iSoul Time has three dimensions

Category Archives: Space & Time

Explorations of multidimensional space and time with linear and angular motion.

Conservations of energy

This post is about the conservation of (space) energy and time energy. I wrote about the conservation of fulmentum here. See also the post on Work, effort and energy.

First, here is a derivation of the conservation of (space) energy from classical physics:

The law of the conservation of (space) energy states that the total (space) energy in an isolated system remains constant over time (distime). The total (space) energy over an arbitrary length of distime, Δt, is constant. Let the total (space) energy at two times be E1 and E2. Then:

(E2E1)/Δt = 0.

Since the total energy equals the kinetic space energy (KSE) plus the potential space energy (PSE), we have

(KSE2 + PSE2KSE1PSE1)/Δt = 0

= (KSE2KSE1)/Δt + (PSE2PSE1)/Δt = 0

= (ΔKSE – ΔPSE)/Δt.

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Transportation symmetry

An experimenter turns on a device and transmits a signal from point A to point B. Two people play catch and toss a ball from one at point A to the other at point B. A truck transports its cargo from the terminal at point A to the terminal at point B. All these are cases of transportation.

Because of translational symmetry the laws of physics are invariant under any translation, that is, rectilinear change of position. But transportation is something more than translation. Motion is outgoing from one point and incoming at the other point. From the perspective of an observer at point A in the above examples, the translation is an outgoing motion. From the perspective of an observer at point B, the translation is an incoming motion.

Time-reversal symmetry (or T-symmetry) is valid in some cases but not in general, so it cannot be the same as transportation symmetry, which is valid in general, A return trip interchanges the sender and receiver but it is a different trip, and has nothing to do with reversing time.

Because of rotational symmetry the laws of physics are invariant under any rotation. If an observer is translated from point A to point B, and then rotated so they’re facing back, that is not the same as a transportation from point A to point B. The perspective must change, not merely the position.

This change of perspective is a physical change. Outgoing and incoming motions are not the same. Transmission of a signal differs from reception of a signal. Throwing a ball differs from catching a ball. Departing a truck terminal differs from arriving at a truck terminal.

But there is a symmetry between these motions. The laws of physics are invariant under a transformation from the perspective of an observer at the sending point A to the perspective of an observer at the receiving point B. This is transportation symmetry. Because of Noether’s theorem, a conservation law corresponds to transportation symmetry.

Spiral/helical motion

The outline below is also available in pdf form here.

Spiral/Helical Motion

A helix is the geodesic of a cylinder; if we develop the cylinder on which the helix is traced, the helix becomes a straight line. Radius r (or a or R or A); velocity v, arc length s, arc time, w, pitch length P; pitch time, M; pitch angle α; pitch time angle β


v = |v| = √(r² + b²)          s = t √(r² + b²)

u = |u| = √(q² + c²)         w = x √(q² + c²)

Pitch and slope

pitch length, P = 2πb     slope, P/S = b/r

pitch time, M = 2πc       time slope, M/T = c/q

Pitch angle

α = atan(P/S) = atan(b/r)       β = atan(M/T) = atan(c/q)

Arc length of one winding    L = √(P² + S²)

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Circular/harmonic motion

The outline below is also in pdf form here.

Angular speed (velocity) and angular pace (legerity)

  • Speed, v = Δst, Pace, u = Δts so u = 1/v and v = 1/u except if u or v are zero
    • Zero speed: no motion but time changes because time is independent
    • Zero pace: no motion but length changes because length is independent

Circular motion in space and time

distance, x; distime, t; radius r (or a or R or A); period radius q; circumference S = 2πr = wavelength λ; period T = 2πq = wavetime μ; angular velocity, v; angular legerity, u; arc length, s; arc time, w

