iSoul In the beginning is reality

Total time

Since time is three-dimensional, what is the total time given the time in each dimension? The answer is exactly like the total distance. Consider the times t1, t2, and t3. If these are the coordinates of three successive movements, then the total time is their sum: t = t1 + t2 + t3. But if the times t1, t2, and t3 are components of one movement, then the total time is the time displacement, which is Euclidean: t² = t1² + t2² + t3². If the times t1, t2, and t3 are the components of the final point in time of a movement, then the total time is the integral of the time path taken to get to that point in time.

The metric for each axis of movement is the hyperbolic metric dsi² = dti² – dri². The total metric is ds² = ∑i dti² – dri² with i = 1, 2, 3.

This raises the question whether space-time is six-dimensional or two three-dimensional geometries. In some sense 3D space and 3D time might combine to form a 6D unity. As Minkowski said, “Henceforth, space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. ” That’s an exaggeration but it’s basically correct.

Dimensions of movement

The instantaneous movement of a particle may be represented by a velocity vector, which describes an instantaneous motion by its magnitude and direction: the magnitude is the ratio of the differentials of the distance and time in each dimension of the movement (dri/dti); the direction is the direction of the instantaneous tangent in each dimension of the movement. The bold point needs to be emphasized because this is what has been missed. Space and time measurements concern the same movement and have the same dimensions as the movement.

Each axis of movement may be projected onto two-dimensions for visualization, with for example axes r1 and t1 like this:

Space-time axes

The velocity vector is thus a three-vector because a particle has three dimensions of movement: v = (v1, v2, v3) = (dr1/dt1, dr2/dt2, dr3/dt3). Similarly, the acceleration vector is a three-vector: a = (a1, a2, a3) = (d²v1/dt1², d²v2/dt2², d²v3/dt3²). These may be represented by component-wise vector division: v = dr/dt and a = dv/dt.

Note that velocity and acceleration may be defined by component-wise division because the denominator of each component is the independent variable and so can and must be non-zero. For example, in the velocity (rb – ra) / (tb – ta) the denominator is the independent variable, which must be non-zero, and the numerator is the dependent variable, which can be any real number. For the legerity it is the opposite: (tb – ta) / (rb – ra) for which the spatial component is the independent variable.

Time on space and space on time

Our culture is oriented toward space, geography, geometry, and spatial relationships. We can easily understand items on a map. Even time can be put on a map, as with an isochrone map such as the contour lines (isochrones) representing equal distances or drive times from an urban center. 3D visualizations extend this to more dimensions. One dimension of time may be projected onto two or three dimensions of space.

But the opposite is more difficult to understand: space projected onto multiple dimensions of time. For this we need a “distorted” map that shows travel times instead of (travel) distances. That is, the background is a kind of map that doesn’t show geography but rather shows temporal relationships. The foreground shows familiar spatial representations except that they may not be where they were on a spatial map. Contour lines showing equal driving distances (isodistances) on top of a temporal map provides another way to see the relationship between time (duration) and space (length).

A familiar situation in some American cities is the city grid. There is a difference between the driving distance and the distance as the crow flies, which should reflect different drive times as well. If driving times are proportional to driving distances, then equal distances from a point on the grid should approximate linearly spaced squares or diamonds. If travel times over equal distances from an urban center are shown over a map, then the drive times will be spaced further apart near the center. The opposite is the case if distances traveled in equal time periods from an urban center are shown over a map: they will be spaced closer together near the center.

A standard speed could be shown by equally spaced circles from a specified point on a map, with each circle representing the distance traveled in a given length of time.

Dual Galilei and Lorentz transformations

I keep going over this because it has been so overlooked for 100 years. The Galilei transformation (GT) is based on three space dimensions and one time dimension (3S+1T). Once it is realized that time is just as dimensional as space, there is a dual Galilei transformation based on one space dimension and three time dimensions (1S+3T). The one dimensional time or space are actually “uni-dimensional” since they combine all their dimensions of time or space together.

In standard configurations all coordinates are zero at one point and only one coordinate in space and one in time are non-zero. That is, in the moving plane r1 = vt, which is the same as saying r1 = vt1 or t1 = r1/v since t = t1 in this case. The GT are then:

3S+1T: t1´ = t1, r1´ = r1 – vt1, r2´ = r2, r3´ = r3.

