iSoul In the beginning is reality

Modes of travel

Travel, that is, the movement of something, includes transporting and signalling. To transport means to take something (e.g., people or goods) from one place to another by means of a vehicle or the like (e.g., a car). To signal means to transmit information or instructions from one place to another through a medium or the like (e.g., sound).

A mode of travel is a means, technology, or technique for moving something. Travel modes may be distinguished by whether they are on or through a solid (e.g., land), on or through a liquid (e.g., water), through a gas (e.g., air), or in a vacuum (e.g., outer space).

A mode of travel has a free-flow speed, which is the speed attained in which there are no impediments to travel in that mode. This is the highest normal speed in the mode but may not be the highest possible speed. If local conditions (e.g., topography) do not exert significant influence, the free-flow speed serves as a reference speed for the mode because it is homogeneous and isotropic.

There are two basic perspectives on travel and the measurement of travel: (1) the most common perspective looks from a state of rest and observes something traveling relative to it; (2) the second perspective looks from a state of travel and observes something that is not moving (e.g., a landscape). The basic measure for perspective (1) is velocity, the change in distance traveled per unit of travel time. The basic measure for perspective (2) is the inverse of velocity, the change in travel time per unit of distance traveled, which could be called invelocity from the words inverse and velocity.

The first perspective (1) is the spatial perspective because it is from a state of rest, which is associated with space, with something that seems to be there apart from time. The second perspective (2) is the temporal perspective because it is from a state of travel, which is associated with the passage of time, with something that takes time. The Galilei and Lorentz transformations apply to the spatial perspective but as we have seen there are similar transformations that apply to the temporal perspective.

Lorentz for space and time

Consider again the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity, v. This time there is a standard constant speed, c. The basic problem is that if they both observe a point event E, how should one convert the coordinates of E from one reference frame to the other?

We return first to the spatial Galilei transformation and include a factor, γ, in the transformation equation for the positive direction of the x axis:

rx′ = γ (rx − vtx)

where rx is the spatial coordinate and tx is the temporal coordinate. Only the coordinates of the x axis are affected; the other coordinates do not change.

The inverse spatial transformation is then:

rx = γ (rx′ + vtx).

The trajectory of a reference particle or probe vehicle that travels at the standard speed in the positive direction of the x axis will follow the equations:

rx = ctx and rx′ = ctx′.

With the spatial transformations we conclude that

rx′ = ctx′ = γ (rx − vtx) = γ tx (c − v),
rx = ctx = γ (rx′ + vtx) = γ tx (c + v).

Multiplying these together and cancelling tx tx leads to:

c2 = γ2 (c − v) (c + v) = γ2 (c2 v2)

so that

γ = (1 – v2/c2) -1/2.

which completes the spatial Lorentz transformation.

We return next to the temporal Galilei transformation and include a factor, ρ, in the transformation equation for the positive direction of the x axis:

tx′ = ρ (tx − rx/v)

where rx is the spatial coordinate and tx is the temporal coordinate. Only the coordinates of the x axis are affected; the other coordinates do not change.

The inverse temporal transformation is then:

tx = ρ (tx′ + rx′/v).

The trajectory of a reference particle or probe vehicle that travels at the standard speed from the origin will follow the equations:

rx/c = tx and rx′/c = tx′.

With the temporal transformations we conclude that

tx′ = rx′/c = ρ (tx − rx/v) = ρ rx (1/c − 1/v),
tx = rx/c = ρ (tx′ + rx′/v) = ρ rx (1/c + 1/v).

Multiplying these together and cancelling rx rx leads to:

1/c2 = ρ2 (1/c − 1/v) (1/c + 1/v) =  ρ2 (1/c21/v2)

so that

ρ = (1 – c2/v2) -1/2.

which completes the temporal Lorentz transformation.

Galilei for space and time

Consider the now-classic scenario in which observer K is at rest and observer K′ is moving in the positive direction of the x axis with constant velocity v. The basic problem is that if they both observe a point event E, how should one convert the coordinates of E from one reference frame to the other?

First assume time is absolute and space is relative with no characteristic speed. Only the spatial coordinates in the positive direction of the x axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the x axis is

rx′ = rx − vtx

where rx is the spatial coordinate and tx is the temporal coordinate in the positive direction of the x axis. The inverse transformation is

rx = rx′ + vtx.

This is called the Galilei (or Galilean) transformation.

Now consider the case in which space is absolute and time is relative with no characteristic speed. Only the temporal coordinates in the positive direction of the x axis are affected. The other coordinates do not change. The transformation equation for the positive direction of the x axis is

tx′ = tx − rx/v

and the inverse transformation is

tx = tx′ + rx′/v.

This could be called the temporal Galilei transformation since only temporal coordinates change. The transformation above would then be called the spatial Galilei transformation.

