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Space and time as opposites

A theme of this blog is that space and time are dual concepts, which means they are two ways of understanding the same thing. But in what ways are space and time opposite concepts?

Space is oriented toward its origin, the place that motion begins. Time is oriented toward its destination, the time that motion ends. Both length and duration are measured from an “origin,” a reference point, which is a zero point for each, but zero speed leaves a body in space at the beginning, whereas zero pace puts the body in time at its destination.

Length in the denominator of speed is a measure of the progress from the origin to the current location in space, whereas time in the denominator of pace is a measure of the lag from the destination to the current location in time. A body at zero speed will remain at its origin and never reach its destination, whereas a body at zero pace will arrive at its destination in literally no time. A body with a small speed will take a long time to reach its destination, whereas a body with a small pace will reach its destination quickly.

Large quantities in space correspond to small quantities in time. Large quantities in time correspond to small quantities in space. A high speed is fast, and a small speed is slow. A small pace is fast, and a large pace is slow. Mass and vass are inverses, as are energy and lethargy.

The origin in space corresponds to the destination in time. Time in space flows from the past toward the future. Space (stance) in time flows from the future toward the past.

Arcloge

An arcloge (arc’-loje) is a continuous, independent measurement of length. That is, it measures an ever-increasing length, which is the stance, similar to how a clock shows the time. The term is a combination of arc (as in arc length) and loge (as in horologe, a clock).

What does an arcloge look like? Start with a sector, which is a geometric figure fixed to the center of a circle that sweeps out an angle and a curved edge:

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Basic definitions

Independent variable is a quantity that is not dependent on another quantity, which is either (a) a quantity chosen before an experiment or race, or (b) an ever-increasing quantity. Dependent variable is a quantity that whose value is a function of another variable.

Space is (1) length; (2) a 3D differentiable manifold of length; (3) the order of events on a stance line; (4) the stance, the reading on an arcloge.

Time is (1) duration; (2) a 3D differentiable manifold of duration; (3) the order of events on a time line; (4) the time, the reading on a clock.

Length is measured by a rigid rod. 3D space is length measured in three directions of motion.

Duration is measured by a stopwatch, timer, or clock. 3D time is duration measured in three directions of motion.

Spacetime is the 6D manifold formed from 3D space and 3D time. Worldline is the path in spacetime traced out by an object in motion.

Displacement is the vector between two points (events) on a worldline. Distance is the magnitude of a displacement.

Distimement is the vector between two points (events) on a worldline. Distime is the magnitude of a distimement.

Reference frame (or frame) is an abstract coordinate system and set of reference points in 3D space that uniquely fix the coordinate system and standardize measurements. Rest frame of a body is the reference frame in which the body is moving at zero speed, which is the time conversion pace.

Reference timeframe (or timeframe) is an abstract coordinate system and set of reference points in 3D time that uniquely fix the coordinate system and standardize measurements. Freeflow frame of a body is the reference timeframe in which the body is moving at zero pace, which is the stance conversion speed.

Proper length is the length of a body measured by a rigid rod moving with it. Proper time is the time of a body measured by a clock moving with it.

Lorentz transformation is a set of equations that relate space and time coordinates of reference frames moving at a constant velocity relative to each other.

Dual Lorentz transformation is a set of equations that relate space and time coordinates of reference timeframes moving at a constant legerity relative to each other.

From racing to relativity

There are three different contexts for 3D time, depending on whether stance is continuously increasing and, if so, whether there is a conversion factor between space and time:

(A) Stance is not continuously increasing. This is the situation of a race or sport in which game time has a definite beginning and ending. For example, in many sports the game lasts a specific time. In a race, the length of the course is set and the time for each contestant ends when they cross the finish line. The average pace of a contestant is their race time divided by the course length.

