Special Relativity

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Lorentz Transformations

Suppose

we have two frames of reference \(S\) with space-time coordinates in

that frame given by \((t,x,y,z)\) and another frame of reference,

\(S'\), that is moving with a constant velocity \(v\) relative to \(S\)

along the \(x\)-axis. Also suppose that the axes of these frames and

their clocks coincide at time \(t=0\) when the origins \(O\) and \(O'\)

are coincident.

Given that the coordinates in frame \(S'\) are \((t',x',y',z')\), then

the Lorentz transformation specifies that the coordinates in the two

frames are related as follows:

\[ t' = \gamma \left(t - \frac{vx}{c^2} \right) \]

\[ x' = \gamma \left( x-vt \right) \]

\[y' = y \]

\[ z' = z \]

where

\[ \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \]

and \(c\) is the speed of light.

When you sit on a hot stove for two minutes, it feels like two hours.

But, when you sit with a nice girl for two hours, it feels like two

minutes. That's relativity!

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