**Coupling
of Electromagnetism and Gravitation**

**in
the Weak Field Approximation**

**Authors:**

**M. Tajmar**, *Austrian
Research Center Seibersdorf, A-2444 Seibersdorf, Austria*, martin.tajmar@arcs.ac.at

**C. J. de Matos**, *ESA-ESTEC,
Directorate of Scientific Programmes, PO Box 299, NL-2200 AG Noordwijk, The
Netherlands*, cdematos@estec.esa.nl

**Abstract: ** Using the weak field
approximation, we can express the theory of general relativity in a
Maxwell-type structure comparable to electromagnetism. We find that every electromagnetic field is
coupled to a gravitoelectric and gravitomagnetic field. Acknowledging the fact
that both fields originate from the same source, the particle, we can express
the magnetic and electric field through their gravitational respective
analogues using the proportionality coefficient *k*. This coefficient depends on the
ratio of mass and charge and the ratio between the electromagnetic and
gravitic-gravitomagnetic permittivity and permeability respectively. Although the coefficient is very small, the
fact that electromagnetic fields in material media can be used to generate
gravitational and gravitomagnetic fields and vice versa is not commonly
known. We find that the coupling
coefficient can be increased by massive ion currents, and electron and nuclear spin-alignment. Advances in material sciences, cryogenic
technology and high frequency electromagnetic fields in material media may lead
to applications of the derived relationships.

1. Introduction

Neglecting
effects due to space curvature and special relativity, the theory of general
relativity, which describes that gravitation can be linearized, unveiling a
structure comparable to the Maxwell equations, which describe
electromagnetism. This weak field
approximation splits gravitation into components similar to the electric and
magnetic field. Although masses attract each other and equal electric charges
repel, the comparable structure between both the gravitational and the
electromagnetic field gives rise to a coupling between them because both fields
originate from the same point source, the particle. In the case of the gravitational field, the source is the mass of
the particle, whereas in the case of the electromagnetic field, the source is
the charge of the particle. Moving the
particle will both create a magnetic and, using the analogy term, a
gravitomagnetic field simultaneously.
Hence, the induction of a gravitational field and (or) a gravitomagnetic
field using electromagnetic fields applied to material media must be
possible. This fact is not commonly
known in nowadays physics.

The
splitting of gravitation into electric and magnetic components originates back
to as early as 1983 when Heaviside^{1} investigated how energy is
propagated in a gravitational field.
His proposed gravitational Poynting vector contained the magnetic
component of gravitation, which is hidden in the tensor equations of Einstein’s
general relativity theory, published more then 30 years afterwards. Forward^{2} first expressed the
linearized general relativity equations in a Maxwell-structure and proposed
experiments to detect gravitomagnetic and non-Newtonian gravitational fields.^{3 } Further experiments were proposed by Braginski^{4}
indicating an increasing interest of detecting effects related to general
relativity using laboratory equipment. However, the proposed experiments
suggested effects too small for being detected. Recently, several approaches to
produce larger gravitomagnetic fields using the unique properties of
superconductors appeared^{5,6,7} following the first approaches from
DeWitt and Ross of modifying the London equation to include the gravitomagnetic
field.^{8,9}

The
main objective of this paper is to show that every electromagnetic field is
coupled to a gravitoelectric-gravitomagnetic field and that induction between both
fields is possible. The derived
coupling is generally valid and does not require special properties like
superconductivity. This enables a new
approach to think about experiments which could modify the gravitational fields
in terrestrial laboratories. Successful
means for producing significant non-Newtonian fields promise many spin-off
applications presently only possible in the microgravity environment of space.^{10}

2. Weak Field
Approximation

In deriving an analogy between
gravitation and electromagnetism, the following assumptions have been made:

1.
all motions are much slower than the speed of light to neglect
special relativity,

2.
the kinetic or potential energy of all bodies being considered
is much smaller than their mass energy to neglect space curvature effects,

3.
the gravitational fields are always weak enough so that
superposition is valid,

4.
the distance between objects is not so large that we have to
take retardation into account.

