iSoul In the beginning is reality

Category Archives: Space & Time

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

Reduced mass and vass

Here we take the reduced mass and show the parallel reduced vass.

In physics, the reduced mass is the “effective” inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem.

Given two object bodies, one with mass m1 and the other with mass m2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass:

where the force on this mass is given by the force between the two bodies.

Given two subject bodies, one with vass n1 and the other with vass n2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of vass:

where the force on this vass is given by the surge between the two bodies.

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Causes for subjects and objects

This continues posts such as the one here related to Aristotle’s four kinds of cause:

final cause formal cause
efficient cause material cause

A subject is a form with purposes. An object is a material with mechanisms. Objects exist in space-time. Subjects exist in time-space.

The upper causes apply to subjects, who have purposes and plans, destinations and routes. The lower causes apply to objects, which have mechanisms and materials, forces and masses. Though subjects can be considered as objects and objects as subjects.

why what
subjects: final cause formal cause
objects: efficient cause material cause

Dynamics is the study of why motion happens, whereas kinematics studies only what motion happens. Kinematics is the material for dynamics. The combination of kinematics and dynamics is called mechanics, but this implies that only objects are considered. If subjects are included, then an alternative term is needed, such as kinedynamics.

Mass and vass

In Isaac Newton’s Principia, Definition 1 states:

Quantity of matter is a measure of matter that arises from its density and volume jointly. (The Principia: The Authoritative Translation and Guide, Bernard Cohen, Anne Whitman, and Julia Budenz. University of California Press, 2016, p.403)

Today density is defined as mass per unit volume, which would make this definition circular. However, when Newton wrote, density was expressed as a relative quantity. (p.90) If we look at mass as the product of density and volume, a complementary measure arises: vass.

Density is a ratio, and ratios may be expressed as fractions in two ways: the ratio of nonzero quantities A:B is equivalent to either A/B or B/A. So instead of density as mass per unit volume we could just as well define its inverse, rarity, as volume per unit of mass. (See Max Jammer’s Concepts of Mass in Classical and Modern Physics, p.27.)

Then the rarity per unit of volume equals the vass, which is the inverse of mass. In SI units, that equates to (m³/kg) / m³ which equals 1/kg.

Mass is also defined as the ratio of force to acceleration, reflecting Newton’s second law. Force is the time rate of change of momentum. A complementary definition would be the space rate of change of fulmentum, which equals the vass.

Inertial mass is the resistance of an object to a change in its state of motion when a net force is applied. A complementary concept is the nonresistance of a subject to a change in its condition of movement when a net surge is applied.

If mass is the “quantity of matter,” what is vass the quantity of? Quantity of matter means how much of a material object there is. Vass answers how much of a material subject there is, which is measured inversely to the mass as subject and object are inverses.

Speed of information

Nowadays, we say that the speed of information is the speed of light. That is justified by the rôle of the speed of light in relativity, in which it is the speed of causation. But it is also justified by the use of electromagnetic waves to transmit information between people.

It was not always so. It took much longer for information to travel in the past.

A day’s journey in pre-modern literature, including the Bible, ancient geographers and ethnographers such as Herodotus, is a measurement of distance. In the Bible, it is not precisely defined; the distance has been estimated from 32 to 40 kilometers (20–25 miles). Wikipedia

A critical fact in the world of 1801 was that nothing moved faster than the speed of a horse. No human being, no manufactured item, no bushel of wheat . . . no letter, no information, no idea, order, or instruction of any kind moved faster. Nothing ever had moved any faster.  Stephen E. Ambrose, Undaunted Courage (Simon & Schuster, 1996), p. 52.

The book A Farewell to Alms includes a table showing how long it took for news of sig­nif­i­cant events to reach London. Faster speeds resulted from the in­ven­tion and de­ploy­ment of the telegraph by 1880:

Speed of Information Travel to London, 1798-1914
Event Year Distance (miles) Days until report Speed (mph)
Battle of the Nile 1798 2073 62 1.4
Battle of Trafalgar 1805 1100 17 2.7
Earthquake, Kutch, India 1819 4118 153 1.1
Treaty of Nanking 1842 5597 84 2.8
Charge of the Light Brigade, Crimea 1854 1646 17 4.0
Indian Mutiny, Delhi Massacre 1857 4176 46 3.8
Treaty of TienSin (China) 1858 5140 82 2.6
Assassination of Lincoln 1865 3674 13 12
Assassination of Archduke Maximilian, Mexico 1867 5545 12 19
Assassination of Alexander II, St. Petersburg 1881 1309 0.46 119
Nobi Earthquake, Japan 1891 5916 1 246

 

6D invariant interval

Since one may associate either the arclength (travel length) or the arctime (travel time) with direction, one might think that the full coordinates for every event are of the form (s, t, ê), with arclength s, arctime t, and unit vector ê. Since the direction is a function of either the arclength or the arctime, the coordinates would be either (s, t, ê(s)) or (s, t, ê(t)).

