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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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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 contra-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 contra-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 contra-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, contra-Galilean, Lorentz, or contra-Lorentz.

Einstein exchanged

Albert Einstein’s book Relativity: The Special and General Theory was originally published in German and translated into English in 1920. In the second chapter he introduces “The System of Co-ordinates”. The following post gives Einstein’s text followed by a revision that exchanges length with duration and space with time. First, Einstein’s text, with alternative wordings in square brackets:

End of Chapter I – If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.

Chapter II – On the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a “distance” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distance S time after time [again and again] until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length.

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Objects and subjects in motion

An object is stable. A rock is an object. Water is an object if it is in a container. A rigid rod is an object.

A subject changes. A person is a subject. Air is a subject since it keeps moving. A clock is a subject.

The grammatical subject and object are distinguished in a sentence, though they both may be things. For example, “The rock rolled down the hill.” Both the rock and the hill are things, that is objects, but in the sentence the rock is the subject and the hill is the object.

Objects are acted upon. A predicate is required to go with an object. An object apart from a sentence is a thing, something passive.

Subjects are active. A verb is required to go with a subject. A subject apart from a sentence is still a potential change agent.

Space is like an object and time is like a subject. If we start with objects and then discuss their motions, we are beginning with the passive objects of space and then adding the active subjects of time. If we start with subjects, we get their motions, too, and may then bring in the objects of space. That is beginning with an active time and adding the passive objects of space.

A reference motion must be active and so include a subject. A comparative motion is passive in relation to the reference motion and so must include an object.

Bodies and things may be subjects or objects, though many are usually one or the other. The difference is in whether they change or move. An object need not move. A subject is usually moving.

Objects are in space. Subjects are in time. Space never moves. Time always moves.

Moving bodies in space and time

Let us compare the motions of two bodies. Let the motion of one body be the reference motion. Let the motion of the other body be the comparative motion. Let the two bodies begin together at one place.

Definitions:

A place is the general term for an answer to Where? A point-place, or simply a point, is the smallest place. A translation is a vector from one point-place to another. Travel distance is the arc length of the trajectory of a motion, which includes any retracing of the trajectory.

Space and time refer to different perspectives of the universe of motion.

Space is the locus of all potential places for the comparative motion, which is said to be “in space.” Displacement is a translation vector from one point to another point of the comparative motion. The travel distance from the beginning point to the ending point of the comparative motion, is the travel length for a motion in space.

Time is the locus of all potential places for the reference motion, which is said to be “in time.” Distimement is a translation vector from one point to another point of the reference motion. The travel distance from the beginning point to the ending point of the reference motion, is the travel time for a motion in time.

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