iSoul Time has three dimensions

Tag Archives: Space & Time

Matters relating to space and time in physics and transportation

Observers in motion

A rigid rod or other device that measures length is at rest relative to itself, even if part moves such as a measuring wheel, because it moves relative to other objects, not relative to itself. A concept of simulstanceity enables an observer to determine length at other times (e.g., they are the same point on the stance line).

A clock measures time, but what is a clock? It is a device with a part that moves relative to a part that is at rest. So a clock is an object in motion relative to itself (yes, this is possible). The part that moves indicates the time. A concept of simultaneity enables an observer to determine time at other places (e.g., they are the same instant on the time line).

Let there be a rigid reference frame associated with each observer or object (e.g., they are attached). An observer may be at rest or in motion relative to their frame. If the observer is at rest, then their frame is a length frame and what they observe is in space. Time is the independent variable and length in three dimensions is the dependent vector variable.

If the observer is going somewhere, they are not at rest but in motion. Their reference frame for rest is not their own frame but a different frame, such as one located on the surface of the earth. In this case the observer and rest frame system are like a clock, that is, a clock frame, and what is observed is in time. A clock frame is moving in the opposite direction of a rest frame. Length is the independent variable and time in three dimensions is the dependent vector variable.

Frames in motion

For Galilean inertial frames the observer is at rest and the moving frame transmits the current stance in an instant of the time line, instantaneously. For contra-Galilean inertial frames the observer is in motion and the rest frame transmits the current time in a point of the stance line, punctstanceously.

The rest frame observer has three dimensions in space. The observed frame in motion is effectively reduced to the one dimension of its motion in time. The moving frame observer is like a clock with space and time exchanged: the dimensions of the observer’s frame are in motion so the three dimensions are in time. The rest frame that is observed appears to move and is effectively reduced to the one dimension of its path in space.

Time and simultaneity

There are several ways of understanding the time of remote events. What follows is a summary of the basic ways of determining simultaneity.

As a way of comparing the different ways consider transmitting a light signal to a remote location where it is reflected back. What is the time when the signal is reflected back?

Observation time is an extension of ordinary perception. When we observe an event, we say that it is happening at the time of observation. So when a light signal is reflected and received back, the reflection observed is considered to have happened when it was observed. In effect the light observed is instantaneous. By implication the one-way speed of light transmitted is c/2 in order for the two-way speed of light to equal c.

Observation time is thus the projection of the time of observation to the entire observable universe. This way of understanding time is characterized by the Galilean transformation.

Transmission time is an extension of the ordinary transmission of light. When we shine a light on an event, we say that it is happening at the time of transmission. So when a light signal is aimed toward a reflector, the event of reflection is considered to have happened when the light was transmitted. In effect the light transmitted is instantaneous. By implication the observed one-way speed of light is c/2 in order for the two-way speed of light to equal c.

Transmission time is thus the projection of the time of transmission to the entire transmittable universe. This way of understanding time is characterized by the contra-Galilean transformation.

Probe time is an extension of measurement by a probe (a “small, unmanned exploratory craft”) to the entire probeable universe. See previous post here. An event is said to occur when intersected by a probe that measures the duration of probe movement since a reference event. So when a probe comes upon the reflection of light, the probe measures the time of reflection as the time of the probe. If the probe is not moving at the speed of light, there may need to be multiple probes.

Consider a series of probes moving at a speed v over a distance d to the reflection event. The probe that leaves at time (d/c) – (d/v) is the probe that intersects the reflection event. If v = c, then the time is zero.

Because probes can measure the length or duration of motion, probe time is characterized by the Lorentz or contra-Lorentz transformation.

Reference frame time measures time by a rigid reference frame that has clocks which were previously synchronized spread throughout. See the Relativity of Simultaneity and Einstein Synchronisation. These synchronizations are characterized by the Lorentz transformation.

