iSoul In the beginning is reality

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.

Measurement of space and time

Here is a roundup of various instruments and methods for measuring space and time that may be stopped or continued indefinitely:

A bematist (from ancient Greek βῆμα bema ‘pace’) was a specialist in ancient Greece who was trained to measure distance by counting their steps.

An odometer for measuring distance was first described by Vitruvius (c. 27 – 23 BC) although the actual inventor may have been Archimedes of Syracuse (c. 287 – 212 BC). The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m) diameter turning 400 times in one Roman mile (about 1400 m).

In 1903 Arthur P. and Charles H. Warner from Beloit, Wisconsin, introduced their patented Auto-meter, which used a magnet attached to a rotating shaft to induce a magnetic pull upon a thin metal disk. Measuring this pull provided automobile drivers with accurate measurements of both distance and speed in a single instrument.

A measuring wheel or surveyor’s wheel is a wheel attached to a handle that can be pushed or pulled along by a person walking to measure distance traveled. It is marked in fractional increments of revolution from a reference position. If the wheel is rotated a full turn, the distance traveled is equal to the circumference of the wheel. Otherwise, the distance the wheel traveled is the circumference of the wheel multiplied by the fraction of a full turn.

A trip meter (tripometer) is an odometer that may be reset to record the distance traveled in a particular journey or part of a journey.

A ruler (aka rule or line gauge) is a straightedge with calibrated lines at specified distances from one edge to measure distances or lengths. A tape measure or measuring tape is a flexible ruler, designed to be rolled up for portability.

A clock (or timepiece) is any device for measuring and displaying the time that is designed not to stop. It is one of the oldest human inventions, meeting the need to consistently measure intervals of time shorter than the natural units: the day, the lunar month, and the year.

All oscillating clocks, mechanical and digital and atomic, work similarly and can be divided into analogous parts. They consist of an object that repeats the same motion over and over again, an oscillator, with a precisely constant time interval between each repetition, or ‘beat’. Attached to the oscillator is a controller device, which sustains the oscillator’s motion by replacing the energy it loses to friction, and converts its oscillations into a series of pulses. The pulses are then counted by some type of counter, and the number of counts is converted into convenient units. Finally some kind of indicator displays the result in human readable form.

A stopwatch is a clock that can be started and stopped easily.

Note that the ability of a measuring device to operate continuously is irrelevant to its utility for measuring the dimensions of or to an object or event. Alternatively, an odometer or other rotation-based distance measuring device could be operated continuously. The conclusion is that there is no necessary connection between time or space and continuous change or movement.

A complete explanation

Who, what, when, where – journalists repeat these adverbial questions to find key factors that explain things. That and the four explanatory factors or “causes” of Aristotle are needed to cover all aspects of a complete explanation.

Consider Aristotle’s example of a statue:

The material factor is what it is made from, “that out of which” it is made, e.g., the bronze of a statue.

The formal factor is the form/design that makes it what it is (“what-it-is-to-be”), e.g., the shape of a statue.

The efficient/mechanism factor is what makes it, “the primary source of the change (or rest)”, e.g., the art of bronze-casting the statue.

The final factor: the end/purpose, what is it made for, “that for the sake of which a thing is done”, e.g., beauty as the end of art, health as the end of walking.

There are also adverbial questions to complete the explanation:

The who factor: who made the statue?

The what factor: what is it? A statue.

The when factor: when was it made?

The where factor: where was it made?

All eight of these factor are necessary for a complete explanation.

Modern natural science looks at the efficient/mechanism factors, the material factors, the what, when and where factors. The who, why, and formal factors are excluded. Thus every explanation of modern natural science is incomplete – and so should be treated as input for others to complete them, which could include changing the partial explanations of science if necessary.

Addendum: There are also what might be called causal metafactors. These come after the causal factor and ask, Why? Alternatives likely exist for each factor. Why was this material selected? Why was this mechanism/force used? Why was this design used? Why was this goal sought?

Lorentz with 3D time

Just as three dimensions of space are combined with one dimension of time, so we can combine three dimensions of time with one dimension of space. The place to start is the Lorentz transformation. Let’s take a common approach, that of spherical wavefronts of light but instead of taking three length coordinates and converting time into length via the speed of light, let’s take three duration coordinates and convert space into duration via the speed of light.

Here’s a revision of the Wikipedia text, using Greek letters for time and Latin letters for space:

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with a velocity v with respect to O in the positive ξ-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance r and r′ and a duration τ and τ′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the radial coordinates to the point P:

τ = r / c and τ’ = r’ / c.

The equation of a sphere of time (duration) in frame O is given by

ξ2 + η2 + ζ2 = τ2.

For a spherical wavefront that becomes

ξ2 + η2 + ζ2 = r2 / c2.

Similarly, the equation of a sphere in frame O’ is given by

ξ’2 + η’2 + ζ’2 = τ’2.

so the spherical wavefront satisfies

ξ’2 + η’2 + ζ’2 = r’2 / c2.

The origin O’ is moving along the ξ-axis. Therefore,

η’ = η and ζ’ = ζ.

Other than these equations the derivation follows as before. A slightly different Lorentz factor is found (call it g):

g2 = v2 / (v2 – c2).

Compare that with the usual Lorentz factor, gamma:

γ2 = c2 / (c2 – v2).

Note that γ2 + g2 = 1. Also note that γ is real only if v < c and g is real only if v > c.

The 3D time Lorentz transformation is then

r’ = g (r – ξ c2 / v)

ξ’ = g (ξ – r/ v)

η’ = η and ζ’ = ζ.

In this Lorentz transformation length is the independent variable, whereas in the usual Lorentz transformation time is the independent variable.

