GRAVIMAGNETISM
CAUSALITY AND JEFIMENKO'S GRAVITY
In establishing his correspondence between gravity and the electromagnetic
field, based primarily on causality and the effects of retardation,
Jefimenko, in *Causality, Electromagnetism, and Gravity*, creates the
correspondence of G to -1/(4*Pi*epsilon_g_0) to -mu_g_0*c^2/(4*Pi). The
term epsilon_g_0 here is the gravitational equivalent to the electrostatic
permittivity of the vacuum epsilon_0, and mu_g_0 is the equivalent to the
magnetic permeability of the vacuum mu_0, as will be explained below.
Jefimenko's version of EM fully accounts for causality, i.e. the fact that
a cause at a distance d can not precede the effect by time delta t which is
less than d/c (or d/c_g in the case of gravity.) Jefimenko shows that
causality justifies invention of the co-gravitational field K, analogous to
B. This will be shown below to make a full gravitational-electromagnetic
field isomorphism possible. Jefimenko demonstrates that B, and thus K, are
merely computed quantities, secondary quantities that necessarily follow
from the only true causes, the interaction of charge upon charge or mass
upon mass. This provides strong evidence for the "real" existence of K, as
"real" as B, i.e. that an (apparent) K can be observed experimentally to
the same extent B can, though it is much more difficult to observe due to
the extreme orders of magnitude involved. In other words, if causal
electromagnetism is correct, then the causal gravity is also necessarily
correct. The isomorphism holds by necessity because the full set of
postulates have already been experimentally verified. However, if it turns
out that causal electromagnetism is incorrect, and B exists in a real
sense, then it does not follow that K can (any longer) be assumed to exist
on the basis that it is merely a computed quantity, like energy.
B and E are variable when the velocity of the observer is taken into
account. This magnitude dependence on observer velocity is fully accounted
for by causality treatment, because the relative velocity of the observer
merely changes the apparent retardation. This aspect even more fully
justifies Jefimenko's treatment of B as an artifact of charge motion.
In Jefimenko's text the world of gravity and electromagnetism are
maintained as separate worlds, and merely corresponded to each other.
Jefimenko thus uses epsilon_0 in the gravity context to mean -1/(4*Pi*G),
and mu_0 to mean -4*Pi*G/c^2. He also uses c to mean the speed of
propagation of gravity.
Here use the new notation epsilon_g_0 to mean the permittivity of space to
gravity, mu_g_0 to mean permeability of space to co-gravity, and c_g to
mean the speed of gravity propagation. So far there is really no change
with the view of Jefimenko, only an extended notation. There are some
immediate advantages to this notation, however. First it provides
corresponding constants which could have been nicely used in the EM to
gravity correspondences on page 104 of Jefimenko's book: *Causality,
Electromagnetism, and Gravity*. Namely we could have the new Table 1,
shown below.
Electric Gravitational
q m
E g
B K
J J_g
epsilon_0 epsilon_g_0
mu_0 mu_g_0
Table 1: Initial Gravity-electromagnetism Isomorphism
Correspondence Table
However, this is still not ideal. We have a problem with signs, as it
appears did Jefimenko, but which he remedies by placing minus signs in the
corresponding formulae. The problem lies in the fact that, to maintain the
convention that a positive force is repelling, we end up with sign problems
between the force equations:
Fg = G*(m1*m2/r)
for gravity and
Fe = k*(q1*q2/r^2) = (1/(4*Pi*epsilon_0))(q1*q2/r^2)
for the Coulomb force.
Jefimenko fixes this problem by making his epsilon_g_0 and mu_g_0 negative.
Thus, in effect he has the gravitational equivalent to the above:
Fg = G*(m1*m2/r) = (-1/(4*Pi*epsilon_g_0))*(m1*m2/r^2)
His gravitational permittivity and co-gravitational permeability thus end
up negative in order to preserve the correct sign on force. This
eventually causes problems. An example is the Poynting vector
correspondence:
S = (1/mu_0) E x B
vs the Jefimenko gravitational version:
P = (c^2/(4*Pi*G)) K x g = (1/mu_g_0) K x g
Note that Jefimenko here reverses K and G instead of using an arbitrarily
placed minus sign.
