Invariance of Space-time intervals

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One useful property of space-time intervals is that they are not changing; they are invariant under Lorentz transformation.  In the IB DP Physics, you need to know this, but not how to derive it.  But the exercises in the end are things you are expected to be able to do.

We will look at the one-dimensional case, but it is easy but tedious to extend it to the three-dimensional case. We have

\displaystyle {{\left( {\Delta s} \right)}^{2}}={{\left( {c\Delta t} \right)}^{2}}-{{\left( {\Delta x} \right)}^{2}}

We shall now show that this is invariant under Lorentz transformation.  To not need too many brackets, we will use s, x, and t for the corresponding delta-expressions. We can just assume we start at (0, 0). In other words, we start with

\displaystyle {{s}^{2}}=c{{t}^{2}}-{{x}^{2}}

We can then substitute the Lorentz transformation expressions into the RHS of the above expression, but for S‘.

\displaystyle {{{{s}'}}^{2}}={{\left( {ct\prime } \right)}^{2}}-{{{{x}'}}^{2}}={{\left( {c\gamma \left( {t-\frac{{vx}}{{{{c}^{2}}}}} \right)} \right)}^{2}}-{{\left( {\gamma \left( {x-vt} \right)} \right)}^{2}}

\displaystyle ={{\gamma }^{2}}\left( {{{{\left( {c\left( {t-\frac{{vx}}{{{{c}^{2}}}}} \right)} \right)}}^{2}}-{{{\left( {x-vt} \right)}}^{2}}} \right)

\displaystyle ={{\gamma }^{2}}\left( {{{c}^{2}}\left( {{{t}^{2}}-2t\frac{{vx}}{{{{c}^{2}}}}+{{{\left( {\frac{{vx}}{{{{c}^{2}}}}} \right)}}^{2}}} \right)-\left( {{{x}^{2}}-2xvt+{{v}^{2}}{{t}^{2}}} \right)} \right)

\displaystyle ={{\gamma }^{2}}\left( {{{c}^{2}}{{t}^{2}}-2tvx+\frac{{{{v}^{2}}{{x}^{2}}}}{{{{c}^{2}}}}-{{x}^{2}}+2xvt-{{v}^{2}}{{t}^{2}}} \right)

But c=x/t, so t=x/c. This gives us

\displaystyle ={{\gamma }^{2}}\left( {{{c}^{2}}{{t}^{2}}\left( {1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}} \right)-{{x}^{2}}\left( {1-\frac{{{{v}^{2}}}}{{{{c}^{2}}}}} \right)} \right)

\displaystyle ={{(ct)}^{2}}-{{x}^{2}}={{s}^{2}}

  1. Two frames of reference, S and S´, have synchronised their clocks as their origin pass each other. In frame S, one event happens 155000 km from the origin at the time 1.50 s. In the other frame, it happens at the time 1.40 s. How far from the origin of S´ does it happen?
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