, where
is the
Kronecker delta,
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
In special relativity, spacetime is described by the Minkowski metric. Here,
it is described by a four-dimensional Euclidean metric,
, where
is the
Kronecker delta,
We introduce now the concept of events in spacetime. These are the analogs
in four-dimensional spacetime of points in three-dimensional space. An event
is something that occurs at a specific place and at a specific time in a
particular reference frame. We represent an event in spacetime by
, where
and
are the
coordinates of the event. If we have a second event
in the
same reference frame, the magnitude of the spacetime separation between the
two events is called the spacetime interval,
, which
is defined as
where 
is
the speed of light. This is, simply, the extension of the Pythagorean theorem
to four dimensions. We have included
to make
the units consistent throughout. It is required that the spacetime interval
be invariant for any inertial observer. That is, any observer moving with
uniform four-velocity will measure the same spacetime interval between the
events. An observer in a primed reference frame might measure the same two
events at 
and 
in his frame. Using the expression for the square of the element of spacetime
interval, 
, we have
We can write the right-hand side of (3.3) in a more general way by using the Einstein summation convention as
where
and similarly for the left-hand side of (3.3). Subscripts are used, exclusively,
to conform with the convention that all indices appear as subscripts in Euclidean
space. The greek subscripts,
and
will
always range from 1 to 4 unless otherwise noted. In comparing measurements,
we will frequently refer to primed and unprimed frames of reference. It will
always be the case that the primed frame (indicated by a prime over the quantity,
for example,
) is in
uniform motion relative to the unprimed frame (indicated by the absence of
a prime over the quantity, for example,
).
Every vectorial quantity in spacetime will be represented by a four-dimensional
vector, or four-vector. We represent an arbitrary four-vector,
, in the
form 
.
The magnitude,
, of
is determined
in the same manner as the magnitude of the spacetime interval
Copyright © 1999-2000 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net