iSoul In the beginning is reality

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.

Subluminal and superluminal Lorentz transformations

This is a re-do of the post Lorentz for space & time both relative? By making a few rearrangements, the contradictions disappear.

Case 1

This case begins with: r´ = r – vt or t´ = t – rv/c².

Consider then a linear function of these: = γ (r – vt) or t´ = γ (t – rv/c²)

along with a standard speed c such that r = ct and r´ = ct´. Combine this with to get

ct´ = r´ = γ(r – vt) = γ  (r – rv/c) = γ (ct – vt) and its inverse

ct = r = γ (r´ + vt´) = γ (r´ + r´v/c) = γ (ct´ + vt´). Multiply these together to get

c² tt´ = rr´ = γ² (rr´ – rr´v²/c²) = γ² (c²tt´ – v²tt´). Divide out rr´ or tt´ and get

γ² = 1 / (1 – v²/c²), which is the same as the subluminal Lorentz transformation.

Alternatively, go back and combine with a standard speed c to get

r´/c = t´ = γ (t – rv/c²) = γ (r/c – rv/c²) = γ (t – tv/c) and its inverse

r/c = t = γ (t´ + r´v/c²) = γ (r´/c + r´v/c²) = γ (t´ + t´v/c). Multiply these together and get

rr´/c² = tt´ = γ² (rr´/c² – rr´v²/c4) = γ² (tt´ – tt´v²/c²), which simplifies to

1/ = γ² (1/c² – v²/c4) or 1 = γ² (1 – v²/c²) so that

γ² = 1 / (1 – v²/c²), which is again the subluminal Lorentz transformation.

Case 2

This case begins with: r´ = r – tc²/v or t – r/v.

Consider a linear function of these: = γ (r – tc²/v) or = γ (t – r/v).

Combine with a standard speed c to get

ct´ = r´ = γ (r – tc²/v) = γ (r – rc) = γ (ct – tc²/v) and its inverse

ct = r = γ (r´ + t´c²/v) =  γ(r´ + r´c) = γ (ct´ + t´c²/v). Multiply these together and get

c²tt´ = rr´ = γ² (rr´ – rr´c²/v²) = γ² (c²tt´ – tt´c4/v²), which simplifies to

1 = γ² (1 – c²/v²) or = γ² (c² – c4/v²) so that

γ² = 1 / (1 – c²/v²), which is the superluminal Lorentz transformation.

Go back and combine with a standard speed c to get

r´/c = t´ = γ (t – r/v) = γ (r/c – r/v) = γ (t – tc/v) and its inverse

r/c = t = γ (t´ + r´/v) = γ (r´/c + r´/v) = γ (t´ + t´c/v). Multiply these together to get

rr´/c² = tt´ = γ² (rr´/c² – rr´/v²) = γ² (tt´ – tt´c²/v²). Divide out rr´ or tt´ to get

γ² = 1 / (1 – c²/v²), which is again the superluminal Lorentz transformation.

Conclusion

By beginning with the correct form of the non-relativistic transformation and its alternate, one may derive the Lorentz transformation and its alternate. Together both subluminal and superluminal velocities are covered.

Complete spatial and temporal Lorentz transformations

The solution to the quandary posed in the previous post, Limitations of the Lorentz transformation, is to begin with an alternate form of relative motion. In the case of the spatial component this is = r – tc²/v. The relative spatial transformation is then:

= γ (r – tc²/v) and its inverse r = γ (r´ + t´c²/v)

along with a standard (characteristic) speed, c, in all reference frames: r = ct, r´ = ct´ implies that

γ² = 1 / (1 – c²/v²), which is the superluminal Lorentz transformation.

The temporal Lorentz transformation is then:

= γ (t – rv/c²) and its inverse t = γ (t´ + r´v/c²)

along with a standard speed, c, in all reference frames: t = r/c, t´ = r´/c implies that

γ² = 1 / (1 – v²/c²), which is the subluminal Lorentz transformation.

So we have both subluminal and superluminal Lorentz transformations for both spatial and temporal components. This is the complete Lorentz transformation, which covers all velocities for both relative spatial and temporal components.

Note that the relative spatial component = γ (r – tc²/v) here and the previously used relative spatial component = γ (r – ut), with u instead of v, are equal if u = c²/v. Similarly the relative temporal component = γ (t – rv/c²) and the previously used relative temporal component = γ (t – t/u), with u instead of v, are equal if u = v/c².

