iSoul Time has three dimensions

Category Archives: Space & Time

Explorations of multidimensional space and time with linear and angular motion.

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 conversion factor between space and time. This is the situation of the theory of relativity and some transportation settings in which the minimum pace is the conversion pace (and maximum speed).

In this case, there is an increasing stance whether or not a positive time is measured. It is like a race that has no finish line. Without an increase in time 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 stance interval is a fast motion, whereas a large amount of time per unit stance interval 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 they are counting down, not up, and so are subluminal.

For the dual Lorentz transformation

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

The cu in λ is u/(1/c), which is the pace of the object divided by the pace of light. While cu is nominally equal to c/v, the denominator for pace is the stance, which is increasing at the conversion rate. As the time increases, the pace increases and the speed decreases from that of light toward zero. So, there is no square root of a negative number here.

(C) Stance is continuously increasing but there is no conversion factor between space and time. This is the situation of transportation in general. Space and time are not proportional, and the fastest route in space and time are in general different.

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').

The invariant interval is

(\Delta w)^2 = (\Delta x)^2 +(\Delta y)^2 +(\Delta z)^2 -(\kappa \Delta r)^2

=\lambda ^2(\Delta x'+ \zeta \kappa \Delta r')^2 +\Delta y'^2 +\Delta z'^2 -\lambda ^2(\kappa \Delta r'+\zeta\Delta x')^2

Expand the squares and cancel the middle terms to get:

=\lambda ^2(\Delta x'^2 (1-\zeta ^2) -\kappa ^2\Delta r'^2(1-\zeta ^2)) +\Delta y'^2 +\Delta z'^2

=\lambda ^2(\Delta x'^2 (1/\lambda ^2) -\kappa ^2\Delta r'^2(1/\lambda ^2)) +\Delta y'^2 +\Delta z'^2

=\Delta x'^2 +\Delta y'^2 +\Delta z'^2 -\kappa ^2\Delta r'^2.


Let Σ (Δri)² = (Δr)² and Σ (Δti)² = (Δt)². Then the two invariant intervals are

(\Delta s)^2 = (c\Delta t)^2 -(\Delta r)^2 \; \textup{and}\; (\Delta w)^2 = (\Delta t)^2 -(\kappa \Delta r)^2

These two invariant intervals are proportional if κ = 1/c:

(\Delta w)^2 =(\frac{\Delta s}{c})^2 = (\Delta t)^2 -(\frac{\Delta r}{c})^2 = (\Delta \tau)^2

In that case, the invariant interval for proper time is the same for both 3+1 and 1+3 dimensions. However, 3D proper time is limited to space-like intervals, while 3D proper distance is limited to time-like intervals. Since space-like intervals require superluminal speeds, they are considered impossible.

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 }.

For the (3+1) Ignatowski transformation there are spatial axes x, y, and, z; temporal axis t (time line), velocity v, and maximum velocity V, with β = v/V:

x'=\frac{x-vt}{\sqrt{1-v^2/V^2}}; y'=y; z'=z; t'=\frac{t-vx/V^2}{\sqrt{1-v^2/V^2}}.

The symmetric form is

x'=\frac{x-\beta Vt}{\sqrt{1-\beta^2}}; y'=y; z'=z; Vt'=\frac{Vt-\beta x}{\sqrt{1-\beta^2}}.

For the (1+3) dual Ignatowski transformation there are temporal axes x, y, and, z; spatial axis r (stance line), legerity u, and maximum legerity U, with ζ = u/U:

x'=\frac{x-ur/U^2}{\sqrt{1-u^2/U^2}}; y'=y; z'=z; r'=\frac{r-ux}{\sqrt{1-u^2/U^2}}.

The dual symmetric form is

Ux'=\frac{Ux-\zeta r}{\sqrt{1-\zeta^2}};\; y'=y;\; z'=z; \; r'=\frac{r-\zeta Ux}{\sqrt{1-\zeta^2}}.

The Galilean transformation has V → ∞:

x'=x-vt; \; y'=y;\; z'=z;\; t'=t.

The dual Galilean transformation has U → ∞:

x'=x-ur; \; y'=y;\; z'=z;\; r'=r.

 

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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Linear clocks and time frames

The idea of a linear clock was mentioned before here, here, and here.

One bar on top of anotherOne bar shifted on another

Consider two bars or rods, one on top of the other (left), each with a zero point aligned at first. The top one moves at a constant rate relative to the other, which is at rest. After a time T, the top bar has moved an interval measured by the difference between the zero points of the bars (right). The length that B moved relative to A measures the time of motion.

Side note: a 12-inch ruler turned into a circle would form the markings for a 12-hour clock. The hours of time would correspond to inches of length.

A time frame of reference (TFR), or time frame, is a frame of reference for time. Like a space frame of reference (SFR) it is composed of rigid bars or rods that can in principle be extended indefinitely.

