Special theory of relativity  (Page 8/9)

$$x\prime =\gamma \left(x-vt\right)$$

Lorentz factor appears in most of the relativistic equations including the calculation of relativistic effects like time dilation, length contraction, mass etc. An understanding of the beahviour of this factor at different relative velocity is intuitive for assessing the extent of relativistic effect. Few values of Lorentz factor are tabulated here.

Lorentz factor begins at 1 and as v->c, y->infinity. It is either equal to 1 or greater than 1. In other words, it is never less than 1. A plot of Lorentz factor .vs. relative speed is shown here.

Lorentz factor .vs. speed plot

Space-time interpretation

We identify an event with spatial (x,y,z) and temporal (t) coordinates. Important point is that an event does not belong to any reference. It is described by different coordinates in different reference system. In classical description, spatial and temporal parameters are essentially independent of each other. The time t of an event can not be dependent on spatial specification (x,y,z,). Now, this independence is not there in relativistic kinematics. In order to imbibe the nature of space time relation, we shall work with few Lorentz transformations here.

We interpret an event in two inertial references which are moving with respect to each other at a velocity say 0.3c in x-direction. We shall consider very small time interval like 0.000005 second so that distance involved is easy to visualize. For convenience, we consider the approximate value of speed of light 300000000 m/s. In time 0.000005 s, the separation of two reference frame at the speed 0.3c works out to be 0.3 X 300000000 X 0.000005 = 450 m.

Lorentz transformation

Here, we calculate both Galilean and Lorentz distance and time of events in two references for events identified in first reference by x and t values. Unprimed values refer to stationary reference, whereas primed values refer to moving reference which is moving right in x-direction with a relative velocity 0.3c. The calculations have been done using Excel worksheet (Reader can also try and verify the results) where distance is in meters and time in seconds.

Since the origins of two references coincide for both Galilean and Lorentz transformations at t= t’=0, the space and time values are all zero as shown in the first row of the table.

Let us now consider the second row of the table. Here, position of event is x=2 m at time, t = 0.000005 s. In this time, primed reference has moved 450 m. According to Galilean transform, the event takes place at -450+2 = -448 m (to the left of origin) in the moving reference. Since time is invariant in Galilean transformation, the time of event is same in moving reference for non-relativistic Galilean transformation. However, when we employ relativistic Lorentz transformation, the event occurs at -469.6317 m (to the left of origin) in the moving reference. Here, the measurement of distance in moving reference is different than that calculated with Galilean transformation. Also, time is not invariant. The event occurs at 0.0000052393 s in this reference instead of 0.000005 s in the unprimed stationary reference. Thus, we see that both space and time are not invariant in Lorentz transformation.