The
procedure of linearizing Einstein’s theory of general relativity is included in
most textbooks.^{11,12} We
start with Einstein’s field equation:

_{
(1)}

Due to assumptions (1) and (3), the metric can be
approximated by

_{
(2) }

where the Greek indices *a*,* b *
= 0, 1, 2, 3 and *h*_{a}_{b}_{ }=
(+1, -1, -1, -1) is the flat space-time
metric tensor, and _{}<< 1 is the perturbation to the flat metric. Using this form of the metric, we can
approximate the Ricci tensor and the Ricci scalar by

_{
(3) }

_{
(4) }

where in obtaining Equation (3) and (4) we choose our
coordinate system so that we have the following "gauge" condition

_{
(5)}

Defining the gravitational potential as

_{
(6) }

we can substitute Equation (3) and (4) into (1) and get

_{
(7) }

3. The
Einstein-Maxwell-Type Gravitational Equations

In this
section we will derive the gravitational analogues to the scalar and vector
potential used to express the Einstein-Maxwell equations. We will cover only the necessary results,
for a detailed analysis the reader is referred to the literature.^{13,14}

In the first approximation (zero
order for the energy momentum tensor), we assume that all quantities are not
varying with time. Then the time
derivative of the gravitational potential is zero and all components of the
energy-momentum tensor are zero except

_{
(8) }

where *r*_{m}
is the mass density. Hence, the weak
field approximation in Equation (7) reduces to the Poisson equation with the
solution

_{
(9) }

where *F*_{g}
is the gravitational analog to the scalar potential and *e*_{g}
the gravitational permittivity which we define using Newton’s gravitational
constant *G* as

_{
(10)}

In the next higher approximation (first order for the
energy-momentum tensor), we still assume that the potential is not varying in
time, but that the masses involved are moving at a certain velocity or rotate.
Then the energy-momentum tensor will have the additional components

_{
(11)}

Using again the weak field Equation (7) we can define the
analog to the vector potential using the mass density flow_{}as

_{
(12)}

where *m*_{g}
is the gravitomagnetic permeability defined by

_{
(13) }

Here we use a different definition of the gravitomagnetic
permeability than various other publications^{2} so that

_{
(14)}

which expresses, that a gravitational wave travels at the
speed of light in analogy to an electromagnetic wave.^{15} Defining the gravitoelectric field _{}and the
gravitomagnetic field _{}by

_{
(15)}

we can rearrange the weak field Equation (7) to reassemble a
Maxwellian structure called the Maxwell-Einstein equations. Our initial assumption of a non time-varying
gravitational potential _{} is not necessary in
this rearrangement allowing a time dependent _{}, however, the identification of the gravitational scalar and
vector potential is much more complicated.
For a detailed derivation the reader is referred to the cited references
at the beginning. We now compare the
obtained Maxwell-Einstein equations to the standard electromagnetic Maxwell
equations:

_{
(16)}

Maxwell Equations
Maxwell-Einstein Equations

(Electromagnetism) (Gravitation)

The volume
integrals in Equation (9) and (12) assume point masses. These integrals would
have to be modified in the case of masses with considerable volumes. However,
in the following we will only concentrate on the ratio between gravitational
and electromagnetic scalar and vector potentials. Hence, the integral will cut
and has no further importance in our outlined coupling concept.

## 4. Coupling between
Electromagnetism and Gravitation

Equation
(16) shows the very similar structure between the electromagnetic and the
linearized gravitational field. We
neglected the effects of space-time curvature from heavy masses which is a
direct result from the strong equivalence principle and special relativistic effects. Those effects would be prominent in the case
of strong gravitational fields or velocities near the speed of light, however,
the structure given by Equation (16) would still apply even with a minor
contribution in extreme cases.

Considering
our definition of the gravitational scalar and vector potential in Equation (9)
and (12), we can propose the following relationship valid for equal charged
particles:

_{
(17)}

_{
(18)}

_{
(19)}

where *e* is the
charge and *m* is the mass of the
particle if we assume the same volume of integration for Equation (12) and its
magnetic analogue. Considering a moving
ion, the *m*/*e* ratio can be several orders of magnitude higher than for a single
electron due to the much higher mass.
If we look at the coefficients between the gravitic-gravitomagnetic and
electromagnetic scalar and vector potentials, we see that they are equal:

_{
(20)}

By defining the proportionality coefficient *k*

_{
(21)}

and putting equations (18), (19) and (21) into equations
(15), we can now express the electromagnetic components by gravitomagnetic ones
and vice versa:

_{
(22)}

Both the
electromagnetic and the gravitational field originate from the same source, the
particle. Hence, moving the source will
create both electromagnetic and at the same time
gravitoelectric-gravitomagnetic fields.
Considering only particles with equal charges, the field lines of the
magnetic and gravitomagnetic field (_{}and _{}) and of the electric and gravitoelectric field
(_{}and _{}) would look completely similar being proportional or
indirect proportional to the coefficient *k*
given by equation (21).