It appears that there is a handy way out of this lack of true isomorphism.
That solution is to specify the sign of the mass charge in terms of i =
(-1)^(1/2), the imaginary number i. Charge has sign, so why not mass?
This then makes the isomorphism complete. We now have
epsilon_g_0 = 1/(4*Pi*G)
mu_g_0 = 4*Pi*G/(c_g)^2
and all the formulae then exactly correspond, including signs. The
disadvantage to this approach is that the imaginary number i must be
carried throughout the gravitational field units. Perhaps this is really
an unexpected advantage though. Gravitational fields are imaginary,
electromagnetic are real. There is then some hidden meaning to this? One
is that the two worlds ARE for the most part disconnected. We have in fact
an indication of field *dis-unification*. Additionally we have that
anti-gravitational matter, if it exists as implied by symmetry, would then
carry sign (-i).
SPEED OF GRAVITY
Jefimenko adapts his theory to account for general relativistic effects by
adjusting the speed of gravity. He notes (p. 135 ff.) that to account for
the precession of the perihelion of Mercury, that the speed of propagation
of gravity must be about 0.3 c. We thus have
c_g = 0.3 * c
and we know that
(c_g)^2 * epsilon_g_0 * mu_0_1 = 1
thus
(0.09 c^2) * epsilon_g_0 * mu_0_1 = 1
and we also have
mu_0 = (4/0.9)*Pi*G/c^2
We now have the full correspondence:
Electric Gravitational
q m * i
E g
B K
J J_g
epsilon_0 epsilon_g_0 = 1.192602x10^9 kg s^2/m^3
mu_0 mu_g_0 = 1.037x10^-25 m/kg
c c_g = = 8.99x10^7 m/s (Jefimenko's estimate)
Table 2: Gravity-electromagnetism Isomorphism
(Without Relativity)
Correspondence Table
where we now (roughly) know epsilon_g_0, mu_g_0, and c_g, and gravitational
mass is expressed in terms of imaginary units i. J_g is mass current.
Inertial mass everywhere in relativistic cases is the relativistic mass
m0*gamma. We have a complete field isomorphism.
This isomorphism implies both a connection, as well as disconnection,
between the electromagnetic and gravitational fields. We have achieved a
form of "field dis-unification." The existence of i in some resulting
equations distinctly and permanently isolates the purely gravitational
fields and masses from electromagnetic components. We also now have
computed fundamental constants: c_g, epsilon_g_0, and mu_g_0, as they must
be according to Jefimenko's theory.
THE PROPOSED ISOMORPHISM IS NOT LIMITED TO JEFIMENKO'S VISION OF EM
Any complete theory of electromagnetism, including electromagnetism within
the framework of relativity, can be used to create an isomorphism between
electromagnetism and gravity, provided B in the theory is not real in the
sense it is simply a byproduct of the other laws of the electromagnetic
theory, and the electromagnetic vector potential function can be be derived
from the (retarded) motion of charge. Jefimenko showed that the law of
causality, if postulated, ensures that B meets this criteria. It is
suggested here that the subject isomorphism can be established by first
measuring or establishing the rate of propagation of gravity, c_g. We then
can compute the permeability of space to co-gravity:
mu_g_0 = 4*Pi*G/(c_g)^2
and the permittivity of space to gravity:
epsilon_g_0 = 1/(4*Pi*G).
It is expected that c_g = c when full relativistic effects are applied,
though, the ratio c/c_g is likley to change within close range to massive
objects, due to the fact gravity and electromagnetism operate in separate
spacial dimensions. A minimum number of dimensions for a full relativistic
gravimagnetism description then is 7.
We now establish the isomorphism by applying the following rules to every
electromagnetic law in order to obtain corresponding gravitational laws.