The speed of spacetime

For each mode of travel there are four speeds to consider: the minimum speed, the typical speed, the maximum speed, and the speed of particular objects. The more that impediments to travel are removed (e.g., other objects, the topography, the network), the more that speed reaches free flow.

In transportation, the free flow speed is slower than the maximum speed. For example, the maximum speed for a highway vehicle might be determined by the fastest speed of a vehicle on the Bonneville Salt Flats. Or by the fastest speed at a NASCAR stock car race. These speeds would be much faster than what is safe on a highway. In physics, the free flow speed and the maximum speed are the same because of the principle of least action.

If there exists a speed that is constant for all particles or vehicles, then there is a way to relate the space and time coordinates of every event. Depending on whether this special speed is the minimum, the maximum, or the typical speed, there will be a kind of Lorentz transformation of the coordinates.

A coordinate system is a map for representing objects in space and/or time. The origin point of a coordinate system represents the reference point for the other points represented.

Maps may be static and represent a rest frame or they may be dynamic and change with their location as with GPS devices. For example, the origin point of a dynamic map may represent the location of a moving vehicle.

A spacetime coordinate system needs a way to relate spatial and temporal coordinates. The relation may be very particular, localized, and complex or it may be general, universal, and simple. The simplest relation between space and time is a constant. Such a constant represents a speed.

All maps have a scale, e.g., one centimetre represents 500 metres. A map with a reference speed also has a time scale, e.g., one centimetre represents one minute. To represent an arbitrary point in spacetime requires two maps: one for the space coordinates and the other for the time coordinates. These may be combined if different axes have different units. Or time coordinates may be put on a space map (and vice versa) as isoline plots, i.e., isochrones or isodistances.

The relation between the scale of space and the scale of time on two related maps is the speed that ties them together. That is the speed of spacetime.

Division of physical vectors

A physical vector is a physical magnitude with a direction that operates as a mathematical vector. As with all physical quantities, it has units of some kind. Both the magnitude and the direction have units. The directional units are called unit vectors.

The units of a magnitude are what it is relative to, for example: metres per second means the distance traveled in metres relative to a travel time in seconds along a particular direction. If we multiply a velocity vector by a time interval, the result is a position vector. What has changed is the units.

The vector formed by the division of two vectors with the same direction has a magnitude that is the ratio of the two magnitudes. Its direction is the same as the direction of the two vectors. So the division of vector s by vector t is the ratio of their magnitudes times their common unit vector:

s / t = (|s|/|t|) (s/|s|) = s / |t|.

That is, the ratio of two vectors equals the numerator vector divided by the magnitude of the denominator vector. Since time is often the denominator vector in which only the magnitude is needed, the vector nature of time has been hidden.

What about vector division for vectors that might be in different directions? What is the magnitude and direction in that case? Look at the vector division above a little differently:

s / t = (|s| s ) / (|s| |t|)

The denominator above gives us a clue: instead of the product of the magnitudes, their dot product can be used, since

s ∙ t = |s| |t| cos θ,

where θ is the angle between vectors s and t. Then put this dot product into the denominator and leave the numerator the same:

s / t = (|s| s ) / (s ∙ t) = s / (|t| cos θ).

So vector division is the denominator vector divided by the magnitude of the projection of the denominator vector onto the numerator vector. Although the direction of the numerator is the direction of the division, the direction applies to the whole magnitude, not just the numerator.

Representations of space and time

Space has been represented with a three-dimensional geometry since ancient times. Descartes added coordinates, which make these dimensions more explicit. We call this Euclidean space R3 = R × R × R, that is, three dimensions of real-number coordinates.

Newton added time but kept it separate since he considered time absolute and space relative. Call this R3 & R. Minkowski integrated time as part of a four-dimensional pseudo-Euclidean geometry, like R4 with signature (+ ‒ ‒ ‒) or (‒ + + +).

What do space and time look like with three time dimensions? I see three possibilities:

(1) Six-dimensional spacetime with signature (+ + + ‒ ‒ ‒) or (‒ ‒ ‒ + + +);

(2) C3 spacetime, with three dimensions of complex numbers;

(3) R3 & R3, with space and time in their own versions of R3.

Since space, time, and spacetime are conventions, which of these representations are chosen will depend on their purpose and use.

Travel in space and time

Objects have measures of length, width, and height. Objects also move, that is, travel and so change their position in one or more of the directions of length, width, and height. The relation of the position before and after movement is measured by the difference of length, width, and height, and these differences are called distances or durations.

Linear dimensions are measured by the circumference of a measuring wheel (or the equivalent) as it moves in a straight line along or parallel to the object being measured, beginning and ending with the extremities of the object. The duration of movement is measured by the angle swept out by measuring wheel (or the equivalent) as it moves in a straight line along or parallel to the object being measured, beginning and ending with the extremities of the object.

