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Electromagnetic Antigravity
L.L. Williams
Colorado Institute for Space Studies, 6009 Olympic, Manitou Springs, CO, 80829
Discovery of a coupling between gravity and electromagnetism could unleash a
phase of technological progress culminating in practical interstellar travel and the
mastery of gravitational force. The classical Kaluza-Klein 1, 2 theory of fivedimensional relativity offers a compelling and elegant theoretical basis for such a
coupling. While the field equations of this theory were studied extensively, the
equations of motion have received less attention. The equations of motion are
revisited here, and it is shown that the theory does predict electromagnetic
corrections to Newtonian gravity which do not couple to electric charge. The scalar
field characteristic of this theory is shown to vary on cosmological lengthscales,
and can be identified with the gravitational constant as in Brans- Dicke 3 theory.
This recognition allows a separation of scales of the dynamics of the
electromagnetic and gravitational fields from the scalar field, preserving the
terrestrial limit of vanishing electromagnetic fields. The gravitational constant
emerges as a universal charge-to-mass ratio, and electric charge is induced by
electromagnetic fields. The Newtonian limit is quantified here, predicting electric
fields of order 1018 volts would neutralize earth's surface gravity.
Only two years after the publication of his general theory of relativity, Einstein
received the manuscript from Kaluza 1 which suggested the equations of
electrodynamics could be recovered from the equations of general relativity, along with
the four-dimensional equations Einstein had recently derived, if only Einstein's new
theory were written in five dimensions.
2
As the greatest minds of that age took notice of and developed Kaluza's original
suggestion, they evidently carried in their work two prejudices. One was an apparent
interest in the unification of the field equations, and less attention paid to that other
aspect of the laws of physics: the equations of motion. The equations of motion describe
how matter is influenced by force, the field equations describe how force is influenced
by matter.
The other prejudice was that of the new quantum theory, emerging at the same
time as Kaluza's paper was published. The assumption was made that relativity and
quantum theory would be united. Electrodynamics did find reconciliation with quantum
theory and the other two quantum forces, and so the interest in the link between
electrodynamics and relativity has waned. We now know that for 80 years, the attempt
to unify relativity and quantum theory has been fruitless. The elegant and
straightforward unification of classical relativity and electrodynamics provided by fivedimensional relativity deserves a closer look, and in particular, its equations of motion.
Recovery of electrodynamics from five dimensional (5D) general relativity
requires two basic assumptions. One is the form of the 5D metric g~ in terms of the
ab
four-dimensional (4D) metric g  and the electromagnetic vector potential A  . This is
given by:
g~  g   k 2 A A / 
g~5  kA
g~55  
g~   g 
g~ 5   kA / 
g~ 55  (1  k 2 A2 /  ) / 
Greek indices run over 4D spacetime, roman indices over all 5 dimensions
and A2  g A A . Tildes denote 5D tensors. k is a constant.
(1)
3
This is essentially the 5D metric studied in a series of classic papers by Kaluza 1 ,
Klein 2 , 4 , Jordan 5 , and Thiry 6 . Reviews are provided by Bergmann 7 and Bargmann 8 .
Applequist et al.
9
provide a historical summary, translation of some of the papers
listed, and further references.
The other basic assumption is that no field depends on the fifth coordinate:
g~ab
0.
x 5
(2)
This constraint, known as the `cylinder condition', eliminates degrees of freedom but
still captures general relativity and electrodynamics. Unlike Klein 2 , we need not make
an interpretation of this relation in terms of the "size" of the fifth dimension.
~
~
~
The 5D vacuum Einstein equations, Gab  Rab  g~ab R / 2  0 , are used to fix the
~
value of k . G contains terms that can be identified with G  as well as with an
electromagnetic stress energy tensor TEM , and k is fixed by demanding correspondence
with the 4D Einstein equations G  (8G / c 4 )TEM , where G is the gravitational
constant and c is the speed of light. In this way we demand k 2  4G / c 4 .
~
The G55 equations provide the field equations for  , as Thiry 4 :
g      1/ 2  k 2 3 / 2 F F 
(3)
where   is the covariant 4D derivative, and F    A   A . The electromagnetic
field is a source for  , but the lengthscale of variation is large. An electromagnetic
energy density similar to the galactic average of one electron-volt per cubic centimeter
would yield a lengthscale approximating the radius of the universe. This is similar to the
arguments developed by Brans and Dicke 9 for their scalar field which they identified
with G , and variation in  can indeed be identified with variation in G / c 4 (Brans and
4
Dicke did not contemplate modifications to the equations of motion). Important for this
development is the assumption that  is approximately constant on terrestrial
lengthscales of interest, and thus forces induced by its derivatives are negligible in the
equations of motion.
The 5D equations of motion arise from the 5D geodesic hypothesis
~
dU a ~ a ~ b ~ c
 bcU U  0
d~
(4)
~
~
where b 2 d~ 2  g~ab dx a dx b , U a  dx a / d~ , bca is the 5D affine connection, and b is a
constant. We are interested in the 4D components of this equation, in terms of
c 2 d 2  g  dx  dx and U a  dx a / d :
 d 
 ~
 d 
2
 dU  ~ 5 2
d 2
~
~   

