iSoul In the beginning is reality

Centers of time measurement

The ancient center of time measurement was the earth, and this is still used in everyday life. The changing positions of the sun and moon relative to the earth make a convenient clock. In this sense, geocentric time makes sense. But the movements of planets are difficult to use in this way; their retrograde movements require ad hoc modifications to a geocentric system.

The proposal to switch to a sun-centered time system was met with resistance but its advantages eventually won out, with Newton’s laws ending the issue. The greater comprehensiveness of heliocentric time (heliochronic system) over geocentric time (geochronic system) proved to be decisive. Nevertheless, the everyday terms noon, morning, afternoon, etc. are still used, showing the naturalness of a geochronic system.

In the 20th century, the atomic clock was invented, which uses an electronic transition frequency of the electromagnetic spectrum of atoms (the signal electrons in atoms emit when they change energy level). This might be called a “phochronic” (light-time) standard. The positions of celestial bodies are not used with this system of time. It is an acentric time standard.

If accuracy is the most important factor, then a phochronic system is best. But it is not surprising that the “24/7” way of life arose since this acentric system was implemented. Time is less and less connected with the rhythms of the sun, the week, the seasons, etc. If the latter are the most important, then the geochronic system is best since it fits well with these rhythms, which are still an important part of the cycles of life.

Directional units

Of the base units of the International System of Units (SI) only two concern movement in a direction: the units of length (metre) and duration (second). The candela measures luminous intensity in a direction without regard for movement. Derived units of movement include speed and velocity (metres per second), force (newton), pressure (pascal), energy (joule), power (watt), acceleration (metres per second squared), momentum (newton second), and action (joule second). Angular measures are directional through the axis of movement: angular velocity (radians per second), angular acceleration (radians per second squared), angular momentum (newton metre second), and torque (newton metre).

All of these units are directed in three dimensions. The units for space (length) and time are the directional base units. The derived units have directions in three-dimensional phase spaces, which follow the directions of space and time. If only one of the components of a derived unit has a direction, then the derived unit has the dimensions of that component. For example, velocity has the direction of length since it uses the time magnitude rather than the time direction.

The second is defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. The metre is the distance traveled by light in a vacuum in 1/299,792,458 second.

The light-year is defined by the International Astronomical Union (IAU) as the distance that light travels in a vacuum in one Julian year (365.25 days). Although it is a unit of length, with the speed of light in a vacuum it may be converted to a unit of time. This conversion is commonly used in astronomy so that looking at the stars is considered looking into the distant past — in three dimensions of direction.

Cycles and orbits

The first clocks were the cycles of daily life, notably the diurnal cycle of light and dark. These continued apparently without end and so provided a measure of change, of ceaseless movement and return. Since ancient times the day has been divided into 24 hours, and since medieval times an hour has been divided into 60 minutes. An old fashioned circular clock mimics these cycles, dividing a day into hours, minutes, and seconds measured by the angular movement of pointers called hands.

Orbits are like raceways with their fixed path for repeated travel over a distance. For a race the goal is to achieve the shortest time to travel the allotted distance. But an orbit continues without apparent end. The distance a planet or satellite travels keeps increasing, providing a consistent movement to compare with other movements. An orbit is like a clock but the distance traveled is the circumferential movement, not the angular movement.

Because cycles were measured in angles first, angular movement was associated with ceaseless movement, which was called time. But circumferential or linear movement can just as well be associated with cycles, especially orbits, whose space could be equally well associated with ceaseless change. But it is change, not time or space, that these ceaseless movements are really about.

Consider the sayings that might result. “Change flies.” “Change and movement wait for no man.” “Change is of the essence.” “A waste of change.” “Change cures all.” “Change is money.” “Change works wonders.”

Converting space and time

To convert a length of space into a corresponding length of time requires a conversion factor. For physical reality that conversion factor is the speed of light: r = ct, where r is a spatial displacement, t is a temporal displacement, and c is the conversion factor. For a mode of transportation the conversion factor between space and time is a typical or conventional speed. One reason for such a convention is to communicate the length of time expected in order to traverse a corresponding length of space.

