iSoul In the beginning is reality.

Tag Archives: Transportation

Space and time as frames

An observer is a body capable of use as a measurement apparatus. An inertial observer is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. Unless specified otherwise, an observer here shall mean an inertial observer.

Space is the structure of places in R3. A place point (or placepoint) is a point in space. The space origin is a reference place point in space. The location of a place point is the space vector to it from the space origin. Trime (3D time) is the structure of times in R3. A time point (or timepoint) is a point in trime. The time origin is a reference time point in trime. The chronation of a time point is the trime vector to it from the time origin

A frame of reference is an imaginary lattice of rigid rods at rest relative to an observer that enables every place point of space to be uniquely identified by coordinates. A timeframe of reference is an imaginary lattice of rigid rods in uniform motion relative to an observer that enables every time point of trime to be uniquely identified by coordinates. A rest frame of observer P is a frame at rest relative to P. A motion frame of observer P is a frame in uniform motion relative to P. Each observer has a rest frame and a motion frame associated with it. An observer’s rest frame is three-dimensional and their motion frame is effectively one-dimensional, that is, only one dimension is used.

A frame of reference is unmarked if there are no units specified for its coordinates. A frame of reference is marked by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A space frame of observer P is a rest frame of P that is marked with units of length. A time frame of observer P is a motion frame of P that is marked with units of duration.

Let there be a frame K1 with axes a1, a2, and a3, that is a rest frame of observer P1, and let there be a motion frame K2 with axes 1, 2, and 3, that is a motion frame of P1 along the coincident a1-a´1 axis. See Figure 1.

Two frames

Figure 1

In the inverse case, the frame K2 is at rest relative to observer P2 and so is a rest frame of P2, and the frame K1 is a motion frame of P2. In other words, the rest frame of P1 is the motion frame of P2, and the rest frame of P2 is the motion frame of P1. The rest and motion frames of K1 and K2 are interchanged. See Figure 2.

Two frames

Figure 2

If observer P1 has a rest frame that is a space frame and a motion frame that is a time frame, then the space and time frames form a space+time framework (3+1) for P1. If an observer P2 has a rest frame that is a time frame and a motion frame that is a space frame, then the time and space frames form a time+space framework (1+3) for P2.

