Crossing the brachistochrone- Inertial mass- gravitational charge equivalence

Throughout this discussion, we have supposed that the acceleration, which we may call a, gained by an object of mass m moving on the brachistochrone, can be identified with the gravitational acceleration g due to the mass M_B of the brachistochrone.

Such an assumption is based on the inertial mass- gravitational charge equivalence. That is, if m is the inertial mass of the object, and q is the object’s gravitational charge, then it will be

In all experiments of free fall it has been measured that the acceleration a of a free falling object is always equal to the acceleration g of the gravitational field, so that the object’s inertial mass m will always be identified with its gravitational charge q,

However this may not be the general case. For example, in the same experiments it has been assumed that the air’s resistance is negligible. This force of resistance is assumed to be proportional to the object’s speed v, where the constant of proportionality is the damping constant b. Including such a force of resistance, we will have an equation of motion of the form

where

If we suppose that the acceleration a of the moving object is equal to the gravitational acceleration g, then it will be

The object’s inertial mass m will be equal to its gravitational charge only if the resistance b=0 is zero, so that the object doesn’t move, v=0.

On the other hand, if we assume that the inertial mass m of the object is always equal to its gravitational charge q, then it will be

so that the object’s acceleration a will be equal to the gravitational acceleration g only if damping is again not present, b=0, v=0.

Thus, as long as the object is moving with some speed v, and damping is present, b≠0, the related parameters will be different, a≠g, m≠q.

What is going on can be seen by acknowledging the following boundary conditions (assuming that the acceleration g of the gravitational field is always present, and constant at least at the boundaries),

and

so that

where, according to these boundary conditions, we recover the damping factor γ.

The meaning of this is that the inertial mass- gravitational charge, or inertial acceleration- gravitational acceleration, equivalence, m≡q, a≡g, will be true on the boundaries. In other words, inbetween the boundaries, some of the properties of the system will be hidden.

In order to show this, supposing that the acceleration of gravity is constant, g≡g₀, we may rewrite the equation of motion in the following form,

This way the gravitational term disappears all together from the equation of motion, and the equation is reduced to that of an object accelerating on its own power. But the gravitational term is still there, although hidden, within the equation.

This may be seen as the simplest case where the equivalence is either expressed or implied. This is the case of the rocket- in- space, which we have already examined. Another case, similar to the ship- in- the- sea example, arises if we consider that true motion in spacetime is fundamentally oscillatory.

Thus we come to the case of the simple harmonic oscillator (without damping, b=0), with an equation of motion

where

and

Similar considerations can be made for the related parameters, as we previously did for damped motion, if we replace the damping force bv by the elastic force ky, so that

and

Choosing the necessary boundary conditions, and supposing that the acceleration of gravity is constant, g≡g₀, from the equation of motion,

we have,

where

Writing this equation of motion in the form,

we take the equation of motion of the simple harmonic oscillator.

The more general case can be examined if we combine the damping term and the elastic term (as well as the implicit gravitational term) into the same equation of motion,

The acceleration of gravity g(t) can be kept constant assuming, for example, the following boundary conditions, as we have seen in the section about the forced and damped harmonic oscillator,

where

Incidentally, such an equation can be used to measure the ratio q/m.

Taking now the equation of the forced and damped harmonic oscillator,

and rewriting it in the following form,

taking also into account the boundary conditions,

we have that,

In that form, the gravitational term becomes explicit, and can be identified with the driving force.

Finally, a general relationship between the inertial mass m and the gravitational charge q can be found, if we take the energy equation of the brachistochrone

and replace in the gravitational term the mass m by the gravitational charge q,

where for the mass M_B of the brachistochrone we have that,

for the inertial mass m of the object moving on the brachistochrone, we have

while for the gravitational charge of the same object we take,

Comparing the ratio of the gravitational charge q to the inertial mass m, we see that

while the speed v of the object and the speed of light c will be given by the following formulas,

Thus, the choice whether we equate the inertial mass m to the gravitational charge q or not, depends on how deep we want to delve into the hidden aspects of the problem.