That this conversion factor is also a speed is extraneous to its status as a conversion factor. Whether or not anything travels at such a speed does not matter to the conversion of space and time. Whether or not one measures something going at such a speed does not matter either. All that matters is that the convention is accepted. The science community has agreed to define the speed of light in a vacuum as exactly 299,792,458 metres per second. A particular map may use a single conversion between space and time; everyone following that map has the same conversion factor.

Does that make the conversion of space and time subjective? Not necessarily, because a convention may be justified by an argument about objective reality. But it could be subjective if someone adopts their own conversion factor which no one else is using. That may reflect their driving style or preference for under or over estimation. The purpose of the conversion is relevant.

A conversion factor applies to a real or virtual phenomena. If people with frame of reference S have the same conversion factor as those with frame S′, then the conversion applies to both. So r = ct in S and r′ = ct′ in S′ because of the status of c as the conversion factor for both S and S′. This is how the Lorentz transformation can apply both to particle physics and everyday transportation.

Actual and potential time and space

I’m made the point that if we’re asked how far away a place is, we can answer in units of space or time. If we’re talking about stable places such as on land, the distance will not change but the length of time depends on a typical or standard speed. In that case the length of time is a potential time it would take assuming the standard speed. The actual speed and thus the actual time may be different.

Here’s a scenario for example: Person X asks, How far is it downtown? Person Y responds, About half an hour, which assumes a typical speed. Person X starts driving downtown and after half an hour they haven’t arrived yet. How much longer will it take? Based on this simple scenario, the time potentially remaining is the distance remaining divided by the typical speed. The actual time is not known yet.

If there is a standard or typical travel speed as for a guideline or a map, then space and time are convertible into one another as potentials. The conversion is actual either after the fact when the actual speeds are known or if the conversion is fixed, as with a vehicle that only goes one speed or light which has a constant speed.

Is there a potential length of space as well as this potential length of time? Yes, if the travel time is predetermined but the distance traveled is not. For example, if someone agrees to go for a walk but only has a certain amount of time, the potential length of the walk is the allotted time divided by the typical pace (e.g., in minutes per mile).

Defining space and time

In order to understand anything we need to have good definitions. Otherwise the words we use will lead us astray — which is what has happened with the word “time”.

I have a copy of the McGraw Hill Dictionary of Physics, Third Edition. Here is its definition of time:

  1. The dimension of the physical universe which, at a given place, orders the sequence of events.
  2. A designated instant in this sequence, as the time of day. Also known as epoch

What’s wrong with this definition? Events can be ordered in various ways, and that’s the gospel truth (compare the differences among the four gospels). Events can also be ordered by their correspondence to space; here’s an example:

“When did you get to the race?” “We didn’t arrive until they’d gone a quarter of the way.”

So time is not an ordering, though it has an ordering. It’s interesting that this dictionary doesn’t define space at all, though it refers to “ordinary space” as if it’s obvious what that is.

Time is not the dimension of ordering; it’s not a dimension at all. Would we call space a dimension? No, we say space has dimensions. It’s the same for time: time has dimensions. What are dimensions? Dimension is a concept that comes to us from geometry, where it is defined as the minimum number of coordinates needed to specify a point on (or in) it.

If time is neither an ordering nor a dimension, what is it? Both time and space should be defined in terms of movement because that’s how we measure them.

Measuring a movement requires comparing it with a parallel movement. There are two basic movements which can be standardized for systematic use in science and other disciplines. One is angular movement and the other is linear movement. Comparison with constant angular movement results in a temporal measurement, a point in time or a length of time. Comparison with constant linear movement results in a spatial measurement, a point in space or a length of space.

This leads to the following parallel definitions of time and space:

  1. Time (space) is what is measured by a standardized angular (linear) movement that begins and ends. In everyday terms, time (space) is what is measured by a stopwatch (measuring wheel) or its equivalent.
  2. A point in time (space) is the beginning or ending of something, an object or event.
  3. A length of time (space) is the difference between the beginning and ending of something, an object or event. Also known as a duration (distance).
  4. A dimension of time (space) is a coordinate for specifying a point in time (space) relative to an origin, a standard reference point. Both angular and linear coordinates are possible.
  5. A position vector of time (space) is an ordered pair of numbers for a point in time (space): the first is the magnitude of the length of time (space) from the origin in time (space) and the second is the direction from the origin in time (space).
  6. As a matter of observation and fact, there are three dimensions of time (space).

