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Category Archives: Relativity

Relativity posts

A theory of time

Speed is the travel distance per unit of travel time. In racing there is a measure of the travel time per unit of travel distance, which is called the pace. These are not exactly inverses since the denominated in different units. Note that a faster motion is indicated by a lower pace since it takes a shorter time to travel the same distance.

Velocity is a vector quantity whose magnitude is a body’s speed and whose direction is the body’s direction of motion. What is the opposite concept, a vector quantity whose magnitude is a body’s pace and whose direction is the body’s direction of motion? The dictionary lacks a word for this concept; I propose calling it lenticity [lentitude] from Latin lentus, slow, since a larger value indicates a slower motion.

Motion is a form of change, and change is characterized by difference. A body at rest does not change. A body in motion changes. But a body is at rest only with respect to another body at rest; they change the same way. A body is in motion only with respect to another body in motion; they change in different ways.

What is a body but something physical with some consistency; some attribute must not change. If something changes completely, it is not a body, or at least not a single body. What is the length of a body? It is the difference between one end and the other end. This difference is a change, a motion, with respect to the body or with respect to an observer at rest with respect to the body.

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Space, time, and dimension

The post continues the ones here, here, and here.

There are three dimensions of motion. The extent of motion in each dimension may be measured by either length or time (duration). There are three dimensions of length and three dimensions of time (duration) for a total of six dimensions.

But there is no six-dimensional metric. Why? Because a metric requires all dimensions to have the same units, which requires a ratio to convert one unit into the other unit. The denominator on a ratio is a one-dimensional quantity, which means either the length or time dimensions need to be reduced by two dimensions.

This ratio is a conversion factor that is either a speed, which multiplied by a time equals a length, or a pace, which multiplied by a length equals a time. In general a speed is the ratio Δdr²/|Δdt²| = Δdr²/(Δt1² + Δt2² + Δt3²)1/2, and a pace is the ratio Δdt²/|Δdr²| = Δdt²/(Δx² + Δy² + Δz²)1/2. The denominator is a distance or distime, which is a linear measure of length or time (duration).

The conversion factors required are the speed of light in a vacuum, c, or its inverse, the pace of light in a vacuum, k. The resulting four-dimensional metric is either c²dt² − dx² − dy² − dz² (with time reduced to one dimension) or dt1² − dt2² − dt3² − k²dr² (with space reduced to one dimension).

These metrics are often simplified by taking c = 1 and k = 1 so that symbolically they are the same. Their units are not the same, however.

Each metric may be further reduced by separating space and time, as in classical physics. Then the space metric is |Δdr²| = (Δx² + Δy² + Δz²)1/2 and the time metric is |Δdt²| = (Δt1² + Δt2² + Δt3²)1/2. In the classical (3+1) of three space dimensions and one time dimension, time is replaced by its metric, and in the classical (1+3) of one space dimension and three time dimensions, space is replaced by its metric.

One and two-way transformations

The transformation of Galileo is a one-way transformation, i.e., it uses only the one-way speed of light, which for simplicity is assumed to be instantaneous. The transformation of Lorentz is a the two-way transformation, which uses the universal two-way speed of light. The following approach defines two different one-way transformations, which combine to equal the two-way Lorentz transformation. Note that β = v/c; 1/γ² = 1 − β²; and γ = 1/γ + β²γ.

Galilean transformation:  {x}' \mapsto x-vt;\; \; {t}' \mapsto t.

Dual Galilean transformation:  {x}' \mapsto x;\; \; {t}' \mapsto t-wx.

These could be combined with a selection factor κ of zero or one:

{x}' \mapsto x - \epsilon vt;\; \; {t}' \mapsto t-(1-\epsilon )wx.

Lorentz transformation (boost): {x}' \mapsto \gamma (x-vt);\; \; {t}' \mapsto \gamma (t-vx/c^{2}).