(B) Stance is continuously increasing and there is a general conversion factor between space and time. This is the situation of the special theory of relativity and some transportation settings in which the conversion pace is the minimum pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time interval is measured. Without an time interval increase the pace is at a minimum (or the speed is a maximum). As the amount of time measured increases, the pace increases (or the speed decreases). Remember that a small amount of time per unit distance is a fast motion, whereas a large amount of time per unit distance is a slow motion.

In this way, the pace increases indefinitely. A pace of infinity would be at rest. A pace of zero is the minimum pace, which in relativity is the speed of light. That is, the speed counts down from the speed of light. This has been misinterpreted as a transformation with superluminal speeds, but because speed decreases as pace increases, object speeds are subluminal.

The dual Lorentz transformation (see here) is

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ur) \; \textup{with}\; \lambda = 1/{\sqrt{1-cu}}

with the understanding that c represents the inverse of the pace of light. The cu in λ is the pace of the object divided by the pace of light, with the stance increasing at the conversion rate. As the time of motion increases, the pace increases (and the speed decreases) from that of light toward the pace or speed of rest. So, the square root never becomes negative here.

(C) Stance is continuously increasing but there is no general conversion factor between space and time. This is the situation of general relativity and transportation in general. Conversion of space and time are local, not global, and the optimal route depends on whether space or time are optimized.

Invariant intervals

The spacetime interval is invariant over the Lorentz transformation (LT). The following is a proof of this for the inverse LT with spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, along with β = v/c and γ = 1/√(1 − β²):

\Delta x=\gamma (\Delta x'+\beta c\Delta t');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; c\Delta t=\gamma (c\Delta t'+\beta \Delta x').

The invariant interval is

(\Delta s)^2 = (c\Delta t)^2 -(\Delta x)^2 -(\Delta y)^2 -(\Delta z)^2

=\gamma ^2(c\Delta t'+\beta\Delta x')^2 -\gamma ^2(\Delta x'+ \beta c\Delta t')^2 -\Delta y'^2 -\Delta z'^2

Expand the squares and cancel the middle terms to get:

=\gamma ^2(c^2 \Delta t'^2(1-\beta ^2) -\Delta x'^2(1-\beta ^2)) -\Delta y'^2 -\Delta z'^2

=\gamma ^2(c^2 \Delta t'^2(1/\gamma ^2) -\Delta x'^2(1/\gamma ^2)) -\Delta y'^2 -\Delta z'^2

=(c\Delta t')^2 -(\Delta x')^2 -(\Delta y')^2 -(\Delta z')^2.


The spacetime interval is invariant over the dual Lorentz transformation (DLT). The following is a proof of this for the inverse DLT follows with temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity κ, along with ζ = u/κ and λ = 1/√(1 − ζ²):

\Delta x=\lambda (\Delta x'+ \zeta \kappa \Delta r');\; \Delta y=\Delta y'\; \Delta z=\Delta z';\; \kappa \Delta r=\lambda (\kappa \Delta r'+\zeta \Delta x').

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Symmetric transformations

What follows are Lorentz and Ignatowski transformations and their duals with symmetric and vector forms for reference.

For the (3+1) Lorentz transformation there are spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity c, with β = v/c and γ = 1/√(1 − β²):

x'=\gamma (x - vt);\; y'=y;\; z'=z;\; t'=\gamma (t - vx/c^2).

The symmetric form is

x'=\gamma (x -\beta ct);\; y'=y;\; z'=z;\; ct'=\gamma (ct - \beta x).

The symmetric Lorentz transformation for vectors is

\mathbf{r'}_\perp = \mathbf{r}_\perp;\; \mathbf{r_\parallel }'=\gamma (\mathbf{r_\parallel } - \boldsymbol{\beta } ct);\; ct'=\gamma (ct - \boldsymbol{\beta } \cdot \mathbf{v}),\; \textup{with } \boldsymbol{\beta } =\mathbf{r_\parallel }/c.

For the (1+3) dual Lorentz transformation there are temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity 1/c, with ζ = cu and λ = 1/√(1 − ζ²):

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; r'=\lambda (r - c^2ux).