 55 (U )  25U 5U   
U U   U  ~ 2  0
d
 d

(5)
The equation for U 5 can be had without recourse to the geodesic hypothesis.
Rather, the cylinder condition (2) implies a conserved quantity
~
dU 5
~
~
~ 5 d

5
U 5  kA U  U  ~ (kA U  U )  ~  0
d
d
(6)
which allows the relation between the 4D and 5D line elements
~
2
b 2 U 52
 d 
 ~  2  2
c c
 d 
(7)
so that (7) can be taken constant under the assumption that  is not varying on
lengthscales of interest. This simplifies (5). Correspondence with the Lorentz force law
requires that the fifth component of the proper velocity must be identified with electric
charge, as originally noted by Kaluza 1 .
5
U5 c
q/m
2 G
(8)
where q and m are particle charge and mass, respectively. The gravitational constant is
seen to be a natural charge-to-mass ratio. We also have the result that electric charge (8)
is not conserved; rather, it is the related quantity described in (6), implying that electric
charge can be induced by an electromagnetic field. As momentum is a manifestation of
motion in space, and energy of `motion' in time, so electric charge is seen to be a
manifestation of motion in the fifth dimension.
Inspecting (5), it is clear that the term quadratic in the 4D proper velocity is the
usual 4D relativistic gravity. The term linear in the 4D velocity, with the identification
(8), is the Lorentz force term. The term quadratic in U 5 is entirely new. For protons and
electrons, that term would dominate the equations of motion in general because the
~
gravitational constant is a relatively small charge-to-mass ratio. However, 55
vanishes
for constant  .
Consider now the case of a body in motion under these forces and having no
electric charge in its rest frame. The metric (1) implies a purely electromagnetic
correction to the standard relativistic 4D field which does not couple to charge:
2G
~


 
 4 g  ( A F  A F )
c
(9)
The Newtonian limit of (5) consists of assuming the spatial components of the
4D proper velocity are much smaller than the time component, that the metric is
approximately Minkowskian, and that there is no time dependence. Under these
assumptions, the three-dimensional low-velocity flat-space equation of motion for an
uncharged body is
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
d 2x  
2G 
   g  2  e2   0
2
x 
dt
c

(10)
where  g is the Newtonian gravitational potential, and  e is an electric potential.
Apparently, the effective gravitational field inside a capacitor could be increased
or decreased by adjusting the direction of the electric field. A steady state voltage of
order 1018 volts, with a characteristic lengthscale of 10 cm, would neutralize the earth's
surface gravity. While enormous, such fields are predicted by (3) to produce
characteristic lengthscales in  of 10 4 km, consistent with the assumption of constant  ,
as well as negligible variations in the 4D metric on the lengthscales of interest. Just as
gravitational acceleration is independent of mass, so is the electromagnetic
neutralization, so that the same electric field would in principle float a battleship as
easily as a paper clip.
1. Kaluza, T. On the unity problem of physics, Sitzungsberichte der K. Preussischen
Akademie der Wissenschaften zu Berlin. 966 (1921). translated and reprinted in 12
2. Klein, O. Quantum theory and five dimensional theory of relativity, Zeitschrift f.
Phys., 37, 895 (1926), reprinted in 12.
3. Brans, C. & Dicke, R. Phys. Rev. 124, 925 (1961).
4. Klein, O. The atomicity of electricity as a quantum theory law, Nature, 118, 516
(1926).
5. Jordan, P., Erweiterung der projektiven Relativitatstheorie, Ann. d. Physik, 1, 219-228
(1947).
6. Thiry, M.Y., The equations of Kaluza's unified theory, Meeting 12 Jan 1948,
translation printed in 12
7. Bergmann, P. Introduction to the Theory of Relativity, Prentice-Hall: New York
(1942).
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8. Bargmann, V., Relativity, Rev. Mod. Phys. 29, 161-174 (1957).
9. Applequist, T., Chodos, A. and Freund, P.G.O., eds., Modern Kaluza-Klein Theories,
Addison-Wesley: Menlo Park, CA (1987).
Correspondence and requests for materials should be addressed to L.L.W. (e-mail: [email protected]).