iSoul Time has three dimensions

Tag Archives: Space & Time

Matters relating to space and time in physics and transportation

Replicating time

In mathematical finance, a replicating portfolio for a given asset is a portfolio of assets with the same properties. Here we replicate time through motions that have the same properties as time.

Step 1. Consider the motion of a rigid body A with a translation and a rotation around the same axis, such that the translation and rotation begin and end together. Measure the displacement of the translation as a multiple of the rigid body length along the axis. Count the number of rotations and any fractional rotation of the rigid body. The assertion here is that the quantity of rotations is a measure of the distimement, that is, the duration of motion around the axis of rotation.

Step 2. Separate the motion of rigid body A into a translation of rigid body B and a rotation of rigid body C such that the displacement of B and the distimement of C are the same as the displacement and distimement of A in step 1. Then the displacement of B is a measure of the displacement of A, and the distimement of C is a measure of the distimement of A.

Step 3. Construct an independent clock as a rotating rigid body that matches the rotation of rigid bodies A and C but runs continuously. Note the marking on the clock when rigid bodies A and C start and stop moving. The quantity of rotations between the start and stop is equal to the duration of motion of rigid bodies A and C. The reading on the clock is a measure of scalar time.

Conclusion. In order to generalize this the clock needs to move at a constant rate that is standardized for all clocks. Then allow another rotation so that the motion of translation and rotation replicates any rigid body motion per Chasles [shahl] Theorem of kinematics.

Chasles [shahl] Theorem states: Every rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis. (Reference)

The independent clock generates a scalar time because it is not associated with any axis or direction. If the clock is associated with an axis of motion, then it generates a vector time, just as a rigid rod along an axis generates a vector length.

Time, space, and order

There are three axes (dimensions) of motion with six degrees of freedom. There are two metrics of motion: a length metric and a duration metric. The length metric is the magnitude of the vector between two points, and is called distance. The duration metric is the magnitude of the vector between two instants, and is called distime.

If one conceives of this as two 3D metric geometries of motion, then there is a 3D space geometry with a distance metric and a 3D time geometry with a distime metric. If the speed of light is an absolute conversion between distance and distime (which is essentially Einstein’s second postulate of special relativity), then there is one 6D spacetime metric geometry.

A 3D space coordinate system is built from an origin point and three orthogonal axes with a distance metric. A 3D time coordinate system is built from an origin instant and three orthogonal axes with a distime metric. A 6D spacetime coordinate system is built from an origin event, three space coordinates, and three time coordinates converted to lengths. An equivalent 6D spacetime coordinate system has three time coordinates and three space coordinates converted to durations.

A stance line represents two opposite linear motions with a constant rate (i.e., inertial motions). The positive direction represents distances to events diverging away from the origin point. The negative direction represents distances to events converging toward the origin point (i.e., destination). Apart from motion a point has a distance but its sign is ambiguous. A stance line represents the stance or scalar space of an odologe.

A time line represents two opposite straight motions with constant rate (i.e., inertial motions). The positive direction represents distimes to events diverging from the origin instant. The negative direction represents distimes to events converging toward the origin instant (i.e., destination). Apart from motion an instant has a distime but its sign is ambiguous. A time line represents the time or scalar time of a clock.

Events may be ordered by the stance or the time. Events ordered by stance are macronological. Events ordered by time are chronological. All events that are equal distances (equidistant) from the origin point are simulstanceous with it. All events that are equal distimes (equidistimed) from the origin instant are simultaneous with it.

6D Galilean spacetime

Here we expand 4D Galilean spacetime into 6D Galilean spacetime, based on section 1.3 Galilean spacetime of The Geometry of Relativistic Spacetime: from Euclid’s Geometry to Minkowski’s Spacetime by Jacques Bros (Séminaire Poincaré 1 (2005) 1 – 45).

[p.3] We start with a representation space whose points are interpreted as the “physical events”. Any motion of a particle which is physically possible between two given events A and B is represented by a certain world-line with end-points A and B. There is an absolute orientation of such worldline, which can be called its “time-arrow”: its physical meaning is that one of the end-point events, e.g. B, is in the future of the other one A.

[p.6] From the viewpoint of mathematical physics, the use of geometry in more than three dimensions turns out to be necessary, if one wishes to represent phenomena whose description necessitates more than three independent quantities. A typical example is the six dimensional space Rab6Ra3 × Rb3 of the positions (a; b) of pairs of material points (or pointlike particles) in mutual interaction. Trajectories of such pairs are represented by curves in R6, described in terms of a parameter t by equations of the form a = a(t); b = b(t).

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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

Constants

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.