, where
is the Kronecker
delta,
Copyright © 1999 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 interval between the two events is called the spacetime
interval, 
. The square of the spacetime interval between these two events in spacetime
is then
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 element of interval,
, we have
To indicate a sum, we will use the Einstein summation convention which states,
and,
for a double sum,
. This
means that whenever an index is repeated in a given term, we are to sum over
that index from 1 to m or 1 to n. The greek subscripts (indices), will range
from 1 to 4 unless otherwise noted. This leads to the general formula for
the element of spacetime
interval
where
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, i.e.
) is in
uniform motion relative to the unprimed frame (indicated by the absence of
a prime over the quantity, i.e.
). We can
simplify things, considerably, by assuming that the first event takes place
at the origin of both reference frames, thus referring to it as
and
in the
two frames, then referring to the second event, simply, as
and
in the
two frames. In this case, the expression for the invariant spacetime interval
becomes
Copyright © 1999 David E. Rutherford
All Rights Reserved
E-mail: drutherford@softcom.net