iSoul In the beginning is reality

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 celerity. Velocity is a speed with direction. Celerity 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 celerity, 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 celerity. 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.

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.