iSoul In the beginning is reality

Time and distance clocks

An ordinary clock measures time. But what if a clock shows a distance instead of a duration? For example, an old-fashioned clock with a face and hands points radially with a constant angular velocity. What if the output came from the circumference instead? A one o’clock reading would be equivalent to 1/12th of the circumference or πr/6, where r is the radius, in units of distance.

That would be using distance to measure time based on a particular clock. That means we can convert time and space based on this clock. Interesting, perhaps, but not very useful since it’s all relative to one particular clock. What if we get everyone to agree on a standard clock, that is, a standard radius for all clocks? Then everyone could specify time by giving a distance: the distance along the circumference of a circular clock.

The point is that clocks can be based on angular or linear movement, so distance and duration can be interconverted based on a standard speed or angular velocity. The circumferential movement is like a wheel on an axle so we could say there are axial (distance) and radial (duration) clocks, just as there are radial (polar) and axial (circumferential) vectors.

[An axial vector (or pseudovector) is a quantity that transforms like a (polar) vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. This is as opposed to a true or polar vector, which on reflection matches its mirror image.]

Angular movement can be used to measure linear distances as well. A measuring wheel traveling at a constant angular velocity along the ground measures the distance and duration between ends of an object or between objects.

What really distinguishes a clock is the fact that it ordinarily doesn’t stop, whether the measure is angular or linear. What distinguishes a ruler is the fact that it ordinarily does stop or has a non-standard movement that isn’t suitable for a standard measure of duration. So we have two kinds of rulers: angular and linear, and two kinds of clocks: angular and linear.

What then is time? Time is a measure associated with constant speed or angular velocity. What is space? Space is a measure that need not have a constant speed or angular velocity. But if there is a constant speed or angular velocity, then the same measure may be used for either time or space.

Actual and default speeds

The actual speed of a particle or vehicle is the local conversion of duration and distance, that is, local time and space. The potential speed of a particle or vehicle is a standard speed for similar particles or vehicles. This standard speed is a default speed, to be used if the actual speed is not known.

A standard speed may be a local default speed, that is, relative to a specific particle or vehicle and not necessarily applicable elsewhere. A standard speed may be relative to someone’s viewpoint, perhaps the way they drive or an estimate of the amount of congestion. A standard speed may be a universal speed, an absolute for all particles and vehicles, and so for time and space in general. In any case, an actual speed over-rides any default speed.

A signal speed used in determining the location or velocity of a remote particle or vehicle is similar to a standard speed. It is as if a generic particle or vehicle were used to bring the news of what actually happened. This speed might be known but if a generic service is used, then the standard speed would be the signal speed.

A standard speed may be the typical speed, which may be subject to change, or a maximum speed, which may never change. There are advantages and disadvantages in either case. If the standard speed is the typical speed, then it errors are minimized if it is substituted for an unknown actual speed. If the standard speed is the maximum or optimum speed, then it shows the extreme case, which may bound the problem.

The speed of light in a vacuum is a maximum or perhaps optimum speed. A typical speed may be the average speed from some speed data or a nominal speed in round units. The specific medium may set the default speed, for example, if the medium is air and sound is used for signaling, the speed of sound may be the logical standard speed. For a specific particle or vehicle type, their typical or optimal characteristics may be decisive for the standard speed.

Time at Mach 1

In a sense every speed is a conversion speed, that is, a way to convert time into space and vice versa because multiplying a time interval (duration) times the speed of an object leads to the corresponding space interval (length or distance). In some contexts, i.e., a transportation mode or physical medium, there is a particular speed, the conversion speed, that reflects the context in general and does not depend on the speed of any particular object in that context. This conversion speed applies to all objects in its context, except for the known speed of an object or signal.

In the context of high-speed jet travel, the speed of sound may be the typical speed of travel. Also, in the context of sound waves in air a standard speed reflecting ideal conditions may be the reference speed. In both these cases the conversion speed is the speed of sound, also known as Mach 1 because in that case the Mach number equals 1.

T. S. Shankara takes a related approach in his article, Tachyons via Supersonics (Foundations of Physics, Vol. 4., No. 1, 1974, p. 94-104). He draws a parallel between acoustic and electromagnetic waves, derives the Lorentz transformations in this way, and shows that the signal velocity is unrelated to its maximality. His goal is to suggest the possibility of tachyons — of which there is a large literature. This is consistent with what we have shown, too.

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).