iSoul In the beginning is reality.

Category Archives: Space & Time

Explorations of multidimensional space and time with linear and angular motion.

Bibliography of 3D time and space-time symmetry

There are a number of references for maps with multiple dimensions of time or a symmetry of space and time. Nothing refers to both.

Maps with multiple dimensions of time

I have argued that isochron(e) maps show time in two dimensions. Such maps have been made for over a century. The Wikipedia article on isochrone maps shows a few older ones: Francis Galton’s isochrone map of travel times in 1881 from London to places around the world and an isochrone map of Melbourne rail transport travel times, 1910-1922. Railroad travel rates from 1800 to 1930 are mapped in the 1932 Atlas of the Historical Geography of the United States. Isochrone maps are used in geology, medicine, and many other disciplines.

A different kind of map shows travel times and distances between selected places such as cities. For example, there’s a map of Interstate Drive Times & Distances for the U.S. Interstate Highway System. Nowadays there are many websites that provide such information interactively, for example travelmath. Since these maps show direction or the websites provide directions, these demonstrate the multidimensionality of travel on the surface of the earth.

It hardly needs mentioning that an astronomy map shows distance in light-years, which are both a travel distance and a travel time for electromagnetic radiation. If light travel distance is 3-dimensional, so is light travel time.

3-dimensional time

Several authors have explored the possible implications of 3D time. I say “possible” because they basically manipulate equations rather than explain phenomena with 3D time. The Italian journal Nuovo Cimento has published many such studies. Authors include E.A.B. Cole, P.T. Pappas, M.T. Teli, and G. Ziino.

See also Information Theory Applied to Space-Time by Henning F Harmuth (World Scientific, 1993). Chapter 4.4 Three Time Dimensions and One Space Dimension, p.100-103.

Symmetry of space and time

J.H. Field has looked at some implications of the symmetry of space and time, especially in “Space–time exchange invariance: Special relativity as a symmetry principle,” American Journal of Physics, 69 (5), 569–575 (2001). He tries to address the “ambiguity” of the time and space dimensions not matching.

Symmetries and relativities

Total energy is conserved because time is homogeneous (time translation invariance). Total linear momentum is conserved because space is homogeneous (space translation invariance). Total angular momentum is conserved because space is isotropic (rotational invariance). These are examples of how symmetries determine the laws of physics.

Another way of looking at it is that linear and rotational movement are relative. There is no absolute reference point or interval or angle.

The symmetry of space and time means that space and time are relative. The laws of physics should reflect this relativity as much as other relativities. Whether space or time are the independent variable should be irrelevant to the laws of physics.

So there should be a transformation of space into time and time into space that preserves the laws of physics, i.e., that is invariant. That transformation in physics is based on the speed of light because light provides an unchanging reference between space and time.

The transformation is this: x’ = ct and t’ = x/c. That is, length and duration may be interchanged with the speed of light as a conversion factor, and the laws of physics will remain unchanged. An example of this is the Lorentz transformation.

J.H. Field has an article on this in the American Journal of Physics, volume 69 (5), May 2001, entitled “Space-time exchange invariance: Special relativity as a symmetry principle.” The difference is that he doesn’t know how time could be 3-dimensional. Now that we see time is 3-dimensional, there is no problem in affirming the symmetry of space and time.

Distance, duration and dimension

There are many kinds of space. The most common space is that of positions, that is, distances and directions. There is also velocity-space. There is force-space. There is duration-space, too.

Particles travel on trajectories, points move on curves, vehicles travel on streets or routes, etc. Trajectories have distances traveled, travel times (durations), velocities, accelerations, etc. These are one-dimensional curves in three-dimensional space.

Trajectories also go in various directions. Travel distances, durations, velocities, forces, etc. have directions, that is, they are vectors.

Thus distances and durations are both properties of trajectories with magnitudes and directions. They are both vectors with dimensions. Travel distance is a measure of space so distance-spaces are a form of space. Duration or travel time is a measure of time so duration-spaces are a form of time.

There is a symmetry between travel distance and travel time. A trajectory or trip can be measured either or both ways. An odometer measures travel distance for example. A clock on a vehicle can measure travel time. Together they measure speed. Or if the speed is known, then the travel distance can be converted into travel time and vice versa.

If distance is 3-dimensional, then so is duration. Since duration is a form of time, the duration-space of 3-dimensions is 3-dimensional time. That is, time has 3-dimensions.

Coordinate lattices

Rindler’s Essential Relativity is a well-written monograph that we can use to explore time and relativity. He describes the coordinate lattice of a single inertial frame of reference (section 2.5). Let us consider it with an eye toward the corresponding temporal coordinate lattice.

Start with the observer at the origin of an inertial frame, with a clock, and measure the distance to any particle by bouncing a light signal off it and multiplying the elapsed time by c/2 (where c is the speed of light). Angle measurements are made with a theodolite to derive 3-dimensional coordinates of the particle. Test particles at rest are placed at all the lattice points — integer multiples of a given length — each with a clock synchronized with the master clock at the origin by a single light signal emitted at time t0; when this signal is received at any lattice clock, that clock is set to read t0 + r/c, where r is the predetermined distance from the origin.

This process is valid for a single inertial frame. With it the coordinates of all events can be read off locally, directly where the events occur, by suitable auxiliary observers. Rindler emphasizes that other types of signal could just as well have been used, for example cannon balls. I would add that probe vehicles on a street grid could also be used to set up such a lattice.

How does this read if we switch space and time? Let’s see.

