Space-Time Interval

The conclusion on existence of invariants we can make from previous information is not optimistic: both the space and time coordinates are different in different frames of reference.

It is desirable to find the physical quantity that is the same in different frames. And such a quantity exists: it is the space-time interval,

.

In system the space-time interval is

.

Using Lorenz’s transformations one can easily prove that

,

i.e. the space-time interval is invariant relative to Lorenz’s transformations.

Fig. 2.6

The combination of space and time coordinates in the form of space-time interval is not changed in different frames of reference although space coordinates themselves are changed and time is changed too.

We can draw a very important conclusion: space and time characteristics do not exist separately but in connection. The idea of four-dimensional space is based on this conclusion.

That is each event which takes place in a given moment of time in given point is to be described by space point in four-dimensional space. This point has three space coordinates and one time coordinate. Space point moves in this four-dimensional space along space lines.

Figure 2.6, as an example, demonstrates space lines for immovable material point, for material point moving at constant velocity and for material point moving with constant acceleration.

Дата добавления: 2015-11-16