The time rate of motion velocity vector is defined in space+time. The space rate of motion lenticity vector is defined in time+space. If only the linear path of motion is considered, as on a railway, then both rates are defined as scalars in a (1+1) framework.

~~~

The coordinates of a space+time framework can be transformed to another space+time framework with a known relative velocity. If observer P1 has space frame K1 with coordinates x1, x2, and x3, and observer P2 has space frame K2 with coordinates x1´, x2´, and x3´, and frame K2 is moving with time velocity v relative to frame K1 along the x1-x1´ axis, then the coordinate transformation is as follows (see Figure 1):

x1´x1 − vt1, x2´ = x2, x3´ = x3t1´ = t1, t2´ = t2, and t3´ = t3.

If frame K2 is moving with space velocity u relative to frame K1 along the t1-t1´ axis, then the coordinate transformation is as follows:

t1´ = t1ux1/c², t2´ = t2, t3´ = t3, x1´ = x1, x2´ = x2, and x3´ = x3.

The coordinates of a time+space framework can be transformed to another time+space framework with a known relative lenticity. If observer P1 has time frame K1 with coordinates t1, t2, and t3, and observer P2 has time frame K2 with coordinates t1´, t2´, and t3´, and frame K2 is moving with space lenticity ℓ relative to frame K1 along the t1-t´1 axis, then the coordinate transformation is as follows (see Figure 2):

t1´ = t1x1/ℓ, t2´ = t2, t3´ = t3, x1´ = x1, x2´ = x2, and x3´ = x3.

If frame K2 is moving with time lenticity w relative to frame K1 along the x1-x1´ axis, then the coordinate transformation is as follows:

x1´x1 − k²t1/w, x2´ = x2, x3´ = x3t1´ = t1, t2´ = t2, and t3´ = t3.

In matrix terms, these are variations on the Galilean transformation:

\begin{pmatrix} t'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -v & 1 \end{pmatrix} \begin{pmatrix} t\\ x \end{pmatrix}           \begin{pmatrix} t'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & -u/c^{2} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} t\\ x \end{pmatrix}

\begin{pmatrix} x'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1/\l & 1 \end{pmatrix} \begin{pmatrix} x\\ t \end{pmatrix}        \begin{pmatrix} x'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & -k^{2}/w \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ t \end{pmatrix}

Since the rates of motion are constant, we may define β = v/c = u/c = k/ℓ = k/w (with c = 1/k = the speed of light in a vacuum), and simplify these as follows:

\begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}          \begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}

\begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} kx\\ t \end{pmatrix}       \begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} kx\\ t \end{pmatrix}

Given that the speed c is a universal maximum and pace k is a universal minimum, the time dilation and length contraction may be derived with the gamma factor (see here and here), then the these transformations can be combined with a gamma matrix as follows:

 \begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & 1/\gamma \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}

\begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} kx\\ x \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & 1/\gamma \end{pmatrix} \begin{pmatrix} kx\\ x \end{pmatrix}

The result in both cases is the Lorentz transformation.

Proper and improper rates

The independent quantity in a proper rate is the denominator. The independent quantity in an improper rate is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is harmonic addition:

\frac{b}{a_{1}}+\frac{b}{a_{2}} \Rightarrow \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right)^{-1}

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the harmonic mean:

\frac{1}{2}\left (\frac{b}{a_{1}}+\frac{b}{a_{2}} \right ) \Rightarrow \left ( \frac{1}{2} \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right ) \right)^{-1}

Time speed is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. Space speed is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the space mean speed. The time mean speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added harmonically. Lenticity normalized by an hypothesized maximum pace is proper, but if the lenticity is divided by the pace of light, it is improper.

Read more →

Harmonic conversion of space and time

As noted here, there are two kinds of mean rates: the time mean and the space mean. If the denominators have a common time interval, the time mean is the arithmetic mean and the space mean is the harmonic mean. If the denominators have a common space interval (stance), the space mean is the arithmetic mean and the time mean is the harmonic mean.

For example, light reflected back from a mirror at known distance forms two successive trips whose mean rate is the space mean pace. Several measurements with the same apparatus have a time mean pace. The mean speed is the inverse of the mean pace.

The general principle is that quantities with independent time (such as velocity) and a common time interval use ordinary algebra but such quantities with a common space interval use harmonic algebra. Alternately, quantities with independent space (stance) such as lenticity and a common space interval use ordinary algebra but such quantities with a common time interval use harmonic algebra.

In other words, quantities over the same interval with independent variables use ordinary algebra but quantities with different independent variables use harmonic algebra to convert between them.

For example, addition of velocities with a common time interval use ordinary vector addition (e.g., u + v) but addition of velocities with a common space interval use harmonic vector addition (e.g., ((1/u) + (1/v))−1 ≡ ((u+v)/u·v)−1 with u, v, u·v, u+v0.

The relativity parameter γ is based on a (3+1) spatial frame. The parameter γ in a temporal frame with a common time interval (k ≡ 1/c and ≡ 1/v) is:

γ² = (1 − v·v/c²)−1 ⇒ (1 − ℓ·ℓ/k²)−1.

The parameter γ in a temporal frame with a common space interval (stance) is:

γ² = (1 − v·v/c²)−1 ⇒ H(1 − ℓ·ℓ/k²)−1 = ((1 − k²/ℓ·ℓ)−1)−1 = (1 − k²/ℓ·ℓ) ≡ (1 − v·v/c²) = 1/γ².

Interchanging space and time

The space-time exchange invariance, as stated by J. H. Field (see here) has an implicit second part. In addition to (1) the exchange (or interchange) of space and time coordinates, there is (2): the exchange of linear and harmonic algebra for ratios. Harmonic algebra is described here.

This is seen in the different averaging methods for velocities that differ spatially vs. velocities that differ temporally. If two vehicles take the same route, their average velocity is their arithmetic mean (u + v)/2, but if one vehicle has velocity u going and velocity v returning, then the average velocity is their harmonic mean 2/(1/u + 1/v). However, if one vehicle has pace u going and pace v returning, then the average pace is their arithmetic mean, but if two vehicles take the same route, their average pace is their harmonic mean.

