iSoul Time has three dimensions

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Duals for Galilean and Lorentz transformations

In Newtonian mechanics inertial frames of reference are related by a Galilean transformation and time is absolute. In the special theory of relativity (STR) inertial frames of reference are related by a Lorentz transformation and the speed of light is absolute. By taking account of the three dimensions of time with a single dimension of space, we may derive a dual mechanics to each of these: (1) a Newtonian dual with a single dimension of absolute space and three dimensions of relative time, and (2) an STR dual with a single dimension of relative space and three dimensions of relative time.

In the usual exposition there is a reference frame S’ moving with constant velocity v in the direction of the x1 space coordinate (with no movement in the x2 and x3 directions) and absolute time t. That changes to constant legerity (inverse of velocity) u in the direction of the t1 time coordinate (with no movement in the t2 and t3 directions) and absolute space r, the travel length. The dual Galilean transformations are then

t1´ = t1 – ur

t2´ = t2

t3´ = t3

r´ = r

What does this mean? It means that there is a constant movement measured by legerity so that as the travel length increases, the duration changes from t to t1´ such that t – t1´ = ur, which is a constant ratio of duration over length multiplied times the travel length.

The Lorentz transformation is analogous to this with the absolute time speed of light, c, replaced by the absolute space speed of light, which is the inverse of c, or 1/c.

t1´ = γ (t1 – ur)

t2´ = t2

t3´ = t3

r´ = γ (r – t1uc²)

in which

γ² = 1 / (1 – u²c²).

What does this mean? It means that time (duration) appears dilated and length appears contracted, which is the same as the standard Lorentz transformation (known as a Lorentz boost). The laws of physics remain the same despite changing to a space reference from a time reference.

Geometric vectors in physics

The concept of a vector in physics is similar to that of mathematics: a geometric object with both magnitude and direction. The magnitude is in units that may be any physical units. The direction is in angular units such as radians or degrees. These are called geometric vectors (also known as Euclidian vectors).

Note that the units for the direction are the same for all vectors. Whether a vector represents force or momentum or current density, the angular units are the same. The directional units do not depend on the units of magnitude. If an observable has magnitude and direction, the units of direction are the same as every other physical vector. What kind of physical vector it is depends on the units of magnitude, not the units of direction.

If the magnitude represents duration in a particular direction, is this a temporal magnitude with a spatial direction? No, like every other physical vector the kind of physical vector it is depends only on the units of magnitude, not on the units of direction, which are the same for all physical vectors. So a vector of duration in a particular direction is a vector of time. A vector of directional lengths or distances is a vector of space.

This is where the different senses of the word “space” can confuse us. There is space as an abstract mathematical concept, space as a directional or orientational concept, and space as a length concept. It is the sense of length or distance that distinguishes space from time in physics, with or without a direction.

There is an underlying geometry that relates to all observables and determines the meaning of “direction” in a geometric vector. In non-relativistic physics, this is an Euclidean geometry. In relativistic physics, the underlying geometry is non-Euclidean.

Speeds and velocities

A common word-problem in arithmetic goes something like this: If someone takes a road trip and for half of the time they go one speed and for the other half they go another speed, how should their average speed be determined? The answer is that the average speed is the arithmetic mean of the two speeds. It is implicitly weighted by the time each speed was driven because the denominator of speed is a unit of time.

In symbols, if the first speed is s1 and the second speed is s2, then the average speed is (s1 + s2) / 2. I call this the mean time-speed, though it is generally known as the time-mean speed since it is averaged in reference to time.

But if someone takes a road trip and for half of the distance they go one speed and for the other half they go another speed, how should their average speed be determined? The answer is that the average speed is the harmonic mean of the two speeds, because then it is weighted by the distance each speed was driven, which is in the numerator.

In symbols, if the first speed is s1 and the second speed is s2, then the average speed is 1/((1/s1 + 1/s2) / 2) = 2 / (1/s1 + 1/s2) = 2 s1 s2 / (s1 + s2). I call this the mean space-speed but it is generally known as the space-mean speed since it is averaged in reference to space.

This becomes clearer if we speak about how slow the vehicle travels rather than how fast, that is, the inverse of speed, which I call the space-speed. This measures the travel time per unit of distance. Since the denominator is now distance, the average of the space-speeds is the arithmetic mean. Inverting this results in an ordinary speed (which I call the time-speed), which equals the harmonic mean of the two speeds in the word problem.

The same procedures apply to angular speed or rotation as well. The arithmetic mean averages two different angular speeds for given time periods. The harmonic mean averages two different angular speeds for given angular distances, i.e., angles or rotations.

What about velocity instead of speed? That is, what happens if the speeds are in different directions? If someone takes a road trip and for half of the time they go one speed East and for the other half they go another speed North, their average speed is the arithmetic mean of the two speeds. But their resultant speed, that is, the magnitude of the resultant vector, is the length of the hypotenuse formed by the triangle with speeds East and North weighted by their duration and divided by the total time. It is in the direction of direct flight from the beginning point to the end point.

