Cavities In A Lambda Vacuum
and
The Meaning Of Matter
Douglas C. George
Eureka, CA
Last Revision
October 29, 2008
Abstract
A hole in the fabric of space itself—a cavitation bubble in a lambda
vacuum—would be gravitationally indistinguishable from an ordinary massive
object and could point to a simple and intuitive description of both matter and
the gravitational field. In addition, the metric of space associated with such
a cavity is found to be completely free of mathematical singularities and
infinities. If space has intrinsic mass and a positive cosmological constant, a
rapid enough expansion of the universe could produce these cavities in great
numbers
Introduction
In this paper, a hypothetical, spheroidal cavity in a lambda vacuum is shown to
be gravitationally equivalent to a normal massive object, a conclusion implying
that all matter may ultimately be, like black holes, constructed of nothing
more than warped regions of space.
The nature of empty space has long been a controversial subject within the
scientific community. Opinions about its nature have, over the years, covered a
broad spectrum of conjecture ranging from it being absolutely nothing—an
“empty box” so to speak—to it being some sort of fluid-like
manifold that is flexible, stretchable and having a mass of its own independent
of the material objects and radiation contained in it.
Whether or not empty space has a mass of its own has emerged, in recent years,
as a subject of both laboratory experiment (the Casimir effect) and
astronomical observations indicating that the expansion of the universe is
accelerating. Because of these and other developments, space is now generally
considered to have an intrinsic mass which is proportional to the cosmological
constant
Λ
(lambda). This paper uses the term “lambda vacuum” for such a
space and explores the possibility and ramifications of a particular type of
discontinuity existing in it, namely: holes or bubbles analogous to the
cavitation bubbles created by underwater propellers.
The concept of discontinuities in space is, in general, not a new or
controversial idea. General relativity allows for many such oddities over a
wide range of realistic models. It is widely accepted that space can have holes
or edges where the mathematical values describing those regions can explode to
infinity or come to an abrupt end.
If the lambda vacuum is considered to be an infinite, perfect-fluid manifold, a
spherical cavity in it would be completely defined by the energy density
gradient in the fluid surrounding the cavity. The gradient would be spherically
symmetrical and would range from zero at the cavity boundary up to the average
universal energy density at infinity. The cavity boundary would demarcate the
edge of space itself.
The spherical symmetry of the gradient means that, by Birkhoff's Theorem, the
exact exterior solution to the Einstein Field Equations for such a cavity must
be the Schwarzschild metric, i.e., the metric of space surrounding a
theoretical cavitation bubble in a lambda vacuum would be gravitationally
indistinguishable from the metric of space surrounding any spherically
symmetric massive object.
We will begin with a brief overview of the Schwarzschild solution to the
Einstein Field Equations for a normal, spherical, uncharged, non-rotating,
massive object. In particular we will focus on those problem areas of the
metric—where the metric displays singularities and infinities—and then show
that the metric of space surrounding a spherical cavity would be completely and
inherently free of all such singularities and infinities.
The Ground Rules
“ ... In a word, matter may be just another form of space. ”
http://www.astronomycafe.net/vacuum/vactext
General relativity treats space as a physical manifold that can expand,
contract or be otherwise distorted by material objects embedded in it. Material
objects are intimately related to the surrounding space and that relationship
is made explicit by the Einstein Field Equations (EFE). A massive object is said to
distort its surrounding space and the distorted space, in turn, determines how
the object will move. In short, the distorted space is the gravitational field.
The physical properties of the vacuum have been, in recent years, a subject of
intense and ongoing investigations. This paper will not speculate on the basic
structure of space other than to accept that it has intrinsic mass and
fluid-like properties. For simplicity, then, we will assume the vacuum to have
a mass of its own and to be an infinite, idealized, compressible fluid with no
viscosity and no heat conduction.
The warpage of a region of spacetime is related by the Einstein Field Equations
to the flux of energy and momentum through the region. The field equations may
be written as
G
ab
=
kT
ab
[1]
where
G
ab
is the Einstein Tensor,
T
ab
is the stress-energy tensor and
k
is a constant. The Einstein tensor represents how space is distorted by the
presence and flux of matter and radiation as accounted for in the stress-energy
tensor.