Limits of the Lorentz transformation

Looking back at the previous posts, we can see that if we begin with the relativity of the spatial component of movement, the Lorentz transformation turns out one way:

= γ (r – vt) and its inverse r = γ (r´ + vt´)

along with a standard (characteristic) speed, c, in all reference frames: r = ct and r´ = ct´ leads to

γ² = 1 / (1 – v²/c²).

By substituting the expression for and simplifying we get

= γ (t – rv/c²) and its inverse t = γ (t´ + r´v/c²).

But if we begin with the relativity of the temporal component of movement, the Lorentz transformation turns out another way:

= γ (t – r/v) and its inverse t = γ (t´ + r´/v)

along with a standard speed, c, in all reference frames: t = r/c and t´ = r´/c leads to

γ² = 1 / (1 – c²/v²).

By substituting the expression for and simplifying we get

= γ (r – tc²/v) and its inverse r = γ (r´ + t´c²/v).

So γ depends on v/c if space is relative, and γ depends on c/v if time is relative. But that also means v < c if space is relative and v > c if time is relative. Plus the converse: space is relative is v < c and time is relative if v > c.

But in fact space can be relative whether or not v < c and time can be relative whether or not v > c. So there is something artificially limiting about the Lorentz transformation.

Lorentz for space & time both relative?

Two Lorentz transformations based on relative space with absolute time and absolute space with relative time were presented here. Now we look at beginning with space and time both relative, in two different ways.

R-R Case 1

This case begins with: r´ = r – vt and t´ = t – r/v.

Consider then a linear function of these:

= γ(r – vt) and = γ(t – r/v)

along with the standard (characteristic) speed c such that r = ct and r´ = ct´. Combine this with to get

ct´ = r´ = γ(r – vt) = γ(r – rv/c) = γ(ct – vt) and its inverse

ct = r = γ(r´ + vt´) = γ(r´ + r´v/c) = γ(ct´ + vt´). Multiply these together to get

c² tt´ = rr´ = γ² (rr´ – rr´v²/c²) = γ² (c²tt´ – v²tt´). Divide out rr´ or tt´ and get

γ² = 1 / (1 – v²/c²), which is the same as the Lorentz transformation.

Go back and combine with the standard speed c to get

r´/c = t´ = γ(t – r/v) = γ(r/c – r/v) = γ(t – tc/v) and its inverse

r/c = t = γ(t´ + r´/v) = γ(r´/c + r´/v) = γ(t´ + t´c/v). Multiply these together to get

rr´/c² = tt´ = γ²(rr´/c² – rr´/v²) = γ²(tt´ – tt´c²/v²). Divide out rr´ or tt´ to get

γ² = 1 / (1 – c²/v²), which is the superluminal Lorentz transformation. Thus we have a contradiction.

R-R Case 2

This case begins with: r´ = r – tc²/v and t´ = t – rv/c².

Consider a linear function of these:

= γ(r – tc²/v) and = γ(t – rv/c²). Combine with the standard speed c to get

ct´ = r´ = γ(r – tc²/v) = γ(r – rc) = γ(ct – tc²/v) and its inverse

ct = r = γ(r´ + t´c²/v) = γ(r´ + r´c) = γ(ct´ + t´c²/v). Multiply these together and get

c²tt´ = rr´ = γ²(rr´ – rr´c²/v²) = γ²(c²tt´ – tt´c4/v²), which simplifies to

1 = γ²(1 – c²/v²) or = γ²(c² – c4/v²) so that γ² = 1 / (1 – c²/v²),

which is the superluminal Lorentz transformation.

Alternatively, go back and combine with the standard speed c to get

r´/c = t´ = γ(t – rv/c²) = γ(r/c – rv/c²) = γ(t – tv/c) and its inverse

r/c = t = γ(t´ + r´v/c²) = γ(r´/c + r´v/c²) = γ(t´ + t´v/c). Multiply these together and get

rr´/c² = tt´ = γ²(rr´/c² – rr´v²/c4) = γ²(tt´ – tt´v²/c²), which simplifies to

1/ = γ²(1/c² – v²/c4) or 1 = γ²(1 – v²/c²) so that

γ² = 1 / (1 – v²/c²), which is the Lorentz transformation. Thus we have a contradiction.

Conclusion

Beginning with both space and time relative leads to a contradiction. We conclude that absolute and relative are jointly required.

