here - Department of Physics, HKU

Chapter 3
Lorentz Transformations
3.1 Lorentz Transformation
One event will have different coordinates on different coordinate systems.
In Newtonian mechanics, if there is a coordinate system moving to the +xdirection
relative to another coordinate, the Galilean transformation between
the two are
x ′ = x − vt (3.1)
t ′ = t (3.2)
where v is the velocity and we have chosen the coordinate systems that their
origins coincide, (x ′ , t ′ ) = (0, 0) if and only if (x, t) = (0, 0). Let emphasize
again that these are the coordinates of the same event, for example, a bomb
explodes at (x, t) relative to a stationary observer and at (x ′ , t ′ ) relative to a
coordinate system fixed with a train moving to the right with speed v.
The coordinate transformation in special relativity is called the Lorentz
Transformation. We have mentioned that the transverse directions do not
change, we will only focus on the direction along the velocity. First, because
two rulers should be two times the length of one ruler in any coordinate
systems, the transformation between the coordinates must be linear,
x ′ = fx + gt (3.3)
t ′ = hx + kt (3.4)
where f, g, h and k are functions of the velocity v. Second, the origin of the
moving system must have coordinates x = vt and x ′ = 0, hence, by Eq. (3.3),
for all t,
0 = fvt + gt
g = −vf . (3.5)
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