ABSOLUTE MOTION IN 3 DIMENSIONAL TIME

In the field below left are shown the formulas that describe the motion of an object m that moves at constant speed v relative fixed points on the surface of an expanding balloon (see previous page). The reader doesn't need to analyze these formulaes to understand the following - they are there for the sake of completeness only.
 

1) The object's constant speed is given by the angular velocity df/dt , that would be constant if the balloon would not expand. It is then also obvious that, no matter how small this velocity is, the object would always reach its destination (B' on the previous page).

It is no different for the expanding surface; the object cannot compensate the recession of a fixed point with any angular speed, as to keep a constant distance to such a point. Naturally, any angular speed finally will complete any given angular distance! Hence, the recession between fixed points is not a speed and consequently such fixed points are at absolute rest! We, as external observers, can clearly see that, because we can consider angular distances and velocities, but for 2-dimensional observers on the surface of the balloon, being unaware of a third dimension in which their world expands, all motion is relative and equivalent and the recession of fixed points appears to them as speed, no different from any other speed.

Nevertheless, as we have the same definition for speed as they have, being change of distance per unit of time, we should understand that our definition is equally wrong. It is an indisputable fact that the distances between fixed points on the surface of an expanding balloon, are increasing as a function of time, also for us in 3d-space and we therefore need to revise our definitions of motion and speed!
(this is crucial understanding; if not clear, read again)

This circumstance alone is almost as good as a prove that time must be 3-dimensional, so it allows us to define motion and speed in terms of time angles, rather than in terms of time intervals. It then likely will show, that the receding distant stars.... are at rest!

2) As the recession appears as a speed, the motion of the moving object m appears to be accelerated. However, it's real motion is not over the surface but along a spiral and at constant speed (it is a logarithmic spiral, as the intersection with the radius of the balloon is at constant angle). For observers on the surface of the balloon, the spiral does not exist and the length, Sarc, of the motion over the observed surface   is always longer, than what it would be, Sapp, for a motion at constant speed during the time of observation (see graphs upper right).

They therefore calculate that the object moves accelerated with the value aobj (formula above). Observe in the lower graph at upper right, that the change of the object's angular velocity, df_m/dt , is not constant. Its differential (curve a) starts at a higher value, that gradually decreases in time and finally approaches (but never becomes) zero. This differential of the angular velocity generates a tangential acceleration in the opposite direction of the motion, just as it also would do in an according circular motion. From calculus follows, that the value of this tangential acceleration a_t is precisely the same as the acceleration in the direction of the object's motion,  a_ob , as shown in the graph below.

As these two accelerations take each other out, observers at rest cannot detect any acceleration of the object's motion; they would measure the constant speed that the object has relative them selves. (Naturally, because the assumption of this whole excercise was that it moves at constant speed over the surface of the balloon; the formulas above, show themselves to be correct).
Moreover, as the object evidentally is not accellerated and with the observed constant speed would travel a shorter distance (Sapp) over the surface of the balloon than what it actually appears to do (Sarc) , the observers at rest must conclude that time goes slower for an observer who travels with the object !

Observe that we have reached a second analogue statement to modern physics, first the invariance of speed as is with fotons, and now the dilation of time! Coincidence only?

Because the moving object follows the curvature of the balloon's surface, there is also a centripetal acceleration, a_c (graph above), similar to that of a normal circular motion, as it varies proportional with the square of the object's angular velocity.

At this point it is crucial to realize that this centripetal acceleration is not relative any observer! Just add an angle for another fixed observer in the formula, and you will see that the curve a_c doesn't change (the derivative of a constant is zero). The centripetal acceleration is a function of the angular velocity and the expansion rate of the balloon only. This circumstance gives the surface of the balloon an absolute character. If no such acceleration would exist, and in lack of a reference angle ( f at zero degrees), observers attached to any object, moving (at constant speed) or not, could consider themselves to be at rest; Scientists on the surface of the balloon would fully accept the relativity principle, as Einstein postulated it. However, if they could detect the centripetal acceleration as a "weight" (the "force" that attaches moving objects to the expanding surface - see previous page), then weightless objects would be the fixed ones at absolute rest and objects with weight (inertia) would be in absolute motion - newtonian bells are ringing!

Unaware of this, observers on the surface of the balloon would consider all motions to be relative, but in fact it are the reference systems at rest (no angular velocity), that are relative ..and inertial. There is absolute motion, but there is no absolute reference system ( f at zero degrees). From this follows that absolute motion can exist, without the need of an absolute frame of reference! (If only Einstein had known about this, ....)