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Tag Archives: Space & Time

Matters relating to length and duration in physics and transportation

Proper and improper rates

The independent quantity in a proper rate is the denominator. The independent quantity in an improper rate is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is harmonic addition:

\frac{b}{a_{1}}+\frac{b}{a_{2}} \Rightarrow \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right)^{-1}

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the harmonic mean:

\frac{1}{2}\left (\frac{b}{a_{1}}+\frac{b}{a_{2}} \right ) \Rightarrow \left ( \frac{1}{2} \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right ) \right)^{-1}

Time speed is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. Space speed is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the space mean speed. The time mean speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added harmonically. Legerity normalized by an hypothesized maximum pace is proper, but if the legerity is divided by the pace of light, it is improper.

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Space with time and their dual

For the first post in this series see here.

Space with time (3+1)

Space is that which is measured by length; time is that which is measured by duration. There are three dimensions of length and one dimension (or parameter) of duration. Direction in space is measured by an angle, which is part of a circle.

Spatial rates are dependent on another variable, usually interval of time (distime).

Time is that which is measured by duration. Events are ordered by time. Time as an independent variable decreases from the past to the present and increases from the present to the future.

Temporal rates are dependent on another variable, usually the interval of space (stance).

Dual: time with space (1+3)

The dual of space with time is time with space. The dual of space is time and the dual of time is space. Space corresponds to time and time corresponds to space.

Time is that which is measured by duration; space is that which is measured by length. There are three dimensions of duration and one dimension (or parameter) of length. Direction in time is measured by a turn, which is part of a rotation.

Temporal rates are dependent on another variable, usually interval of space (stance).

Space is that which is measured by length. Events are ordered by length (stance). Space as an independent variable decreases from a past there to here and increases from here to a future there.

Spatial rates are dependent on another variable, usually the interval of time (distime). Read more →

Harmonic conversion of space and time

As noted here, there are two kinds of mean rates: the time mean and the space mean. If the denominators have a common time interval, the time mean is the arithmetic mean and the space mean is the harmonic mean. If the denominators have a common space interval (stance), the space mean is the arithmetic mean and the time mean is the harmonic mean.

For example, light reflected back from a mirror at known distance forms two successive trips whose mean rate is the space mean pace. Several measurements with the same apparatus have a time mean pace. The mean speed is the inverse of the mean pace.

The general principle is that quantities with independent time (such as velocity) and a common time interval use ordinary algebra but such quantities with a common space interval use harmonic algebra. Alternately, quantities with independent space (stance) such as legerity and a common space interval use ordinary algebra but such quantities with a common time interval use harmonic algebra.

In other words, quantities over the same interval with independent variables use ordinary algebra but quantities with different independent variables use harmonic algebra to convert between them.

For example, addition of velocities with a common time interval use ordinary vector addition (e.g., u + v) but addition of velocities with a common space interval use harmonic vector addition (e.g., ((1/u) + (1/v))−1 ≡ ((u+v)/u·v)−1 with u, v, u·v, u+v0.

The relativity parameter γ is based on a (3+1) spatial frame. The parameter γ in a temporal frame with a common time interval (k ≡ 1/c and ≡ 1/v) is:

γ² = (1 − v·v/c²)−1 ⇒ (1 − ℓ·ℓ/k²)−1.

The parameter γ in a temporal frame with a common space interval (stance) is:

γ² = (1 − v·v/c²)−1 ⇒ H(1 − ℓ·ℓ/k²)−1 = ((1 − k²/ℓ·ℓ)−1)−1 = (1 − k²/ℓ·ℓ) ≡ (1 − v·v/c²) = 1/γ².

Interchanging space and time

The space-time exchange invariance, as stated by J. H. Field (see here) has an implicit second part. In addition to (1) the exchange (or interchange) of space and time coordinates, there is (2): the exchange of linear and harmonic algebra for ratios. Harmonic algebra is described here.

This is seen in the different averaging methods for velocities that differ spatially vs. velocities that differ temporally. If two vehicles take the same route, their average velocity is their arithmetic mean (u + v)/2, but if one vehicle has velocity u going and velocity v returning, then the average velocity is their harmonic mean 2/(1/u + 1/v). However, if one vehicle has pace u going and pace v returning, then the average pace is their arithmetic mean, but if two vehicles take the same route, their average pace is their harmonic mean.

Space and time are related to each other as covariant and contravariant components. If space is covariant, then time is contravariant, and if time is covariant, then space is contravariant.

The equations of space-time (3+1) and time-space (1+3) physics are symmetric to one another with the interchange of space and time dimensions. The equations of spacetime (4D) physics is self-symmetric. The interchange of space and time dimensions produces equivalent 4D equations.

