Physics Lesson 18.3.2 - Inertial System in the Special Theory of Relativity and Synchronization of Clocks

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Inertial System in the Special Theory of Relativity and Synchronization of Clocks

Let's consider again the concept of inertial system, especially in regard to the measurement of time interval between two physical events. A physical event is an event that occurs somewhere in space, at a given (time) instant. These two concepts (space and time) have a meaning only when we provide a sufficient information about the frame of reference in which the questions "Where?" and "When?" get answer. Based on the first postulate, all inertial systems are equivalent.

Like in the classical theory of relativity, we have a three-dimensional of origin O and three perpendicular Cartesian axes X, Y and Z, which allow us to determine the position of any object by measuring the three basic positions using a standard of measurement (a known unit). Obviously, the same physical event occurs in different coordinates to the system S when measured in another inertial system S'. Now, we cannot say anymore that the time of event occurrence is the same in both systems, as in the case of the Newtonian system. Hence, a procedure for measuring the time that is typical for a given system, is necessary. This time is t for the system S and t' for the system S' where t' ≠ t.

In Einstein's theory, the system of synchronized clocks is used to describe one or more relativistic event/s. These are perfect imaginary clocks, placed at every point of space (of the universe); yet, every inertial system has its own local clocks, which are stationary relative to the system itself. In this sense, we must imagine that the clock systems of two inertial frames of reference intertwine with each other during the relative motion. When we say a particle is "born" somewhere, physically this means the particle is generated in the point of space of coordinates (x, y, z) for the system S and (x', y', z') for the system S'. It is obvious that in the place where the even occurs, there are two standard clocks: one is the clock of the system S and the other is the clock of the system S' which record the times t (for S) and t' (for S') for the same event. All the fixed clocks of the system S show the same time t, as well as those of the system S' (which all show all the time t'), as all clocks within the same system are synchronized. The question that arises here is: "How the synchronization of clocks inside the same inertial system is done?" Let's use the following figure to answer this question.

In the figure above, a central clock fixed at centre O of the system S and another clock at the point P(x, y, z) are shown. All clocks of the inertial system S (including the two clocks shown in the figure), are stationary relative to each other.

The central clock contains a mechanism which when operated (starting from t = 0) emits a light signal in all directions. On the other hand, the clock at the point P is settled forward to show the time τ = l / c prior to the event occurrence. This clock contains a mechanism that makes it start working when receiving a light signal (in this case, the light signal comes from the origin O). It is obvious that the clock located at P start recording time (from the value τ) when the light signal moves by τ seconds from the origin. The two clocks will henceforth show the same value (we say they are synchronized). This procedure is used to synchronize all clocks of the system S, as well as those of the system S' (using a central clock located at O').

Everything discussed so far is simply imagination and theory; however principally they have a great importance to explain the time difference between two events or the same event when viewed from two inertial frames of reference. We should keep in mind that all events are absolute as objective facts, however, their time (and place) of appearance is different in different inertial frames of reference. Thus, the same event is shown at coordinates (x, y, x, t) in the system S and (x', y', z', t') in the system S'.

Example 1

The closest star to the Earth is Proxima Centauri, which is 4.26 light years (a unit of distance showing the distance travelled by the light in one year through vacuum) away from us. Suppose that just in the instant you are reading this text, we observe a process in this star. What time would a clock located at the surface of Proxima Centauri show at the instant of event's occurrence if this clock is synchronized with our clock?

Solution 1

The process we observe now, has occurred in the given star 4.26 light years ago (the time needed to the light to reach our sight). Since our clock shows the value 0 at the occurrence of the event, this value is also shown by the clock located at the star. However, to achieve this, the clock located at the star must be settled forward by

In other words, when the event occurs, the clock at the star must show the value t' = - 4.26 years (minus 4.26 years).

This means the time measurement using the clock at Proxima Centauri shows the value 0 when light reaches the Earth (4.26 years after the event's occurrence).

You have reached the end of Physics lesson 18.3.2 Inertial System in the Special Theory of Relativity and Synchronization of Clocks. There are 3 lessons in this physics tutorial covering Space and Time in Einstein Theory of Relativity, you can access all the lessons from this tutorial below.

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