__On
Gravitomagnetism
__

**Abstract****.**

Discusses parallels between Maxwell/Lorentz charge-charge interactions and suggested mass-mass interactions. Examples are Mercury orbiting the Sun, stars rotating in galaxies, and satellites passing the Earth.

__1.Introduction.__

Scientists
have long noted that Newton's Law of Universal Gravitation and Coulomb's Law are
both inverse square laws. A difference is that there is only one type of mass in
the world of everyday experience, but there are two types of electric charge.
Since the gravitational force between two masses is attractive, whereas the
electric force between like-sign charges is repulsive, it has been customary to
say that the gravitational field, **g**,
points *toward* the
source positive mass, whereas the electric field, **E**,
points *away* from a source *positive* charge
(and toward a negative charge). This results in the similar equations:

**F**_{gravity} =
m**g**, (1-1)

**F**_{electric} =
q**E**. (1-2)

Note
in Eq. (1-2) that the force is attractive when the source of **E** and
the subject charge, q, are of opposite sign.

__2.
Why Gravity and Gravitational Mass should be Mathematically Imaginary.
__

A
convenient way to have the gravitational field point *away* from
a mass (and not toward it) is to have the field and whatever mass it acts upon
be mathematically *imaginary*.
That way Eq. (1-1) still results in a real *attractive* force.

There
is another compelling reason for making the gravitational field imaginary. One
of the elegant results in classical electromagnetic theory is that, when two
separated, like-signed charges are restrained to have their separations changed
at a constant velocity, then *the
change in their field energy is precisely equal to the work expended by the
restraining agent*.

Now
whereas the electric field *energy* is
positive in the case of electric charge, it is presumably *negative* in
the case of electrically neutral, mathematically imaginary mass. For negative
work must be done on the restraining agent in order to bring two masses closer
together. And the change in gravitational field energy is *negative *by
the same amount; it goes from a smaller magnitude *negative* value
to a larger magnitude negative value.

In
electromagnetic theory the electric field is positive and proportional to E^{2}.
We would like the gravitational field energy to be proportional to g^{2},
the square of the gravitational field. And g^{2} is
indeed negative if g is imaginary.

__3.
When
Gravity Balances the Lorentz Force
__

Fig
(3-1) illustrates two resting, hollow spheres of positive charge viewed from
their "rest" inertial frame, K. Each charge is held together by some
restraining agent, which also holds them at a constant distance from one
another. Since the charges are at rest, the total force on each charge must be
zero.

__Figure
3-1__

__ __

__Two
Spherical Shells of Charge Held at Rest__

Viewed
from any other inertial frame, the charges move at a *common,
constant velocity*, which among other things means that (a) *the
distance between them is unchanging* in
all inertial frames and (b) by
Newton 1 *the total force
on each charge is zero in
every inertial frame*. Of course if frame K' moves to the left (so that the
charges move to the right) then

Let
us now fill the spheres with neutral matter, at a density such that the
gravitational attractive forces precisely cancel the electrostatic repulsive
forces. Under these circumstances an agent need *not* hold
the charges at a constant distance from one another. The spheres are in
equilibrium. We know that the electromagnetic force, experienced by each charge,
varies from frame to frame. But since each sphere moves with a constant velocity
in every inertial frame, we also know that the total force on each charge must
be zero in every inertial frame. In effect, then, the attractive force between
the masses must transform from frame to frame precisely as the electromagnetic
force does.

__4.
Gravitomagnetism
__

The simplest way to obtain the equilibrium of the spheres in Fig. (3-1), in every inertial frame, is to postulate that there is a field analogous to the charge's magnetic field, and to invoke a force law that is analogous to the Lorentz force law. This field has been dubbed the gravitomagnetic field (symbolized by the letter "O") and the force law would be

**F** =
m(**v** x **O**),
(4-1)

where
m and **O** are
imaginary (so that **F** is
real). It is important to note that, if m and **O** are
both mathematically imaginary, then a *left* hand
rule must be employed to determine the direction of the gravitomagnetic force.

In
general the equations that parallel those of electromagnetic theory can be
obtained by substituting m for q, G for 1/4pe_{0}, **g** for **E**,
and **O** for **B**.
m, g and O are imaginary.