The dual symmetric form is

x'=\lambda (x - ur);\; y'=y;\; z'=z;\; \frac{r'}{c^2}=\lambda (\frac{r}{c^2} - ux).

The dual symmetric Lorentz transformation for vectors is

\mathbf{t'}_\perp = \mathbf{t}_\perp;\; \mathbf{t_\parallel }'= \lambda (\mathbf{t_\parallel } - \boldsymbol{\zeta }\frac{r}{c});\; \frac{r'}{c}=\lambda (\frac{r}{c} - \boldsymbol{\zeta } \cdot \mathbf{u}),\; \textup{with } \boldsymbol{\zeta }=c \mathbf{t_\parallel }.

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Ignatowsky relativity

Vladimir Ignatowski (1875-1942) was a Russian physicist. “In 1910 he was to first who tried to derive the Lorentz transformation by group theory only using the relativity principle (postulate), and without the postulate of the constancy of the speed of light.” K M Browne gave a simplified derivation in the European Journal of Physics, 39 (2018) 025601, from which the key steps are presented below, followed by the corresponding steps for a dual transformation, switching space and time.

This is a derivation of the Ignatowsky transformation in which the axes x, y, and z are taken to represent space axes rx, ry, and rz with time t. The relativity postulate is taken to be: a valid relativistic transformation must be identical in all inertial frames.

Step 1. To find a valid transformation, we take the usual inertial reference frames S and S′ (the latter moving at velocity v in the +x direction relative to the former) for which intra-frame space is Euclidean but inter-frame space (measured from one frame to the other) may be non-Euclidean. Linear equations are necessary so that an event in one frame appears as a single event, without echoes, in the other. Initial conditions are x′ = x = 0 when time t′ = t = 0. We expect the generalised x equation to be the Euclidean equation x′ = xvt with an added multiplier, and if time is the fourth dimension, then the time equation will be similar but with two additional multipliers. The second of these, n, having the dimension of inverse velocity squared, is required to make the equation dimensionally correct. The y and z coordinates are not expected to be affected by x and t. The generalised transformation and its inverse are then

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Length contraction and time dilation

These derivations follow that in ‘Hyperphysics’ here.

Length Contraction

The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The length is maximum in the frame in which the object is at rest.

Fixed and moving reference frames

If the length L0 = x2´ − x1´ is measured in the moving reference frame, then L = x2x1 in the rest frame can be calculated using the Lorentz transformation.

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Length and duration in space and time

The following derivations are based on the exposition by G. G. Lombardi here.

Time Dilation

Time dilation with a light clock

A clock is made by sending a pulse of light toward a mirror at a distance L and back to a receiver. Each “tick” is a round-trip to the mirror and back. The clock is shown at rest in the “Lab” frame in Fig. 1a, or any time it is in its own rest frame. Consequently, it also represents the clock at rest in Rocket #1. Figure 1b is the way the clock looks in the Lab when the clock is at rest in Rocket #1, which is moving to the right with velocity v and legerity u, hence speed v and pace u.

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Lorentz from light clocks

Space and time are inverse perspectives on motion. In space length is measured by a rigid rod at rest, whereas duration is measured by a clock that is always in motion. In time duration is measured by a clock at rest relative to the time frame, whereas length is measured by a rigid rod in motion that counteracts time as it were.

This is illustrated by deriving the Lorentz factor for time dilation and length contraction from light clocks. The first derivation is in space with scalar time and the second is in time with scalar space.

light clock with height hlight clock in motion

The first figure above shows a light clock in space as a beam of light reflected back and forth between two mirrored surfaces. The height that the light beam travels between the surfaces is h. Let one time cycle Δt = 2h/c = 2 or h = cΔt/2 = Δt/(2¢), with mean speed of light c and mean pace of light ¢.

The second figure shows the light clock as observed by someone moving with velocity v and pace u relative to the light clock; the length of each leg is d; and the length of the base of one triangle-shaped cycle is b.

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