frames to be t
and such that t
= 0 at the moment of coincidence when
w is instantaneously at rest in I
and at the aforementioned (c
2
=g; 0; 0) point of I
. Clearly as I
=0
= I
then t
= 0 when = 0. All the frames in the collection are freely falling toward W and when any one of
them is instantaneously at rest with w it is instantaneously falling with acceleration g = g
w
= xg with
respect to w in the direction of W. It is not dicult to realize that w appears in those frames as uniformly
accelerated and hence performing a hyperbolic motion with constant proper acceleration g
w
.
This situation is equivalent to that of an object w accelerating with respect to an ensemble of I
reference frames in the absence of gravity. In that situation, the concept of the ensemble of inertial frames,
I
, each with an innitesimally greater velocity (for the case of positive acceleration) than the last, and each
coinciding instantaneously with an accelerating w is not dicult to picture. But how does one picture the
analogous ensemble for w held stationary with respect to a gravitating body?
We are free to bring reference frames into existence at will. Imagine bringing a reference frame into
existence at time = 0 directly adjacent to w, but whereas w is xed at a specic point above W, we
let the newly created reference frame immediately begin free-falling toward W. We immediately create a
replacement reference frame directly adjacent to w and let it drop, and so on. The ensemble of freely-falling
local inertial frames bear the same relation to w and to each other as do the extended I
inertial frames
used in the case of true acceleration of w.
For convenience we introduce a special frame of reference, S, whose x-axis coincides with those of the
I
frames, including of course I
, and whose y-axis and z-axis are parallel to those of I
and I
. This frame
S stays collocated with w which is positioned at the (c
2
=g; 0; 0) point of the S frame. For I
(and for the
I
frames) the frame S appears as accelerated with the uniform acceleration g
w
of its point (c
2
=g; 0; 0). We
will assume that the frame S is rigid. If so, the accelerations of points of S suciently separated from the
w point (c
2
=g; 0; 0) are not going to be the same as that of (c
2
=g; 0; 0). This is not a concern however since
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we will only need in all frames (I
, I
, and S) to consider points in a suciently small neighborhood of the
(c
2
=g; 0; 0) point of each frame
The collection of frames, I
, as well as I
and S, correspond exactly to the set of frames introduced in
[1]. The only dierences are, rst, that now they are all local, in the sense that they are only well dened for
regions in the neighborhood of their respective (c
2
=g; 0; 0) space points; and second, that now I
and the I
frames are all considered to be freely falling toward W and the S frame is xed with respect to W. Similarly
to [1], the S frame may again be considered to be, relative to the viewpoint of I
, a Rindler noninertial
frame. The laboratory frame I
we now call the Einstein laboratory frame, since now the \laboratory" is
local and freely falling. We call the collection of inertial frames I
the Boyer family of frames as he was the
rst to introduce them in SED [9].
The relativity principle as formulated by Einstein when proposing SR states that \all inertial frames are
totally equivalent for the performance of all physical experiments."[6] Before applying this principle to the
freely-falling frames I
and I
that we have dened above it is necessary to draw a distinction between these
frames and inertial frames that are far away from any gravitating body, such as W. The free-fall trajectories,
i.e. the geodesics, in the vicinity of any gravitating body, W, cannot be parallel over any arbitrary distance
owing to the fact that W must be of nite size. This means that the principle of relativity can only be
applied locally. This was precisely the limitation that Einstein had to put on his innitely-extended Lorentz
inertial frames of SR when starting to construct GR [6,8,10].
We adopt the principle of local Lorentz invariance, (LLI) which can be stated, following Will [6], as
\the outcome of a local nongravitational test experiment is independent of the velocity of the freely falling
apparatus." A non-gravitational test experiment is one for which self-gravitating eects can be neglected. We
also adopt the assumption of space and time uniformity, which we call the uniformity assumption (UA) and
which states that the laws of physics are the same at any time or place within the Universe. Again, following
Will [6] this can be stated as \the outcome of any local nongravitational test experiment is independent of
where and when in the universe it is performed." We do not concern ourselves with physical cosmological
theories that in one way or another violate UA, e.g. because they involve spatial or temporal changes in
fundamental constants [11].
Locally, the freely falling local Lorentz frames which we now designate with a subscript L | I
;L
and
I
;L
| are entirely equivalent to the I
and I
extended frames of [1]. The free-falling Lorentz frame I
;L
locally is exactly the same as the extended I
. Invoking the LLI principle we can then immediately conclude
that the electromagnetic zero-point eld, or electromagnetic quantum vacuum, that can be associated with
I
;L
must be the same as that associated with I
. From the viewpoint of the local Lorentz frames I
;L
and
I
;L
the body w is undergoing uniform acceleration and therefore for the same reasons as presented in [1] an
acceleration-dependent drag force arises.