By
way of an example, we might consider a solid sphere of charge Q and radius R_{0},
centered on the origin of a rectangular coordinate system and spinning around
the z-axis at rate w.
At points r>R_{0} and
in the sphere's equatorial plane, the formula for B** **is

B=m_{0}wQr^{2}/20pe_{0}R_{0}^{3}.^{ }(4-2)

Bearing
in mind that m_{0}=1/e_{0}c^{2} we
find that

O=4pG^{2}wMr^{2}/5R_{0}^{3}c^{2}.
(4-3)

__5.
A
Computation of Mercury's Orbit in the Sun's Gravitomagnetic Field.
__

In
this section we imagine that Mercury is the sole planet in the solar system. In
this case Newton would have believed that the only force acting on Mercury is
the Sun's gravitational force, whose magnitude is specified in his law of
universal gravitation. Kepler deduced that Mercury orbits the sun in a *closed
ellipse*, repeating the same path in space forever. This result might be expected if the sole
force acting on the planet is the gravitational attraction to the Sun. However,
it is now known that the Sun spins, and according to gravitomagnetic theory *gravity
is not the sole force acting on Mercury*. For the rotating Sun should also
engender a nonzero gravitomagnetic field, and the total force acting on Mercury
should be the sum of the attractive Newtonian gravitational force, plus a very
much weaker *repulsive* gravitomagnetic force.*left*-hand rule must be applied to cross products of
two imaginary values. We model with the Sun (a) at the origin of rectangular
coordinates, and (b) spinning counterclockwise in the xy-plane. In this case O_{z}
points in the negative z-direction in the Sun's equatorial plane (and in
Mercury's equatorial plane)..Mercury is assumed to orbit the Origin
counterclockwise.)

In
this section we report display the orbit of Mercury in a very much *magnified*
gravitomagnetic field. (The Sun's gravitomagnetic field in equation 4-3 is
multiplied by c^{2}, thereby making the gravitomagnetic force comparable
to the gravitational force.)

*two* forces:
(1) the ever-present gravitational force toward the Sun, and (2) the
gravitomagnetic force. The latter is due to a dipolar gravitomagnetic field
(denoted as the O field) engendered by the spinning Sun. As previously
mentioned, O is analogous to the B field in electromagnetism.

It
is assumed that Mercury's orbit lies in the Sun's equatorial plane (not quite
true in reality). The coordinate axes are set such that this plane is the xy-plane.
The formula used for the *imaginary* z-component
of O at points in the xy-plane is therefore

O_{z}(r)=-4pG^{2}wMr^{2}/5R_{0}^{3}c^{2}.^{ }(5-1)

(In
this equation w
is
the Sun's rotation rate, M is its mass, r is Mercury's distance from the Sun,
and R_{0} is the Sun's
radius. As previously mentioned, the equation is obtained by substituting G for
1/4pe_{0},
M for Q, and 4pG/c^{2} for m_{0} in
the formula for the B field of a spinning sphere of *charge*.)

Figure 5.1 depicts Mercury's nearly complete orbit in the field of a non-rotating Sun. Slightly less than one complete orbit is computed, in order to illustrate that multiple orbits do indeed overlay one another. Figure 5.2 displays the result of computing four consecutive orbits.

__Figure
5-1__

__Computed
Partial Orbit of Mercury, Gravitational Force Only__

__Figure
5-2__

__Four
__**Computed
Orbits of Mercury, Gravity Only**

We
now consider cases where Mercury experiences the omnipresent gravitational
field, *plus* an *enhanced* gravitomagetic field. Again we display the
computed partial orbit in the first case, and then four successive orbits, to
determine whether the orbits again overlay one another.

__Figure
5-3__

__Computed
Partial Orbit of Mercury, Gravitational and Enhanced Gravitomagnetic Forces__

__Figure
5-4__

__Four
__**Computed
Orbits of Mercury, All Theoretical Forces Acting**

It
is clear in Figures 5-2 (no enhanced O field) and 5-4 (enhanced O field) that
the perihelion of Mercury has shifted off of the x-axis in the latter case. It
is also clear in Figure 5-4, however, that (according to gravitomagnetic theory)
*the aphelion has precessed even further in the same direction!* As a
consequence the more or less perfect ellipse of Figure 5-2 has become distorted
in Figure 5-4. Noteworthy in Figures 5-1 and 5-3 is the increase in the orbital *period*
after the greatly magnified gravitomagnetic force has been "turned
on."

The
result that the orbits in Figure 5-4 overlay suggests that the O_{z}
enhancement factor ("multiplier" in the program) must itself gradually
increase with each orbit. There are actually three variables in Equation 4-3 (O_{z}),
viz. M, w and r. (The Sun's mass attenuates as it
radiates prodigious quantities of radiant energy.)

