iSoul In the beginning is reality

Mechanics in multidimensional time

As (spatial) velocity and acceleration are vectors, so are their temporal analogues. This perspective makes sense because of the multidimensionality of time. There is an implicit recognition that time has directionality since time is commonly considered as any real number, and not a non-negative real number, which it would be if time were merely a magnitude. This “reverse time” is an example of time’s directionality — which however has nothing to do with reverse causal sequences.

The (spatial) distance or length (especially of an object) is a magnitude that is used to represent physical space. Similarly, the duration (or length of time) is a magnitude that is used to represent physical time. We speak of the location or position in space of an object or a place. Similarly, we speak of the point in time or temporal position of an action or event.

A point in space or time is “that which has no part” (Euclid) whose location is represented by a position vector. A point itself is an abstraction that is zero dimensional but makes up all multidimensional abstract ‘spaces’ (which may represent space or time or whatever). If s is the distance of a point from a specified origin point in space, then its position may be represented by a position vector whose magnitude equals s. If t is the duration of a point in time from a specified origin point in time, then its position may be represented by a temporal position vector whose magnitude equals t.

The movement of a point through space in time may be represented by a vector function of temporal position t whose value is the spatial position at each temporal position t. The movement of a point through time in space may be represented by a vector function of spatial position s whose value is the temporal position at each spatial position s.

During the time interval (duration) Δt = t2 – t1, the position vector of an object changes from r1 = r(t1) to r2 = r(t2), with a displacement vector Δr = r2r1 (boldface represents vectors). The rate of change of the displacement vector is the average (time) velocity vector over the time interval, vavg = Δr / Δt. The rate of change of the average velocity vector is the average acceleration vector aavg = Δv / Δt.

Similarly, while traversing the space interval (distance) Δs = s2 – s1, the position vector of an object changes from p1 = p(s1) to p2 = p(s2). The rate of change of the displacement vector is the average space velocity vector over the length of space, uavg = Δp / Δs. The rate of change of the average space velocity vector is the average space acceleration vector bavg = Δu / Δs.

Instantaneous velocity is considered to be measured over a differential of time (duration), dt. In that case the instantaneous (temporal) velocity is defined as v(t) = ds/dt and the instantaneous (temporal) acceleration as a(t) = dv/dt = d2s/dt2.

Similarly, the coincidental spatial velocity may be measured over a differential of space (length), ds. The coincidental spatial velocity is defined as u(s) = dt/ds and the coincidental spatial acceleration as b(s) = du/ds = d2t/ds2.

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