Mass Real or Imaginary?
Or equivalently, Is it
inertial or gravitational mass in a given equation? An obvious question in
gravito-magnetic equations containing mass terms is: “should a term be Real
(inertial) or Imaginary (gravitational)? The answer is provided by the rule that
all forces are Real. In any
equation with force on the left side, the term(s) on the right side must be
Real. Let us consider two celebrated examples (courtesy of Isaac Newton).
(1) Newton’s mechanics
equation, F=ma. m must be Real or Inertial, since kinematic variables consisting
of space and time (“a” in the present case) are Real.
(2) Newton’s gravitational
force law, F=GMm/r2. M and m must be Imaginary (or gravitational),
since the force is attractive/negative and the product of two Imaginary terms is Real. Notice that the product of mgravitational
and Mgravitational is negative.
Hence F is an attractive force.
But what about the physics
of Maxwell? The key rule is that everything in Maxwellian theory is Real. In
Coulomb’s law, for example, F=(1/4peo)Qq/r2,
Q and q are Real and either positive (same sign) or positive and negative. In
the first instance the force is positive or repulsive. In the second case F is
of course attractive.
In gravito-magnetic theory every Maxwellian equation has an equivalent Newtonian one, provided certain equivalence rules are implemented. For example, consider the Maxwellian/Lorentzian equation that specifies the magnetic force experienced by a moving charge: F=qvB. The mechanical equivalent would be F=mvO, where O is a new concept for a “gravito-magnetic” field equivalent to the Maxwellian magnetic field, B. O fields (which must be Imaginary) were not considered in Newton’s world picture. But the inclusion of O in Newtonian equations solves several long-standing “paradoxes” like the subtle "precession" of Mercury’s quasi-elliptical orbit. It would be remiss not to acknowledge in this regard the success of Einstein’s theory of warped space-time in General Relativity. The simple truth is that gravito-magnetic theory is mathematically easier to comprehend and to work with than is the General Theory of Relativity.