# 5.5 The lorentz transformation

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• Describe the Galilean transformation of classical mechanics, relating the position, time, velocities, and accelerations measured in different inertial frames
• Derive the corresponding Lorentz transformation equations, which, in contrast to the Galilean transformation, are consistent with special relativity
• Explain the Lorentz transformation and many of the features of relativity in terms of four-dimensional space-time

We have used the postulates of relativity to examine, in particular examples, how observers in different frames of reference measure different values for lengths and the time intervals. We can gain further insight into how the postulates of relativity change the Newtonian view of time and space by examining the transformation equations that give the space and time coordinates of events in one inertial reference frame in terms of those in another. We first examine how position and time coordinates transform between inertial frames according to the view in Newtonian physics. Then we examine how this has to be changed to agree with the postulates of relativity. Finally, we examine the resulting Lorentz transformation equations and some of their consequences in terms of four-dimensional space-time diagrams, to support the view that the consequences of special relativity result from the properties of time and space itself, rather than electromagnetism.

## The galilean transformation equations

An event    is specified by its location and time ( x , y , z , t ) relative to one particular inertial frame of reference S . As an example, ( x , y , z , t ) could denote the position of a particle at time t , and we could be looking at these positions for many different times to follow the motion of the particle. Suppose a second frame of reference $S\prime$ moves with velocity v with respect to the first. For simplicity, assume this relative velocity is along the x -axis. The relation between the time and coordinates in the two frames of reference is then

$x=x\prime +vt,\phantom{\rule{0.5em}{0ex}}y=y\prime ,\phantom{\rule{0.5em}{0ex}}z=z\prime \text{.}$

Implicit in these equations is the assumption that time measurements made by observers in both S and $S\prime$ are the same. That is,

$t=t\prime \text{.}$

These four equations are known collectively as the Galilean transformation    .

We can obtain the Galilean velocity and acceleration transformation equations by differentiating these equations with respect to time. We use u for the velocity of a particle throughout this chapter to distinguish it from v , the relative velocity of two reference frames. Note that, for the Galilean transformation, the increment of time used in differentiating to calculate the particle velocity is the same in both frames, $dt=dt\prime .$ Differentiation yields

${u}_{x}={u}_{x}^{\prime }+v,\phantom{\rule{0.5em}{0ex}}{u}_{y}={u}_{y}^{\prime },\phantom{\rule{0.5em}{0ex}}{u}_{z}={u}_{z}^{\prime }$

and

${a}_{x}={a}_{x}^{\prime },\phantom{\rule{0.5em}{0ex}}{a}_{y}={a}_{y}^{\prime },\phantom{\rule{0.5em}{0ex}}{a}_{z}={a}_{z}^{\prime }.$

We denote the velocity of the particle by u rather than v to avoid confusion with the velocity v of one frame of reference with respect to the other. Velocities in each frame differ by the velocity that one frame has as seen from the other frame. Observers in both frames of reference measure the same value of the acceleration. Because the mass is unchanged by the transformation, and distances between points are uncharged, observers in both frames see the same forces $F=ma$ acting between objects and the same form of Newton’s second and third laws in all inertial frames. The laws of mechanics are consistent with the first postulate of relativity.

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