These formal arguments demonstrate that the analyses of of [1,2] which found the existence of a RF in
an accelerating reference frame translate and correspond exactly to a reference frame xed above a gravitating
body. In the same manner that light rays are deviated from straight-line propagation by a massive gravitating
body W, the other forms of electromagnetic radiation, including the electromagnetic zero-point eld rays
(in the SED approximation) are also deviated from straight-line propagation. Not surprisingly this creates
an anisotropy in the otherwise isotropic electromagnetic quantum vacuum. This is the origin of the RF in
the gravitational case.
In [1] we interpreted the drag force exerted by the RF as the inertia reaction force of an object that is
being forced to accelerate through the electromagnetic zero-point eld. Accordingly, in the present situation,
the associated nonrelativistic form of the inertia reaction force should be
= m
i
g
w
(1)
where g
w
is the acceleration with which w appears in the local inertial frame I
. As shown in [1] the
coecient m
i
is
V
0
c
2
Z
h!
3
2
2
c
3
d!
m
i
=
(!)
(2)
where V
0
is the proper volume of the object, c is the speed of light, h is Planck's constant divided by 2 and
(!), where 0 (!) 1, is a function that spectralwise represents the relative strength of the interaction
4
f
zp
between the zero-point eld and the massive object which acts to oppose the acceleration. If the object
is just a single particle, the spectral prole of (!) will characterize the electromagnetically-interacting
particle. It can also characterize a much more extended object, i.e. a macroscopic object, but then the
(!) will have much more structure (in frequency). We should expect dierent shapes for the electron, a
given quark, a composite particle like the proton, a molecule, a homogeneous dust grain or a homogeneous
macroscopic body. In the last case the (!) becomes a complicated spectral opacity function that must
extend to extremely high frequencies such as those characterizing the Compton frequency of the electron
and even beyond.
Now, however, what appears as inertial mass, m
i
, to the observer in the local I
;L
frame is of course
what corresponds to gravitational mass, m
g
, and it must therefore be the case that
V
0
c
2
Z
h!
3
2
2
c
3
d!
m
g
=
(!)
:
(3)
As done in [1], Appendix B, it can be shown that the right hand side indeed represents the energy of the
electromagnetic quantum vacuum enclosed within the object's volume and able to interact with the object
as manifested by the (!) coupling function. A more thorough, fully covariant development can also be
implemented to show that the force expression of eqn. (1) can be extended to the relativistic form of the
inertia reaction force as in [1] , Appendix D. (This development also served to obtain the nal form of m
i
given above in eqn. (3) eliminating a spurious 4/3 factor.) * Summarizing what we have shown in this
section is that if a force f is applied to the w body just large enough to prevent it from falling toward the
body W, then in the non-relativistic case that force is given by
f = mg
(4)
where we have dropped the nonessential subscripts i and g and superscripts, because it is now clear that
m
i
= m
g
= m follows from the quantum vacuum inertia hypothesis.
3. CONSISTENCY WITH EINSTEIN'S GENERAL RELATIVITY
The statement that m
i
= m
g
constitutes the weak equivalence principle (WEP). Its origin goes back to
Galileo and Newton, but it now appears, as shown in the previous section, that this principle is a natural
consequence of the quantum vacuum inertia hypothesis. The strong equivalence principle (SEP) of Einstein
consists of the WEP together with LLI and the UA. Since the quantum vacuum inertia hypothesis and its
extension to gravity allow us to obtain the WEP assuming LLI and the UA, this approach is consistent with
all theories that are derived from the SEP. In addition to GR, the Brans-Dicke theory is derived from SEP as
are other other lesser known theories [6], all distinguished from each other by various particular assumptions.
All theories that assume the SEP are called metric theories. They are characterized by the fact that they
contemplate a bending of spacetime associated with the presence of matter. Two important consequences of
the LLI-WEP-UA combination are that light bends in the presence of matter and that there is a gravitational
Doppler shift. Since the quantum vacuum inertia hypothesis is consistent with this same combination, it
would also require that light bends in the presence of gravitational elds. This can, of course, be interpreted
as a change in spacetime geometry, the standard interpretation of GR.
4. A RECENT ALTERNATIVE VACUUM APPROACH TO GRAVITY
The idea that the vacuum is ultimately responsible for gravitation is not new. It goes back to a proposal
of Sakharov [12] based on the work of Zeldovich [13] in which a connection is drawn between Hilbert-Einstein
* We use this opportunity to correct a minor transcription error that appeared in the printed version of
Ref. [1]. In Appendix D, p. 1100, the minus sign in eqn. (D8) is wrong. It should read
= 1
(D8)
and the corresponding signature signs in the line just above eqn. (D8) are the opposite of what was written
and should instead read (+ ).
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