Event horizons in the PV Model

Abstract

2/27/2020 - In the paragraph leading up to equation (13), the formation was using R.A. d’Inverno's text, where he uses the 1st approximation in the path integration. As such, I expressed the speed of light as being inversely proportional to 1/sqrt(K(r, M)). This only holds for a 1st approximation, where sqrt(K(r, M)) ~ K(r, M) and are both approximately equal to 1. In general, throughout the rest of the paper, K(r, M) is correctly expressed as 1 / (1 - 2M/r), and the speed of light is proportional to 1/K(r, M). This use of the square root has caused some confusion regarding which form is correct. The correct form is found by solving the light-like line element for the coordinate speed of light, dr/dt, not dr/dtau. ______________________________________

The Polarizable Vacuum (PV) Model representation of General Relativity (GR) is used to show that an in-falling particle of matter will reach the central mass object in a finite amount of proper time, as measured along the world line of the particle, when using the PV Metric. It is shown that the in-falling particle passes through an event horizon, analogous to that found in the Schwarzschild solution of GR. Once it passes through this horizon, any light signal emitted outward by the in-falling particle will be moving slower than the in-falling particle, due to the reduced speed of light in this region. Therefore the signal can never escape this horizon. However, the light emitted by a stationary object below the horizon is exponentially red-shifted and can escape along the null geodesics, as was originally predicted by the PV Model. A static, non-rotating charge distribution is added to the central mass and the PV equivalent to the Reissner-Nordstrom metric is derived. It is illustrated that the dipole moment induced in a neutral, polarizable body, reduces the effects of gravity in the strong field region. We demonstrate the existence of the event horizon and how it may be affected by the presence of electric charge. v1 Final Release

Full text

PACS numbers: 03.03.+p, 03.50.De, 04.20.Cv,
04.25.Dm, 04.40.Nr, 04.50.+h
Event horizons in the PV Model
Todd J. Desiato
1
, Riccardo C. Storti
2
April 15, 2003 v1
Abstract
The Polarizable Vacuum (PV) Model representation of General Relativity (GR) is used to show
that an in-falling particle of matter will reach the central mass object in a finite amount of proper time, as
measured along the world line of the particle, when using the PV Metric. It is shown that the in-falling
particle passes through an event horizon, analogous to that found in the Schwarzschild solution of GR.
Once it passes through this horizon, any light signal emitted outward by the in-falling particle will be
moving slower than the in-falling particle, due to the reduced speed of light in this region. Therefore the
signal can never escape this horizon. However, the light emitted by a stationary object below the horizon is
exponentially red-shifted and can escape along the null geodesics, as was originally predicted by the PV
Model.
A static, non-rotating charge distribution is added to the central mass and the PV equivalent to the
Reissner-Nordstrom metric is derived. It is illustrated that the dipole moment induced in a neutral,
polarizable body, reduces the effects of gravity in the strong field region. We demonstrate the existence of
the event horizon and how it may be affected by the presence of electric charge.
v1 Final Release
1
tdesiato@deltagroupengineering.com Delta Group Research, LLC. San Diego, CA. USA, an affiliate of
Delta Group Engineering, P/L, Melbourne, AU.
2
rstorti@deltagroupengineering.com Delta Group Engineering, P/L, Melbourne, AU, an affiliate of Delta
Group Research, LLC. San Diego, CA. USA,

2
1 INTRODUCTION
The Polarizable Vacuum (PV) Model has a long history as a heuristic tool for describing
gravitation.[1,2,3] In the PV Model, it is assumed that the speed of light and the Refractive Index are
variables of the coordinates in the PV medium. As demonstrated in [4,5,6], the PV Model passes four
crucial tests of General Relativity (GR). Three of which are the predictions of gravitational red-shift, the
bending of light by a star and the advance of the perihelion of the planet Mercury. It can also be cast in a
metric representation that leads to the Schwarzschild vacuum solution of GR as an approximation to an
exponential metric coefficient. The inclusion of charge also leads to a representation of the Reissner-
Nordstrom metric. [4,5,6]
The success of the PV Model is remarkable given that it is not a geometric model like GR. It is a
model based on the polarizability of the classical vacuum and described by variations of the classical
constants of permeability, “
o
µ
” and permittivity, “
o
ε
” as a function of the coordinates, along the lines of
“
TH
ε
µ
” methodology. Note that in the Gaussian system of units, the values of inductance and capacitance
are derived from geometrical units of length and time. Hence, historical geometrical representations also
exist.
In the PV Model, the vacuum of space-time is endowed with a variable Refractive Index “
K
”, whose
value as measured in a locally inertial reference frame is
1K =
. The local value is always unity because
the rulers and clocks which are used to measure the speed of light, “ c ” are in effect calibrated by the local
value of “
K
”, such that “ c ” and “
K
” remain constant, in all locally inertial reference frames.
However, a distant observer at infinity introduces the value of “
K
” by assuming that the speed of
light is different, in the reference frame of the distant Gravity Well he observes.
Local Observer Distant Observer
o
µ
o
K
µ
o
ε
o
K
ε
c
/cK
Table 1: Constants as defined by local and distant observers.
Relative to the distant observer, who uses a globally flat coordinate system in his local rest frame,
variations in the gravitational field manifest as variations in the value of “
K ”. Its value may be determined
by measurements of gravitational red shift, as seen by the distant observer. We may then restore the value
to unity by choosing the appropriate coordinate map and by assuming the constant value of “ c ”, thereby
restoring the geometric interpretation of GR, [7] in the weak field approximation.
2 EVENT HORIZONS
In the strong field, the PV Model[4,5,6] does not predict the event horizon of a black hole.[7,8] A
light signal emitted by a stationary object at any distance from the central mass object, can escape along a
null geodesic. The frequency of the light emission, “
o
ω
” will be red-shifted exponentially as it emerges
from the Gravity Well,
()
()
2
exp
,
o
o
GM
K
rc
KrM
ω
ωω

