Single-Particle Electrodynamics - Assassination Science

The use of this signature of metric means that raising or lowering a time
component index of a quantity does not change the value of the quantity; on
the other hand, it does mean that the position of placement of Latin indices
on the symbols of Lorentz-covariant quantities is important. However, the
position of placement of indices on non-Lorentz-covariant three-vectors is
defined to be in all cases irrelevant, just as it is for mathematically non-
Lorentz-covariant three-vectors (see Section A.9). For example, with x α the
symbol for the four-position, x i and x i are different (indeed, x i ≡ −x i ); on
the other hand v i ≡ d t x i requires no distinction, as it is a non-covariant
quantity; it may, if convenient, be denoted v i , but in all cases v i ≡ v i . This
implicit convention is of use, conceptually, when one side of an equation or
definition involves (implicitly or explicitly) a non-covariant three-vector, and
the other the spatial part of a covariant quantity (see, e.g., the definition of v
below). In practice, however, subscripts are generally used on non-covariant
quantities in explicit expressions.
A.8.13
Timelike, spacelike, lightlike
If A is a four-vector, then it is timelike if A 2 > 0; it is spacelike if A 2 < 0; it
is lightlike if A 2 = 0.
If, in some particular Lorentz frame, A = 0, then A is purely timelike in
that frame; if A 0 = 0, then A is purely spacelike in that frame.
A.8.14
Four-position of a particle
The Lorentz-covariant four-position of a classical particle is denoted z α .
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