However, since s = ∫ || r′(τ) || , where the integral is from 0 to t, and t = ∫ || w′(σ) || , where the integral is from 0 to s (see here), this reduces to either (t, r) or (s, w).

But science seeks unification and so must combine these forms into one. In that case, both s and t are redundant, and the full coordinates for every event are of the form [r, w]. That is, there are three dimensions of space and three dimensions of time. The arclength and arctime are implicit, and may be made explicit through integration.

The standard exposition of special relativity looks at one dimension of space and one dimension of time. This is convenient and makes Δs = Δx and Δt = Δw1. But in general Δs and Δt will either be measured directly or found through integration.

What is the distance-like invariant interval then between two events? The interval in length units (proper length) is (dσ)² = (cdw)² – (dr)²,  where c is the speed of light. The interval in time units (proper time) is (dτ)² = (dw)² – (dr/c)².

This appears different from special relativity because it substitutes the vector dw for the scalar dt. However, the scalar (dt)² = (dw1)² + (dw2)² + (dw3)² so there is no discrepancy.

In order to demonstrate that this interval is invariant for two observers traveling at different rates, one must either convert dw to dt or convert dr to ds, which reduces the six dimensions to four.

The intervals above may be generalized for general relativity with the relation L = cP √(–gμν dxμ dxν), where P is the path, gμν is the metric tensor, and there are six coordinates xμ and xν.

Physics of subjects

If a stone rolls down a hill, we would say it is simply following the law of gravitation. It is not “going somewhere” as if it had a destination – that would require nature to have a soul, a view that died out in the early modern period. But if a person or an animal or even a seed pod moves down a hill, we expect it to be going somewhere, to have a destination or purpose.

That is the difference between a subject in motion and an object in motion. At a minimum, an object must have some starting point, at least from our observation, but need not have a destination or purpose for all we know. On the other hand, a subject need not have a known starting point but at a minimum there must be some movement toward a destination or end, else they would not be a subject.

This simple difference leads to a different formulation of space, time, and matter for subjects and objects. Modern physics has been entirely focused on bodies as objects, particles, or waves. In contrast, the physics of subjects will focus on bodies as subjects (somebodies), transicles, and networks.

Since there is a destination, something about its location must be known. At a minimum there must exist a route or path for the subject to traverse to reach their destination. Even if the length of the path is not known, one can at least measure the progress made toward reaching the destination by measuring the space rate of movement, called the pace.

The difference between speed, the time rate of motion, and the pace is the difference between taking space or time as the independent variable. For objects their motion from a point in time is what is given and so time is the independent variable. For subjects space is the independent variable since their movement toward a destination in space is given.

That means for subjects the dependent variable is time, which is measured along with the direction of movement, which results in three dimensions of time. Space is confined to the path of movement, which may be rectified as a line for linear referencing. Examples of a linear reference are the milepoint (MP) and kilometric point (PK) on a map or sign.

Objects have chronologies. Subjects have a destinations. But subjects are like objects in some ways, and objects are like subjects in some ways. For example, a projectile is an object that has been launched by a subject toward a destination.

Mechanistic sciences such as physics study objects. Teleological sciences such as economics study subjects. The physics of subjects is physics for the social sciences.

For more, see the other posts on this website about time-space, with 3D time and 1D space.

Space and time as references

A clock provides a linear reference to measure duration of motion. Similarly, there is a linear reference to measure length of movement. What is this linear reference?

In mapping and geographic information systems (GIS) a linear referencing system (LRS) “is a method of spatial referencing, in which the locations of features are described in terms of measurements along a linear element, from a defined starting point, for example a milestone along a road.”

This may be extended to 3D space by a reference frame, “a space-time coordinate system and set of reference points in space-time that assigns unique space positions and reference durations.” From such a reference frame, one can derive linear references from path lengths.

Alternately, one may attach an odometer (cyclometer, pedometer) to each vehicle or subject in motion, and measure their transit length directly.

Thus there is a strict parallel between the reference provided by a clock and a linear reference such as an odometer. As the former is said to constitute time, an ordering by duration, so the latter constitutes an ordering of space by length.