Replace c, mu_0 and epsilon_0 with corresponding terms c_g, mu_g_0, and
epsilon_g_0 above. Co-gravity K is defined as the gravitational equivalent
to (corresponds under the isomorphism to) B, the magnetic field intensity
B. Gravity g is defined as the gravitational equivalent of the
electrostatic field E. Wherever charge is used, gravitational mass
(gravitational charge) is substituted, with the sign of the charge removed
(if ordinary matter is involved, i.e. not anti-gravitational matter) and
replaced by the imaginary number i. J_g is the mass current vector
corresponding to current density vector J.
When relativity is included, we then have the full correspondence:
Electric Gravitational
q m * i
E g
B K
J J_g
epsilon_0 epsilon_g_0 = 1.192602x10^9 kg s^2/m^3
mu_0 mu_g_0 = 1.037x10^-25 m/kg
c c_g = c
Table 3: Gravity-electromagnetism Isomorphism
Correspondence Table
NOTATION AND NOMENCLATURE RELATED TO GRAVITATION
The EM-GK isomorphism provides analogs to a vast quantity of physical laws,
formulae and terms. This can cause much confusion in the process of
attempting to assign names and symbols the gravitational analog items.
To be consistent, and end terminology confusion, when discussing or
expanding the isomorphism proposed here between the electromagnetic (EM)
and gravikinetic (GK) fields, when referring to a gravitational feature the
analogous term borrowed from the EM universe should be prefixed with
"gravi" to indicate that that analogous feature is in the GK universe. If
it is not appropriate to prefix a term with "gravi" then it can be preceded
with the adjective "gravitational".
Under the proposed EM-GK isomorphism every variable, every formula, every
unit in EM has a corresponding value, a gravitational analog. The formulas
and variables from the EM world should be used faithfully, and simply
subscripted where necessary with a g to designate the GK analog.
The exceptions to these rules are the variables g, and G, and
co-gravitational field K, which is hereby now called the gravimagnetic
field K, which are symbols that already have specific meanings.
Based on the above principles, the following are sample correspondences:
electrostatic field E: gravitational field g
magnetic field B: gravimagnetic field K
electromagnetic (EM) : gravikinetic (GK) (a necessary rule exception)
charge: gravicharge (an imaginary quantity in units of +i kg, or
possibly -i kg, not to be confused with mass)
current: gravicurrent (an imaginary quantity in units of +i kg/s)
magnet: gravimagnet
monopole: gravimonopole
Poynting vector P: gravitational Poynting vector P_g
ohm (omega): graviohm (omega_g)
permittivity (epsilon) : gravipermittivity (epsilon_g)
permeability (mu) : gravipermeability (mu_g)
lightspeed (c): gravispeed (c_g)
impedance of the vacuum (nu): graviimpedence of the vacuum (nu_g)
Maxwell's laws of electromagnetism: Maxwell's laws of gravimagnetism
Gauss' Law of electric flux: Gauss' Law of gravitational flux
Laplace's Law of Electrostatic potential: Laplace's Law of Gravitational
Potential
Similar terminology should be used when applied to the laws of Lenz,
Biot-Savart, Ampere, Ohm, etc. The theory itself, the EM-GK Isomorphic
Theory, can thus simply be called a theory of gravimagnetism.
This approach to nomenclature puts an end to the need for all kinds of
special terms and variables. Also, when the meaning is clear, one can
simply dispense with the g subscripts, and thus incur no notation overhead
whatsoever. Note that this approach would not work well if the isomorphism
were not complete.
SOME COMPUTATIONS
Let's try a sample calculation using the isomorphism.
