1000 Solved Problems in Modern Physics

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 of X 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)