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.

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”:

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.

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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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.