iSoul In the beginning is reality

# Parametric time and space

Parametric equations are equations in terms of a single parameter. An example is the parametric equation for a circle: x = r cos(t), y = r sin(t), for a circle with radius r with rectilinear coordinates x and y. It’s best if the parameter represents a causal variable but time is often used whether or not time causes anything.

One advantage of parametric equations is that the arc length is also a function of the parameter so the parameter may be considered as a position on the curve. This has been seductive with time, so much that time is thought of parametrically as if it were a one-dimensional variable. To see the multidimensional character of time one must de-parameterize it and see it in terms of coordinates.

We could parameterize space and de-parametrize time to switch the places of the two. For example, on a road trip the odometer measures the trip length, which would be the parameter. If the travel time in the E-W and N-S directions is measured with the trip length, the trip may be plotted as two dimensions of time and one dimension of space. Before and after would be associated with smaller and larger values of the odometer.

If the fuel gauge were precise enough, the fuel level could be used as the parameter for both travel time and distance. Because of causal connections, it may well make sense to treat some variable parametrically, but either time, space, or some other variable can serve the purpose. Much of the character of time comes from its parametric association.

# Speed and its inverse

Physics and transportation have many things in common, in particular they both deal with objects that move or travel. The case of an object that happens to stay in place should be considered as well. An object that may move can be called a vehicle or particle. Let the path, route, or course of an object that may move be called its trajectory.

The speed of an object that may move is defined as the rate of travel distance, that is, the distance traveled per time of travel. The inverse speed, or pace, of an object that may move is defined as the rate of travel time, that is, the travel time per distance traveled.

Note that if you have a non-zero travel distance and a non-zero travel time, then either the speed or the pace may be determined, and the travel distance or the travel time may be in the numerator and or the denominator. However, usually either a travel time or a travel distance is given as an independent variable, with the other as the dependent variable. Then the independent variable should be in the denominator, with the other in the numerator. There are two reasons for this: (1) the independent variable can always be non-zero, whereas the dependent variable might be zero, and (2) the numerator shows what is measured, whereas the denominator shows the units that are used.

For example, 10 metres per second means that in a second of time the object moves 10 metres. A pace of 10 minutes per kilometre means that in a kilometre of space the object takes 10 minutes.

The mean speed is the total distance travelled by all objects in the measurement region, divided by the total time spent in this region. The mean pace is the total travel time by all objects in the measurement region, divided by the total distance travelled in this region. How the mean speed and pace are calculated depends on whether the region is defined spatially or temporally.

The space-mean speed is either the arithmetic mean of the speeds within a spatial region or the harmonic mean of the speeds within a temporal region. The time-mean speed is either the arithmetic mean of the speeds within a temporal region or the harmonic mean of the speeds within a spatial region.

The time-mean pace is either the arithmetic mean of the pace within a temporal region or the harmonic mean of the speeds within a spatial region. The space-mean pace is either the arithmetic mean of the paces within a spatial region or the harmonic mean of the paces within a temporal region.

A basic question is whether or not there is a maximum speed or minimum pace. In transportation there are posted speed limits but whether or not they are obeyed, each vehicle type has a speed beyond which it is physically impossible for it to go. Or, what is equivalent, speed data with values beyond some maximum will be impossible to believe, and so will be ignored or discarded. In physics it has been determined that the speed of light is a maximum speed. So in either case, there is some maximum speed.

# Symmetry of space and time

The duality between space and time leads to many dual principles. For example, Fermat’s principle says that light travels between two given points along the path of shortest time. The dual to this is that light travels between given points in time along the path of shortest length in space.

But as long as there are only single particles with constant velocities, the dual principle will be just another form of the principle. It’s when multiple particles are averaged or their acceleration is non-zero that the dual may make a difference.

Is there a complete symmetry between space and time? I have entertained the thought but hesitated because that seems to say space and time are the same. The symmetries of translation and rotation are like that: move there or turn and the physics is just the same as before. But it’s not surprising that reflection symmetry is sometimes violated since one cannot move an object into its reflection.

Can space and time be switched by a movement? Distance and time can be measured together. Consider a distance measuring wheel traveling at constant speed; it measures distance and duration. Which is which — which is the independent variable and which is the dependent variable, which goes in the numerator and which goes in the denominator, distance or duration? It’s an arbitrary choice. So yes, space and time are symmetric; they may be exchanged without changing the laws of physics.

