316 6 Special Theory of Relativity
Inverse transformations
The inverse transformations (6.12), (6.13), (6.14), and (6.15) are immediately writ-
ten with the aid of inverse matrix
Λ−^1 =Λ ̃
Λ−^1 =
⎡
⎢
⎢
⎣
γ 00 −iβγ
010 0
001 0
iβγ 00 γ⎤
⎥
⎥
⎦ (6.23)
Four vectors
Ifs=√
Σμxμ^2 =invariant,μ= 1 , 2 , 3 , 4 (6.24)under Lorentz transformation, thens is said to be a four vector,s can be
positive or negative or zero. Examples of Four vectors are
X=(x 1 ,x 2 ,x 3 ,ict) (6.25)
cP=(cPx,cPy,cPz,iE) (6.26)In (6.25) the first three space components ofX define the ordinary three-
dimensional position vectorxand the fourth, a time componentict. The four-
momentum in (6.26) has the first three components of ordinary momentum and
E is the total energy of the particle. The four-vectors have the properties which are
similar to those of ordinary vectors. Thus, the scalar product of two four-vectors,
A.B=A 1 B 1 +A 2 B 2 +A 3 B 3 +A 4 B 4
Consequences of Lorentz transformations
Time Dilation
Δt′=γΔt (6.27)
Rule: Every clock appears to go at its fastest rate when it is at rest relative to
the observer. If it moves relative to the observer with velocityν, its rates appears
slowed down by the factor
√
1 −ν^2 /c^2. No distinction need be made between the
stationary and moving frame. Each observer will think that the other observer’s
clock has slowed down. What matters is the only the relative motion.
The Lorentz contraction
l′=l√
1 −β^2 (6.28)