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.

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.

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 R_ab⁶ = R_a³ × R_b³ 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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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:

(E₂ – E₁)/Δt = 0.

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

(KSE₂ + PSE₂ – KSE₁ – PSE₁)/Δt = 0

= (KSE₂ – KSE₁)/Δt + (PSE₂ – PSE₁)/Δt = 0

= (ΔKSE – ΔPSE)/Δt.

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