Space and time are inversely symmetric. They have opposite signs in the spacetime metric. They have opposite (complementary) Lorentz transformations. The limiting speed of light has opposite meaning: it is the maximum travel distance for a given travel time, and it is the minimum travel time for a given travel distance.

Language and culture make space and time seem more different than they are. Physically, they are very similar, and form a finite symmetry.

# An introduction to co-physics, part 2

According to the McGraw-Hill Dictionary of Physics, mass is “A quantitative measure of a body’s resistance to being accelerated; equal to the inverse of the ratio of the body’s acceleration to the acceleration of a standard mass under otherwise identical conditions.” This is because the ratio of two masses equals the inverse ratio of the magnitudes of the accelerations produced by the same force applied to each mass. By employing a reference mass, the mass of other bodies may be determined. That is:

m(X) / m(R) = a(R) / a(X)

for body X, reference R, acceleration a, and mass m. The reference mass may be set to one for simplicity. For the dual, we need to use the inverse of these quantities:

(1/m(X)) / (1/m(R)) = (1/a(R)) / (1/a(X))

= n(X) / n(R) = b(R) / b(X)

where b and n represent the co-acceleration and co-mass, respectively.

The duals of Newton’s laws of motion are referenced to space instead of time in the denominator, with the numerator’s directions in three dimensions of time:

(1) Every object with uniform co-velocity tends to remain in that state of motion unless an external co-force is applied to it.

(2) The co-force equals the co-mass times the co-acceleration, with the direction of the co-force vector the same as the temporal direction of the co-acceleration vector.

(3) If a given body A acts on a body B with a co-force, then B will also act on A with a co-force equal in magnitude but opposite in temporal direction.

# An introduction to co-physics, part 1

In order to keep things as simple as possible, I’m starting to name the dual to standard physics with the prefix “co-“, so that co-physics, co-mechanics, co-speed, co-velocity, etc. refer to their dual terms. Like tangent and cotangent, there is an inverse relationship between physics and co-physics. Here is the beginning to a systematic presentation of this dual physics, starting with classical mechanics.

The simplest mechanics concerns a point object with finite but negligible mass, called a particle. A particle is described by its position in space, which may vary in a time series. Dually, a co-particle is described by its position in time, which may vary over a series of places, i.e., over a path or route. A particle might not change its position in space but it must exist over multiple positions in a time series. Similarly, a co-particle might not change its position in time but it must exist over multiple positions in a series of places.

Note: This is an unusual perspective, but it will make more sense as we go along. Have patience.

The definition of speed is, “The time rate of change of position of a body in space without regard to direction; in other words, the magnitude of the velocity vector.” (McGraw-Hill Dictionary of Physics, 3rd edition, used throughout with slight modification). Implicit in this definition is that there must be a non-zero change in a time series, which is nominally the unit in the denominator, e.g., “metres per second” refers to the spatial change in metres over one second of a time series.

The definition of co-speed or pace then is “The space rate of change of position of a body in time without regard to direction; in other words, the magnitude of the co-velocity vector.” Implicit in this definition is that there must be a non-zero change in a place series, which is nominally the unit in the denominator, e.g., “seconds per metre” refers to the temporal change in seconds over one metre of a place series.

The definition of velocity is, “The time rate of change of position of a body in space; it is a vector quantity, having direction as well as magnitude.” The definition of co-velocity is, “The space rate of change of position of a body; it is a vector quantity, having co-direction as well as magnitude.”

Direction in space is based on distance and co-direction is based on duration using triangles and trigonometry. If light or anything with a constant speed is used to measure both distance and duration, then these directions will be equivalent. But otherwise they may be different, for example with travel on city streets.

The definition of acceleration is, “The rate of change of velocity with respect to a time series.” The units are, e.g., metres per second per second. Similarly, the definition of co-acceleration is, “The rate of change of co-velocity with respect to a place series.” The units are, e.g., seconds per metre per metre.

# Terminology for space and time, part 1

There are several senses of the words space and time that need to be carefully distinguished in order to avoid confusion. Let’s start with natural philosophy in the tradition of Aristotle:

Space is “the feature of physical being according to which each such being can be identified as occupying a place — and, as such, can be located and measured in relation to other such beings.” (John W. Carlson, Words of Wisdom: A Philosophical Dictionary for the Perennial Tradition)

The author notes that this conception of space is different from the idealized expanse of early modern science. It is a more relational conception of space, which fits better with the relativisitic manifolds of late modern physics.