The Schwarzschild vacuum solution to the field equations describes the geometry
of space and time exterior to a spherical, non-rotating, uncharged massive
object. It describes the warpage of the space as a function of the proper
radial distance
r
from the center of the object. The geometry
1
is described by the following metric (in units where the speed of light
is
equal to one):
[2]
The quantity
ds
is the invariant spacetime interval,
t
is the time coordinate,
r
is the radius of a sphere,
Ω
(Omega) is the standard interval of spherical solid angle and
r
s
is the Schwarzschild radius.
The Schwarzschild radius is the critical size for an object of a given mass at
which gravity overwhelms all other known forces. It depends on the mass of the
object according to the following equation:
r
s
=
2 GM /c
2
,
[3]
where
G
is Newton's gravitational constant,
M
is the object's mass and
c
is the speed of light. In the discussion of black holes that follows below,
the term “Schwarzschild radius” is used interchangeably with the
terms “critical radius” and “event horizon”.
The invariant interval
ds
represents the absolute measure of distance between two events in space and
time and is the same for all comoving observers.
At a given instant of time, the proper radial distance, the actual distance
between two points separated by an interval
dr
, as measured by a stationary observer positioned at radius
r
, is the square root of the radial component of the above metric and is equal
to
( 1
−
r
s
/
r )
− 1/2
dr
,
[4]
which is greater than the same interval as it would be measured in flat,
Euclidean geometry. It is this radial component of the metric that will be
important in the arguments that follow because it is what determines the
variations in the energy density surrounding a hypothetical cavity in space.
Black Holes
“ ... space and time may exhibit 'holes' or 'edges': spacetime
singularities.”
http://www.einstein-online.info/en/spotlights/singularities/index
Under certain conditions, the metric described in equation [2] in the preceding
section implies the existence of black holes. If an object with mass
M
is compressed to its Schwarzschild radius, its gravity becomes so strong that
no known force can halt its complete collapse. The mass will implode, crushing
itself out of existence and leaving in its place a black hole.
As viewed by outside observers at rest with respect to the Schwarzschild
metric, the space surrounding a black hole, like the space around any massive
object, is well-behaved and the mathematics describing it contain no infinities
or singularities. There are, however, two places where the metric appears
ill-behaved: at the event horizon (where
r
=
r
s
) and at the origin (where
r
=
0
).
The Event Horizon
The event horizon
r
s
of an object depends, as expressed by equation [3] above, on its mass
M
. As
r
approaches
r
s
from the outside, a number of mathematical problems appear that demonstrate
how the geometry is “ill-behaved”. Most relevant for the purposes
of this paper, the geometry of space is stretched in the radial direction as
described by equation [4] where, at the Schwarzschild radius, the stretching
becomes infinite.
Although they can't be actually eliminated, the problems at the event horizon
can be made “analytically removable”
2
by a change of coordinate system which allows a mathematical analysis to
continue smoothly through the region.
Ill-behaved geometry at the event horizon can be considered to be artifacts of
the particular frame of reference—the Schwarzschild coordinate system
itself—and, as mentioned, can be made analytically removable. In a more
appropriate reference frame—called the free-fall coordinate
system—spacetime appears well-behaved all the way through the event horizon
and down to (but not including) the central singularity. Using this special
coordinate system, the Schwarzschild geometry appears to be that of ordinary,
flat Euclidean space. Unfortunately, as we will see, this analytical taming of
the metric comes with its own set of problems.
The free-fall coordinate system is the view of things as seen by an observer
falling freely from infinity toward the black hole. Its distinctive feature (as
well as the crux of its difficulties) is that space, itself, must be considered
to be flowing radially inward at the Newtonian escape velocity:
v
= (
2 GM
/
r
)
1/2
.
[4]
This infall velocity of space starts out at zero at infinity, reaches the speed
of light at the event horizon and continues to increase all the way to the
central singularity where the velocity becomes infinite.
Because relativity theory limits the relative velocity of comoving objects to
be less than the speed of light, if the infall velocity of an object reaches
and then exceeds the speed of light, the argument asserts that it must be
because space itself is flowing radially inward and carrying the coordinate
system along with it.
So, in order to avoid the mathematical singularities and infinities at the
event horizon, a coordinate system is used that drifts passively with a space
that is assumed to be flowing toward the central singularity like water down a
drain and which reaches infinite velocity just as it disappears from our
universe.
While this coordinate system does circumvent the problems at the event horizon,
it still leaves us with the problems at the central singularity and adds to
those a new problem: if space has an intrinsic energy content, we are faced
with the failure of the conservation of energy and momentum as space disappears
at the central singularity.
