Relativity Series Part 2 - Lorentz Contraction and Time Dilation

Where we left off in the previous part, we discussed Lorentz transformations. We will now use these transformations to explain Lorentz contraction and time dilation.

Lorentz transformations give rise to several new phenomena. Suppose a rod of length $l$ is at rest in an inertial frame $S$. Its length will be measured differently in another inertial reference frame $S'$ moving at velocity $v$. This difference in length is called Lorentz Contraction.

Lorentz Contraction

The coordinates of the endpoints of this rod, as measured in frame $S$, are given by $A(x_A,0,0)$ and $B(x_B,0,0)$. Thus, in frame $S$, the length of the rod is measured as $l=x_B-x_A$.

$$l=\frac{l'}{\sqrt{1-\frac{v^2}{c^2}}}$$

We have therefore shown that it is given by this expression. As can be seen here, the length measured in the moving reference frame is smaller than the length measured in the rest frame.

Time Dilation

Another consequence of Lorentz transformations is the concept of Time Dilation. Let us consider a clock located at the coordinates $(x,0,0)$ in the rest frame $S$.

$$t'_2-t'_1=\frac{(t_2-t_1)}{\sqrt{1-v^2/c^2}}$$

is measured. As can be seen from this:

$$t'_2-t'_1 \ge t_2-t_1$$

Thus, for the moving reference frame, the clock operating in the rest frame runs more slowly. This phenomenon is called time dilation.

Velocity and Acceleration Transformations

Unlike in Newtonian mechanics, when we wish to calculate relative velocities in special relativity, we cannot obtain the result simply by adding the velocities.

$$u'_x=\frac{u_x-v}{1-(vu_x/c^2)}$$ $$u'_y=\frac{u_y}{\gamma(1-(vu_x/c^2))}$$ $$u'_z=\frac{u_z}{\gamma(1-(vu_x/c^2))}$$

If the $S'$ frame were moving at the speed of light $c$, and the moving particle were also travelling at speed $c$ relative to the $S'$ frame:

$$u_x=\frac{c+c}{1+(cc)/c^2}=\frac{2c}{2}=c!$$

We have thus arrived at a result consistent with Einstein's postulates.

Matrix Representation of Lorentz Transformations

A Lorentz transformation is a linear coordinate transformation and can be represented by a 4x4 matrix.

$$L_v= \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0\\ -\gamma \beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$

That brings this week's topic to a close. In the next part, we will look at spacetime diagrams. Until then, see you, and keep reading us :)