Terms for motion again

Previous posts deal with terms for motion, such as here. Further thoughts are below.

When someone asks about the length of a trip, they are not asking for the distance between the origin and destination of the trip – that is the magnitude of the displacement. They are asking about the length of the route taken. Mathematically, travel length is the arc length of the curve of the route.

The length of a trip in time, or travel time, is the duration of a trip. Time is a kind of length, not a distance; an arc length, not a straight-line distance.

The magnitude of a displacement is the distance between two points. Call the magnitude of a distimement between two points in time the distime. This is the shortest-length travel time between them, which depends on the mode of travel.

We have the expression “as the crow flies” to distinguish the straight-line distance between two points from the travel length. Physicists would say “as light travels” to indicate the straight-line (geodesic) distance or time between two events.

While physicists may convert time and space dimensions (by multiplying time by the speed of light, or dividing length by the speed of light), this does not change the character of the dimensions. Only if the time and space dimensions are switched does their character change.

Tempo in music

Tempo in a piece of music is often stated with conventional Italian terms (recall that the Renaissance began in Italy). Since the invention of the metronome, tempo is also given in beats per minute (bpm). However, a slow tempo is one in which a beat takes more time, and a fast tempo is one in which a beat takes less time. So it would make sense to measure tempo in units of time per beat.

Since bpm ranges from about 20 to 200, it would be appropriate to convert it to units of jiffies per beat, with a jiffy defined as one-sixtieth of a second (so 60 jiffies make one second). To convert from X bpm to Y jpb, divide X into 3600; that is, Y jpb = 3600 / X.

Here are suggested jpb tempo values from slowest to fastest based on the list here:

  • Larghissimo – very, very slow (24 bpm and under) = 150 jpb and over
  • Grave – very slow (25–45 bpm) = 80 to 144 jpb
  • Largo – broadly (40–60 bpm) = 60 to 90 jpb
  • Lento – slowly (45–60 bpm) = 60 to 80 jpb
  • Larghetto – rather broadly (60–66 bpm) = 54 to 60 jpb
  • Adagio – slowly with great expression (66–76 bpm) = 47 to 54 jpb
  • Adagietto – slower than andante (72–76 bpm) = 47 to 50 jpb
  • Andante – at a walking pace (76–108 bpm) = 50 to 33 jpb
  • Andantino – slightly faster than andante (80–108 bpm) = 33 to 45 jpb
  • Marcia moderato – moderately, in the manner of a march (83–85 bpm) = 42 to 43 jpb
  • Andante moderato – between andante and moderato (92–112 bpm) = 32 to 39 jpb
  • Moderato – at a moderate speed (108–120 bpm) = 30 to 33 jpb
  • Allegretto – by the mid 19th century, moderately fast (112–120 bpm) = 30 to 32 jpb
  • Allegro moderato – close to, but not quite allegro (116–120 bpm) = 30 to 31 jpb
  • Allegro – fast, quickly, and bright (120–156 bpm) = 23 to 30 jpb
  • Vivace – lively and fast (156–176 bpm) = 20 to 23 jpb
  • Vivacissimo – very fast and lively (172–176 bpm) = 20 to 21 jpb
  • Allegrissimo or Allegro vivace – very fast (172–176 bpm) = 20 to 21 jpb
  • Presto – very, very fast (168–200 bpm) = 18 to 21 jpb
  • Prestissimo – even faster than presto (200 bpm and over) = 18 jpb and under

Music tempo measured by jpb is similar to legerity in physics previously discussed (e.g., here).

Measurement of space and time

The various ways of measuring space and time are parallel.

Measuring space:

  1. A ruler measures length, that is, the distance between two points in space (A to B).
  2. An ruler turned upside-down measures length backwards (B to A).
  3. A tripmeter measures the travel distance of a vehicle trip.
  4. An odometer measures the cumulative travel distance.Odometer 12,000
  5. A measuring wheel measures the travel distance of a wheel being pushed.
  6. A road map measures travel distance of a standard vehicle. See Geodistance.

Measuring time:

  1. A stopwatch measures time, that is, the duration between two points in time (A to B).
  2. A timer measures the time counting down from a set time, i.e., backwards (B to A).
  3. A GPS watch or time clock measures the duration of an activity, such as running or working. GPS watch
  4. A GPS watch (or smartphone app) measures cumulative travel time (or flight time).
  5. A measuring wheel with a stopwatch measures the travel time of a wheel being pushed.
  6. A clock measures travel time synchronized with a standard motion.

Note that #2 shows time can be measured backwards. Space and time can both be counted up or counted down. There’s nothing magical about it.