4
Special frames
The definition and interpretation of a physical measurement are better achieved
if reference is made to a system of Cartesian axes which matches an instantaneous
inertial frame. With respect to such a frame, a measurement is expressed in terms
of the projection on its axes of the tensors or tensor equations characterizing the
phenomenon under investigation. Two different Cartesian frames are connected
by a general Lorentz transformation; hence the properties of the Lorentz group
play a key role in the theory of measurements. In Newtonian mechanics the local
rest spaces of all observers coincide as a result of the absoluteness of time but in
relativistic mechanics the local rest spaces do not coincide, because of relativity of
time; hence a comparison between any two quantities which belong to the local
rest spaces of different observers requires a non-trivial mapping among them.
This is what we are going to discuss now.
4.1 Orthonormal frames
Let u be the 4-velocity of an observer; at any point of his world line let us choose
in LRS
u
three mutually orthogonal unit space-like vectors e(u)
ˆa
with ˆa =1, 2, 3.
In what follows these vectors will be simply denoted by e
ˆa
whenever it is not
necessary to specify the time-like vector u which is orthogonal to them. They
satisfy the relation
e
ˆa
· e
ˆ
b
= δ
ˆa
ˆ
b
. (4.1)
It is clear that the triad {e
ˆa
}
ˆa=1, 2, 3
is an orthonormal basis for LRS
u
.
If we write u ≡ e
ˆ
0
, then the set {e
ˆα
} with ˆα =0, 1, 2, 3 itself forms an orthonor-
mal basis for the local tangent space and is termed a tetrad (Pirani, 1956b). The
vectors e
ˆα
satisfy the condition
e
ˆα
· e
ˆ
β
= η
ˆα
ˆ
β
≡ diag [−1, 1, 1, 1], (4.2)