1S+3T: r1´ = r1, t1´ = t1 – r1/v, t2´ = t2, t3´ = t3.

There are various derivations of the Lorentz transformation (LT). Take Rindler’s from his 2006 book Relativity. The key step is this (p.44):

Next, suppose x´ = γx + Fy + Gz + Ht + J. By the choice of coordinates, x = vt must imply x´ = 0, so γv + H, F, G, J all vanish and x´ = γ(x – vt).

Let’s add the subscripts as above:

Next, suppose r1´ = γr1 + Fr2 + Gr3 + Ht1 + J. By the choice of coordinates, r1 = vt1 must imply r1´ = 0, so γv + H, F, G, J all vanish and r1´ = γ(r1 – vt1).

We can also proceed in terms of t1´ instead of r1´:

Next, suppose t1´ = γt1 + Ft2 + Gt3 + Hr1 + J. By the choice of coordinates, t1 = r1/v must imply t1´ = 0, so γ/v + H, F, G, J all vanish and t1´ = γ(t1 – r1/v).

That leads, as we’ve shown several times (e.g., here) to a dual LT, not differentiated by dimension but by speed:

|v| < c: r1´ = γ (r1 – vt1), r2´ = r2, r3´ = r3, t1´ = γ (t1 – r1v/c²), t2´ = t2, t3´ = t3, with γ = √(1 / (1 – v²/c²));

|v| > c: t1´ = γ (t1 – r1/v), t2´ = t2, t3´ = t3, r1´ = γ (r1 – t1c²/v), r2´ = r2, r3´ = r3, with γ = √(1 / (1 – c²/v²)).

Otherwise, |v| = c: r1´ = r1, r2´ = r2, r3´ = r3, t1´ = t1, t2´ = t2, t3´ = t3, with γ = 1.

In summary, there are dual transformation for both GT and LT, and the total number of dimensions is six. In GT the transformations are distinguished by collapsing the six dimensions to four in either of two ways. In LT the transformations are distinguished by the relationship between the speed of the object (or the frame) and the standard speed.

Measurement of space and time

To measure means to compare with a standard. A physical movement may be measured in terms of the most direct movement between its beginning and ending. There are two kinds of measures of movement, magnitude and angle, which each have two aspects, spatial and temporal.

First, the magnitude of movement:

(1) Spatial measurement of the magnitude of a movement is by comparison with a standard linear movement, which is divided into units of magnitude and placed along the movement. The result is a number of linear units, called a length or distance.

(2) Temporal measurement of the magnitude of a movement is by comparison with a standard cyclic movement, which is divided into units of magnitude simultaneous with the movement. The result is a number of cyclic units, called a time or duration.

Second, the angle of movement:

(3) Spatial measurement of the angle of a movement is by comparison with a standard circle whose center is placed at the spatial starting-point of movement and divided into for angular units (e.g., a protractor). The result is a number of angular units, called a direction in space.

(4) Temporal measurement of the angle of a movement is by comparison with a circular movement that is simultaneous with the movement along lines from the center and divided into units of angle (e.g., a clock face). The result is a number of angular units, called a direction in time.

The difference between these two angular measurements is the difference between a circle with radii that don’t movement and a circle with a radius that moves at a standard rate. This may also be understood in terms of rectilinear components, each of which is either a spatial measure of magnitude or a temporal measure of magnitude.

Some will likely say that direction is a property of space, not time, but direction is a property of movement, which can be measured by spatial or temporal means. The difference between space and time are in what they measure. They are similar in that they measure aspects of movement, including magnitude and direction.

Invariant interval check

It’s a good exercise to check the invariant interval for both subluminal and superluminal objects. Let’s do this with the delta form of the Lorentz transformations:

Subluminal case:

This is a check that c²(Δ)² – (Δ)² – (Δ)² – (Δ)² = c²(Δt)² – (Δx)² – (Δy)² – (Δz)².

The Lorentz transformation is

= γ (cΔt – vΔx/c), Δx´ = γx – vΔt), Δ = Δy, Δz´= Δz.