In-between
The mathematics describing the region between the event horizon and the central
singularity are, likewise, ill-behaved. There, space becomes
“time-like”, the mathematics become imaginary (i.e., the geometry
is represented by the square root of a negative number) and moving objects must
be considered to travel backward in time.
As will be shown, a spherical cavitation bubble in space exhibits none of these
problems but, first, let us examine more closely the problems at the central
singularity.
The Central Singularity
Unlike at the event horizon, the central singularity is a true mathematical
singularity. The problems there cannot be eliminated or made removable. It is
the point where the mathematics of general relativity fail and, worse, until
now there has been no way to get rid of it. At the central singularity, the
Riemann tensor values representing the tidal stretching of space in the radial
direction and the accompanying, orthogonal tidal compressions, become infinite.
According to the presently accepted view, the application of the metric to the
space within the event horizon of a normal but highly compressed massive object
leads to the mathematically inescapable conclusion that the object will implode
and all of its matter will be crushed out of existence at the central
singularity. The original imploding matter would be gone, completely. At the
singularity, matter is stretched infinitely in the radial direction and
squeezed infinitely in the orthogonal directions. Not only would the original
matter be destroyed but any new matter falling past the event horizon in the
future would be crushed out of existence at the central singularity.
The extraordinary thing here is that, even though all matter and radiation are
completely destroyed, there would remain an enduring, pulsating shell of
deformed space—the black hole—with none of the original matter left inside.
All that remains is an empty shell of warped space. Even more remarkable is
that, for outside observers, this empty shell of warped space still has all of
the original mass!
To quote Kip Thorne
3
, the imploded matter is transformed into “... an object made wholly and
solely from the warpage of space itself.”
The original matter has apparently converted itself completely into
deformations of space and it is those warpages, and nothing else, that
subsequently possess all the mass.
So, here we have a massive object that appears to the outside world to be
normal matter but has, in the usual sense, no material matter in it. That an
empty shell of deformed space, devoid of matter, can be indistinguishable from
an ordinary massive object, provides the first clue that matter itself might
consist of nothing more than deformations of space. It is the position of this
paper that this may actually be the case. We will argue that stable
deformations of space may, in fact, not be limited to black holes but may be,
in general, what both matter and its gravitational fields are made of
A Hole in Space
The aim of the above discussion was to show briefly where and how the
mathematics describing the space associated with a black hole displays
well-known infinity and singularity problems. These problems have been known to
physicists and mathematicians since shortly after the theory of general
relativity was published in 1915 and are still seen as serious, if not fatal,
flaws in the theory.
This paper introduces a theoretical structure, a pattern of warpage in space,
that is not only gravitationally equivalent to normal matter (and compatible
with the concept of black holes) but avoids all of the infinity and singularity
problems associated with its metric and leads to a simple and intuitive
description of both matter and its gravitational fields.
This structure—a cavity in space—may be more accurately described as a
three-dimensional, spheroidal singularity in space whose surface has the
property of geodesic incompleteness and which demarcates the edge of space
itself.
Since general relativity is a description of how space and matter are
interrelated, it would be inappropriate to apply the mathematics of general
relativity to the region inside such a cavity simply because space literally
ends at the surface. Within the cavity, there would be—as Gertrude Stein
said—no “there” there.
To see how such an object would be constructed and how it might come to exist,
we turn now to the topic of energy density gradients in a lambda vacuum. This
portion of our discussion is aimed at those who have a fundamental knowledge of
basic general relativity and will be recognized to contain nothing new or
unusual in the mathematical arguments. It is simply basic general relativity as
applied to the admittedly unusual proposal of a cavity in the fabric of space.
Nonetheless, a surprising result emerges from these simple arguments: we find
that the metric of space surrounding such a cavity is completely free of
singularity and infinity problems.
Vacuum Density Gradients
“ ... you can't really separate the structure of space-time from the
particles ...”
Michael Green (1988)
The energy density of the vacuum has, historically, referred to the
distribution of normal matter and radiation dispersed in an otherwise mass-free
space and is used in calculations predicting the evolution of the universe at
large. For a lambda vacuum, however, where space itself has intrinsic mass, the
energy density refers to the energy content of a region of space that is
completely free of normal matter and radiation.