At least two caveats are in order: (a) In reality Mercury's orbital plane does not quite coincide with the Sun's equatorial plane, and (b) Different latitudes of the Sun spin at different rates. It is also worth noting that the gap in Figure 5-3 is considerably greater than the one in Figure 5-1. The same initial components of r and v are modeled in the two cases, but the gravitomagnetic force opposes the gravitational force, and Mercury requires significantly more time to complete an orbit in the Figures 5-3 and 5-4 cases.

Appendix 5-1 provides the Power Basic program used in the modeling exercises.

__Appendix
5-1. Program that computes Figs. 5-1 and 5-2.__

#COMPILE EXE

#DIM ALL

'Compute the orbit of Mercury

FUNCTION PBMAIN () AS LONG

DIM orbits AS LONG

'Specify number of consecutive orbits to compute.

orbits=1

'orbits=4

DIM n AS LONG

n=2500000

DIM pi AS DOUBLE 'Needed for Oz.

pi=3.14159

DIM G AS DOUBLE

G=6.67408E-11

DIM omega AS DOUBLE 'Sun angular velocity.

omega=2*pi/(27*24*60*60)

DIM M AS DOUBLE 'Sun mass.

M=1.9891e30

DIM R0 AS DOUBLE 'Sun radius.

R0=695.51e6

DIM perihelion AS DOUBLE

perihelion=46e9

DIM r(orbits*n) AS DOUBLE

DIM rx(orbits*n) AS DOUBLE

DIM ry(orbits*n) AS DOUBLE

rx(0)=-perihelion

ry(0)=0

r(0)=SQR(rx(0)^2+ry(0)^2)

DIM c AS DOUBLE 'Needed for Oz.

c=3e8

DIM Oz AS DOUBLE

Oz=-4*pi*G^2*omega*M*r(0)^2/(5*R0^3*c^2)

'Choose one.

DIM Multiplier AS DOUBLE

multiplier=0 'Choose 1.

'multiplier=c^2/.7

Oz=multiplier*Oz

DIM mhg AS DOUBLE 'Mass of Mercury.

mhg=3.30104E23

DIM tau AS DOUBLE 'Orbital period of Mercury.

tau=88*24*60^2

DIM dt AS DOUBLE

dt=tau/n

DIM t(orbits*n) AS DOUBLE

DIM Fx(orbits*n) AS DOUBLE

DIM Fy(orbits*n) AS DOUBLE

DIM ax(orbits*n) AS DOUBLE

DIM ay(orbits*n) AS DOUBLE

DIM vx(orbits*n) AS DOUBLE

DIM vy(orbits*n) AS DOUBLE

DIM v(orbits*n) AS DOUBLE

t(0)=0

vx(0)=0

vy(0)=-38.86e3

Fx(0)=G*M*mhg/r(0)^2*(-rx(0))/r(0)

Fx(0)=Fx(0)-mhg*vy(0)*Oz

Fy(0)=G*M*mhg/r(0)^2*(-ry(0))/r(0)

Fy(0)=Fy(0)+mhg*vx(0)*Oz

ax(0)=Fx(0)/mhg

ay(0)=Fy(0)/mhg

vx(0)=ax(0)*dt/2

vy(0)=(-38.86e3+ay(0))*dt/2

'Then iterate.

DIM rep AS LONG

rep=0

DIM index AS LONG

repeat:

FOR index=1+rep*(n-1) TO (rep+1)*(n-1)

t(index)=index*dt

rx(index)=rx(index-1)+vx(index-1)*dt

ry(index)=ry(index-1)+vy(index-1)*dt

r(index)=SQR(rx(index)^2+ry(index)^2)

Fx(index)=G*M*mhg/r(index)^2*(-rx(index))/r(index)

Fx(index)=Fx(index)-mhg*vy(index)*Oz

Fy(index)=G*M*mhg/r(index)^2*(-ry(index))/r(index)

Fy(index)=Fy(index)+mhg*vx(index)*Oz

ax(index)=Fx(index)/mhg

ay(index)=Fy(index)/mhg

vx(index)=vx(index-1)+ax(index)*dt

vy(index)=vy(index-1)+ay(index)*dt

NEXT

rep=rep+1

IF rep<orbits THEN GOTO repeat:

OPEN "c:\\users\Marjorie Dixon\Documents\plotvals.dat" FOR OUTPUT AS #1

FOR index=0 TO orbits*n-1 STEP orbits*1000

WRITE #1, rx(index),ry(index)

NEXT

CLOSE #1

MSGBOX("ready for plotting")

END FUNCTION

__6.
__**Galaxies
and Gravitomagnetism
**

The
peripheral charge increments in a spinning disc of *charge* experience
Lorentz magnetic forces that reinforce the electric forces of repulsion. The
spinning disc is more likely to fly apart owing to magnetic forces on its charge
increments (as well as centrifugal forces).