The mass of the earth is m_t_earth = 5.98x10^24 kg. The radius of earth is
6378 m. The moment of inertia for a sphere of radius r and mass M is (2/5)
M r. For estimating purposes, considering the iron core, we might assume
the mass is located in a ring of radius 2300 km, rotating once every day,
i.e. at 2*Pi*2300 km/day = 167 m/s. The gravicurrent is (5.98x10^24 kg i
kg)/day = 6.92x10^41 i kg/s. Note that i here is the imaginary number
(-1)^(1/2). The gravimagnetic dipole moment mu_k of the earth's
gravicurrent is thus the gravicurrent times the area of the current loop,
or (6.92x10^41 i kg/s)(Pi*(2300 km)^2) gives:
mu_k_earth = 1.15x10^55 i kg m^2/s
Let us now compute the gravimagnetic dipole moment of a spinning steel ball
of radius 1 cm spinning at 30,000 rpm. The volume of the ball is 4/3 Pi (1
cm)^3 = 4.189 cm^3, so its mass is about (7.14 g/cm^3)(4.189 cm^3) = 29.9
gm = .0299 kg. This is effectively spinning at a radius of .004 m, so has
a velocity of (30000*2*Pi*.4 cm)/(60 s) = 12.56 m/s. The mass turns 30000
rpm/(60 s/m) = 550 rps and thus gravicurrent of (.0299 i kg)(500/s) = 14.95
i kg/s. The gravimagnetic dipole moment is thus the gravimagnetic dipole
moment mu_k of the steel ball's gravicurrent is thus that gravicurrent
times the area of the gravicurrent loop, or (14.95 i kg/s)(Pi*(.004 m)^2)
gives:
mu_k_ball = 7.51x10^-4 i kg m^2/s
To properly calculate the gravimagnetic force between the earth and
spinning steel ball, we might use a computer program to fully integrate the
gravimagnetic field in the presence of the steel ball considering the
density of the earth at each radius. However, for the sake of a first try
at estimating the gravimagnetic field force we can simply assume the
earth's gravimagnetic field to be due to a gravimagnetic dipole located at
the center of the earth in the plane of the equator.
The EM force between two magnetic dipoles mutually aligned on their axes is:
F_mu = [-3 mu0/(2 Pi)] Mu_1 Mu_2 / r^4
Note that a positive force is repelling. Two mutually aligned gravicurrent
coils (rings) generate an attracting force because their fields are aligned
N-S ... N-S.
For convenience, let's use Jefimenko's value of mu_g_0 = 1.037x10^-25 m/kg
in order to convert the above into the isomorphic gravitational analog for
the earth and ball:
F_mu_g = [-3 mu_0_g/(2 Pi)] Mu_1 Mu_2 / r^4
F_mu_g = [-3 mu_0_g/(2 Pi)] mu_k_earth mu_k_ball / r^4
where Mu_1 and Mu_2 are gravimagnetic dipole moments expressed in units of
[i kg/(m s)] and the separation distance r is in meters. For simplicity's
sake let's assume the experiment is performed at the north pole. This
gives r = 6378 m and the force is:
F_mu_g = [-3 (1.037x10^-25 m/kg)/(2 Pi)]
* (1.15x10^55 i kg m^2/s) * (7.51x10^-4 i kg m^2/s) /
(6378 km)^4
= [4.95x10^-26 m/kg] * (1.765x10^51 kg^2 m^4/s^2) /
1.655x10^27 m^4
= .0528 N = 5.38 gf
and we seem to be way off. However, if we use c_g = c we have:
mu_g_0 = 4*Pi*G/c^2 = 9.3295973x10^-27 m/kg
and the ratio of the two gravipermeabilities is (9.3295973x10^-27
m/kg)/(1.037x10^-25 m/kg) = 0.09 and we get a force of (5.38 gf)*0.09) =
0.484 gf, which is still too high. This gives an acceleration due to
kinetic force of (0.484 gf)/(29.9 gm) = 0.01619 g. However, the assumptions
provide only a crude estimate of the force. A careful integration of the
co-gravitational field of the earth is required. Also, the approximation
to the force between dipoles used here is only valid at much larger
(relative) distances. It is not valid up close to the earth. A finite
element approach may be the best way to get accurate results for this kind
of problem.
Regards,
Horace Heffner