Time defined anew

“Time is that which is measured by a clock” wrote Hermann Bondi in Relativity and Common Sense (p.65), though the idea goes back to Albert Einstein, and ultimately to Aristotle.

“A space is that which is measured by a ruler; time is that which is measured by a clock.” (George Lundberg, quoted in Abrahamson, 1981: p.256)

I think the truth of the matter is somewhere between a ruler and a clock. Let’s start with what a clock is.

“Almost any clock consists of three main parts: (1) a pendulum or other nearly periodic device, which determines the rate of the clock; (2) a counting mechanism, which accumulates the number of cycles of the periodic phenomenon; and (3) a display mechanism do indicate the accumulated count (i.e., time).” (Clocks, Atomic and Molecular)

The key component is a periodic movement, that is, a cycle. What is measured is a number of these cycles but notice that our clock nomenclature includes part of a cycle, too. The standard clock cycle is one hour, subdivided into minutes and seconds. These parts of a cycle are naturally associated with angles; they are angular divisions of a cycle. A clock is essentially a way of measuring a constant angular velocity.

Consider a vehicle; it travels on wheels and the turning of the wheels (or the axle mechanism) allows it to measure its distance traveled. If the wheels are turning at a constant angular velocity, they are a kind of clock — with this important difference: it is the movement of the circumference that is significant for the distance, not the angular movement.

A ruler is a tool to measure length. We do not associate movement with a ruler but in fact a ruler must be moved into position to measure anything. A measuring wheel shows that a ruler need not be a linear object, though length is a linear measure. It is the circumference of the measuring wheel that is significant; its angular movement is a means to linear movement. The odometer on a vehicle uses the circular movement of the wheels, but the result is a linear measure because the vehicle is moving linearly.

All this shows that movement is required for measuring length and time. The only difference is that we associate time with a continually moving system, a clock; whereas we associate length (and space) with a temporarily moving system. When the measuring wheel stops, the length ends; but time goes on, so we say.

This is so confused! The measuring wheel that measures length can also measure time (duration) if it is going at a constant rate. Where the wheel ends, the length ends; when the wheel ends, the duration (time) ends. The length (space) and the duration (time) go together.

If we wanted to have devices that measure length, and keep on going without stopping, we could do that. We could use a vehicle moving at constant speed around a racetrack. “What length is it?” would mean how far has the vehicle traveled. It would be exactly like “what time is it?” The only difference is whether the linear measure or the angular measure is taken.

The conclusion is that time (duration) is just like length: a device with constant cyclic movement is related to what is being measured; for the length, take the circumferential movement and for the duration, take the angular movement. After the measure is taken the device may be turned off or it may be left running. If it is left running, it is called a clock; otherwise not.

What’s the difference between a stopwatch and a clock? One stops and the other doesn’t. What’s the difference between an express train and a local train? One stops (more) and the other doesn’t. Movement can be measured by angular devices or linear devices. Either way, it’s still movement but we distinguish between them.

There are various reasons for a standard reference movement, as clocks provide. But the standard movement could be linear; it could be measured in units of length. It could be like a satellite circling the earth: its position over the earth could be taken as a measure of its movement; or if it were in a polar orbit, the corresponding longitude could be the measure. It doesn’t stop so it’s like a clock but it could just as well measure distance as an angle.

I could go around and around about this but the bottom line is that time is a difference in angular movement and space is a difference in linear movement. In the case of length, we’re only concerned with the difference; with time we’re more concerned with the movement. But it could just as well be the other way around. Clocks could cease and constant linear movements could be kept going. Time is that which is measured by a constant angular movement that stops when the measurement is complete.

When a measuring wheel stops, does space stop? No. When a clock stops, does time stop? No. Is it possible to have a zero length while a stopwatch is going? Yes, it’s called stationary. Is it possible to have a zero time (duration) while a measuring wheel is going? Yes, and that is also stationary.

We see both linear and angular movement with light. Frequency and wavelength, and the distance it travels and the time it takes to travel are proportional. The special properties of light make it an ideal standard for relating angular and linear movement, space and time.

Lorentz interpreted

The question is how to interpret the Lorentz transformation. In a previous post, Lorentz generalized, a modest generalization of the Lorentz transform was derived. Absolute reference speeds were combined with a relative actual speed.

Let’s step back and look at a map of space and time:

Interstate Drive Times & Distances Sample

Interstate Drive Times & Distances Sample

This map of nodes and links on the U.S. interstate highway system displays travel distances and driving times between cities. If you look closer, you can see that it is based on a standard travel speed of about 50 mph (with some local variations). So each point on the network represents a travel distance and a travel time: in other words, space and time are in sync.

Now compare this map with some actual travel experience, say, one traveler going at 40 mph and another at 60 mph. If they start together, after one hour of travel they will have gone 40 and 60 miles respectively, compared to the standard of 50 miles. After one hour, the standard “map” distance is 50 miles but the actual distances are 40 and 60, so space and time are not in sync with these travelers.

The problem is space and time can no longer be mapped together: either the distance traveled or the travel time can be mapped but not both. At most all the distances for one travel time or all the travel times for one distance can be mapped.

A physicist approaching this situation might ask, is there some function of space and time that can still be mapped? Is there a quantity that is invariant no matter what the travel speed is? Can an alternate map be constructed?

The answer is yes and the key is the Lorentz transformation. Note that this is for an alternate map: if travel speeds equal the standard speed, no new map is needed. So we’re looking at speeds u and u’ that differ from the standard speed, c.

The alternate map has one limitation: it’s from the point of view of one traveler. But an alternate map can be constructed for any traveler and the principles of its construction are the same for all travelers. That’s the best that can be done.