iSoul In the beginning is reality

Six dimensions of space-time

If one travels a distance X east, then goes a distance Y north, that is the same as going a distance √(X² + Y²) northeast. But if one travels for a time X east, then goes for a time Y north, is that the same as going for a time √(X² + Y²) northeast? No, the travel time would be (X + Y) in that case. This is because time is conceived as a magnitude, without regard for direction, and so is cumulative.

I’ve mentioned before that there are apparently six dimensions of space-time (or time-space) but there is more to it. For one thing we need to distinguish two ways that a point event relates to multiple dimensions. The first way is that a point event has a location in multi-dimensional space-time. The second way is that a point event may be the resultant of a series of motions in different dimensions. For space these two ways are equivalent but for time they are different.

The first way has been developed in detail: for 3D space the components are combined with a Euclidean metric and for 3D space + 1D time a hyperbolic metric. The spacetime (invariant) interval between two point-events is:

s² = Δr² – c²Δt² = Δr1² + Δr2² + Δr3² – c²Δt².

If time is a vector, it should have components with a Euclidean metric, too:

s² = Δr² – c²Δt² = Δr² – c²Δt1² – c²Δt2² – c²Δt3².

But this is misleading because we don’t ordinarily think of time that way. Instead, we think of time as something flowing from one motion to the next, which would mean time is cumulative. So a time vector would be understood similar to a taxicab metric:

s² = Δr² – c²Δt² = Δr² – c²(Δt1 + Δt2 + Δt3)²,

where the Δ quantity is understood as a distance (and so is non-negative). Otherwise the absolute value would be taken:

s² = Δr² – c²Δt² = Δr² – c²(|Δt1| + |Δt2| + |Δt3|)²,

But this is misleading, too, since it is a series of motions and their resulting time displacement rather than the components of a space-time location. So we should go back to the Euclidean metric and think of the time components differently.

What do the components of time mean if they aren’t the flow of time for a series of motions? Temporal components should be considered like distances measured by time with a constant speed. For a vehicle traveling at constant speed (or pace) space and time are very similar. Multidimensional time isn’t the cumulative flow of time but the dimensions of duration by direction of a vehicle traveling in space-time.

In the end, the six dimensional space-time (invariant) interval is what would be expected:

s² = Δr² – c²Δt² = Δr1² + Δr2² + Δr3² – c²Δt1² – c²Δt2² – c²Δt3².

It’s just that we need to be careful not to confuse time here with a cumulative flow of time.

Miracles and uniformity

The week before Christmas is a good time of year to write about miracles because it’s a time to be reminded of the meaningfulness of miracles. But what about their truth? Doesn’t the uniformity of nature make miracles impossible?

Thomas Aquinas said a miracle is ‘beyond the order commonly observed in nature’ (Summa Contra Gentiles III), but David Hume went further and defined a miracle as ‘a violation of the laws of nature’ (Of Miracles, 1748). Hume also claimed that scientific induction required the uniformity of nature, so on his telling, miracles undermined science.

However, Hume failed to establish the uniformity of nature on rational grounds. The future does not necessarily resemble the past. The most he could say was that the uniformity of nature is a matter of custom and habit. (There’s a convenient summary of his argument here: Probable reasoning has no rational basis.)

Others have also been unable to establish the uniformity of nature on rational grounds. This failure led to Karl Popper’s argument that induction is merely not untrue, and that one counterexample can falsify any induction. However, the history of science shows an unwillingness to abandon well-accepted science because of one or a few anomalies.

Does scientific induction really require the uniformity of nature? No, that is a misunderstanding of science that goes back to Scholasticism, which was revived in the 19th century by Richard Whately and John Stuart Mill. See John P. McCaskey’s writings on The History of Induction.

Induction is based on classification, not a principle of uniformity. Observation and experiment lead to the definition of a class by a uniformity. Then by definition other objects or events in the same class possess the same uniformity, whether in the past, present, or future. As I wrote here, science studies uniformity but that is far from requiring uniformity everywhere at all times.

