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Relativity posts

# Lorentz transformation derivation

The following derivations are similar to here.

Lorentz transformations for space with time

Let unprimed x and t be from inertial frame K and primed x′ and t′ be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, C, and D:

x′ = Ax + Bt
t′ = Ct + Dx

A body at rest in the K′ frame at position x′ = 0 moves with constant velocity v in the K frame. Hence the transformation must yield x′ = 0 if x = vt. Therefore, B = −Av and the first equation becomes

x′ = A (x – vt).

Using the principle of relativity

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing v with −v:

x = A (x′ − (−vt′)) = A (x′ + vt′).

# Equivalence principle

Inertia is the property of a body that resists changes in its motion. Inertial mass of a body is the ratio of the applied force divided by the body’s acceleration. Gravitational mass is the mass of a body as measured by its gravitational attraction to other bodies.

The Equivalence Principle takes several forms. The Newtonian version of the equivalence principle:

The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing. Then, gravitational force is proportional to inertial mass, and the proportionality is independent of the kind of matter. Ref.

The Einsteinian version of the Equivalence Principle:

All objects fall the same way under the influence of gravity; therefore, locally, one cannot tell the difference between an accelerated frame and an unaccelerated frame. Ref.

These can be expressed in terms of facial vass and levitational vass as follows:

Facilia is the property of a body that does not resist changes in its motion. Facilial vass of a body is the ratio of the applied release divided by the body’s retardation. Levitational vass is the vass of a body as measured by its levitational attraction to other bodies.

The Equivalence Principle then states:

Facilial vass and levitational vass are the same thing. Levitational release is proportional to facilial vass, and the proportionality is independent of the kind of matter.

All bodies rise the same way under the influence of levity; therefore, locally, one cannot tell the difference between a retardated frame and an unretardated frame.

# Speed of light

Speed is defined as “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (Dictionary of Physics, 3rd edition, McGraw-Hill, 2002.

This is ambiguous, however. Consider a light beam reflected off a surface:

(1) Since the light returns to its starting point, the total travel distance is zero, so the overall velocity is zero and the speed is zero.

(2) However, the interest is in each leg of the journey. In that case, in the first leg light travels +L in time t, and in the second leg light travels –L in time t. The mean velocity in the first leg is v1 = +L/t, and the mean velocity in the second leg is v2 = –L/t. The mean velocity for both legs is the harmonic mean of these two velocities because what is fixed and independent is the length, not the duration.

1/((1/v1) + (1/v2)) = 1/((1/L) – (1/L)) = 1/0 = ∞.

Thus the mean velocity is infinite, and the mean speed of light is infinite.

(3) Another approach looks at length of each leg apart from direction. In that case, in the first leg light travels L in time t, and in the second leg light travels L in time t. The speed in each leg is L/t, so the mean speed of light is L/t. This is the best known approach to the speed of light.

It’s interesting that (2) leads to the Galilean transformation, and (3) leads to the Lorentz transformation.

# A theory of 6D space-time

Note: as the research develops this post will be updated.

Introduction

Experience shows motion takes place in three dimensions. There are two measures of the extent of motion: length and duration. The length of motion in three dimensions comprises three-dimensional space. The duration of motion in three dimensions comprises three-dimensional time. Length and duration are symmetric concepts, as will be shown below.

Introduction

An independent variable is specified prior to measuring any dependent variable, so an independent variable is the domain of a functionally-related dependent variable. The independent variable is commonly an interval of time. Distance is the independent variable of an inverse square law. In Hooke’s law the independent variable is mass.

A date-time or time-stamp is a combined time-of-day and date on the calendar, which is of interest in history and astronomy. A time interval or elapsed time is the difference between two date-times, which is of interest in science.

A linear reference is of interest in geography and transportation. The stance interval or distance is the difference between two linear references, which is of interest in science.

Distance is an equivalence relation between pairs of points in space. Distime is an equivalence relation between pairs of instants in time.

An elapsed time or distime is the date-time that changes during an event or motion. A travel stance or distance is the change in linear reference during an event or motion.

Variables of time periods and distances are fixed. Variables of elapsed values are increasing from a starting point. Intervals are deltas of elapsed values. E.g., time periods are deltas of time. Distances are deltas of stances, that is, stations or points along a line or curve. Elapsed time and elapsed distance, or stance, are increasing variables.

Given that there are three dimensions of motion, and that every motion is measured by its length and duration, then motion requires three dimensions of length and three dimensions of duration. Three dimensions of length are called three-dimensional space. Three dimensions of duration are called three-dimensional time.

For example, motion on a two-dimensional surface can be presented as a two-dimensional map scaled in units of length or as a two-dimensional map scaled in units of duration. [The latter time maps are …]

# 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:  ${x}'&space;\mapsto&space;x-vt;\;&space;\;&space;{t}'&space;\mapsto&space;t.$

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

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

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

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

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

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

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

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

# 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:

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.

# Space as time and time as space

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 time interval bc“. Galileo uses a distance to measure a time interval, 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.

# 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.

# 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.