Space and time are related to each other as covariant and contravariant components. If space is covariant, then time is contravariant, and if time is covariant, then space is contravariant.

The equations of space-time (3+1) and time-space (1+3) physics are symmetric to one another with the interchange of space and time dimensions. The equations of spacetime (4D) physics is self-symmetric. The interchange of space and time dimensions produces equivalent 4D equations.

To interchange the space and time coordinates, take these steps: For the equations of classical physics, (1) ensure either space or time is a parameter, (2) interchange one dimension with the parameter, and (3) expand the single dimension into three dimensions. For the equations of relativistic physics, (1) ensure there is a symmetry between space and time dimensions, (2) interchange one space and time dimension but leave dimensionless quantities unchanged, and (3) expand the single dimension into three dimensions.

These steps reflect the difference between Galileo’s and Einstein’s relativity. Galileo transforms one frame into another frame but does not combine frames as Einstein’s does. For example, Einstein requires all frames to have the same orientation, but Galileo accepts frame-specific orientations such as the right-hand rule.

The Galilean transformation represents the addition and subtraction of velocities as vectors. The dual Galilean transformation represents the addition and subtraction of lenticities as vectors. The Lorentz transformation represents the combination of Galilean and dual-Galilean transformations, as previously shown.

Motion as a parade

Have you ever seen a parade? Have you ever been in a parade?

parade

A parade is basically a linear (1D) motion. It begins at a point in space and time and ends at a point in space and time. It is planned to progress in a certain order.

The view from the side sees the parade pass by. The parade participants change in time but the location does not. The parade represents the diachronic perspective of motion in time.

The bird’s-eye view from above (as of a camera on a drone) sees the parade as a whole. The time keeps changing but the general location does not. The parade represents the synchronic perspective of motion in time.

The view from a parade participant sees the streetscape pass by. The parade watchers change in space but the chronation does not. The parade represents the diachoric perspective of motion in space.

The view from the plan for the parade sees the parade as a whole. The stance keeps changing but the chronology does not. The parade represents the synchoric perspective of motion in space.

Places, spaces, and times

Time is like a river that flows on indefinitely, as observed from a place on its bank. The flow of time is downstream. Place does not change in this way but the time keeps changing.

Space is like a river that flows on indefinitely, as observed from a platform floating down the river. The flow of space is upstream, as places on the bank recede from view. Time does not change in this way but the place keeps changing.

Places have spaces between them. Spaces are distances measured as lengths (length of space). Places are also called stations, as in railroad stations, if they are places along a route (stance and station are related etymologically). Spaces are located by the places at their beginning and end points. “What station is it here?” could be asked by a passenger in a train at a stop.

Times have time intervals between them. Time intervals are distances measured as durations (length of time). Times are chronated (positioned) in 3D time. Time intervals are chronated by the times at their beginning and end instants. “What time is it now?” could be asked in many contexts.

Spacetime is a place-based metric. Timespace is a time-based metric.

In classical physics there is a conversion factor between space and time that is adopted as a convention by all observers and is measured by a uniform motion relative to each observer. In relativity physics there is a uniform motion that is absolute, that is, the same as measured by every observer, and functions as a conversion factor between space and time.

Terminology contexts

This post continues the one here. While I avoid coining new terms or new definitions, some have been necessary. To have a consistent vocabulary, I try to imagine contexts in which they easily fit.

Some words are simply variations of words in use: distime is like distance; dischronment is like displacement; chronation is like location; elaphrance is like mass; levitation is the opposite of gravitation; and oldtons are the units for release, analogous to newtons for force. Metreloge is like horologe, which is a clock.

One context is racing. The term pace is used, particularly in running and (bi)cycling to mean the time interval per unit distance, which is the inverse of speed. The direction is ignored or assumed to follow the course of the race so a new term is needed to indicate the vector version of pace. A term that has been used is lenticity, from Latin lentus, slow. [Note: previously used legerity, which is an old literary term for lightness of movement.]

The second context is transport, such as package delivery. Consider an order to expedite a delivery. That means to reduce the time of transport, analogous to de-retardation. Release is analogous to a force applied. A package stamped with “RUSH” gets a greater effort to reduce the time of delivery, analogous to a negative release. Drawing means a release over a distance, analogous to a force applied over time (which is called impulse). Repose is a release applied over a dischronment, and is the inverse of work. Lethargy is the capacity for repose, which is analogous to energy.

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 stanceline).

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 timepoint 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 timepoint of the time line, instantaneously. For dual Galilean inertial frames the observer is in motion and the rest frame transmits the current time in a spacepoint of the stanceline, puncstancialy.

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.

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 target 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 target motion, which is said to be “in space.” Displacement is a translation vector from one point to another point of the target motion. The travel distance from the beginning point to the ending point of the target 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.” Dischronment 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.

Read more →

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.