It is better to define a miracle by what it is – unique – rather than what it is not – uniform. A miracle is a highly unique event or result, especially one attributed to divine agency. Since science studies uniformity, not uniqueness, it doesn’t have much to contribute about miracles. But uniqueness is studied by other disciplines such as history, philosophy, theology, and literature – that is, the humanities, not the sciences.

Miracles are by their nature very unique and significant. They fall outside of uniformity but since there is no valid principle of uniformity, that is not a problem.

Time scale maps

Maps of travel times have various names, e.g., time scale maps, isochrone maps, etc. Often a geographic map forms the background so that travel time is superimposed on distance traveled. Occasionally, the time scale replaces the distance scale and the map looks distorted from a geographic perspective, but is correct from a travel time perspective. For example, here is the Washington, DC, Metro (subway) map in two forms, the left in geographic form and the right in time scale form:

Here is a map of Europe, on the left in geographic form, then representing rail travel times, and on the right, high speed rail travel times:

Next is an outline of travel times by train from Paris superimposed on an outline of France:

Just as maps of distance show space in two dimensions, so maps of travel time show time in two dimensions.

A new geometry for space and time

This blog has described how as the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. One way of looking at this is as a map with the distance and duration given between places, such as this from the Interstate Drive Times and Distances:

There are two numbers for each leg or link in the map; one number in red for the distance in miles and another in blue for the time in hours and minutes. The durations are not proportional to the distance, which reflects the differing local conditions and topography.

It is common to adopt a speed that represents the typical speed for the mode of travel, which I’m calling the modal speed (or rate). This serves various purposes, one of which is for pre-trip planning to estimate the travel time to a destination. It can also serve as the conversion of distance and duration for the mode, much as the speed of light serves for relativity.

Divergence from the modal rate may be represented through a third dimension (z), similar to a topographic map. The modal rate then would be represented by a flat topography. A link with greater distance than the modal one would be like a positive z coordinate, and a link with less than the modal one would be like a negative z coordinate. The resultant link would be determined by the Pythagorean theorem for positive values of z, and by a hyperbolic version (as if z were imaginary) for negative values of z.

That is, |A|² = x² + y² + z² if z > 0 (Pythagorean), and |A|² = x² + y² – z² if z < 0 (hyperbolic). For positive values of z, the magnitude of the resultant link is greater than unity, that is, greater than the modal rate. But for negative values of z, the magnitude of the resultant link is less than unity, that is, less than the modal rate.

The result may be mapped like a contour or isoline map, except that negative values have a hyperbolic geometry. The simplest way to see this is to take Δt = 1 and Δr = > 1, which uses the Pythagorean theorem. If either the time interval changes to Δt < 1 or the space interval is reduced to Δr < 1, the hyperbolic version is used for the link.

Why time is three dimensional

The case for 3D time is very simple: space is based on the measurement of distance and time is based on the measurement of duration. As the distances between places cover three dimensions of space, so the durations between events cover three dimensions of time. As distance may be measured going from or toward a space-time point, so duration may be measured going from or toward a point in time-space.

In everyday life we speak of how far away a place is as a length of space or a length of time for a mode of travel. For example:

Can this informal way of speaking be formalized in the language of physics? Yes.

The difference between space and time is that distance is measured asynchronously, whereas duration is measured synchronously. One must synchronize a motion with a standard motion – called a clock – in order to measure time. A clock measures one dimension of time in one direction (or without regard to direction). A rod or rule is used asynchronously to measure a motion before, during, or after the fact – in one direction.

The context requires a mode of travel in which there is a modal rate of travel that can be used to convert space and time into one another. This is commonly done in physics with the speed of light. Length can be defined in terms of the distance traveled for a specified time at the speed of light (and the standard meter is defined that way, see here). Time can be defined in terms of the travel time for a specified distance at the speed of light. The independence of this modal rate from any particular speed of travel leads to relativity theory.

Direction is determined by the relation of a motion to a spatio-temporal point, which is either a point from which the motion comes or toward which the motion is going. For example, magnetic north is a point on earth toward which compasses point. This point moves slowly, so for accuracy surveyors must specify the year a measurement was taken.

