In the previous handout we encountered the notion of
4-vectors, the prototype of which is the space-time
vector, 
,
where the ``zeroth component'' 
is time
multiplied by the speed of light: 
and
the remaining three components are the three ordinary
spatial coordinates. [The notation is new but the idea is the same.]
In general we assume that a vector with Greek indices
(like 
) is a 4-vector, while a vector with
Roman indices (like 
) is an ordinary spatial 3-vector.
It can be shown
that the inner or scalar product of any two
4-vectors has the agreeable property of being
a Lorentz invariant - i.e. it is unchanged by
a Lorentz transformation - i.e. it has the
same value for all observers. This comes in
very handy in the confusing world of Relativity!
We write the scalar product of two 4-vectors
as follows:
where the first equivalence expresses the Einstein summation convention - we automatically sum over repeated indices. Note the - sign! It is part of the definition of the ``metric'' of space and time, just like the Pythagorean theorem defines the ``metric'' of flat 3-space in Euclidean geometry.
Our first Lorentz invariant was the proper time
of an event, which is just the square root
of the scalar product of the space-time 4-vector
with itself:
We now encounter our second 4-vector, and probably the last one we need for our purposes here; the energy-momentum 4-vector,
where 
is the total relativistic energy
and 
is the usual momentum 3-vector
of some object in whose kinematics we are interested.
[Check for yourself that all the components of this vector
have the same units, as required.]
If we take the scalar product of 
with itself,
we get a new Lorentz invariant:
where 
is the square
of the magnitude of the ordinary 3-vector momentum.
It turns out
that the constant value of this particular Lorentz invariant
is just the 
times the square of the rest mass
of the object whose momentum we are scrutinizing:
or 
.
As a result, we can write
which is a very useful formula relating the energy E, the rest mass m and the momentum p of a relativistic body.
Although there are lots of other Lorentz invariants we can define by taking the scalar products of 4-vectors, these two will suffice for my purposes; you may forget this derivation entirely if you so choose, but I will need Eq. (E=mc2.14) for future reference.
