Lorentz transformations Lorentz transformation
Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that 



(

x
′

,
c

t
′

)


{\displaystyle (x',ct')}

 coordinates of 



(
0
,
1
)


{\displaystyle (0,1)}

 in the primed coordinate system transform to 



(
β
γ
,
γ
)


{\displaystyle (\beta \gamma ,\gamma )}

 in the unprimed coordinate system. Likewise, 



(

x
′

,
c

t
′

)


{\displaystyle (x',ct')}

 coordinates of 



(
1
,
0
)


{\displaystyle (1,0)}

 in the primed coordinate system transform to 



(
γ
,
β
γ
)


{\displaystyle (\gamma ,\beta \gamma )}

 in the unprimed system. Draw gridlines parallel with the 



c

t
′



{\displaystyle ct'}

 axis through points 



(
k
γ
,
k
β
γ
)


{\displaystyle (k\gamma ,k\beta \gamma )}

 as measured in the unprimed frame, where 



k


{\displaystyle k}

 is an integer. Likewise, draw gridlines parallel with the 




x
′



{\displaystyle x'}

 axis through 



(
k
β
γ
,
k
γ
)


{\displaystyle (k\beta \gamma ,k\gamma )}

 as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between 



c

t
′



{\displaystyle ct'}

 units equals 





(
1
+

β

2


)

/

(
1
−

β

2


)




{\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}}

 times the spacing between 



c
t


{\displaystyle ct}

 units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as 



β
→
1.


{\displaystyle \beta \to 1.}