If someone takes a road trip and for half of the distance they go one speed East and for the other half they go another speed North, their average speed is the harmonic mean of the two speeds. But their resultant speed, that is, the magnitude of the resultant vector, is the length of the hypotenuse formed by the triangle with speeds East and North weighted by their distance and divided by the total time. It is in the direction of direct flight from the beginning point to the end point.

We can easily picture a map representing the route in space, but it is more difficult to picture a time map representing the route in time. Yet the velocities in space and the velocities in time are vectors which can each be represented on a map, that is, geometrically.

Direction and dimension

What does it mean to say that space has three dimensions? It means that space has directions that have three dimensions, that is, three degrees of freedom. The dimensions are the directions in the space.

It’s not that there are some dimensions that are spatial and others are something else but that space is characterized by a certain number of independent directions. That is why three coordinates are needed to specify locations in space.

Saying that time also has three dimensions does not mean time has three different dimensions. It means that time also has directions that have three dimensions, that is, three degrees of freedom. The dimensions of time are the directions in time.

Vectors have magnitude and direction. A position vector specifies a location in multiple dimensions. Whether that location is in time or in space depends on the units of the magnitude. If the units are length, the vector is in space. If the units are durations, the vector is in time. If the units are lengths per unit time, the vector is in spacetime. If the units are newtons, the vector is in a force field.

Vectors have directions, and directions have dimensions.

No change in time per distance

Speed can be zero, that is, the change in spatial position per unit of duration can be zero. Can the change in temporal position per unit of distance be zero, too? Let’s see.

First, the denominator cannot be zero. We cannot simply invert a zero speed because that would lead to a zero denominator, which is disallowed mathematically. The denominator is non-zero no matter what the measured quantity is.

Second, the units in the denominator are the reference for what the numerator is measured against. It’s as if the units keep ticking away while the numerator is measured. Since time is often in the denominator, the seconds, minutes, hours, etc. seem to be ticking away no matter what the value of the numerator is.

Third, in this case the distance units are in the denominator. The context is that distance units are ticking away while the duration is measured.

Here’s an example of what this means. Suppose you’re on a train going at a steady speed. The click clack of the train reminds you that it’s making distance. In your mind the click clack measures the distance away from your departure and closer to your destination.

Suppose a train comes up beside yours and goes at the same speed. You aren’t moving relative to the other train. But in units of distance what is the change in time? Since your motion is synchronized, there is no relative change in temporal position between the two trains. The relative change in time is zero, while the distance ticks off, click clack click clack.

Yes, a change in time per unit of distance can be zero.

The flow of time and space

Marcus Aurelius wrote, “Time is like a river made up of the events which happen, and a violent stream; for as soon as a thing has been seen, it is carried away, and another comes in its place, and this will be carried away too.” Time flows, and keeps flowing day and night, whether anything is happening or not.

But a river flows in space as well as time. Heraclitus is reported to have said, “No man ever steps in the same river twice”. This is because a river moves, and the water that was here is now there. The spatial position of water in a river keeps changing. Space is like a river, too.

There is a sense that time is always moving because clocks are always moving. But on the world’s roads vehicles are always moving somewhere, and odometers are always measuring distances. So in a similar sense space is always moving.

Clocks sometimes stop or are stopped intentionally. If we stop measuring time, time does not stop. The same is true with distance. If we stop measuring distance, space does not stop. We have not reached the edge of the universe.

We cannot measure time without movement so movement must have some special relation to time. But with a ruler we move our eyes from one place to another even if we don’t move the ruler. With longer distances we must move the ruler. A rolling distance measuring wheel can move indefinitely. If we measure distance with light as surveyors or astronomers do, the light moves. We cannot measure distance without movement as well.

As with duration, so with distance in a parallel manner. As with time, so with space. Time flows? So does space.

Is time three-dimensional?

This post is a companion with the post “Is space one-dimensional?“. As we can compare the alleged one-dimensionality of time with how we think about space, so we can compare the three-dimensionality of space with how we think about time. In both cases the comparison is instructional. Space and time are parallel in both cases but in the latter case which we are examining here, the answer to the question is Yes.

A world line is the path of an object through spacetime. It has both spatial and temporal components, so its simplest representation is with one dimension of space and one dimension of time on a two-dimensional surface as a graph. But then we say that space has two other dimensions which are not represented in such a case.

These two other dimensions of space are movements in different directions, so there are components of distance in a total of three dimensions. Does it take time for an object to traverse these components of distance in the other two dimensions? Yes, it takes time to move in each dimension, that is, an object has a measurable component of duration in each of the dimensions.

So there are three components of distance in three dimensions, and there are three components of duration in three dimensions. Since we speak of these three dimensions as space (because of the three components of distance), should we not also speak of time as having three dimensions (because of the three components of duration)? Yes, we should.