The speed of travel depends on the mode of travel (e.g., the type of propulsion, the medium, the vehicle, etc.). Each mode of travel likely has a maximum possible speed and a typical speed relative to a that mode of travel and perhaps other distinctions.

Thus far we have spoken about measurement and said nothing about space or time. That is because the measurement of distances and durations does not depend on a concept of space or time or space-time.

As we place objects into a conceptual realm, we need concepts of space and time as the conceptual context for linear and angular measures. This allows measures to be related to one another and inductive inferences to be made.

So linear measures are put into space and angular measures are put into time. Space and time may be conceived to extend indefinitely or to have a definite beginning and/or ending. Space and time may be given an origin to relate the position of all objects.

All of these concepts are conventions. So space and time and space-time are conventions. But the measurement of distance and duration are not conventions.

Proof of three time dimensions

The proof that there are three time dimensions is based on showing that a temporal position is a vector, that is, it has magnitude and direction. That may be shown by considering three orthogonal movements of an object. Let the position of the object be represented by a point (as on a corner) relative to an origin point.

Consider the travel times of three separate orthogonal movements from their distances traveled divided by their average speeds, times their unit vectors (e1, e2, and e3):

(s1 / v1) e1, then (s2 / v2) e2, then (s3 / v3) e3.

The resultant duration vector t is found by adding each of the three orthogonal vectors together by vector addition:

t = (s1 / v1) e1 + (s2 / v2) e2 + (s3 / v3) e3.

That is, the resultant duration vector t has three components:

t = t1 e1 + t2 e2 + t3 e3.

Since t1, t2, and t3 can be different, they represent three different components of the vector t. That is, t1, t2, and t3 are orthogonal components of a temporal vector t:

t = (t1, t2, t3).

Thus we have demonstrated three dimensions of time.

These three dimensions are based on the same directions as displacement (spatial), velocity, force, and other physical vectors.

Velocity with three-dimensional time

What is the average velocity if the distance vector is s and the duration vector is t? Let the vectors be represented with rectilinear components:

s = (s1, s2, s3) = e1 s1e2 s2 + e3 s3,

t = (t1, t2, t3) = e1 t1e2 t2 + e3 t3.

where e1, e2, and e3 are orthogonal unit vectors. Then is the velocity vector the distance vector divided by the magnitude of the time vector? That would be equal to the ratio of the magnitudes times the unit direction vector of the distance:

s / | t | = (| s | / | t |) (s / | s |).

Or should the direction of the duration vector be taken into account as well? If the velocity were the resultant of three orthogonal movements, we would have:

v = (v1, v2, v3) = (s1 / t1, s2 / t2, s3 / t3) = e1 s1 / t1 + e2 s2 / t2 + e3 s3 / t3.

Then consider that every vector is the sum of its component vectors and we have the answer. So vector division is a division of parallel components. If t1 = t2 = t3, then the velocity vector equals the spatial vector divided by a scalar.

Note that if a vehicle moves East a distance sa in time ta, then moves North a distance of sb in a time tb, the resulting position could have distance and duration that are not parallel. The distance angle would be atan(sb/sa) and the duration angle would be atan(tb/ta). The angle of the velocity could be different from either. This is because they are in different ‘spaces’ because they have different units. The space of distance is commonly taken as the real space even though they all are real.

Dimensions of dimension

The word dimension has many dimensions of meaning. The basic meaning of a dimension is an independent component of something. For example, color is often considered to have three dimensions: chroma, hue, and value.

Mathematically, a dimension is one of a minimal set of independent components of an abstract space. There are many applications of this concept. For example, principle components analysis (PCA) is a technique that takes multivariate data and finds the uncorrelated components, which are the dimensions of the data.

In everyday usage a dimension is one of a minimal set of directional components, that is, linear measures from a center point. In this sense there are three dimensions in the world we inhabit. Time is often added as “fourth dimension” but it is not a fourth direction since there is no fourth direction. If time adds any dimensions to space, they must be some other kind of dimension.

A ruler can be oriented in three dimensions. So can a clock. You say a clock is the same clock in each dimension, so there’s no difference? That’s also true for a ruler: it’s the same ruler oriented in each dimension. As a ruler cannot be oriented in three dimensions simultaneously, so a clock cannot be oriented in three dimensions simultaneously. The simple conclusion is that there are three dimensions for rulers and for clocks.

Both space and time are three-dimensional, and they are the same three dimensions. These three dimensions may be characterized linearly or planarly because of the duality between lines and planes. Space is associated with linear (interval) measures. Time is associated with planar (circular) measures. No additional dimensions are needed to characterize both.