As mentioned previously, a lambda vacuum has an energy content of its own that
is proportional to the cosmological constant
Λ
. It is equivalent to saying that the vacuum has a certain mass per unit volume
(currently thought to be about 10
-29
g/cm
3
). Collectively, this mass is said to comprise about seventy percent of all the
mass in the universe, i.e., most of the mass of the universe is associated with
the mass content of “empty” space.
The cavitation bubbles proposed in this paper are defined as energy density
gradients in such a space. So, the question here is whether or not it is
possible for such energy density gradients to exist in a lambda vacuum.
Energy density gradients are, in fact, a normal part of general relativity. To
see this we look at the relationship between the vacuum energy density of a
region and the flux of momentum through the region as expressed by the
stress-energy tensor of the field equations.
The illustration at the right shows the components of the stress-energy tensor
T
ab
in equation [1] above.
The energy density term in the upper left corner represents the energy density
of a region and the three “pressure” terms arrayed along the
diagonal highlighted in green represent the flux of momentum through the region
in the x, y, and z directions. For an idealized fluid with no viscosity and no
heat conduction, all of the other components of the tensor are zero.
This stress-energy tensor normally includes only the flux of energy and
momentum associated with matter and radiation dispersed in an otherwise empty
space. It does not include contributions from the space itself. If the space is
considered to have a mass of its own, a new term, containing the cosmological
constant, is added to the field equations.
The field equations with the cosmological constant added are:
G
ab
+
Λg
ab
=
kT
ab
,
[6]
where the new term represents the contribution from empty space to the overall
flux of momentum. This term can be moved to the right hand side of equation [6]
and be considered as just another contribution to the stress-energy tensor
shown in the illustration. For an empty universe (containing no material
objects or radiation), the
kT
ab
term vanishes and only the stress-energy tensor representing the vacuum
remains on the right hand side.
The net result is that the diagonal components in the illustration now
represent the relationship between the energy density of the vacuum itself and
the flux of energy through the region.
Furthermore, a simple analysis shows that, for the vacuum to not pick out a
preferred notion of rest, the energy density term must be the negative of the
three pressure terms. According to theoretical mathematician John Baez
4
“ ... , this amounts to giving empty space an energy density equal to
Λ
and pressure equal to -
Λ
.”
Put another way by physicist Sean M. Carroll
5
, for a universe free of ordinary material objects but still having a uniform
vacuum energy density, “... the vacuum can be thought of as a perfect
fluid ... with
p
vac
= −
ρ
vac
”, i.e., the pressure
p
of the vacuum is equal to the negative of the energy density
ρ
(rho).
Thus we have the basic inverse relationship between the vacuum energy density
in a region of space and the pressure of the region (the flux of momentum
through the region):
Vacuum energy density (
ρ
) = - pressure (
p
)
[10]
Formation of Cavities in Space and The Resulting Density Gradients
If the cosmological constant is positive, the associated negative pressure as
expressed in equation [10] would drive an accelerated expansion of empty space.
Recent astronomical observations indicating that the expansion of the universe
is, indeed, accelerating strongly implies that the cosmological constant is
positive and that the universe is presently under such a negative pressure
condition.
If the negative vacuum pressure described above were to exceed the tensile
strength of space, the vacuum would be brought to a foaming boil by the
expansion in the same way that any fluid would boil under similar conditions.
Cavitation bubbles would form in the fluid and the energy density would no
longer be uniform. Around each cavity, an energy density gradient would form
that starts out at zero at the cavity boundary and increases, in some fashion,
back to the average density of the universe.
The Density Gradient Curve
The specific shape of the curve representing this density gradient would depend
on the exact solution to the EFE for a cavitation bubble in a lambda vacuum. We
know of no such solution that has been derived directly but, as mentioned
above, because Birkhoff's theorem states that
any
exterior, spherically symmetric solution of the vacuum field equations must be
given by the Schwarzschild metric, the exact solution in our case, likewise,
has to be the Schwarzschild metric.
Taking, then, the Schwarzschild metric for the fluid surrounding such a cavity,
what would be the mathematical form of the energy density gradient curve in the
fluid?
The radial component of the Schwarzschild vacuum metric (equation [4]) tells us
that the space around each cavity would be stretched in the radial direction as
a function of the distance from the center of the cavity.
Consider the example shown in Figure 1. which shows a cavity located at the
origin of the coordinate system. The cavity has the mass equivalence of
M
because the external metric is identical to that for a mass with the same
Schwarzschild radius.