General Lorentz boost (see here): {x}' \mapsto \gamma (x-vt);\; \; {t}' \mapsto \gamma(t-k^{2}vx)

with \gamma =\left (1-\frac{v^{2}}{c^{2}} \right )^{-1}  and k = 1/c for the Lorentz boost.

General dual Lorentz boost:  {x}' \mapsto \gamma_{2} (x-kwt);\; \; {t}' \mapsto \gamma_{2} (t-wx)

with \gamma_{2} =\left(1-\frac{w^{2}}{k^{2}} \right)^{-1}and k = 1/c.

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Light clocks with multidimensional time

A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979).

A light clock is a device with an emission-reflection-reception cycle of light that registers the current timeline point and placeline point in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´. Then, as the light clock in K´ moves transversely relative to K with uniform motion at velocity v (right), from K one observes the following:

transverse light clock

The illustration above shows one cycle length of the light path (i.e., wavelength), X, on the left and one cycle duration (i.e., period), T, on the right at rest in reference frames K, K´. For the reference frame K´, in motion relative to reference frame K, call the arc length of one cycle of the light path x. Call the distance between the beginning and ending place points of one cycle x. For the reference frame K´ relative to reference frame K, call the arc time of one cycle of the light path t. Call the distime between the beginning and ending timepoints of one cycle t.

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Space as time and time as space

Galileo parabola

Galileo used the distance of uniform motion as a measure of the distime, i.e., time interval (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914):

Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time-interval bc finds itself at the point i. p.199

Without getting into the details of the Figure 108, notice the shift of language: “the body moves from b to c” [i.e., a distance], then “the distime bc“. Galileo uses a distance to measure a distime, which is justified since the motion is “with uniform speed” and so they are proportional.

The point to make here is that a distance and a distime can be interchanged if the motion is uniform. That is exactly the function of a clock: to provide a standard distime for a corresponding distance of motion. The change from distance to distime is basically a change of units. So, the line with a to e and beyond is a linear clock: it measures elapsed distime or “elapsed distance”.

Let there a ball be dropped out the window by a passenger on a train in uniform motion. Consider the following four scenarios, in which the distance or distime of a uniform motion is measured: (1) looking down above the moving ball, measuring the distance of fall; (2) looking down above the moving ball, measuring the (uniform) distime of fall; (3) looking from the side, measuring the distance of motion in two dimensions; and (4) looking from the side, measuring the (uniform) distime of motion in two dimensions.

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Simultaneity and simulstanceity

Max Jammer’s book Concepts of Simultaneity (Johns Hopkins UP, 2006) describes the significance, meaning, and history of simultaneity in physics. Here are a few excerpts from his Introduction:

… Einstein himself once admitted: “By means of a revision of the concept of simultaneity in a shapable form I arrived at the special relativity theory.” p.3

That not only temporal but also spatial measurements depend on the notion of simultaneity follows from the simple fact that “the length of a moving line-segment is the distance between simultaneous positions of its endpoints,” as Hans Reichenbach … convincingly demonstrated. Having shown that “space measurements are reducible to time measurements” he concluded that “time is therefore logically prior to space.” p. 4-5

P. F. Browne rightly pointed out that all relativistic effects are ultimately “direct consequences of the relativity of simultaneity.” p.5

One might give the dual to the second statement as: That not only spatial but also temporal measurements depend on the notion of simulstanceity follows from the simple fact that “the duration of a moving line-segment is the time interval between simulstanceous chronations of its endpoints. Space is therefore logically prior to time.

In the next chapter, Terminological Preliminaries, Jammer clarifies the relevant concepts. It is ironic that he gives an early example of the metonym “of spatial terms to denote temporal relations that is frequently encountered both in ancient and in modern languages.” (p.9) Space has priority in language.

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From spacetime to space and time

This relates to the post here.

There are three dimensions of motion with two measures of the extent of motion, which makes a total of six metric dimensions of motion. But these six metric dimensions collapse into two structures of one and three dimensions as the conversion factor approaches infinity.

