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Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
105
Variation of Gravitational Mass in Electromagnetic Field
Z.-H. Weng1 and Y. Weng2
1
School of Physics and Mechanical & Electrical Engineering
Xiamen University, Xiamen 361005, China
2
College of Chemistry & Chemical Engineering
Xiamen University, Xiamen 361005, China
Abstract— A fraction of energy density is theoretically predicted to be captured from the
electromagnetic field to form a gravitating mass density, when a particle enters the strong field
from a region of no electromagnetism. In this paper the mass variation has been calculated for
the uncharged particle on free-fall in the electromagnetic field. It shows that there is an evident
effect to the variation in mass density when the uncharged particle is in the electromagnetic field.
1. INTRODUCTION
The equality of the inertial and gravitational masses is being doubted all the time. The equality of
masses remains as puzzling as ever. But the existing theories do not explain why the gravitational
mass has to equal the inertial mass, and then do not offer compelling reason for the empirical fact.
The paper attempts to reason out why there exists the equality of masses in most cases, even in
the electromagnetic field.
Some tests for the equality of masses have been performed by L. Eötvös, H. Potter, R. H. Dicke [1],
and V. B. Braginsky [2], etc. And more precise experiments have been carried out by I. I. Shapiro [3],
K. Nordtvedt [4], and J. H. Gundlach [5] etc. Presently, no deviation from this equality has ever
been found, at least to the accuracy 10−15 . But all of these verifications are solely constrained to
be in the range of weak gravitational strength, and have not been validated in the strong gravity
or the electromagnetic field. Therefore this puzzle of the equality of masses remains unclear and
has not satisfied results.
The paper puts forward a theoretical model to study the variation of gravitational mass density
in the electromagnetic field. It carries out the result that the mass density will be varied in the
strong strength by capture or release the energy density of electromagnetic field.
2. ELECTROMAGNETIC FIELD
The electromagnetic field can be described with the algebra of quaternions, which was invented
by W. R. Hamilton [6]. In the treatise on electromagnetism, the quaternion was first used by
J. C. Maxwell [7] to demonstrate the electromagnetic field. The gravitational field can be described
by the quaternion also, and worked out the variation of the gravitational mass density in the
gravitational field.
The gravitational field and electromagnetic field both can be illustrated by the quaternion,
but they are quite different from each other indeed. In some special cases, the electric charge is
combined with the mass to become the electron or the proton, etc. So the electric charge and the
mass have the same radius vector.
Some physical quantities are the functions of coordinates r0 , r1 , r2 , and r3 in these fields. But
the basis vector, Eg = (1, i1 , i2 , i3 ), of the gravitational field differs from the basis vector, Ee =
(e, j1 , j2 , j3 ), of the electromagnetic field. These two kinds of quaternion basis vectors combine
together to become the octonion basis vector E = Eg + Ee , with Ee = Eg ◦ e. The octonion was first
introduced by A. Cayley. The symbol ◦ denotes the octonion multiplication. The octonion basis
vectors satisfy the multiplication characteristics in Table 1.
In the quaternion space, the radius vector is R = r0 +Σ(rj ij ), and the velocity is V = c+Σ(vj ij ).
Where, r0 = ct; c is the speed of the light beam; t denotes the time; j = 1, 2, 3.
The gravitational potential is Ag = (a0 , a1 , a2 , a3 ), and the electromagnetic potential is Ae =
(A0 , A1 , A2 , A3 ). They constitute the potential A = Ag + keg Ae , with keg being the coefficient.
And the strength B = ♦ ◦ A consists of the gravitational strength Bg and the electromagnetic
strength Be .
B = Bg + keg Be
(1)
where, ♦ = ∂0 + Σ(ij ∂j ); ∂i = ∂/∂ri , i = 0, 1, 2, 3.
PIERS Proceedings, Beijing, China, March 23–27, 2009
106
Table 1: The octonion multiplication table.
1
i1
i2
i3
e
j1
j2
j3
1
1
i1
i2
i3
e
j1
j2
j3
i1
i1
−1
i3
−i2
j1
−e
−j3
j2
i2
i2
−i3
−1
i1
j2
j3
−e
−j1
i3
i3
i2
−i1
−1
j3
−j2
j1
−e
e
e
−j1
−j2
−j3
−1
i1
i2
i3
j1
j1
e
−j3
j2
−i1
−1
−i3
i2
j2
j2
j3
e
−j1
−i2
i3
−1
−i1
j3
j3
−j2
j1
e
−i3
−i2
i1
−1
In the above equation, we choose the following gauge conditions to simplify succeeding calculation.
∂0 a0 + ∇ · a = 0, ∂0 A0 + ∇ · A = 0.
where, ∇ = Σ(ij ∂j ); a = Σ(ij aj ); A = Σ(ij Aj ).
