iSoul In the beginning is reality

Superluminal Lorentz transformation again

This is a revision to the derivation of the superluminal Lorentz transformation, as presented previously such as here. The problem with the previous derivation is that it did not distinguish between speed and pace, velocity and celerity.

Consider Case 2 again.

This case begins with: r´ = r – tc²/v or t – r/v. These can be written as: r´ = r – ct (c/v) or c = ct – r (c/v). If β = c/v, then the equations are: r´ = r – ctβ or c = ct – rβ.

Note that v ≠ 0 in these equations since it is in the denominator. But the restriction should be c ≠ 0 instead. More precisely, β should be a ratio of paces, not speeds. That is, the rate of change should be measured as the change in time per change in space (length or distance).

If we notate the pace by an underline, then β should be defined as v/c. That allows the definition of β as follows:

β = v/c if |v| < |c|, in the spatial direction of v,

β = v/c = c/v if |v| > |c|, i.e., |v| < |c|, in the temporal direction of v (for emphasis this could be notated as β),

β = 0 if |v| = |c|, without direction.

Then for the superluminal Lorentz transformation: γ² = 1 / (1 – v²/c²) = 1 / (1 – β²),

This matters even more with the vector version of β, i.e., β:

β = v/c if |v| < |c|, with the spatial direction of v,

β = v/c if |v| > |c|, with the temporal direction of v (for emphasis this could be notated as β),

β = 0 if |v| = |c|, without direction.

The physics of a trip

Movement in its simplest form is a trip from A to B. There are two ways one can look at such a movement: (1) as the line segment from A to B, or (2) as the angle between the lines from A to a reference point and A to B. Each of these may be spatial or temporal. Let’s look at these in more detail.

Linear measures:

(1a) Spatially, a line segment from A to B is measured with a co-extensive unit of length by a calibrated rod. This linear distance is the basis of space, which has a Euclidean metric, in which the locus of points equidistant from a point forms a sphere.

(1b) Temporally, a cycle period from A to B is measured with a co-extensive unit of time by a stop watch. This cyclic duration is the basis of time, which has a Euclidean metric, in which the locus of point events equal in time from a point event forms a cycle (or sphycle since it is 3D).

Angular measures:

(2a) Spatially, the angle between the lines from A to a reference point and A to B is measured with a co-extensive unit of angle by two lines and a protractor or two line segments and trigonometry.

(2b) Temporally, the turning angle between the directions from A to a reference point and A to B is measured with a co-extensive unit of angular speed by a clock and a stop watch. The clock provides the unit of angular speed and the stop watch provides the unit of measure, though they may be combined into one device. The temporal angle might be called a tangle.

There are two kinds of length and two kinds of angle: spatial and temporal. Both the spatial and the temporal measures have three dimensions because they have three degrees of freedom. Temporal geometry is comparable to spatial geometry but all measures are temporal: linear and angular durations of movement.

Space-time is formed from a union of these measures and has a hyperbolic metric. The locus of space-time points equal in space-time from a space-time point forms a space-time hyperbola.

From generalizations to universals

John P. McCaskey’s Key to Induction shows what scientific induction is all about:

“[Scientists] want to know not only what is generally true but what is universally so, what is true without any possible exception. Below are three cases in which scientists were able to begin with general statements and progress to exceptionless universal ones. In each of the cases, scientists’ definitions evolved from being merely descriptive to identifying causes. That transition was crucial.”

He goes on to detail what happened with the science of cholera, electrical resistance, and tides, in which generalizations led to inductions. More famous examples would be the passage from the generalizations of Ptolemy and Kepler to the inductions of Galileo and Newton, or from the ad hoc transformations of Lorentz to Einstein’s principles.

As McCaskey puts it in Induction Without the Uniformity Principle, “The whole project of mature abstract thought is to identify similarities and differences, uniformities and changes, and to classify accordingly. And that—to Aristotle and followers such as Bacon and Whewell—is what induction is.”

Science progresses from data collection, to generalizations, to universals, to deductive hypotheses, and then new data collection to repeat the cycle. Sometimes “induction” is considered generalization (or testing) and universals are guessed by “abduction” (cf., C.S. Peirce) but the process of developing universals is the key element of induction.

In the series of posts on space and time, I have tried to show how approximate generalizations in transportation are exact inductions in physics.

Measuring movement

The dimensions of an object are measured by movements, whether by moving a measuring device or moving one’s eyes while a measuring device stays in the same position. Movements themselves are measured by comparing them with standard movements, such as a movement with constant velocity. A movement compared with a standard linear movement generates a spatial component of the movement. A movement compared with a standard cyclic movement generates a temporal component of the movement.

Movements may also be compared with standard movements that do not begin or end but move continually. A standard cyclic movement that is continual is called a clock. One can mark a virtual beginning and ending on a clock so it acts as a stopwatch.

A standard movement has a constant linear or angular velocity. If the movement it is compared with changes in velocity, the standard movement keeps going at the same rate. If changes to velocity are ignored, the result is a length (distance). If distance is ignored, the result is a time (duration).

A clock is a standard movement that doesn’t go anywhere. It’s all angle and no distance. A standard speed is all distance and no angle. It goes everywhere at the same speed, like light.

A velocity is the ratio of a change in distance with a given change in time, which is a non-zero duration such as a unit of time. A celerity is the ratio of a change in duration with a given change in distance, which is a non-zero distance such as a unit of length.

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 celerity 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 (distance).

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.