Reference probes and systems

A reference frame is in principle a rigid structure embodying a 3D coordinate system. It represents an observer at rest with complete access to rods and clocks to measure length and duration in any direction:

Such a reference frame may be the framework or infrastructure for a reference probe moving like a miniature aerial tram in any direction. A probe is a “small, unmanned exploratory craft”. Such a reference probe compared with a target motion can measure either the extent of the framework crossed by the target, which is the length, or the extent of the framework crossed by the reference probe, which is the duration. The rate of the target motion is the ratio of the length to the duration or the ratio of the duration to the length.

Alternatively, the reference frame may be the framework or infrastructure for a reference system of probes jmoving in all directions. The motion of such a system can be given by a table of changes, which are the intersections of consecutive trips, called “times”, and consecutive stations, called “stances”:

Table of Changes

Times

Trip 1

Trip 2

Stances

Location 1

change 1,1 change 1,2

Location 2

change 2,1 change 2,2

A target motion can be measured as the number of stances, which is the length, or as the number of times, which is the duration. The rate of the target motion is the ratio of the length to the duration or the ratio of the duration to the length.

What if one reference framework is moving with respect to another reference framework? The motion of a framework is no different than the motion of an object as observed by a reference framework. How can one compare the observation of an object from one framework with that of another framework? That requires applying the appropriate transformation, Galilean, contra-Galilean, Lorentz, or contra-Lorentz.

Einstein exchanged

Albert Einstein’s book Relativity: The Special and General Theory was originally published in German and translated into English in 1920. In the second chapter he introduces “The System of Co-ordinates”. The following post gives Einstein’s text followed by a revision that exchanges length with duration and space with time. First, Einstein’s text, with alternative wordings in square brackets:

End of Chapter I – If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.

Chapter II – On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a “distance” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distance S time after time [again and again] until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length.

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Objects and subjects in motion

An object is stable. A rock is an object. Water is an object if it is in a container. A rigid rod is an object.

A subject changes. A person is a subject. Air is a subject since it keeps moving. A clock is a subject.

The grammatical subject and object are distinguished in a sentence, though they both may be things. For example, “The rock rolled down the hill.” Both the rock and the hill are things, that is objects, but in the sentence the rock is the subject and the hill is the object.

Objects are acted upon. A predicate is required to go with an object. An object apart from a sentence is a thing, something passive.

Subjects are active. A verb is required to go with a subject. A subject apart from a sentence is still a potential change agent.

Space is like an object and time is like a subject. If we start with objects and then discuss their motions, we are beginning with the passive objects of space and then adding the active subjects of time. If we start with subjects, we get their motions, too, and may then bring in the objects of space. That is beginning with an active time and adding the passive objects of space.

A reference motion must be active and so include a subject. A comparative motion is passive in relation to the reference motion and so must include an object.

Bodies and things may be subjects or objects, though many are usually one or the other. The difference is in whether they change or move. An object need not move. A subject is usually moving.

Objects are in space. Subjects are in time. Space never moves. Time always moves.

Moving bodies in space and time

Let us compare the motions of two bodies. Let the motion of one body be the reference motion. Let the motion of the other body be the comparative motion. Let the two bodies begin together at one place.

Definitions:

A place is the general term for an answer to Where? A point-place, or simply a point, is the smallest place. A translation is a vector from one point-place to another. Travel distance is the arc length of the trajectory of a motion, which includes any retracing of the trajectory.

Space and time refer to different perspectives of the universe of motion.

Space is the locus of all potential places for the comparative motion, which is said to be “in space.” Displacement is a translation vector from one point to another point of the comparative motion. The travel distance from the beginning point to the ending point of the comparative motion, is the travel length for a motion in space.

Time is the locus of all potential places for the reference motion, which is said to be “in time.” Distimement is a translation vector from one point to another point of the reference motion. The travel distance from the beginning point to the ending point of the reference motion, is the travel time for a motion in time.

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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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