Absolute vs relative space, time, and dimension

In Aristotle’s physics there exists a prime mover that is in a state of absolute rest so that the position in space and time of everything else is understood relative to this place of absolute rest. Consequently, the reference frames of two observers are the same.

We still use this framework from the perspective of a trip or trajectory, measuring the distance traveled and travel time with length and time measures that are taken along for the ride. Such a one-dimensional perspective uses an absolute space and absolute time.

Galileo famously criticized Aristotle’s physics and said instead that the laws of physics
are the same for every observer moving with a constant speed along a straight line, called an inertial observer. Galileo regarded space as relative but left time absolute, which began the development of classical physics using three dimensions of relative space and one dimension of absolute time.

This partially changed with Einstein’s interpretation of the Lorentz transformation in the special theory of relativity (SR). This related both space and time but required an absolute framework of clocks to measure simultaneity. SR might be better called the theory of semi-relativity. It is still based on three dimensions of space and one of time.

If space is absolute but time is relative, one may develop an alternative with one dimension of space and three dimensions of time, as shown here. It remains to investigate fully relative space-time with three dimensions of space with three of time.

Complete Lorentz group

The complete Lorentz transformation may be written as

r′ = γ (r − ct(v/c)), ct′ = γ (ct – rv/c), and γ = (1 – v2/c2)–1/2,

which applies only if |v| < |c|, and

r′ = γ (rct(c/v)), ct′ = γ (ct − r(c/v)), and γ = (1 − c2/v2)–1/2,

which applies only if |v| > |c|. If |v| = |c|, then r′ = r and t′ = t.

In order to express this more easily, define β and γ as follows:

β =

  1. v/c if |v| < |c|
  2. c/v if |v| > |c|
  3. 0 if |v| = |c|

Based on this define γ = 1 / √(1 – β²) for all v. Then the complete Lorentz transformation may be expressed as

r′ = − ctβγ and ct′ = ctγ – rβ,

which may be displayed in matrix form as:

This is formally identical to the Lorentz transformation, which forms a multiplicative group, and so the complete Lorentz transformation forms a group as well.

Complete Lorentz transformation

This is a continuation of a series of posts that began with Lorentz for space and time.

The standard Lorentz transformation applies only if |v| < |c|. The complete transformation for all real values of v is presented here based on both the relative space, absolute time (R-A) Galilei transformation as well as the complementary absolute space, relative time (A-R) Galilei transformation. The absolute is associated with one dimension and the relative with three dimensions but there is no necessary connection.

R-A Lorentz transformation

Consider the relative space and absolute time Galilei transformation and include a factor, γ, in the transformation equation for the positive direction of one axis:

r′ = γ (rvt) and t′ = γ (trv/c²)

where r is one spatial coordinate (the others are unchanged) and t is the temporal coordinate. The inverse R-A transformations are then:

r = γ (r′ + vt′) and t = γ (t′ + r′v/c²).

The trajectory of a reference particle or probe vehicle that travels at the standard speed in the positive direction of the x axis follows the equations:

r = ct and r′ = ct′.

Combined with the Galilei transformation for space this leads to

ct′ = r′ = γ (rvt) = γr (1 – v/c) = γt (cv),

ct = r = γ (r′ + vt′) = γr′ (1 + v/c) = γt′ (c + v).

Alternately, the Galilei transformation for time combined with the standard speed leads to

r′ = ct′ = γ (ctrv/c) = γr (1 – v/c) = γt (c – v),

r = ct = γ (ct′ + r′v/c) = γr′ (1 + v/c) = γt′ (c + v).

Multiplying these pairs together for space and dividing out rr′ yields:

1 = γ2 (1 – v2/c2).

Or multiplying these pairs together for space and dividing out tt′ yields:

c2 = γ2 (c2v2).

Multiplying these pairs together for time and dividing out rr′ leads to:

γ2 (1 – v2/c2) = 1.

Or multiplying these pairs together for time and dividing out tt′ leads to:

c2 = γ2 (c2v2).

Whichever way is done yields

γ = (1 – v2/c2)–1/2,

which is the standard Lorentz transformation and applies only if |v| < |c|.

A-R Lorentz transformation

Consider the absolute space and relative time Galilei transformation and include a factor, γ, in the transformation equation for the positive direction of one axis:

r′ = γ (rc2 t/v) and t′ = γ (tr/v).

where r is the spatial coordinate and t is one temporal coordinate (the others are unchanged). The inverse A-R transformations are then:

r = γ (r′ + c2 t′/v) and t = γ (t′ + r′/v).