So we have

γ² (cΔt – vΔx/c)² – γ²x – vΔt)² – (Δy)² – (Δz

= γ² (c²(Δt)² – vΔtΔx + v²(Δx)²/c² – (Δx)² + vΔtΔx  – v²(Δt)²) – (Δy)² – (Δz

= γ² ((c² – v²)(Δt)² – (1 – v²/c²)(Δx)²) – (Δy)² – (Δz

= γ² (1 – v²/c²)(c²(Δt)² – (Δx)²) – (Δy)² – (Δz

= c²(Δt)² – (Δx)² – (Δy)² – (Δz)².

Superluminal case:

This is a check that c²(Δ)² – c²(Δt1´)² – c²(Δt2´)² – c²(Δt3´)² = (Δr)² – c²(Δt1)² – c²(Δt2)² – c²(Δt3)².

The Lorentz transformation is

Δ = γr – c²Δt/v), cΔt1´γ (cΔt1cΔr/v), cΔt2´ = cΔt2, cΔt3´ = cΔt3.

So we have

γ²r – c²Δt/v)² – γ² (cΔt1cΔr/v)² – c²(Δt2)² – c²(Δt3

= γ² ((Δr 2ΔrΔt1/v + c4t1/v² – c²(Δt1)² + 2Δt1Δr/v – c²(Δr)²/v²) – c²(Δt2)² – c²(Δt3

= γ² ((Δr)²(1 – c²/v²) – c²(Δt1)²(1 –  c²/v²)) – c²(Δt2)² – c²(Δt3

= (Δr)² – c²(Δt1)² – c²(Δt2)² – c²(Δt3)².

Six dimensional space-time

Because of the directionality, symmetry, and convertibility of space and time, there could be three dimensions of both (3S+3T). However, the formation of rates, notably velocity and legerity, effectively reduces the dimensionality to either three dimensions of space and one of time (3S+1T) or one dimension of space and three of time (1S+3T). Furthermore, the on-board or proper measures of space and time form one dimension each of space and time (1S+1T). So the six dimensions of potential space-time are commonly reduced to four or less dimensions.

If there are six dimensions, a point-event of space-time would be indicated by its coordinates: (ct1, ct2, ct3, r1, r2, r3) = (ct, r1, r2, r3) = (ct1, ct2, ct3, r) = (ct, r), where c is the speed of light in a vacuum and lowercase bold indicates a three-vector. If either or both three-vectors is replaced with its distance (or duration) from the origin, the dimensionality is reduced to four, (ct, r1, r2, r3) or (ct1, ct2, ct3, r), or two, (ct, r), in which t = |t| and r = |r|.

The spacetime interval (or invariant interval) between two point-events would be defined as

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

This is commonly given in its differential form, which in six dimensions would be

ds² = dr² – c²dt² = dr1² + dr2² + dr3² – c²dt² = dr² – c²dt1² – c²dt2² – c²dt3² = dr1² + dr2² + dr3² – c²dt1² – c²dt2² – c²dt3².

The proper time, dτ², would be defined similarly:

dτ² = ds²/c² = dr²/c² – dt² = (dr1² + dr2² + dr3²)/c² – dt² = dr²/c² – dt1² – dt2² – dt3² = (dr1² + dr2² + dr3²)/c² – dt1² – dt2² – dt3².

Note that (dτ/dt)² = 1 – (dr/dt)²/ = 1 – v²/c², with v the speed of the object. And so dt/dτ = γ and = dt/γ, where γ is the factor from the Lorentz transformation.

The six dimensions of potential space-time are reduced to four in order to represent rates of time (speeds) or rates of distance (paces). For this purpose time (duration) or space (length) are converted into a scalar:

:= r1² + r2² + r3² or := t1² + t2² + t3².

The four-vector for velocity is thus:

V = ds/dτ = γ ds/dt = γ (v1, v2, v3, 1), where the uppercase bold indicates a four-vector.

This shows “four-theory as a special case” of the potential six-dimensions of space-time.

Two one-way standard speeds

The conventionality thesis in physics concerns the conventionality of simultaneity, which states that the choice of a standard synchrony is a convention, not an observable. This arises because the speed of light in a vacuum can only be measured as a two-way speed, so the one-way speeds are either taken to be the same (the standard answer) or two speeds whose harmonic mean speed is the constant c. This is expressed as c / (1±κ), where κ is between 0 and 1 (see here and references), and the standard value for κ is 0.

As pointed out in Lorentz generalized, different observers (or travelers) may have different standard (modal) speeds for various reasons. If one accepts that the standard round-trip speed is a constant for all travelers (or observers), that restricts the standard one-way speeds but still allows different possibilities.