The gravitational strength Bg includes two components, g/c = ∂0 a + ∇a0 and b = ∇ × a,
while b = 0 in Newtonian gravity. And the electromagnetic strength Be involves two parts,
E ◦ e∗ /c = ∂0 A + ∇A0 and B ◦ e = ∇ × A; the ∗ denotes the conjugate of the octonion.
The linear momentum density Sg = mV is the part source of gravitational field, and the
electric current density Se = qV ◦ e is the source of electromagnetic field. They combine together
to become the source S . For the sake of including the energy density of the particle and of the
fields simultaneity, the source S is defined as,
µS = (B/c + ♦)∗ ◦ B = µg Sg + keg µe Se + B ∗ ◦ B/c
(2)
where, m is the mass density; q is the electric charge density; µ, µg , and µe are the constants;
2 = µ /µ ; ♦2 = ♦∗ ◦ ♦.
keg
g
e
Due to incorporate the different kinds of applied force density, the applied force density F is
defined from the linear momentum density P = µS /µg . The latter is the extension of the Sg .
F = c(B/c + ♦)∗ ◦ P
(3)
where, the above includes the density of gravity, inertial force, Lorentz force, and force between
magnetic moment with magnetic field, etc.
In the electromagnetic field and gravitational field, the low-mass uncharged particle is on equilibrium state, when the applied force density is equal to zero. And the related equilibrium equations
deduce the free-fall motion of the low-mass uncharged particle.
3. GRAVITATIONAL MASS
Considering the free-fall motion of a low-mass uncharged particle along the earth’s radial direction
i1 in the electromagnetic field and gravitational field of the earth. The radius vector R and velocity
V of the particle are respectively,
R = r0 + ri1 . V = c + V; V = V i1 .
(4)
and the strength exerted on the low-mass uncharged particle is
B = g/c + keg B. g = gi1 ; B = Bi1 ◦ e.
By Eq. (3), the equilibrium equation of applied force density is
¡
¢∗ ¡
¢
g/c2 + keg B/c + ♦ ◦ m0 c + mV = 0
where, m0 = m + 4m, 4m = B ∗ ◦ B/(µg c2 ); ♦ = ∂/∂r0 + i1 ∂/∂r.
(5)
Progress In Electromagnetics Research Symposium, Beijing, China, March 23–27, 2009
The above can be rearranged according to the gravitational field,
¡
¢∗ ¡
¢
g/c2 + ♦ ◦ m0 c + mV = 0
107
(6)
and it can be decomposed along the direction i1 further.
¡
¢
∂(mV)/∂t − m0 g + f M 2 / πR5 i1 = 0
where, M is the earth’s mass, and f is the gravitational constant; the ∂(mV)/∂t is written as
d(mV)/dt, when the time t is the sole variable.
If we consider the ∂(mV)/∂t and m0 g as the inertial force density and gravity density respectively, then the m and m0 will be the inertial mass density and gravitational mass density
correspondingly. And
¡
¢
¡
¢
4m = f M 2 / 4πc2 R4 + B 2 / c2 µe
(7)
where, R is the particle’s distance to the earth’s center.
From the above, we find the 4m has a limited effect on the motion of the large-mass particle,
because the 4m is quite small. Therefore the equality of masses is believed to be correct in most
cases. However, when there exists a very strong strength of electromagnetic field, the 4m can
become very huge, and then has an impact on the motion of the low-mass uncharged particle
obviously.
4. CONCLUSIONS
The gravitational mass density changes with the electromagnetic strength, and has a small deviation
from the inertial mass density. This states that it will violate the equality of masses in the fairly
strong strength of the electromagnetic field.
The definition of the gravitational mass equates the gravitational mass density m0 with the sum
of the inertial mass density m and the variation of gravitational mass density 4m. This inference
means that the strong strength of the electromagnetic field will cause the distinct variation of
gravitational mass density also. But this problem has never been discussed before. In other words,
the equality of masses has not been validated in the strong electromagnetic field.
It should be noted that the investigation for the gravitational mass density has examined only
one simple case, of which the low-mass uncharged particle is on the free-fall motion in the electromagnetic field and gravitational field. Despite its preliminary character, this study can clearly
indicate that the gravitational mass density is always changed with the electromagnetic strength.
For the future studies, the research will concentrate on only the predictions about the large variation
of gravitational mass density in the fairly strong strength of the electromagnetic field.
ACKNOWLEDGMENT
The author is grateful for the financial support from the National Natural Science Foundation
of China under grant number 60677039, Science & Technology Department of Fujian Province of
China under grant number 2005HZ1020 and 2006H0092, and Xiamen Science & Technology Bureau
of China under grant number 3502Z20055011.
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352–355, 1968.
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balance with angular acceleration feedback,” Phys. Rev. Lett., Vol. 85, No. 14, 2869–2872,
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