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Observers in motion

A rigid rod or other device that measures length is at rest relative to itself, even if part moves such as a measuring wheel, because it moves relative to other objects, not relative to itself. A concept of simulstanceity enables an observer to determine length at other times (e.g., they are the same point on the stance line).

A clock measures time, but what is a clock? It is a device with a part that moves relative to a part that is at rest. So a clock is an object in motion relative to itself (yes, this is possible). The part that moves indicates the time. A concept of simultaneity enables an observer to determine time at other places (e.g., they are the same instant on the time line).

Let there be a rigid reference frame associated with each observer or object (e.g., they are attached). An observer may be at rest or in motion relative to their frame. If the observer is at rest, then their frame is a length frame and what they observe is in space. Time is the independent variable and length in three dimensions is the dependent vector variable.

If the observer is going somewhere, they are not at rest but in motion. Their reference frame for rest is not their own frame but a different frame, such as one located on the surface of the earth. In this case the observer and rest frame system are like a clock, that is, a clock frame, and what is observed is in time. A clock frame is moving in the opposite direction of a rest frame. Length is the independent variable and time in three dimensions is the dependent vector variable.

Frames in motion

For Galilean inertial frames the observer is at rest and the moving frame transmits the current stance in an instant of the time line, instantaneously. For dual Galilean inertial frames the observer is in motion and the rest frame transmits the current time in a point of the stance line, punctstanceously.

The rest frame observer has three dimensions in space. The observed frame in motion is effectively reduced to the one dimension of its motion in time. The moving frame observer is like a clock with space and time exchanged: the dimensions of the observer’s frame are in motion so the three dimensions are in time. The rest frame that is observed appears to move and is effectively reduced to the one dimension of its path in space.

Time and simultaneity

There are several ways of understanding the time of remote events. What follows is a summary of the basic ways of determining simultaneity.

As a way of comparing the different ways consider transmitting a light signal to a remote location where it is reflected back. What is the time when the signal is reflected back?

Observation time is an extension of ordinary perception. When we observe an event, we say that it is happening at the time of observation. So when a light signal is reflected and received back, the reflection observed is considered to have happened when it was observed. In effect the light observed is instantaneous. By implication the one-way speed of light transmitted is c/2 in order for the two-way speed of light to equal c.

Observation time is thus the projection of the time of observation to the entire observable universe. This way of understanding time is characterized by the Galilean transformation.

Transmission time is an extension of the ordinary transmission of light. When we shine a light on an event, we say that it is happening at the time of transmission. So when a light signal is aimed toward a reflector, the event of reflection is considered to have happened when the light was transmitted. In effect the light transmitted is instantaneous. By implication the observed one-way speed of light is c/2 in order for the two-way speed of light to equal c.

Transmission time is thus the projection of the time of transmission to the entire transmittable universe. This way of understanding time is characterized by the dual Galilean transformation.

Probe time is an extension of measurement by a probe (a “small, unmanned exploratory craft”) to the entire probeable universe. See previous post here. An event is said to occur when intersected by a probe that measures the duration of probe movement since a reference event. So when a probe comes upon the reflection of light, the probe measures the time of reflection as the time of the probe. If the probe is not moving at the speed of light, there may need to be multiple probes.

Consider a series of probes moving at a speed v over a distance d to the reflection event. The probe that leaves at time (d/c) – (d/v) is the probe that intersects the reflection event. If v = c, then the time is zero.

Because probes can measure the length or duration of motion, probe time is characterized by the Lorentz or dual Lorentz transformation.

Reference frame time measures time by a rigid reference frame that has clocks which were previously synchronized spread throughout. See the Relativity of Simultaneity and Einstein Synchronisation. These synchronizations are characterized by the Lorentz transformation.

Reference probes and systems

A reference frame is in principle a rigid structure embodying a 3D coordinate system. It represents an observer at rest with complete access to rods and clocks to measure length and duration in any direction:

Such a reference frame may be the framework or infrastructure for a reference probe moving like a miniature aerial tram in any direction. A probe is a “small, unmanned exploratory craft”. Such a reference probe compared with a target motion can measure either the extent of the framework crossed by the target, which is the length, or the extent of the framework crossed by the reference probe, which is the duration. The rate of the target motion is the ratio of the length to the duration or the ratio of the duration to the length.

Alternatively, the reference frame may be the framework or infrastructure for a reference system of probes jmoving in all directions. The motion of such a system can be given by a table of changes, which are the intersections of consecutive trips, called “times”, and consecutive stations, called “stances”:

Table of Changes

Times

Trip 1

Trip 2

Stances

Location 1

change 1,1 change 1,2

Location 2

change 2,1 change 2,2

A target motion can be measured as the number of stances, which is the length, or as the number of times, which is the duration. The rate of the target motion is the ratio of the length to the duration or the ratio of the duration to the length.

What if one reference framework is moving with respect to another reference framework? The motion of a framework is no different than the motion of an object as observed by a reference framework. How can one compare the observation of an object from one framework with that of another framework? That requires applying the appropriate transformation, Galilean, dual Galilean, Lorentz, or dual Lorentz.