Let there be observer-travelers going in the same direction but in different vehicles (or trains, boats, etc.). Distinguish them by their frame of reference, unprimed or primed. Call their frames S and S’, their positions in space r and , in time t and , the actual speed of the second frame relative to the first v, and their reference travel speeds a and b respectively.

Consider only the path/trajectory followed, i.e., one dimension of space and time each. Then we have: r = at and r´ = bt´ as time-space conversions for each frame. To proceed like the previous derivations of the Lorentz transform, let there be a factor γ for each equation:

= γ (r – vt) = γt (a – v) = bt´, and

r = γ (r´ + vt´) = γt´ (b + v) = at.

Multiply these together and divide out tt´ to get:

γ² (a – v)(b + v) = ab, so that

γ² = ab / ((a – v)(b + v)).

Now let a = c/(1–κ) and b = c/(1+κ). Then

γ² = / ((c + κv – v)(c + κv + v)).

If κ = 0, then a = b = c, and there is only one reference speed for both traveler-observers, which is the requirement of the Lorentz transformation:

r´ = γ (r – vt) and t´ = γ (t – rv/c²) with γ² = 1 / (1 – v²/c²).

On the other hand, if κ → 1, then a → ∞ and b → c/2, so γ² = 1 / (1 + 2v/c) with

r´ = γ (r – vt) and t´ = γt, which is the Galilei transformation with a factor.

Thus two observer-travelers don’t have to agree on a standard one-way speed.

Movement and dimensions

The movement of an object is a change in its spatial and temporal location. The measurement of a movement by a ratio apart from direction is either a speed or a pace. Speed is a change in distance per a given duration. Pace is a change in duration per a given distance. If direction is combined with the ratio, it is either a velocity or a legerity. Velocity is a speed with direction. Legerity is a pace with direction.

There is a relation between measures of movement and dimensions. If movement is measured by velocity, the denominator is a vector of space, which means space is considered multidimensional. But if movement is measured by legerity, the denominator is a vector of time, which means time is considered multidimensional.

Max Tegmark in his 1997 letter On the dimensionality of spacetime gives his judgment concerning the number of space and time dimensions in a chart:

TegmarkThe chart indicates that one space dimension with three time dimensions (1+3D) includes only tachyons (objects traveling at more than the speed of light) and not bradyons (or tardyons, objects traveling at less than the speed of light). Bradyons exist in 1+3D, see Subluminal and superluminal Lorentz transformations.

The statement “We are here.” for 3+1D reflects its status as a cultural commonplace and the use of velocity. There is however equal justification for using 1+3D and legerity. Minkowski showed the way to use either 3+1D or 1+3D by using hyperbolic geometry.

Insights on the complete Lorentz transformation

There are several insights in the previous post Subluminal and superluminal Lorentz transformations to explore here.

Case 1 begins with r´ = r – vt or t´ = t – rv/c². The equation for comes straight out of the Galilei transformation with the equation for allowed to change. So the ghost of Galilei lives on in the Lorentz transformation.

What if we began with the Galilei transformation for ? Then t´ = γt and t´ = r´/c leads to r´ = γct so the reference frames are simply proportional. Space and time are equivalent. This would be the case if space and time were both scalars, essentially one dimensional. That is the case if v = c.

Look again at Case 1:

r´ = r – vtr´/c = r/ct (v/c) = = t (1 – v/c) and

t´ = t – rv/c²ct´ = ctr (v/c) = = r (1 – v/c),

which shows the parallelism between the two beginnings for the subluminal Lorentz transformation.

Look again at Case 2:

r´ = r – tc²/vr´/c = r/ct (c/v) = = t (1 – c/v) and

t´ = t – r/vct´ = ctr (c/v) = = r (1 – c/v),

which shows the parallelism between the two beginnings for the superluminal Lorentz transformation. It also shows that the superluminal Lorentz transformation may be derived from a form of the Galilei transformation. So much depends on pre-Einstein mechanics, which is called non-relativistic although it includes Galilei relativity.

What is the difference between (v/c) and (c/v)? Both are dimensionless. In the first case v is denominated in units of c and in the second case c is denominated in units of v. They are just slightly different perspectives, which lead to the two main parts of the complete Lorentz transformation.