To interchange the space and time coordinates, take these steps: For the equations of classical physics, (1) ensure either space or time is a parameter, (2) interchange one dimension with the parameter, and (3) expand the single dimension into three dimensions. For the equations of relativistic physics, (1) ensure there is a symmetry between space and time dimensions, (2) interchange one space and time dimension but leave dimensionless quantities unchanged, and (3) expand the single dimension into three dimensions.

These steps reflect the difference between Galileo’s and Einstein’s relativity. Galileo transforms one frame into another frame but does not combine frames as Einstein’s does. For example, Einstein requires all frames to have the same orientation, but Galileo accepts frame-specific orientations such as the right-hand rule.

The Galilean transformation represents the addition and subtraction of velocities as vectors. The dual Galilean transformation represents the addition and subtraction of lenticities as vectors. The Lorentz transformation represents the combination of Galilean and dual-Galilean transformations, as previously shown.

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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Invariant intervals

Let’s begin with the space-time invariant interval r´² − ct´² = r² − ct². Then let us solve the equations:

r´ = Ar + Bt and t´ = Cr + Dt.

r´ = 0 = Ar + Btr = −tB/A = vt where v = −B/A {or} B = −Av

r´ = Ar + Bt = A(rvt)

A²(rvt)² − c²(Cr + Dt)² = r² − c²t²

A²r² − 2A²vrt + A²v²t² − C²c²r² − 2CDc²rtD²c²t² = r² − c²t²

⇒ (A² − C²c²)r² = r² {or} A² − c²C² = 1

⇒ (A²v² − D²c²)t² = −c²t² {or} D²c² − A²v² = c²

⇒ (2A²v + 2CDc²)rt = 0 {or} CDc² = −A²v

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2D light clock

The famous Michelson-Morley experiment used what could be described as a 2D light clock since their interferometer combined two light clocks at right angles. Their hypothesis was that this would show the Earth moving through the aether, but they failed to detect any motion. Einstein explained this failure as a feature of relativity. In other words, the expected difference between the two light clocks was “corrected” by relativity.

The reasoning of the Michelson-Morley experiment went like this:

Light is sent from the source and propagates with the speed of light c in the aether. It passes through the half-silvered mirror at the origin at T = 0. The reflecting mirror is at that moment at distance L (the length of the interferometer arm) and is moving with velocity v. The beam hits the mirror at time T1 and thus travels the distance cT1. At this time, the mirror has traveled the distance vT1. Thus cT1 = L + vT1 and consequently the travel time T1 = L / (cv). The same consideration applies to the backward journey, with the sign of v reversed, resulting in cT2 = LvT2 and so T2 = L / (c + v). The longitudinal travel time T|| = T1 + T2 is:

T longitudinal

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Lorentz in spacetime and timespace

This post builds on the previous posts here and here, and follows the approach of J.-M. Levy here.

First, consider the Lorentz transformation in spacetime along the x axis per Levy’s section III:

Let us now envision two frames in ‘standard configuration’ with having velocity v with respect to K and let x, t (resp. x´, t´) be the coordinates of event M in the two frames. Let O and be the spatial origins of the frames; O and cöıncide at time t = = 0.

Here comes the pretty argument: all we have to do is to express the relation

OM = OO´ + O´M (equation 5)

between vectors (which here reduce to oriented segments) in both frames.

In K, OM = x, OO´ = vt and O´M seen from K is /γ since is O′M as measured in . Hence a first relation is:

x = vt + /γ (equation 6)

In , OM = x/γ since x is OM as measured in K, OO′ = vt´ and O′M = . Hence a second relation:

x/γ = vt´+ x´ (equation 7)

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Clock-rods

A clock-rod is a linear or planar clock with a parallel rod attached to it. A mechanical or electronic clock-rod might look like this:

clock-rod

Three clock-rods mutually perpendicular would measure length and duration in all directions.

A light clock-rod is conceptually like this:

light clock-rod

The clock and rod are parallel to each other so that parallel or perpendicular motion would change either the measurements of either the clock or the rod but not both. A complete harmonic cycle is not affected by motion:

light clock moving longitudinally

The +vt/2 increased distance of the first half-cycle is offset by the −vt/2 decreased distance of the second half-cycle. Likewise for the times +2d/v and −2d/v.

Kinematic derivations

Displacement with distime: displacement s, distime (time interval) t, velocities v1 and v, acceleration a:

To prove: v = v1 + at

a = (vv1) / t    by definition

at = (vv1)     multiply by t

v = v1 + at

To prove: s = v1t + ½ at2

vavg = s / t          by definition

vavg = (v1 + v) / 2        by definition

s / t = (v1 + v) / 2       combining these two

s = (v1 + v) t / 2         multiply by t

s = (v1 + (v1 + at)) t / 2       from above

s = (2v1 + at) t / 2

s = v1t + ½ at2

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