  • Circle in space
    • space angle θ, arc length s, radius r
    • angle in space: θs/r; r = s/θ; s = rθ; 1/θ = r/s; 1/r = θ/s
    • angular time rate: ωθ/t; t = θ/ω; θ = ωt; 1/ω = t/θ; 1/t = ω/θ
  • Circle in time
    • time angle φ, arc time w, period radius q
    • angle in time, φw/q; q = w/φ; w = ; 1/φ = q/w; 1/q = φ/w
    • angular space rate: ψφ/x; x = φ/ψ; φ = ψx; 1/ψ = x/φ; 1/x = ψ/φ

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Geometry of motion

Kinematics, the geometry of motion, studies the positions of geometric objects parameterized by time. This is a 3D space with functions representing the path or trajectory as the locus of places occupied by points. It has a dual mathematics of 3D time with functions representing the course of motion as the locus of times occupied by events. Below is an introduction to both, following the exposition in Principles of Engineering Mechanics: Kinematics by Millard Beatty Jr.

1.3 Motion and Particle Path

To locate an object in space, we need a reference system. The only reference we have is other objects. Therefore, the physical nature of what we shall call a reference frame is an assigned set of objects whose mutual distances do not change with [dis]time – at least not very much. …

We define a three-dimensional Euclidean reference frame φ as a set consisting of a point O of space, called the origin of the reference frame, and a vector basis {ei} ≡ {e1, e2, e3}. That is, φ = {O; ei}. We shall require for convenience that the basis is an orthonormal basis, i.e., a triple of mutually perpendicular unit vectors.

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Helical motion

This continues the previous post here.

The parametric equation for a circular helix around the x1-axis with radius r and slope b/a (or pitch 2πb) is x1(t) = bt, x2(t) = r cos(t), x3(t) = r sin(t). Its arc length equals t · sqrt(r² + b²).

The parametric equation for a circular helix around the t1-axis with radius q and slope c/q (or pitch 2πc) is t1(x) = cx, t2(x) = a cos(x), t3(x) = q sin(x). Its arc length equals x · sqrt(q² + c²).

The linear motion along the x1-axis measures length, s, with velocity b. The circular motion around the x1-axis measures time, t, as an angle. The linear motion along the t1-axis measures length, w, with velocity c. The circular motion around the t1-axis measures length, x, as an angle.

(1) If the circular motion around the x1-axis is independent, it measures time, t, as an angle. If the linear motion along the x1-axis is dependent, it measures length, s, as a parameter, and the axis is a space axis, xs.

(2) If the linear motion along the x1-axis is independent, it measures length, x, as a parameter, with velocity b. If the circular motion around the x1-axis is dependent, it measures time, t, as an angle, and the axis is a time axis, xt.

(3) If the motion is helical, that is both circular and linear, then if it is measured by length, it forms a curve in 3D space. If it is measured by time, it forms a curve in 3D time.

Galilean transformation with independent time: xt´ = xt, xs´ = xs – vx. xtys´ = ys, , zs´ = zs.

Simplified notation: t´ = t, x´ = xvt, y´ = y, z´ = z.

Galilean transformation with independent space: xs´ = xs, xt´ = xtux. xsyt´ = yt, , zt´ = zt.

Simplified notation: s´ = s, t´ = tux, t2´ = t2, t3´ = t3.

Motion and axes

This post continues from the previous post here.

All motion is axial. It is a principle of kinematics that every motion is composed of the simple motions of translation and rotation. These simple motions are either along an axis (translation) or around an axis (rotation).

Here is a table of these two simple motions:

Motion: Motion Along Motion Around
Moving: Straight Turning
Movement: Rectilinear Angular, Circular
Simple Motion: Translation Rotation
Measurement: Length Revolution
Manifold: Space Time

Length measurement is the ratio of the target length to a standard unit of length, such as one metre. Time measurement is the ratio of the target angle to a standard unit of angle, such as one revolution.

Space is a manifold of three dimensions of translation. Time is a manifold of three dimensions of rotation, which are three axes. These comprise six axes of motion, or six degrees of freedom.

The space and time exchange postulate means that the kinematic variables of translation and rotation are symmetric in the equations of motion. That is, a dual equation of motion is formed by exchanging space and time variables.