As space has direction in three dimensions, so does time. For more about 3D time, there is a list of posts here.

Necessary and possible dimensions

In everyday life 1D space and 1D time are typically used. We are concerned with how far away something is (travel distance), how long will it take to get there (travel time), what the speed is (apart from direction). Unless we are doing something in which size or direction are important, 2D is all we need.

Only two dimensions are necessary: 1D space and 1D time. Anything beyond this is possible but not necessary – up to a limit of 3D space and 3D time. Thus it is possible to define, measure, and adopt conventions in which there is exactly 3D space and 1D time. But it is also possible to define, measure, and adopt conventions in which there is exactly 1D space and 3D time. And it may be possible to define, measure, and adopt conventions in which there is exactly 3D space and 3D time.

So it is no objection to 3D time that it can be defined to be 1D. It is also no objection to 1D space that it can be defined to be 3D. Other conventions are possible. And a 3D space and 1D time convention is symmetric with a 1D space and 3D time convention.

Geometric and temporal unit systems

A geometric (or geometrized) unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity. Sometimes Coulomb’s constant, ke, and the electric charge, e, are also set to unity.

The General Conference on Weights and Measures defines the meter as the length of the path traveled by light in vacuum during time interval of 1  ⁄ 299792458 of a second. (17th CGPM; 1983, Resolution 1, CR, 97) In geometric units, every time interval is interpreted as the distance traveled by light during the given time interval so time has the geometric units of length. This is consistent with the notion from special relativity that time and distance are on an equal footing.

One could do the opposite and define a temporal unit system in which every length is interpreted as the travel time of light in vacuum over the given length so length has the temporal units of time. That would show another way in which time can be multidimensional.

Time conventions

The natural concept of time is solar time, which is based on the Sun’s position in the sky. But local solar time or local mean time varies by longitude. With the spread of railroads in the 19th century, there was a need for time zones to standardize time and simplify east and west travel.

The 1884 International Meridian Conference adopted a universal day beginning at Greenwich midnight, but did not specify any local times or time zones. Over the years since then each country has determined their own time zones, usually offset by one hour from Greenwich mean time (GMT), but in some cases offset by one-half or one-quarter hour.

If one travels far enough in an east or west direction, a time zone will be crossed. In general this differs from travel north or south. A traveler going west is said to gain time as the clock is turned back, and a traveler going east is said to lose time as the clock is moved forward. The difference in time between east-west travel and north-south travel shows that time is capable of having two dimensions.

Coordinated universal time (UTC) is based on International Atomic Time, which is a weighted average of the time kept by over 400 atomic clocks in over 50 national laboratories worldwide. This time convention does not vary by location.

Posts on space and time chronologically

Here is a chronological listing of the titles of my posts on space and time (without hyperlinks):

Read more →

Direction in three-dimensional time, part 2

This is a continuation of what I wrote on this topic here.

IV

Apart from time zones and daylight savings time, noon means midday, the time that the sun is directly overhead, when the sun crosses the meridian. Is noon the time when the sun is directly overhead or is vertical the direction of the sun at noon? Which is first, noon or vertical, time or space? They’re together.

The direction north means toward the North Pole. For practical reasons magnetic north is often used instead. Surveyors use magnetic north to orient their surveys but they have to specify the year because magnetic north moves slowly on the earth. So magnetic north is a place in space and time, not space only. That means north is a direction in space and time.

One could say something similar about the direction toward the North Pole from an astronomical perspective as the earth moves in its orbit around the sun, i.e., the celestial north pole. Polar north is a direction in space and time.

What are the poles but the intersections of the surface of the Earth with its axis of rotation?

Here are helpful definitions from the Sundial Primer:

North: the intersection of the local meridian with the horizon, in the direction of the north celestial pole.

South: one of the cardinal points of the compass, in the direction opposite north, in the direction of the south celestial pole.

East: the point on the horizon 90º (measured clockwise) from the North. The Sun appears to rise from the East point on the equinoxes.

West:  the point on the horizon 90º (measured anti-clockwise) from the North. The Sun appears to set at the West point on the equinoxes.