However, the coupling is very
small. For the case of an electron moving in a vacuum environment, the
proportionality coefficient *k*=4.22x10^{-32}
T^{-1}s^{-1}. Hence,
the magnetic field produced by the movement of equal charges is associated with
a very weak gravitomagnetic field being more than 32 orders of magnitude
below. This coefficient can be
increased if we think about a higher *m*/*e* ratio, for example using protons (*k*_{proton }= -7.6x10^{-29} T^{-1}s^{-1}),
ions (e.g. single charged positive lead ions *k*_{lead
}= -1.6x10^{-26} T^{-1}s^{-1}) or electrically
charged clusters with even higher masses.

We find
that every electromagnetic field is associated with a gravitic-gravitomagnetic
field. However, due to the possibility
of having neutral matter, only the gravitomagnetic field can exist without an
associated electromagnetic field.

##

## 5. Non-Newtonian
Gravity using Electromagnetic Fields

The divergence part of Equation
(16) describes the well-known fact, that a mass density (or an electric charge
density associated with its mass) produces a gravitational field. The remaining
rotational part, however, unveils an additional way of creating gravitational
fields similar to Faraday’s electromagnetic induction principle (non-Newtonian
gravity). Using the relationship between the magnetic and gravitomagnetic
fields in Equation (22), we can rewrite this part of the Einstein-Maxell-type
equations leading to

_{
(23)}

Coupled Maxwell-Einstein Equations

(Gravitation®Electromagnetism)

_{
(24)}

Coupled Maxwell-Einstein Equations

(Electromagnetism®Gravitation)

Equation (23) can be used to detect gravitational waves
(assuming single charged matter) whereas Equation (24) indicates an induction
of non-Newtonian gravitational fields using electromagnetic fields. Due to the small value of *k*, present day technology capable of constantly
producing magnetic fields in the order of 10 Tesla using large coils with high
electron currents at a frequency of 1 GHz would only create maximum
gravitoelectric fields of _{}= 2.65x10^{-21} ms^{-2}. Hence, it looks very discouraging using the
obtained relationships to design experiments, which create measurable
non-Newtonian gravitational fields.

Looking at the coefficient *k*, we suggest two possible ways of increasing the
coupling, which may result in stronger non-Newtonian gravitational fields:

1. __Increasing
the __*m*/*e* ratio: As already
outlined, massive ions instead of electrons can increase this ratio by 6 orders
of magnitude (e.g. single charged lead ions). This could be achieved using
dense plasmas, which are accelerated by electrostatic/magnetic fields to gain
high velocities.

2. __Gravitomagnetic
analogue to magnetism:__ Similar to
para-, dia-, and ferro-magnetism, the angular and spin momentums from free
electrons in material media could be used to propose a gravitomagnetic relative
permeability *m*_{gr} which
could increase the gravitomagnetic field _{}. If we define the
gravitomagnetic electron spin momentum _{}and the corresponding
gravitomagnetic magnetization _{}as:

_{
(25)}

where _{}is the electron spin
momentum and _{}is the magnetization,
we can express the gravitomagnetic susceptibility *c*_{g}
using

_{
(26)}

If we substitute equation (26) with (25) and (22), we get

_{
(27)}

We see that the gravitomagnetic field will be increased by
the momentum alignment in materials through the presence of an external
magnetic field with the same susceptibility than in the case of magnetism. On the other hand, the gravitomagnetic
susceptibility of matter, if only
exposed to gravitomagnetic fields, is negligible.

**Para- and
Diamagnetism**

If we align the spin momentums
only with gravitomagnetic fields, we can’t use the correlation between the
magnetic and gravitomagnetic momentums given by Equation (25). In this case, we
have to look at the expression for the magnetic susceptibility using spin
moments as described in solid-state physics textbooks.^{16} We then exchange the electromagnetic terms
using the analog gravitic-gravitomagnetic terms to express the gravitomagnetic
susceptibility. We find that for para- and dia-magnetism, the magnetic
susceptibility scales as

_{
(28)}

We can then estimate the value of *c*_{g}
using Equation (25)

_{
(29)}

Considering electron spins, the gravitomagnetic
susceptibility relates to the magnetic susceptibility as *c*_{g}
/ *c* = 2.39x10^{-43}.

**Ferromagnetism**

The temperature associated with
spontaneous magnetic spin ordering is proportional to the square of the
magnetic (in this case gravitomagnetic) momentum.^{16} Therefore applying the previous result for
gravitomagnetic spin ordering, we expect temperatures in the range of 300 K*m^{2}/e^{2}
which is 9.7x10^{-21} K for an electron, extrapolating from normal
ferromagnetic phenomena at room temperatures.