Why haven’t we seen the three dimensions of time? Time has been associated with movement and space with stasis. But the movement of an object involves change in both space and time. The movement of an object is measured by its distance and its duration. For example, speed is a ratio of change in spatial position (distance) over change in temporal position (duration).

The conception of spacetime that comes from relativity theory is ready-made for this recognition of the three-dimensions of time. Space and time are parallel and intertwined and so might be considered together, as a six-dimensional spacetime.

Is space one-dimensional?

While the answer is No, space is three-dimensional, it is instructional to compare space with time as people speak about it.

The philosopher JME McTaggart made a well-known distinction between an A-series, which is “the series of positions [in time] running from the far past through the near past to the present, and then from the present to the near future and the far future,” and a B-series, which is the “series of positions [in time] which runs from earlier to later” (Mind 17 (1908), p.458).

In general usage, a world line is “the sequential path of personal human events (with time and place as dimensions) that marks the history of a person”. So a personal world line shows places where we were before the present, the place we are presently located, and (perhaps) places we may be in the future. It also shows the dates we were at each location. From this it may be determined the times at and between locations, and the distances too.

A world line exhibits a temporal B-series and, if the present time is indicated, an A-series as well. But a world line also exhibits spatial versions of A-series and B-series: the series of positions in space “running from the far past through the near past to the present, and then from the present to the near future and the far future” as a spatial A-series and the series of positions in space “which runs from earlier to later” as a spatial B-series. A world line shows that the path we take through space is analogous to the path we take through time. Since we “know” that time is one-dimensional, space must be one-dimensional, too.

The problem is that time is confused with change. McTaggart wrote, “It would, I suppose, be universally admitted that time involves change”. No, that should not be admitted. Both space and time involve change and stasis. For example, speed is the change in spatial position divided by the change in temporal position. Both space and time may involve stasis as well. Even an object that does not change spatial position has parallel cases in which an object does not change temporal position.

This may happen in several senses. The strongest sense is that of relativity theory, in which time at a certain speed (that of light) or in certain cases (depending on forces) does not result in measurable duration and so time stops. Another sense is that when we measure duration (as with any other measurement) there is a beginning to that measurement so if the measurement of time has not begun, then there is no change in time. For example, before a stopwatch is started, no duration is measured. A third sense in which time stops is that time is measured by cycles and at the end of each cycle, time has returned to its starting point and so has not changed position.

Neither space or time are one-dimensional.

Time in spacetime

Consider a worldline in one dimension of space and one dimension of time that tracks the position of a point that moves from position 20 to 10 to 15. This could represent the movement of a point in the E-W dimension. Another worldline could track the movement of the same point in the N-S dimension. All would agree that the two diagrams together represent two dimensions of space. But the case with time is completely analogous; the two diagrams together represent two dimensions of time.

To see this consider someone traveling on city streets arrayed in a grid oriented N-S and E-W with two stop watches. To keep it simple say they are traveling only north and east. They use one stop watch when they travel north and the other stop watch when they travel east. So we would have two travel times: one going north and the other going east, which would correspond to two dimensions of travel distance. As we would all agree that the travel distances are associated with two dimensions, we should agree that the two travel times are associated with two dimensions.

One objection might be that the dimensions here are all “spatial” rather than “temporal”. But the travel times are measured in units of time independently of the travel distances (which might not even be known). It seems arbitrary to say that there are two dimensions in the case of travel distances but not in the case of travel times.

There is a tendency to associate dimensionality with space rather than time (although one strange dimension is granted to time). But dimensionality is a mathematical concept that can be applied to many things, as multivariate analysis shows. As we apply concepts of scalar and vector to spatial quantities, so we can apply these to temporal quantities. Both space and time are multi-dimensional.

Space, time and causality

If we drop a stone into a calm body of water, it sends out circular waves. As the waves move outward, the clock is ticking and we say the dropped stone caused the waves, which are an irreversible process in time. But we would also say the waves are moving in space, so why isn’t it an irreversible process in space? It is, we just don’t ordinarily speak that way.

What is the difference between the “now” and the “here”? The now is the present, which seems to move with us in time. But the here moves with us in space, like a webcam that follows us everywhere.

Is the past where we were or what we were? It’s both. Is the future where we will be or what we will be? Again, it’s both. Events in times past can cause events in the present time, and events in places past can cause events in the present place. There is an exact parallel.

Causality is transmitted through time and space. We’ve heard this in relativity theory but we don’t need relativity to realize it’s true. The world line of an object in space and time is subject to causality in space and time.

But just as space has three dimensions (directions), so does time. So causality has what — six dimensions? No, causality has three dimensions because they are the same three dimensions. This is no different from saying that force has three dimensions, which are the same three dimensions as space. Dimensions are a property that applies to vector quantities.

We’re so used to thinking that dimensions are spatial but they are just as much temporal — and dynamical (having to do with force and torque) and a property of every other vector quantity in physics. Dimensions are an abstraction that applies to many physical quantities.