Before cavitation takes place, space has a uniform density proportional to
Λ
. After the formation of a cavity, the density at a radial distance
r
from the center of the cavity would be proportional to the radial component of
the metric. Therefore, in the space surrounding the cavity, we have an energy
density gradient curve of the form:
,
[12]
where the cosmological constant
Λ
represents the average universal density of space,
M
is the virtual mass and
K
is a constant of proportionality dependent on the system of units used. The
curve in Figure 1. is a plot of Equation [12].
The Crux of the Argument
As can be seen in Figure 1, the slope of the density gradient curve increases
as
r
approaches the origin from the outside and the amplitude of the curve goes to
minus infinity at the event horizon. The plot shows that the energy density of
space thins out ever more rapidly as
r
approaches
r
s
.
Most notably—and what is the crux of our argument concerning infinities and
singularities in the metric—we note that (as Figure 1. shows) the energy
density gradient curve crosses the horizontal axis (where the energy density
value is zero)
before
it reaches the event horizon. As long as lambda has a positive value, the
energy density of space surrounding such a cavity must go to zero before
r
reaches
r
s
A value of zero for the energy density of a region in a fluid means, logically,
a cavity in the fluid. In our case, the fluid is the vacuum of space and,
therefore, the cavity represents a spherical hole in the fabric of space whose
radius is slightly larger than the critical radius of a mass that would produce
the same energy density gradient in its surrounding space.
Because the density of space goes to zero just
outside
of the event horizon, the metric avoids all of the singularity and infinity
problems at and within the event horizon. The upshot is that, for a cavitation
bubble in a lambda vacuum, there are no singularity or infinity problems with
the metric anywhere in space.
Motion of other masses
There are two ways to think about the effect on other objects exerted by the
warped space surrounding these cavities.
In general relativity, for any specific region of space, the energy density and
the flux of momentum are locally conserved. For a test particle moving through
the area, its motion would be the only possible contribution to the flux of
momentum. Consequently, by equation [10], the changing momentum of the test
particle would be proportional to the negative of the radial component of the
Schwarzschild metric. Its momentum, then, would just be the mirror image of the
density gradient curve associated with the cavity
M
. In other words, the test particle's mass must increase and it must accelerate
as it approaches the cavity.
The acceleration of the test particle would be proportional to the time
derivative of the momentum divided by the particle's mass and would be
calculated in the usual way from the metric and the field equations. Since the
metric is the Schwarzschild metric, all calculations of motion would be the
same as that for a normal material object moving through a gravitational field.
The other way to look at it is that since, by Birkhoff's Theorem, the space
surrounding a cavity is distorted in the same way as it is by a normal massive
object, the motion of a test particle must be identical to the motion of an
object moving through a normal gravitational field.
To sum up the situation, if space undergoes an expansion rapid enough to
produce cavitation bubbles and if the metric around those resulting cavities is
the Schwarzschild metric the conservation law for energy and momentum tells us
the resulting energy density gradients would have the same effect on other
massive objects as gravitational fields produced by ordinary matter.
Gravitational fields may, therefore, be seen simply as energy density gradients
in the vacuum: the simple thinning out (or thickening) of the density of space
itself.
Black Holes vs. Ordinary Matter
We have, so far, drawn the similarities between a normal massive object, a
black hole and a hypothetical cavitation bubble in a lambda vacuum. We have
shown that they would be indistinguishable as viewed by outside observers and
that the cavities would have problem-free metrics.
The basic cavities produced by an expanding universe would be tiny indeed. A
cavity with the mass equivalence of an electron, for instance,
would—according to equation [3]—have a Schwarzschild radius of ten to the
minus fifty-seven meters which is twenty-two orders of magnitude smaller than
the Planck distance. Quark-sized cavities would, according to their masses,
have roughly similar dimensions.
The link between our theoretical cavitation bubbles and the plethora of known
particles would depend on the details of the mechanical and electrical
structure of the vacuum itself and is beyond the scope of the present paper. It
is hoped that the reader will forgive us in what follows as we engage in some
wild and shameless speculation about how these cavities in space would behave
and interact.
We proposed, tentatively, that all matter may be constructed in the same way as
black holes: as warpages of space and nothing more; the basic argument being
that matter and black holes are both manifestations of cavities in space. Black
holes would be large holes in space and the basic constituents of matter would
be tiny ones. However, there is an obvious difference between a black hole and
an ordinary massive object such as a planet or a star. Those objects, being
full of ordinary particle matter, are definitely not empty shells of deformed
space. This problem is resolved by recognizing that planets and stars would be
comprised of sub-nuclear, foam-like agglomerations of un-coalesced cavities
while black holes would result from the forced coalescence of those small
cavities to form ever larger holes in space.