Does this mean there are six dimensions in all? Perhaps. Both time and space have three dimensions, and their dimensions might be combined, as Minkowski combined four dimensions.

Equality of space and time

“How far is it to X from Y?” That everyday question can be answered either by a distance or a duration with a mode of travel (e.g., walking, driving, flying). The interchangeability of a length of space and a length of time leads to two simple conclusions: (1) time has as many dimensions as space does, and (2) space and time are symmetric. In short, space and time are equals in an almost political way: we should not discriminate against one or in favor of the other if possible.

“What is the length of a coastline?” This question is used to point out that the coastline of a landmass does not have a well-defined length. If someone is walking the coastline or examining high-resolution aerial photographs, they will find a longer coastline than someone flying along the coastline or looking at low-resolution photographs. This reinforces that mode of travel or resolution are needed to specify a distance properly. It is similar for travel on land: flying between two cities is likely more direct than driving on a highway network. Which is the correct distance? It depends on the mode of travel.

In physics the “mode of travel” is a light wave — but there may be exceptions. For example, a sound studio or study of whale sounds would be more concerned with the distance that sound travels. Or a phase space might have its own kind of distance. In any case, the mode is there whether it is specified or implicitly understood.

Another aspect of distance and duration is the path that is taken. That may be clear from the mode or there may be alternate routes within a mode. For example, there are many ways to travel from X to Y on a highway network. One may be the shortest distance of travel and another the shortest travel time. So the question of “how far is it?” leads to the response, “with the shortest distance or travel time?”

There is a desire in physics to describe physical laws independently of any observer. If distance and duration aren’t convertible, e.g., by a conversion factor, then how can this be done? The answer is via knowledge of a field, which in transportation terms corresponds to a transportation network. Then physical laws can specify which path will be taken. The key way to do this is via the principle of least action (or stationary action).

Kinds of relativity

A simple way to look at the world is to assume that space and time are absolute: the locations, the distances, the durations, speeds, and so forth as measured by one person are the same for everyone. That is, if my automobile speedometer shows 50 mph (80 kph), then the police with a laser gun at the side of the road will show the same speed.

For many purposes of everyday life, that works just fine. But for those who think about it more or those who perform experiments, that breaks down. Galileo Galilei was the first express a principle of relativity in his 1632 work Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity on a smooth sea: any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. He still accepted absolute time, however.

We can call Galilean relativity “spatial relativity” since it applies only to space. Since we have seen the symmetry between space and time, we could develop a similar “temporal relativity” in which time is relative but space is not. This may seem odd at first but it is as consistent (and limited) as spatial relativity. For reference, here are the transformations for spatial and temporal relativity, given two reference frames, S and S’, with an event having space and time displacements r and t (r’ and t’) respectively, with S’ moving at constant velocity v relative to S, then:

r’ = r – vt and t’ = t for spatial (Galilean) relativity, and

r’ = r and t’ = t – r/v for temporal relativity.

Both of these relativities are nonconvertible: knowing the spatial displacement tells us nothing about the temporal displacement and vice versa. Other relativities are convertible; these have finite conversion factors between space and time. The transformation for finite relativity was given here and here:

r´ = (1 – v/c) r and = (1 – v/c) t for finite relativity.

What is that conversion factor between space and time? In everyday life we may use a typical travel speed to tell others, for example, that “it’s two hours from Baltimore to Philadelphia,” which assumes an average speed of 50 mph (80 kph) and so is equivalent to 100 mi (160 km). However, a typical speed is relative to a particular time and place, and perhaps a particular driver or opinion of typical traffic conditions. What is the conversion between time and space for everyone, for all times and places, for the whole physical world?

Einstein was the first to answer the question by combining the principle of relativity with the speed of light as absolute. This led to his derivation of the Lorentz transformation which in addition to the finite speed of light includes the property that the speed of light is the same for all observers traveling with constant speed:

= (1 – v/c) γr and= (1 – v/c) γt,

where the Lorentz factor γ2 = 1 / (1 – (v/c)2). This could be called “isoluminal relativity” because the conversion factor between time and space is the constant speed of light.