Again the standard speed is

r = ct and r′ = ct′.

Combine these together to get for space

ct′ = r′ = γ (rc2 t/v) = γr (1 − c/v) = γt (cc2/v),

ct = r = γ (r′ + c2 t′/v) = γr′ (1 + c/v) = γt′ (c + c2/v).

and for time

r′/c = t′ = γ (t − r/v) = γr (1/c − 1/v) = γt (1 − c/v),

r/c = t = γ (t′ + r′/v) = γr′ (1/c + 1/v) = γt′ (1 + c/v).

Multiplying these pairs together for space and dividing out rr′ yields:

1 = γ2 (1 – c2/v2).

Or multiplying these pairs together for space and dividing out tt′ leads to:

c2 = γ2 (c2c4/v2).

Multiplying these pairs together for time and dividing out rr′ yields:

1/c2 = γ2 (1/c2 – 1/v2).

Or multiplying these pairs together for time and dividing out tt′ leads to:

1 = γ2 (1 – c2/v2).

Whichever is done, this yields

γ = (1 − c2/v2)–1/2,

which is the complementary Lorentz transformation that applies only if |v| > |c|.

Complete Lorentz transformation

The complete Lorentz transformation is then

r′ = γ (r − vt), t′ = γ (t – rv/c²), and γ = (1 – v2/c2)–1/2,

which applies only if |v| < |c|, and

r′ = γ (rc2 t/v), t′ = γ (t − r/v), and γ = (1 − c2/v2)–1/2,

which applies only if |v| > |c|. If |v| = |c|, then r′ = r and t′ = t.

The story of nothing

Mathematics is the study of nothing. We make something out of nothing, acting the creator in a world of nothing. Here’s the story:

In the beginning is nothing. Not totally nothing because we’re there. But a blank page, a clear slate, a tabula rasa.

We draw a distinction, a part of nothing. The indistinct blankness of nothing gains a something. We indicate the something. We indicate the original nothing. We develop a logic of nothing.

We draw a place of nothing, a point. We draw a line of points, then a plane, and a solid. We select an original nothing, an origin. We develop a geometry of nothing.

We draw a number of nothing, a zero. We add it, subtract it, and multiply it. We raise numbers to its power. We develop an arithmetic and algebra of nothing.

We reduce a number to nothing, an infinitesimal. We define a function of it. We take its tangent, its sum, and its mean. We develop a calculus of nothing.

We take the set of nothing, the null set. We intersect it, union it, complement it. We take the power set of it. We find its cardinality. We develop a set theory of nothing.

In the end we have nothing, nothing but mathematics.

Four perspectives on space and time

There are four perspectives on space and time depending on whether the observer is internal or external to space or time. The four perspectives are internal space with internal time, external space with internal time, internal space with external time, and external space with external time.

The internal perspective is that of an observer traveling along a route or trajectory. It does not track direction because direction refers to the external perspective. So the internal perspective on movement is one-dimensional, a curve or world line, which has an arc length in space or time. The internal perspective makes space or time absolute because they travel with the observer.

The external perspective is that of one who observes an object move through a space or time with all its directional possibilities. So the external perspective on movement is three-dimensional, an abstract space or manifold, which contains a curve or world line that has direction at every point. Note that an internal perspective is external to the movement of another observer. The external perspective makes space or time relative because the observer and observed move relative to each other.

Each of these perspectives may be “classical” or “relativistic” depending on whether a Lorentz-type transformation is applied to them. The internal-internal perspective is absolute so its space and time are Lorentz invariant. Time and distance are measured along the world line; they are the proper time and proper distance (not to be confused with the comoving distance).

Otherwise the ratio of movement should be one that is relative and three-dimensional in the numerator and absolute and one-dimensional in the denominator so that a vector is divided by a scalar. Thus the velocity should be used with 3D space and 1D time and the pace should be used with 1D space and 3D time.

The external-external perspective requires a different approach. Velocity and acceleration are defined relative to one-dimensional time so they cannot be used if time is three-dimensional. An inverse quantity (e.g., pace) is no better: that would be relative to one-dimensional space, which cannot be used if space is three-dimensional. One way would be to go back and forth between 3D space with 1D time and 1D space with 3D time.

A better way is via Minkowski’s approach to space-time: a new geometry. One advantage of this is that the invariant interval is defined without reference to velocity (though it includes the speed of light, a scalar). The Lorentz transformation can be represented as a hyperbolic rotation in a Minkowski space-time.