==−


(1)
Where “
()
,KrM ” is the Refractive Index at the location of emission at a distance “ r ” from the central
mass, “
M
”. The frequency measured by the distant observer is “
()
K
ω
”.
In Geometric units, 1cG== and the mass defined by,
2
/MGMc≡ has units of length. The
radial component of the Schwarzschild line element of GR is then follows,[8]

3
()
22 2
1
(1 2 / )
12 /
ds M r dt dr
Mr
=− −
−
(2)
In the PV Model, the equivalent line element is,
()
()
222
1
,
,
ds dt K r M dr
KrM
=−
(3)
Where,
()
2
12212
exp 1 ...
,2
MMM
KrM r r r
−
 
=−++


 
 (4)
Using GR, d’Inverno[8] explains how a free particle falling into a black hole will follow a time
like geodesic. The equations are,
2
1
Mdt
rd
κ
τ

−=


(5)
22
21
11
2
1
Mdt dr
M
rd d
r
ττ
 
−− =
 

 
−


(6)
Where “
τ
” is the proper time along the world line of the particle. Different initial conditions are
represented by the constant “
κ
”. Choosing conditions where the particle starts from infinity at rest, the
value of
1
κ
=
.[8] Asymptotically, at large values of “ r ”,
t
τ

. This leads to the result,
2
2
dr
dr M
τ

=


(7)
If the particle is at “
o
r ” at time “
o
τ
” then the particle will reach the singularity in a finite proper
time,[8] as follows,
33
22
2
32
oo
rr
M
ττ

−= −


(8)
The event horizon in this representation is at 2rM=
. For values of
2rM<
the velocity of the in-falling
particle along its world line exceeds the speed of light.
1
dr
d
τ
> (9)
If the particle were a source of light, then after it falls below the event horizon, any light signal emitted
outward by the particle cannot be seen outside that horizon.
Similarly, the PV Model also requires that
t
τ

asymptotically at large values of “ r ”. Therefore,
the same derivation may be applied. The coefficients are replaced in (5) and (6) as follows,
()
1
1
,
dt
KrM d
τ
=
(10)

4
()
()
22
1
,1
,
dt dr
KrM
KrM d d
ττ
 
−=
 
 
(11)
This leads to a result analogous to equation (8), that an in-falling particle in the PV Model will reach the
central mass in a finite amount of proper time.
()
0
0
1
2
1
1exp
,
o
o
r
o
r
dr dr
M
KrM
r
ττ
−=− =
−

−
−


∫∫
(12)
The solution of this integral is presented graphically in Fig. 1, for a normalized central mass,
1M =
where
()
r_0
o
ττ τ
−=
as shown. The following illustrations were created using MathCAD.
Fig. 1
In the PV Model, the variable speed of light must be considered when finding the equivalent
representation of equation (9). As the in-falling particle is accelerated toward the center of gravity, the
speed of light decreases due to the increasing value of “
()
,KrM
”. The speed of light at any distance may
be expressed as,
()
()
1
v, exp
,
M
rM
r
KrM
−

==


(13)
Where as, the speed of the in-falling particle along its world line is,
()
1
1
,
dr
dKrM
τ
=− (14)
It is illustrated in Fig. 2, for
1M =
that the event horizon occurs at approximately, 3
h
rM . In
Fig. 2 “ /dr d
τ
” is represented as “dr(r_0,M)”. The GR escape velocity “ 2/Mr” has been included for
reference.