Since the building blocks of ordinary matter, in our scenario, would be
constructed from sub-nuclear cavities in space, it is easy to think that, under
sufficient gravitational pressure, they could be forced to coalesce into larger
cavities. The coalescence of small cavities to form ever larger black holes is
conceptually and theoretically simpler than crushing matter out of existence at
the central singularity of a black hole or having space flow down a cosmic
drain at infinite velocity. It would amount, simply, to gravity overcoming
electrical and nuclear forces of repulsion between cavities.
Now consider the test particle mentioned above to, itself, be a small cavity.
Its leading edge would always be in a steeper region of the density gradient
produced by the large cavity than would its trailing edge. This means the
leading edge would have a greater acceleration, causing the small cavity to be
stretched radially and compressed orthogonally as described by the
Schwarzschild metric in equation [2] (the term “spaghettification”
applies here). If the leading edge were to reach the event horizon of the large
cavity, the radial stretching of space there would be infinite. Fortunately, it
would never get there. As we saw above, the energy density gradient curve must
cross the zero value at the horizontal axis before the curve gets to minus
infinity at the event horizon. The space would simply thin out and disappear
completely before the event horizon is reached.
The leading edge of the test cavity, then, would coalesce with the near edge of
the large cavity at the common point where the density values of both go to
zero and this would happen, depending on the initial value of the cosmological
constant, somewhere outside the event horizon. A tunnel would open up between
the two cavities and they would merge. After any transient oscillations die
out, the large hole would be a bit larger than it was before and the small
cavity would be gone. In our scenario, therefore, everything would appear the
same as in the old paradigm but without the usual mathematical problems
associated with the metric.
Furthermore, in the free-fall coordinate system, the infalling observer would
never reach the velocity of light at the event horizon because the the small
cavities making up the observer would merge with the black hole just outside of
the event horizon. This, in turn, means that it is unnecessary to think that
space is flowing radially inward, disappearing into a cosmic sink-hole and, so,
the conservation laws for energy and momentum can be preserved.
Summary
“ What else is there out of which to build a particle except spacetime
itself?”
John Wheeler (1964)
If space is taken to be an infinite, ideal, fluid-like manifold with a positive
energy density and if cavitation bubbles are somehow induced in this fluid, the
resulting energy density gradients surrounding those cavities would be warpage
complexes in space that are indistinguishable from ordinary material objects.
If these sub-nuclear cavitation bubbles prove to be the basic building blocks
of matter, a massive object and its associated gravitational field would be
amalgamated into a single, complex pattern of warpage in space and nothing
more. The idea of massive particles as entities distinct from the space in
which they are imbedded would, therefore, be rendered superfluous.
This paper has argued that ordinary matter may be, like black holes, made
wholly and solely from the warpage of space itself. The advantages of this
paradigm are considerable. The entire structure of reality could be reduced to
just one thing: the vacuum of space. It would be unnecessary to think that
matter is crushed out of existence at the central singularity of a black hole;
the conservation laws would be preserved; space wouldn't have to drain away
into a cosmic sink-hole at infinite velocity; and there would be no
mathematical problems associated with the metric anywhere in space.
1. For an excellent explanation of the geometry of space around massive
objects, go to:
http://casa.colorado.edu/~ajsh/schwp.html#radius
. The “Ground Rules” section of this paper is based almost entirely
on Mr. Hamilton's web site.
(back)
2. Being “analytically removable” is not the same as being
completely eliminated. The singularity is removable, in the sense that all the
invariant measures of the field smoothly approach fixed finite values as
r
approaches
r
s
. For the mathematically inclined, a detailed discussion of the issues
surrounding the event horizon is given at:
http://www.mathpages.com/rr/s8-07/8-07.htm
(back)
3. Kip Thorne, “Black Holes & time Warps: Einstein's Outrageous
Legacy”, W. W. Norton & Company, 1994.
(back)
4.
http://www.math.ucr.edu/home/baez/einstein/node8.html
(back)
5. See equation 13 in section 1.3
Vacuum Energy
(in the sidebar table of contents) on this site:
http://nedwww.ipac.caltech.edu/level5/Carroll2/frames.html
(back)
PROLOGUE (A brief history of this paper)
|