# In-Class Activities

By

Dr Lim Zhi Han

,

Koh Zhi Yong

,

Gerald Kang Joon Kiat

●

Published 2022-09-25

In the low speed limit where v\ll c, \beta\approx0 and \gamma\approx1. What do the time dilation (Eq TD (3)), length contraction (Eq LC (5)) and Lorentz transformation equations (Eq LT (1)) become? Does this make intuitive sense?

Let us visualise how the space-time coordinates of two inertia observers in uniform relative motion are related to each other in a space-time diagram. For simplicity, we consider two inertia frames (S and S') where the relative motion between them occurs only along the x-axis. Specifically, suppose S' moves with a constant speed v in the +x-direction relative to S.

On the grid below, draw a vertical axis and label it ct. Draw a horizontal axis and label it x. This set of axes represent the inertia frame S. With this construction, a space-time event from the perspective of an observer in S is simply a point located in the plane of the two axes. The coordinate position and coordinate time (ct,x) can be read off the axes directly. Notice that position is represented on the horizontal axis while time is represented on the vertical axis. In relativity, it is conventional to represent time on the vertical axis. Moreover, the space-time diagram gives both axes the same units by scaling the vertical axis by the speed of light, c.

Next, we represent the space-time diagram, (ct',x'), for frame S' on the same grid. To draw the ct' axis, we examine various coordinate points where x'=0. First, using Lorentz transformation LT (1), show that x=\gamma\beta and ct=\gamma when {x'=0,ct'=1}.

Repeat the previous step to find the corresponding x and ct coordinates for {x'=0,ct'=2} and so on. Plot the points on the (ct,x) graph for \beta=0.6. Draw a line through these points and label it as ct'.

To draw the x' axis, find the corresponding x and ct coordinates for {x'=1,ct'=0}, {x'=2,ct'=0} and so on. Plot the points on the (ct,x) graph for \beta=0.6. Draw a line through these points and label it as x'.

Suppose an observer in S observes an event occurring at {x=3\gamma,ct=4\gamma}. With the help of your space-time diagrams, what are the corresponding ct' and x' coordinates for the same event from the perspective of an observer in S' when \beta=0.6?

How will the ct' and x' axes change for higher or lower values of \beta ?

What happens if \beta=1?

Use the Lorentz transformation equations (Eq LT 1) to show the invariance of the space-time interval between two events at (t,x,y,z) and (t+dt,x+dx,y+dy,z+dz) when transforming from one inertia frame S to another inertia frame S', i.e. ds=ds'.

From Lorentz transformation (Eq LT 1), we have

\begin{align}
cdt'=\gamma(cdt-\beta dx) \nonumber \\
dx'=\gamma\left(dx-\beta cdt\right), \nonumber 
\end{align}

where \gamma=\frac{1}{\sqrt{1-\beta^{2}}} and \beta=\frac{v}{c}.

Consider inertia frame S' which moves at a constant speed (comparable to the speed of light) relative to inertia frame S. On the same Minkowski space-time diagram, sketch a pair of (ct',x') axes and (ct,x) axes for frames S' and S respectively. Mark out 2 events on (ct',x') axes that are simultaneous in frame S'.

Show that two events that are simultaneous in one inertia frame is not simultaneous in another inertia frame.