The same author defines time this way: “a measure of the physically changing as such, numbered as to before and after.” This would be called a “B-series” in the philosopher John McTaggart’s terminology: a static, tenseless series of events in before-after relationship. An “A-series” is a time series from past events to what’s happening now, on to future events. This is a dynamic, tensed conception of time.

Either way, time is considered a series or a location in a series (as in a point in time), whereas space is considered a place or a locus of places (as in the definition of a circle as the locus of points equidistant from a point). However, this obscures how a route through space is a series of points that have a before and after, similar to time. It also obscures how places in time can be in different directions from one another.

To keep all this straight, I suggest speaking of a time series for a serial conception of time (whether A or B series). The corresponding term for space would be a place series, which is similar to a world line in spacetime. A locus of points in time, not necessarily in a single series, could be called a chronus of points in time. The corresponding term for spacetime is a manifold.

We think of physical objects as having spatial properties but they also have temporal properties — e.g., they are constructed, used, wear out, fall apart. So the word object should not be considered merely something with spatial extent. The length of an object in space corresponds to the length of time of an event (or the length of an event in time), and the distance between objects in space corresponds to the duration between events in time or between the beginning and ending of one or more events.

In a time series one may speak of going forward or back in time but this should not be considered as reversing the chain of causality. It is either considering a time series in the opposite direction (i.e., a change of perspective) or taking a return trip along the same route. The term “time travel” should be avoided.

# Direction and units of magnitude

I want to clarify the statement in the previous post that “the three dimensions of direction are the same for space and time”. I have made the point that vectors in physics have various units of magnitude but direction is the same for all of them. That is accurate in the sense that directionality is the same concept in all cases. But that does not mean that the particular directions are necessarily the same. They are not.

Here’s a simple example: if someone travels 10 miles east in 17 minutes, then 17 miles north in 10 minutes, the distance direction will be about 60 degrees northeast but the duration direction will be about 30 degrees. Their directions are different but east and north are the same in both cases. This is no different than vectors with other units (velocity, acceleration, etc.). We don’t notice these other differences because we almost always relate them to an underlying distance space.

Multidimensional time has its own “space” as it were, even if directionality is the same concept. That’s why mapping the travel time between cities as map distances proportional to durations results in a “distorted” map. We’re so used to a distance map that anything else looks distorted. But if travel time is more important to us than travel distance (as it often is), a map of travel times is more useful.

# Six dimensional spacetime

First consider the dual to Minkowski spacetime. Recall that the invariant interval of Minkowski spacetime has one dimension of time with three dimensions of space:

(ds)² = (c dt)² – (dx1)² – (dx2)² – (dx3)² = (c dt)² – (dr

where t is the time coordinate and x1, x2, and x3 are space coordinates of r.

This could be called temporal spacetime since speeds and other ratios are referenced to duration, i.e., they have units of time in the denominator. The dual could be called spatial spacetime since measures of movement are referenced to distance, i.e., they have units of length in the denominator. In that case the dual invariant interval is:

(ds)² = (dr)² – (c dt1)² – (c dt2)² – (c dt3)² = (dr)² –  (c dt

where r is the space coordinate and t1, t2, and t3 are time coordinates of t.

Here’s how one might put together the two four-dimension spacetimes into one six-dimension spacetime invariant interval:

(ds)² = (c dt1)² + (c dt2)² + (c dt3)² – (dx1)² – (dx2)² – (dx3)² = (c dt)² – (dr

= (c dt1)² – (dx1)² + (c dt2)² – (dx2)² + (c dt3)² – (dx3

where the three dimensions of direction are the same for space and time. The six dimensions are in two groups of three dimensions, i.e., there are 2 × 3 dimensions or three complex dimensions.

Minkowski and dual Minkowski spacetime have 10 symmetries each. Six-dimension spacetime has 6 translations (one for each dimension), 6 rotations (along the x1-x2, x2-x3, x3-x1, t1-t2, t2-t3, and t3-t1 planes), and 3 Lorentz boosts (about the t1-x1, t2-x2, and t3-x3 planes) for a total of 15 symmetries.

# 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 argosity (inverse of velocity) w 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 – wr

t2‘ = t2

t3‘ = t3

r’ = r

What does this mean? It means that there is a constant movement measured by argosity so that as the travel length increases, the duration changes from t to t1 such that t – t1‘ = wr, 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 – wr)

t2‘ = t2

t3‘ = t3

r’ = γ (r – t1wc²)

in which

γ = 1 / sqrt(1 – w²c²)

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.