Although the use of 4vectors is not necessary for a full understanding of Special Relativity, they are a
most powerful and useful tool for attacking many problems. A 4vectors is
just a 4tuplet A = (A_{0}, A_{1}, A_{2}, A_{3}) that transforms under a Lorentz
Transformation in the
same way as (cdt, dx, dy, dz) does. That is:
A_{0} = γ(A_{0}' + (v/c)A_{1}')   

A_{1} = γ(A_{1}' + (v/c)A_{0}')   

A_{2} = A_{2}'   

A_{3} = A_{3}'   

As we saw in the
minkowski diagrams, Lorentz transformations
are very much like rotations in 4dimensional spacetime. 4vectors, then, generalize the concept of rotations
in 3space to rotations in 4dimensions. Clearly, any constant multiple of
(cdt, dx, dy, dz) is a 4vector,
but something like
A = (cdt, mdx, dy, dz) (where
m is just a constant) is not a 4vector because
the second component has to transform like
mdxâÉáA_{1} = γ(A_{1}' + (v/c)A_{0}')âÉáγ((mdx') + vdt') from the definition of a 4vector, but also like
mdx = mγ(dx' + (v/c)dt'); these
two expression are inconsistent. Thus we can transform a 4vector either according to the 4
vector definition given above, or using what we know about how the
dx_{i} transform to transform each
A_{i} independently. There are only a few special vectors for which these two methods yield the same
result. Several different 4vectors are now discussed:
Velocity 4vector
We can define a quantity τ = which is called the proper time, and
is invariant between frames. Dividing out original 4vector ((cdt, dx, dx, dz)) by dτ gives:
V = (cdt, dx, dy, dz) = γc,,, = (γc, γ 

This arises because
= γ.
Energymomentum 4vector
If we multiply the velocity 4vector by m we get:
P = mV = m(γc, γ 

This is an extremely important 4vector in Special Relativity.
Properties of the 4vector
What gives 4vectors their usefulness in Special Relativity is their many nice properties. First, they are
linear: if A and B are 4vectors and a and b are any constants, then C = aA + bB is also a
4vector. Even more importantly, 4vectors have inner product invariance.
We define the inner product of two 4vectors A and B to be:
A.BâÉáA_{0}B_{0}  A_{1}B_{1}  A_{2}B_{2}  A_{3}B_{3}âÉáA_{0}B_{0}  

It is not hard to verify by direct computation that this inner product is the same
no matter which
frame it is calculated . This is a crucial result. Just as the usual dot product is invariant under
rotations in 3dimensions, the inner product defined here is invariant under rotations in our 4space. The
unusual minus signs arise because of the form of the Lorentz Transformations; this is just the
way the math comes out in order for the inner product of two 4vectors to be invariant under the Lorentz Transformations. We can also use this inner product to define the norm, or length, of a 4vector as:
 A^{2}âÉáA.A = A_{0}A_{0}  A_{1}A_{1}  A_{2}A_{2}  A_{3}A_{3} = A_{0}^{2}   bfA^{2} 

We can now begin to see the usefulness of 4vectors: they can, given an arbitrary combination of 4vectors, we can immediately produce a quantity that is independent of reference frame, enabling us to
draw immediate conclusions about what is going on in the particular frame we are interested in. One
example is that if we take the combination
P.P, the inner product of the momentum 4vector with
itself we have
P.P = E^{2}/c^{2}  , which we know must be invariant. However, it is not obvious what constant value this is. But the invariance of the
4vector allows us to choose
any frame; we can choose the one where
. Here the
inner product becomes
P.P = E^{2}/c^{2}. But for a particle at rest we know
E = mc^{2}, thus
E^{2}/c^{2} = m^{2}c^{2} and hence
P.P = E^{2}  c^{2} in every frame. Thus we have
derived the same relationship between momentum and energy that we saw in Section 1, this
time by using inner product invariance.