Motion measurements

As described in the previous post here, the three dimensions of motion are axes for traveling along (length) or revolving around (time).

A measure of motion may be either (1) dependent on the the target motion, or (2) independent of the target motion. A measure that is independent is either available prior to or separately from the target motion. For example, an independent measure may be determined by agreement, such as the length of a race, or it may measure another motion, such as the motion of a clock, which is then correlated with the target motion.

A standard clock measures time because it measures rotations around an axis as an angle. A length clock measures rotations about an axis as a length. With constant rates of rotation constant, there is a fixed ratio between the two kinds of clock.

A device that measures its own internal motion may be called an autometer. A clock is an example of an autometer. The internal motion of an autometer can be correlated with a target motion. For clocks this is called synchronization. For a length clock this is called symmacronization.

An odometer is a measurement device that depends on its target motion. The standard odometer measures length of travel. A time odometer, or trip-timer, measures time of travel. A trip-timer is a stopwatch that is on only while the target motion takes place. If there is a stop in the target motion, then the trip-timer also stops. So the trip-timer measures time of motion rather than elapsed time.

A device that measures a quantity of motion need not be attached to the moving body. The theory of relativity deals with the remote measurement of quantities of motion. A device that is attached to the moving body produces proper measures such as proper length or proper time.

Motion coordinates

As a thought experiment, consider a rifle bullet, conceived of as an inertial projectile, fired at a target. Let the bullet itself be a source of measurement units: there is the length of the bullet and the rotation of the bullet. The extent of the motion of the bullet to the target could then in principle be measured as a number of bullet lengths and a number of rotations.

Length is the number of bullet lengths. Time is the number of bullet rotations. Thus length is essentially a linear measure and time is essentially a rotational measure. Length is generalized as the correspondence of the motion to a linear object, a rigid rod or ruler, which forms the basis of space. Time is generalized as the correspondence of the motion to a rotational object, a clock, which forms the basis of (abstract) time.

A rifle bullet provides a way to conceive of motion coordinates. Consider individually labeled rifle bullets continually fired from a common origin toward three orthogonal directions. The coordinates of a particular motion are then the labels on the coordinates that correspond to the motion. This means there are three pairs of coordinates: two for each bullet. These axes are the six degrees of freedom.

Motion conceived as a length function of time means that each for each rotational coordinate there corresponds its paired length coordinate. Motion conceived as a time function of length means that for each length coordinate there corresponds one paired rotation coordinate.

In conclusion, there are three dimensions of mobility. There are three dimensions for each measure of the extent of motion, which totals six dimensions. For ordinary purposes, the three dimensions of motion are sufficient, with space and time kept separate. But for science, which seeks a unified treatment, space and time should be united into six dimensions.

Dual Lorentz Transformation

Victor Yakovenko has a derivation (see here) of the Lorentz Transformation (LT) in which he uses “only the equivalence of all inertial reference frames and the symmetries of space and time.” Because of the use of (spatial) reference frames and velocity, this is not completely symmetric. As we have seen, there is a dual Lorentz Transformation. Let us follow Yakovenko’s derivation but with reference timeframes and legerity (matrix forms omitted).

1) Let us consider two inertial reference timeframes P and P´. The reference timeframe P´ moves relative to P with legerity u along the t1t axis. We know that the coordinates t2 and t3 perpendicular to the legerity are the same in both reference timeframes: t2 = t2´ and t3 = t3´. So, it is sufficient to consider only transformation of the coordinates x and t from the reference timeframe P to = fx(x; t) and t´ = ft(x; t) in the reference timeframe P´.

From translational symmetry of space and time, we conclude that the functions fx(x, t) and ft(x, t) must be linear functions. Indeed, the relative distances between two events in one reference timeframe must depend only on the relative distances in another timeframe:

 t´1t´2 = ft(x1x2, t1t2),     x´12 = fx(x1x2, t1t2).          (1)

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