5
Fig. 2
After the in-falling particle crosses this horizon, any signal emitted outward by the in-falling particle cannot
escape, because the speed of the outgoing light signal is now less than the speed of the in-falling particle.
Furthermore, if the classical orbital escape velocity “ v
esc
” is calculated, it is found to be equal to
equation (14).[8]
()
()
2
2
1
v , 1
2
12
v1
,
1
v1
,
esc
esc
esc
GMm
mcG
r
M
KrM r
KrM
==
→=−
→=−


(15)
This means that, for a mass “ m ” to orbit the central mass object “
M
” at distances less than “
h
r ”, then
()
v/,
esc
cKrM>
in that region. Therefore, these orbits cannot exist and a mass below the horizon cannot
escape and will fall all the way to the central mass object, in a finite proper time.
3 THE EFFECT OF CHARGE ON THE POLARIZABLE VACUUM
Consider a static, non-rotating electrically charged central mass object, such that the space-time
surrounding it may be represented by the Reissner-Nordstrom metric.[8] The radial part of the metric as in
(2) is,
2
22 2
2
2
2
21
1
2
1
MQ
ds dt dr
rr
MQ
rr

=− + −



−+


(16)
Where “
Q
” is interpreted as the electrical charge on the central mass. The PV version of this would then
be,
()
()
222
1
,,
,,
ds dt K r M Q dr
KrMQ
=− (17)
Where,
()
22
22
12 2
exp 1 ...
,,
MQ MQ
KrMQ r rrr


=− − −++





(18)

6
The speed of light, the speed of the in-falling neutral particle and the event horizon are each
modified by the presence of charge on the central mass object. This is illustrated in Fig. 3.
Fig. 3
The traces from Fig. 2 (blue, red) are included for reference. The bottom most (magenta) trace
represents the velocity of the in-falling particle and is reduced by the charge. The upper (black) trace
represents the increased velocity of light in the new Refractive Index, while the geometric radius of the
event horizon has decreased.
Additional charge will eliminate the horizon completely as shown in Fig. 4.
Fig. 4
These figures are analogous to those usually represented by the Reissner-Nordstrom metric, outside a
charged black hole.[8] The potential well in Fig. 4 will allow a neutral, uncharged object to rest stationary
above the central mass, at a distance incrementally less than
2rM=
. This is never possible with an
uncharged central mass.
Using the line element for the PV Model (3), the effects of gravity on a material rod of unit length
contracts the length of the rod.[4,5,6] The addition of electric charge in (17) and therefore the presence of
radial electric fields have the opposite effect. That of stretching the length of the rod and thereby reducing
the effects of gravity.
This may be understood in terms of a classical electric field acting on any neutral, polarizable
material. The field polarizes the rod by attracting opposite charge and repelling like charge, inducing a
dipole moment within the material. This inherently leads to a stretching of the molecular bonds that hold
the material together. But does this stretching also affect "time"?
According to the Reissner-Nordstrom metric it does. By polarizing matter in an electric field, its
proper length and proper time may be affected. The well-known Stark Shift demonstrates that atomic
spectra are modified by an external electric field.[9] This may be interpreted to support this conjecture,
however more research is required in this area.

7
The induced dipole moment in the material also reduces its energy density. Using a dipole
approximation, a density of dipoles has a dipole moment “
()
ω
d
”. A dipole moment for one electric dipole
may be usefully approximated by,
dqx
=⋅
(19)
In terms of the Fourier components of the electric field “
()
ω
E
”, of frequency "
ω
",[10]
() ()()
ωαω ω
=dE
(20)
Where “
()
αω
” is the polarizability of the material, at the frequency "
ω
". The energy "W ", or Work,
associated with the induced dipole moment on matter is, [9]
() () () () ()
2
11
22
W
ωωωαωω
=− =−dE Ei (21)
Where “
()
2
ω
E ” may be interpreted as the Intensity of the field.
For any real positive value of polarizability "
()
αω
", the energy density is negative and
proportional to
()
2
1
2
ω
− E . This result is consistent with the violation of the Weak Energy Condition
(WEC) required to raise the speed of light in all Faster than Light (FTL) conjectures.[11,12,13,14]
4 CONCLUSION
The existence of event horizons in the PV Model has been demonstrated and compared to the
analogous horizons in GR. It was shown that electric charge can affect the Refractive Index and change the
radial position of the event horizon in the Reissner-Nordstrom metric space-time. The electric field
polarizes an in-falling body of material and reduces the effects of gravity. From this we infer that space-
time geometry may be modified or engineered by purposeful design, utilizing modern electromagnetic
theory[10] and technologies. We anticipate that these examples may hold the key to developing new
technologies that may one day allow these conjectures to become widely accepted possibilities.
Acknowledgement
This research was made possible by the collaborative efforts of Delta Group Engineering, P/L,
Melbourne, AU, and Delta Group Research, LLC. San Diego, CA. USA.

8
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