Because the Lorentz factor is not a real number if v > c, we either have to assume that this never happens or we have the alternate situation described in Lorentz with 3D time, which we can express in a symmetric way as:

r’ = (1 – c/v) gr and t’ = (1 – c/v) gt,

where g2 = 1 / (1 – (c/v)2).

Note that γ2 + g2 = 1. Also note that γ is real only if v < c and g is real only if v > c. The latter is called superluminal motion if c is the speed of light. It is controversial whether such speeds exist (in contrast to subluminal motion). But if c is just a typical speed used to relate space and time in a transportation mode, it is not an absolute and actual speeds may easily be larger or smaller.

As v and c diverge, the Lorentz transformations lead to those of finite relativity. This implies that speeds greater than the conversion speed also lead to an alternate transformation in general:

r’ = (1 – c/v) r and t’ = (1 – c/v) t.

In conclusion, there are several kinds of relativity principles: spatial relativity (in which time is absolute), temporal relativity (in which space is absolute), finite relativity (in which a finite conversion factor relates time and space), and isoluminal relativity (in which the conversion between time and space is the absolute speed of light).

Symmetric laws of physics

Because of the symmetry of space and time, the laws of physics should be symmetric in space and time, or at least show their symmetry. Granted, one must either use the speed (change in position per unit of time) or the pace (change in time per unit of length). But other than such choices, the form of a law of physics should show the symmetry of space and time.

I have written on Galileo revised, in which the symmetry of space and time leads to a modification of the Galilean (or Galilei) transformation. The addendum includes the need to make the transformations for space and time similar. If the spatial displacement is r, the temporal displacement is t, the relative velocity v, and the conversion constant from time to space is c, then the following transformations fulfill those requirements:

= (1 – v/c) r and= (1 – v/c) t.

The same requirements may be applied to the Lorentz transformation as well, with its inclusion of the Lorentz factor, γ:

= (1 – v/c) γr and= (1 – v/c) γt,

where γ2 = c2 / (c2v2) = 1 / (1 – (v/c)2).

The similarity between these transformations is remarkable. Since (v/c)2 approaches zero faster than (v/c), the Lorentz factor approaches one and the Lorentz transformation approaches the revised Galilean transformation for relatively small velocities. Both transformations include a standard conversion between time and space, that is, an absolute speed, contrary to the original Galilean (and Newtonian) assumption of an absolute time.

Diachronic and synchronic physics

Diachronic, 1857, from Greek dia “throughout” + khronos “time” means something happening over time, particularly the historical development of something such as a language through time.

Synchronic, 1775, means “occurring at the same time,” from Late Latin synchronus “simultaneous,” means the analysis of something such as a language over a wide area at a point or period in time.

The terms diachronic and synchronic may be used to distinguish two approaches to the analysis of anything with spatial and temporal aspects. The diachronic approach stays with one place or people and focuses on the development through time. In transportation it is the perspective on a moving vehicle or data gathered from inside moving vehicles. The synchronic approach looks at a wide area or multiple places at a point in time or within a particular time period. In transportation it is the perspective from the side of the road, on the earth.

Diachronically, the pace of each vehicle is measured from as the ratio of its travel time over a road segment. The (arithmetic) average pace or harmonic average speed is the space mean traffic speed for the length of roadway.

Synchronically, vehicle speeds (spot speeds) are measured from sensors at a location on the road over a period of time. The (arithmetic) average is the time mean traffic speed for a given period of time.

Physics normally uses speeds, not paces, combined with the time displacement and so is synchronic. If the pace is used instead of the speed, combined with the length displacement, physics is diachronic. The laws of physics are the same in either case: space and time are symmetric. For example, the Lorentz transformation:

x’ = γ (x – vt), y’ = y, z’ = z, and t’ = γ (t – vx/c2)

may be interpreted as spatial coordinates x, y, and z, time displacement t, and speeds c and v; or as temporal coordinates x, y, and z, length displacement t, and paces c and v.