For
the sake of discussion it seems reasonable to model a spiral galaxy (such as our
own Milky Way) as a spinning disc of *matter*.
Of course it is not as uncomplicated as a rotating disc of uniform charge; parts
of the galaxy move at different angular speeds, etc. Nonetheless such a simple
analogy might at least provide qualitative insight into the following
astronomical fact: the galaxy should not hold together solely under the
influence of gravity. Some additional attractive force is needed to keep
the peripheral stars from flying off into deep space.

Theorists
have wrestled with this problem ever since it was first realized. The postulate
that the rotating stars of a galaxy would engender a gravitomagnetic field,
perpendicular to the galactic plane, is simply too straightforward to be lightly
dismissed. As previously pointed out, the force law for the gravitomagnetic
force requires application of a *left* hand
rule, and the gravitomagnetic forces on peripheral stars would *reinforce* the
gravitational forces.

__7.
Gravitomagnetic
Waves
__

The analogies between electromagnetism and grav-gravitomagnetism suggest that masses with nonzero jerks (or time-varying accelerations) should emit waves quite as charges do. Such waves would usually be of extremely low frequency and power, and would have enormous wavelengths.

A
mass with nonzero jerk will hypothetically experience a *gravitational
radiation reaction* force
proportional to m^{2}d**a**/dt. But given the imaginary nature of m,
this force may point in the *same* direction
as the mass' velocity. In the act of moving, the mass may *absorb* positive
power from its field. To the extent the mass is in equilibrium with its
environment, it will also obtain negative power in the waves from other masses
and the net power gain and loss should be zero.

Given the extremely low frequencies (and enormous wavelengths) of such radiation, the detection of such waves would be very difficult. Their existence is mentioned primarily to complete the analogy between charge/charge and mass/mass interactions.

__8
. The
Pioneer Anomalies
__

The
following quotation of a NASA Internet article is instructive:

"NASA
scientists are baffled when they discovered the "Pioneer Anomaly,"
where unexplained forces acting on spacecraft as they fly past Earth on their
way out of the solar system:

*Mysteriously,
five spacecraft that flew past the Earth have each displayed unexpected
anomalies in their motions.*

*These
newfound enigmas join the so-called "Pioneer anomaly" as hints that
unexplained forces may appear to act on spacecraft.*

*A
decade ago, after rigorous analyses, anomalies were seen with the identical
Pioneer 10 and 11 spacecraft as they hurtled out of the solar system. Both
seemed to experience a tiny but unexplained constant acceleration toward the
sun.*

*A
host of explanations have been bandied about for the Pioneer anomaly. At times
these are rooted in conventional science — perhaps leaks from the spacecraft
have affected their trajectories. At times these are rooted in more speculative
physics — maybe the law of gravity itself needs to be modified.*

The spinning Earth should engender a gravitomagnetic field quite as the Sun does. Let us suppose that we are viewing the Earth from its negative spin axis. From that perspective let the planet be spinning clockwise. At external points in Earth's right equatorial plane a spacecraft whizzes vertically upward in the diagram. It moves through a gravitomagnetic field that points out of the diagram. Applying the left hand rule we determine that a subtle gravitomagnetic force draws the spacecraft toward the Earth (in addition to the much greater gravitational force). Mystery solved? Perhaps.

We
can use the formula in Eq. (4-1), multiplied by an appropriate factor, to
determine the magnitude of this tiny component of attractive force. Note that if
a spacecraft passed the Earth in a downward direction, then the gravitomagnetic
force would *oppose* gravity.

__9.
Final
Thoughts
__

Two other theories that warrant mention are The Theory of General Relativity and The Theory of Dark Matter. As the author understands it, General Relativity has accounted for the precession of Mercury, but has not successfully explained why galaxies do not fly apart. Dark matter can account for the latter problem, but to date its existence has evidently been an ad hoc conjecture.

Perhaps the theory of gravitomagnetism relates to General Relativity like Newtonian mechanics relates to Lagrangian mechanics. The first is easier to understand, whereas the second might be better able to address certain problems. Perhaps gravitomagnetism will in time become the paradigm for the majority of physics aficionados, whereas General Relativity will be the province of the relative few able to master its subtleties.