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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:

Fgravity = mg,    (1-1)

Felectric = qE.    (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 E2. We would like the gravitational field energy to be proportional to g2, the square of the gravitational field. And g2 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 F'E, the combined electric and magnetic forces in K', is somewhat less than FE, the electrostatic force in K. This fact is consistent with the way d(mv)/dt transforms relativistically.

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 1/4pe0, 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 R0, centered on the origin of a rectangular coordinate system and spinning around the z-axis at rate w. At points r>R0 and in the sphere's equatorial plane, the formula for B is

B=m0wQr2/20pe0R03.   (4-2)

Bearing in mind that m0=1/e0c2 we find that

O=4pG2wMr2/5R03c2.    (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. (We recall that the 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 Oz 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 c2, thereby making the gravitomagnetic force comparable to the gravitational force.)

In the modeling program it is assumed that Mercury is acted upon by 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

Oz(r)=-4pG2wMr2/5R03c2.   (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 R0 is the Sun's radius. As previously mentioned, the equation is obtained by substituting G for 1/4pe0, M for Q, and 4pG/c2 for m0 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 Oz enhancement factor ("multiplier" in the program) must itself gradually increase with each orbit. There are actually three variables in Equation 4-3 (Oz), 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

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 m2da/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.