Start with the observer at the spacetime origin of an inertial frame, with a ruler, and measure the duration to any particle by bouncing a light signal off it and dividing the distance traveled by 2c (where c is the speed of light). Angle measurements are made with a theodolite to derive 3-dimensional temporal coordinates of the particle. Test events at rest are placed at all the temporal lattice points — integer multiples of a given time — each with an event synchronized with the master event at the spacetime origin by a single light signal emitted at point p0; when this signal is received at any lattice event, that event is set to read p0 + ct, where t is the predetermined duration from the origin event.

The result is a lattice built out of durations, not distances. As these are related to the distances via the factor c, it is equivalent to a lattice of distances. But conceptually it is different and the result is that distance or length is the independent variable and time is the dependent variable.

Independent and dependent time

We are so accustomed to having time as the independent variable that it takes effort to think of it as the dependent variable. This reflects the Newtonian absolute time. But either time or space can be an independent variable. The Einsteinian the use of simultaneity is another reflection of the independence of time.

If time is a dependent variable with space, that is, length is the independent variable all this changes. Either space becomes the absolute or colocation replaces simultaneity. And clocks are not imagined to be everywhere, ticking off the relentless time of change. Instead, rulers are imagined to be always there, unceasingly measuring change. Where are the clocks? They must be positioned in relation to the rulers.

Independent space is parameterized; it is the trajectory of a particle, vehicle, or other object that is moving. Dependent time is no longer parameterized; it is the full range of geometric possibilities in 3-dimensional time. Everything is switched around.

What does absolute space or time mean in practice except that they are independent variables? If they are independent, then they are not related to other variables, so they must be absolute.

Every measureable value that goes into the denominator is implicitly independent, and so absolute. How can we prevent this from misleading us? We should allow either spatial or temporal variables in the denominator. By switching between them, we can show how they are completely relative.

Claims about time

It’s time to list the claims about time made in this series of blog posts.

  1. Time is 3-dimensional. This is the over-arching claim which is explained and expanded by the other claims.
  2. Time is duration with direction. That is, time is a vector variable similar to a position vector in space.
  3. Time as ordinarily conceived is parametric time. That is, time along a trajectory, curve, path, route, trip, etc.
  4. Time has continuous symmetries of homogeneity and isotropy.
  5. Time and space are symmetric with one another.
  6. Minkowski spacetime should be expanded to six dimensions. That is, the invariant distance should be: (ds)² = (c dt1)² + (c dt2)² + (c dt3)² – (dx1)² – (dx2)² – (dx3.
  7. Replacing time with its negation produces a duration in the opposite direction. It does not reverse time or switch past and future.

I am working on a paper that explains and defends these claims.

Reality and relativity

As a realist I respect what is often called “common sense.” This means that our faculties of discerning reality in everyday life are basically correct. Yes, we make some mistakes, we can get fooled by a illusionist, but we almost always agree what it is that happens when things happen to us or in front of us. These are the faculties we use to determine if a mistake has been made. Our human faculties show us reality.

The corolary to this in the sciences is twofold: (1) observation is a reliable method to gather information about the universe, and (2) the observer who is closest to an object or event is the one who is in the best position to describe the object or event. This could be called the View from Everywhere (as opposed to The View from Nowhere).

One thing this means is that an observer who is at rest with respect to some object or phenomena is in the best position to measure it properly. So the proper time, length, mass, etc. of something is the measurement at rest. What other observers measure may be different but should be related to these rest measurements. That is the origin of transformations that are made between observers, or frames of reference.

At this point there are two ways to proceed: (a) either the laws of physics are defined with respect to observers at rest, or (b) the laws of physics are independent of any particular observer. Option (b) has been chosen because option (a) does not consider how observers should relate their observations to one another. So while we might say that the real time, length, mass, etc. of something is what the observer at rest measures, nevertheless it is best to transform all observations into ones that are invariant with respect to each observer.

The result is to affirm the relativity principle (RP) that Albert Einstein articulated: “all inertial [reference] frames are totally equivalent for the performance of all physical experiments.” But it also affirms that, for example, the rest mass is the real mass of a particle (not to be confused with what is called the relativistic mass).

What is single-value time?

Time is commonly expressed by a single value, a numerical expression of a point in time. A sequence of these points is also called time. Because of this, time is commonly considered one-dimensional.

Since time actually has three dimensions, what does this single value for time represent? It must be some function of the three-dimensional time. It could be, for example, their sum, their average, the magnitude of their resultant vector, or something else.

The answer is that a single value for time represents the (temporal) length of the path taken in time. If someone travels for 30 minutes east and 40 minutes north, the total duration is 70 minutes. That is what time it is relative to the starting point in time.

One could also say that effectively they’ve gone 50 minutes in the arctan(40/30) direction (about 53 degrees from the east). That is true but it is not what people mean by the time. How long it took to get to a point in time is all that a single value of time represents.

Once again space could be treated the same way. If someone travels for 30 kilometres east and 40 kilometres north, the total trip length is 70 kilometres. We could call that the “what space it is” relative to the starting point in space. But space is usually treated geometrically so the resultant distance and direction from the origin are often more significant than the path taken to get there.

To distinguish the two ways of looking at time (or space), let us call the single value perspective as the parametric and the three-dimensional perspective as the geometric. So time commonly refers to the parametric time and space commonly refers to geometric space.

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 multiple dimensions of time one must 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 body or particle or vehicle. Let the path, route, or course of an object in motion be called its trajectory.

The speed of a body is defined as the rate of travel distance, that is, the distance traveled per time of travel. The inverse speed, or pace, of a body 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 in a ratio is always 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.