With the pace of light, k, the invariant proper length, , is:

dσ² = dr²dt²/k² = dr1² + dr2² + dr3² – dt²/k² = dr² – dt1²/k² – dt2²/k² – dt3²/k² = dr1² + dr2² + dr3² – dt1²/k² – dt2²/k² – dt3²/.

As the pace of light approaches infinity this becomes

dσ² = dr² = dr1² + dr2² + dr3².

That is, the time coordinates separate from the invariant length, which becomes the Euclidean distance of three dimensional space. Time is left as an invariant scalar called the time.

Similarly, with the speed of light c, the invariant proper time, , is:

dτ² = dσ²/c² = dr²/c² – dt² = (dr1² + dr2² + dr3²)/c² – dt² = dr²/c² – dt1² – dt2² – dt3² = (dr1² + dr2² + dr3²)/c² – dt1² – dt2² – dt3².

As the speed of light approaches infinity this becomes

dτ² = – dt² = – dt² = – dt1² – dt2² – dt3².

That is, the length coordinates separate from the invariant time, which becomes the Euclidean distime of three dimensional time. Space is left as an invariant scalar called the stance.

The result is that six dimensional spacetime collapses into 3D space with scalar time or 3D time with scalar space.

Duality as a convention

Is color an absorption phenomena or an emission phenomena? The answer is that it’s both. But absorption works subtractively whereas emission works additively. The question then is whether color is subtractive or additive. Again the answer is that it’s both. Color is a duality.

Does an artist work with subtractive colors or additive colors? Here the answer is one or the other. A painter works with pigments that are subtractive, whereas a glass artist works with stained glass that is additive. Even though absorption and emission are operating in both cases, working with color requires picking one or the other (except for mixed media).

A simultaneity convention can also be a duality. What has been called apparent simultaneity is the convention that the backward light cone is simultaneous. But it is possible to adopt a complementary convention in which the forward light cone is simultaneous (see here). Either of these is something of a combination of Newton’s and Einstein’s physics.

One could recover Newtonian physics by adopting a combination of the backward and forward light cone simultaneity conventions. For an absorption event the backward light cone is simultaneous. For an emission event the forward light cone is simultaneous. This is like half-duplex communication (push to talk, release to listen). Such a duality convention recovers Newtonian physics because it is as if the speed of light is instantaneous in all directions.

Mean speed of light postulate

Einstein stated his second postulate as (see here):

light is propagated in vacant space, with a velocity c which is independent of the nature of motion of the emitting body.

Since the one-way speed of light cannot be measured, but only the round-trip (or two-way) speed, let us modify this postulate to state:

The two-way mean speed of light in vacant space is a constant, c, which is independent of the nature of motion of the emitting body.

This is the most that can be empirically verified. Then for convenience sake, let us adopt the following convention:

The final observed leg of the path of light in empty space takes no time.

Since the (harmonic) mean speed of light is c, the speeds of the other legs of light travel are at least c/2 such that the mean speed equals c. In this way, the Galilean transformation is preserved for the final leg. And interchanging length and duration leads to an alternate version of the Galilean transformation.

This accords with common ways of speaking. Even astronomers speak of where a star is now, rather than pedantically keep saying where it was so many years ago. Physical theory should be in accord with observation of the physical world as much as possible. This is an example of how amateur scientists can help re-integrate science and common life.

Half-duplex relativity

Galilean relativity requires the speed of light to be instantaneous (i.e., zero pace). Because the one-way speed of light is not known, it may be instantaneous as long as the mean speed of light is finite. Such a situation is possible if light is conceived as in half-duplex telecommunications: one direction at a time is observed or transmitted, but never both simultaneously.

Consider a light clock in this context:

light at restSaw-tooth light path

Let Δt be the time for one cycle of light at rest (top diagram). Let Δt’ be the time for one cycle of light traveling at relative velocity v (bottom diagram). The mean speed of light is c. Then

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