**Nuclear Magnetism**

Another contribution to gravitomagnetism
can result from nuclear momentums. In
electromagnetism, nuclear magnetism is very small compared to para-, dia-, and
ferro-magnetism due to the higher nuclear mass compared to electron’s mass. However, nuclear gravitomagnetic momentums
must have the same order of magnitude than gravitomagnetic electron spins
considering its definition in Equation (25).
Therefore, nuclear magnetism will influence gravitomagnetism at the same
order of magnitude than normal magnetism at very low temperatures required to
align the nuclear spins (typically at 10^{-9} – 10^{-3} K) by
an applied magnetic field.

Due to the absence of negative
masses, there is no gravitomagnetic analogue to the polarization _{}which defines the
relative permittivity *e*_{r}.
Hence, the relative gravitational
permittivity in material media is always equal to 1. Therefore, the construction of a gravitational “Faraday-Cage” to
shield gravitational fields is impossible.

## 6. Conclusion

The presented coupling of
electromagnetism and gravitation in the weak field approximation shows that
both fields can be converted into each other.
We find that every electromagnetic field is associated with a
gravitomagnetic field, only gravitomagnetic fields can exist without
electromagnetic ones in the case of neutral matter. Due to the small coupling, experiments that modify the
gravitational field in a terrestrial laboratory will require a lot of
engineering to achieve measurable results.

Comparing the non-Newtonian
gravitational acceleration to the one of a 1 kg point mass at a radius of 10 cm
(_{}=6.7x10^{-9} ms^{-2}), we see that the fields
associated with electromagnetism are usually at least 10 orders of magnitude
below the gravitational fields produced by standard masses. However,
electromagnetism is associated with available wide ranges of frequencies,
currents and potentials and is therefore more "engineerable" than the handling of heavy masses.
Hence, using electromagnetic fields to induce non-Newtonian gravitational fields
is an issue closely related to material sciences, cryogenic technology and
extremely high frequency problems.

Additionally to this, we find
that massive ion currents, electron and nuclear spin-alignment in materials can
lead to a higher coupling coefficient. The presented relationship encourages
further research into practical applications towards experimental gravitation.

Our proposed coupling concept
between gravitation and electromagnetism shall help to understand the
similarity of both force fields despite their different mathematical
description in general relativity and Maxwell theory. This is necessary along
the goal of physics to unify all forces in nature.

##

## References:

1. Heaviside, O., "A Gravitational and
Electromagnetic Analogy," The Electrician, **31**, 281-282 and 359 (1893).

2. Forward, R.L. "General
Relativity for the Experimentalist," Proceedings of the IRE, **49**, 892-586 (1961).

3. Forward, R.L., "Guidelines
to Antigravity,"
American Journal of Physics, **31**,
166-170 (1963).

4. Braginski, V.B.,
Caves, C.M., Thorne, K.S., "Laboratory Experiments to Test Relativity Gravity," Physical
Review D, **15**(5), 2047-2068 (1977).

5. Li, N., and Torr,
D.G., "Effects
of a Gravitomagnetic Field on Pure Superconductors," Physical Review D, **43**(2), 457-459 (1991).

6. Li, N., and Torr,
D.G., "Gravitational
Effects on the Magnetic Attenuation of Superconductors," Physical Review B, **46**, 5489 (1992).

7. Li, N., Noever,
D., Robertson, T., Koczor, R., Brantley, W., "Static Test for a Gravitational Force Coupled
to Type II YBCO Superconductors," Physica C, **281**, 260-267 (1997).

8. DeWitt, B.S., "Superconductors
and Gravitational Drag," Physical Review Letters, **16**, 1092 (1966).

9. Ross, D.K., "The London
Equation for Superconductors in a Gravitational Field," *J.
Phys. A: Math Gen*, **16**, 1331-1335
(1983).

10. Walter, H.U., Fluid Sciences and Materials
Science in Space: A European Perspective, (Springer Verlag, 1987).

11. Møller, C., The
Theory of Relativity, (Oxford University Press, London, 1952).

12. Misner, C.W., Thorne,
K.S., and Wheeler, J.A., Gravitation, (W.H.Freeman, San Francisco, 1973).

13. Campbell, W.B.,
Morgan, T.A., "Maxwell
Form of the Linear Theory of Gravitation," American Journal of Physics, **44**, 356-365 (1976).

14. Peng, H., "On Calculation of
Magnetic-Type Gravitation and Experiments," General Relativity and
Gravitation, **15**(8), 725-735 (1983).

15. Jefimenko, O.D.,
Causality,
Electromagnetic Induction and Gravitation, (Electret Scientific Company,
1992).

16. Ali Omare, M., Elementary Solid-State Physics,
(Addison-Wesley Publication Company, 1996).