Elements of Modern Physics
14
The statement in Eq. (1.40) is the reappearance of the assumption that the
speed of light is the same in all inertial frames of reference. Eq. (1.41) has been
considered for particles with speeds greater than the speed of light, known as
tachyons. It is interesting to note that for tachyons, the equation u x v = c
2
, is
possible in which case u′ tends to infinity. Observation of tachyons would be of
great interest since it would imply that information can be transmitted at a speed
greater than the speed of light. However, so far, tachyons have not been observed
experimentally.
1.9 LORENTZ FOUR-VECTORS
The postulates of the special theory of relativity require that the physical laws
retain the same form under Lorentz transformations. This requirement is satisfied
if the laws are stated as equalities between terms which have similar
transformation properties. For ordinary three-dimensional transformations, these
terms are three-dimensional vectors or their generalizations. However, the
Lorentz transformations mix the space and time coordinates so that we need to
generalize our ideas to four-dimensional vectors.
A relativistic four-vector may be defined to be a set of four quantities
Aµ (µ = 1, 2, 3, 0) ≡ (A x , A y , A z , A 0 ) which transform like the space-time
coordinates x µ = (x, y, z, ct) under Lorentz transformations (the factor c is
included to give the same dimension to all the components), i.e.,
0
1/ 2
2
2
1
1
′ =
−
−
x
x
v
A
A
A
c
v
c
(1.42)
A y ′ = A y
A z ′ = A z
0
0
1/ 2
2
2
1
1
′ =
−
−
x
v
A
A
A
c
v
c
The scalar product between two Lorentz vectors, A µ = (A, A 0 ) and
B µ = (B, B 0 ) can be defined as
A⋅B ≡ A 0 B 0 – A⋅B
(1.43)
which can easily be shown to be equal to A′·B′, and hence is called a Lorentz
scalar. In particular, the ‘length’ of a vector is defined (A·A)
1/2
which has the
same value in all inertial frames:
(A⋅A)
1/2
= (A 0 2 – A⋅A)
1/2
(1.44)
14
The statement in Eq. (1.40) is the reappearance of the assumption that the
speed of light is the same in all inertial frames of reference. Eq. (1.41) has been
considered for particles with speeds greater than the speed of light, known as
tachyons. It is interesting to note that for tachyons, the equation u x v = c
2
, is
possible in which case u′ tends to infinity. Observation of tachyons would be of
great interest since it would imply that information can be transmitted at a speed
greater than the speed of light. However, so far, tachyons have not been observed
experimentally.
1.9 LORENTZ FOUR-VECTORS
The postulates of the special theory of relativity require that the physical laws
retain the same form under Lorentz transformations. This requirement is satisfied
if the laws are stated as equalities between terms which have similar
transformation properties. For ordinary three-dimensional transformations, these
terms are three-dimensional vectors or their generalizations. However, the
Lorentz transformations mix the space and time coordinates so that we need to
generalize our ideas to four-dimensional vectors.
A relativistic four-vector may be defined to be a set of four quantities
Aµ (µ = 1, 2, 3, 0) ≡ (A x , A y , A z , A 0 ) which transform like the space-time
coordinates x µ = (x, y, z, ct) under Lorentz transformations (the factor c is
included to give the same dimension to all the components), i.e.,
0
1/ 2
2
2
1
1
′ =
−
−
x
x
v
A
A
A
c
v
c
(1.42)
A y ′ = A y
A z ′ = A z
0
0
1/ 2
2
2
1
1
′ =
−
−
x
v
A
A
A
c
v
c
The scalar product between two Lorentz vectors, A µ = (A, A 0 ) and
B µ = (B, B 0 ) can be defined as
A⋅B ≡ A 0 B 0 – A⋅B
(1.43)
which can easily be shown to be equal to A′·B′, and hence is called a Lorentz
scalar. In particular, the ‘length’ of a vector is defined (A·A)
1/2
which has the
same value in all inertial frames:
(A⋅A)
1/2
= (A 0 2 – A⋅A)
1/2
(1.44)