iSoul In the beginning is reality

Angles in space and time

In a previous post on Different directions for different vectors I gave this example, for which I’m switching North and East:

Suppose someone drives 30 miles North in 50 minutes, then turns East and drives 40 miles in 50 minutes. Overall, they have driven 70 miles in 100 minutes but as the crow flies they ended up 50 miles from where they started (302 + 402 = 502). And a crow flying at the same speeds would have taken only 71 minutes to get there (502 + 502 = 712).

It may be surprising that the angle the crow should fly is different in space than in time. Actually, it is the same angle measured two ways: one by distance and the other by duration. Let’s work out the details:

The spatial angle for the crow is arctan(40/30) = 53 degrees clockwise from the North. The temporal angle for the crow is arctan(50/50) = 45 degrees. But 45 degrees of duration is equivalent to 53 degrees of distance in this case.

How does this work? The temporal angle is like a clock with hands: 45 degrees means for example that the minute hand has moved 60*45/360 = 7.5 minutes or the second hand has moved 7.5 seconds. Since this corresponds to 53 degrees in space, the rate is 53/7.5 = 7.1 spatial degrees per minute. Divide this by 60 to get 0.12 revolution per minute or 7.1 cycles per second, known as Hz.

So the crow’s angle is 53 degrees clockwise from North, which is equal to 7.1 spatial degrees per minute times 7.5 minutes. Or to be faster, it’s equal to 7.1 spatial degrees per second times 7.5 seconds. Or 7.5 * 53/45 = 8.8 seconds of a second hand.

In summary to convert a temporal angle to a spatial angle, multiply it times the angular conversion factor, i.e., the frequency, which is the ratio of the spatial and temporal angles.

Basis for the symmetry of space and time

The symmetry of space and time is based on the existence of a constant of proportionality that converts space into time and time into space. This number is a fixed speed for travel under free-flow conditions for the mode of travel. Such a constant makes it arbitrary whether one answers “how long” with a length of space or a length of time, since they are interconvertible.

This constant, called c in the case of light travel in a vacuum, is an absolute within the space-time defined by a mode of travel. Given that the physical universe is the largest observed medium and the mode of electromagnetic transmission is the fastest mode, the free-flow speed of light in a vacuum provides a standard constant of proportionality for space and time in general.

Since a mode of travel may be contextualized to a particular medium or network or conditions such as congestion, space-time within a particular context is a conventional scheme. This accords with Poincaré’s conventionalism.

With a constant of proportionality between space and time, for every property of one there is a corresponding property of the other. For example, space is three-dimensional and so time is three-dimensional. Time on a trajectory is one-dimensional and so is space within that context. Space is isotropic (the same in all directions) and so time is isotropic. And so forth.

Lorentz without absolutes

The post Lorentz for space and time derived the standard (spatial) Lorentz transformation and also the temporal Lorentz transformation. It is surprising in this age of relativity how the standard Lorentz transformation is dependent on absolute time. While time is relativized in the sense of Lorentz and Einstein, it remains absolute in the sense of Galilei and Newton. This is why clock synchronization is still so important.

The temporal Lorentz transformation is based on the temporal Galilei transformation in which space is absolute. This is at least a different absolute, and so provides an alternative and comparison.

Consider the temporal coordinate of the spatial Galilei transformation and include a factor, γ, in the transformation equation for the positive direction of the x axis:

ctx′ = 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:

ctx = 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

ctx′ = rx′ = γ (rx − vtx) = γ (ctx − vtx)
ctx = rx = γ (rx′ + vtx) = γ (ctx′ + vtx).

Multiplying these together and dividing out txtx leads to:

γ2 (c − v) (c + v) = c2

so that

γ = (1 v2/c2)-1/2

which is the spatial Lorentz transformation. So the temporal coordinate can lead to the spatial transformation as well.

One may similarly consider the spatial coordinate of the temporal Galilean transformation:

ctx′ = rx′ = ρ (rx − c2 tx/v) = ρ (ctx − c2 tx/v)

and its inverse

ctx = rx = ρ (r′x + c2 t′x/v) = ρ (ct′x + c2 t′x/v).

Multiplying these together and dividing out c2txtx leads to:

ρ2 (v − c) (v + c) = v2

so that

ρ = (1 c2/v2)-1/2

which is the temporal Lorentz transformation. So the spatial coordinate can lead to the temporal transformation as well.

Optimizing travel time routes

It is not unusual to seek the route in space that minimizes travel time, for example, a drive from point A to point B may go out of the way to include a high-speed facility that reduces travel time even if it increases distance traveled.

But what about routes in time? Does it ever make sense to minimize the distance traveled? Yes, for example, when a resource cost is related to the distance traveled, as with some taxi fares, or the wear on tires, or for railroad track access. In other cases, minimizing time and distance go together, as with the great circle routes of ships or aircraft.

A race could be delimited by an amount of time rather than a distance. The goal would be to maximize the distance traveled in a fixed time period, rather than to minimize the travel time over a fixed distance. For example, walk-a-thon participants may seek pledges of support for every mile they travel within a specified time period.

An indirect example would be those sports that take place over a fixed time period, such as basketball, football, and hockey: the goal is to score the most points, which usually involves moving the ball or puck the greatest distance (though there are strategies to control the ball and run out the clock).

Commuters seek to minimize the travel time rather than the distance traveled, so a map with distances is not as important as a map with travel times during rush hour. There are apps that show (or speak) the route with the shortest time. Restaurants near businesses need to take the fixed lunch hours of their potential customers into account; short travel time routes may lead through walkways, highways, or public transit stops.

In all these cases, the route through time is more important than the route through space.

Different directions for different vectors

One surprising result is that an object’s velocity and its inverse, which I’m calling celerity, may have different directions. Here’s what I mean:

Suppose someone drives 30 miles East in 50 minutes, then turns North and drives 40 miles in 50 minutes. Overall, they have driven 70 miles in 100 minutes but as the crow flies they ended up 50 miles from where they started (302 + 402 = 502). And a crow flying at the average of the speeds would have taken only 71 minutes to get there: 30/50 = 36 mph, 40/50 = 48 mph, 50/((36+48)/2) = 71.

It may seem strange at first, but the direction of this hypothetical crow is different for the space, time, and velocity vectors. The reason is that their units are different, and so their vector spaces are different. The meaning of direction is different with different units for the magnitude.

But what about the inverse velocity, the celerity – does that have the same direction as the velocity? No. Again, that’s because the units are different, and so their vector spaces are different.

There’s another reason they’re different: because the direction of the velocity is that of the distance interval (relative to a unit of time), but the direction of the celerity is that of the duration interval (relative to a unit of space).

There’s really not much new here. We usually take the spatial directions to be the only ones that matter, which is fine, but we could as well take the temporal directions as the ones that matter.

Claims about time, updated

Here is an updated list of claims about time made in this series of blog posts:

  1. Time has 3-dimensions. 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 space vector (a distance with a direction).
  3. The magnitude of time is that which is measured by a stopwatch, similar to a length, which has beginning and ending points.
  4. Division of two vectors is a vector with the direction of the numerator, which is why time in the denominator has not been recognized as a vector.
  5. The spatial and temporal perspectives are complementary opposites.
  6. Time has continuous symmetries of homogeneity and isotropy (as does space).
  7. Time and space are symmetric with one another, and so may be conceptually interchanged.
  8. Minkowski spacetime could be expanded to six dimensions, three for time and three for space. The invariant distance could be: (ds)² = (c dtx)² + (c dty)² + (c dtz)² – (drx)² – (dry)² – (drz.
  9. 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.

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.