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(This is part 2 of a lengthy post on four-vectors.)

                       IV.  Physics

All this becomes useful in relativity when you start expressing
physical quantities as four-vectors.  One such quantity is
the energy-momentum four-vector associated with an object: its
time component is the total energy, and its spatial components
are the components of momentum.  These are all in the same
units if we set c = 1.  The vector's invariant squared
"length,"

    a                 2    2         2
   P  P       =      E -|p|      =  m
       a                                 ,

is mass squared, which does not change under Lorentz transformations.
E and p do, though, since they're just components of the vector.
In the rest frame, though, p = 0 and E = m, so m is also
the rest energy, E = mc^2, if we don't set c = 1.
                  0

For a photon, m = 0, so the vector is a "null vector," one with
zero length; but because of the minus signs in the metric, that
doesn't mean that the components are all zero, just that E = |p|.
In the units we're used to, that relation becomes E = pc.

The coordinates of events in spacetime can also be represented
by four-vectors, and this is where kinematics comes in.  In
this case the time component is the time of the event and the
spatial components give the position, in some reference frame.
The same rules used to transform the energy-momentum vector
now give the prescription for finding the position and time
of the event in any frame.

Subtracting the coordinates of two ordinary vectors in
three-space gives the displacement of one position from
another, and the length of the vector is the distance
between the points.  Likewise, if a vector D = (t,x,y,z)
is the difference between two events' coordinates, we can
define an invariant quantity

   a              2    2    2    2
  D  D     =     D  - D  - D  - D
      a           t    x    y    z  ,

which can be positive, negative, or zero!  If it's positive,
then its square root is the proper time experienced by an
object traveling at constant speed from one event to the
other.  If it's negative, then the events are too far apart
to travel from one to the other, and the square root of minus
this quantity is sometimes called "proper distance."  If it's
zero, then the path from one to the other is called "lightlike,"
and a light beam traveling at c can get from one event to the
other, experiencing zero proper time.

If a four-vector has zero length in one frame, it will
have zero length in *all* frames, since length is invariant
under Lorentz transformations.  This is the expression in
the four-vector formalism of the postulate that the speed
of light is the same for all observers.

Intervals with positive squared lengths (and real proper
times) are called "timelike," and intervals with
negative squared lengths (and real proper distances) are
called "spacelike."  Information can only travel along
timelike or lightlike intervals.  The proper time experienced
by a high-velocity traveler (or, of more experimental
importance, a high-velocity subatomic particle) who
accelerates in arbitrary complicated ways may be
found by integrating up the proper times for short
segments of the path.

Kinematics is properly handled by taking derivatives with
respect to *proper* time, at least for things traveling at
less than c.  The velocity four-vector V is the derivative
of the displacement vector with respect to proper time.  It
always has an invariant length of 1.  Its spatial 
components are not the three-dimensional velocity, but the
velocity times the time-dilation factor gamma; the time
component is gamma.  We then have a very nice-looking
relation between four-vectors

       P = mV ,

where P is the four-momentum, V is the four-velocity, and
m is the invariant, unchanging mass.  The velocity four-
vector becomes ill-defined for things traveling at the
speed of light, along intervals that always have a 
length of zero; this is consistent with the fact that
for these objects m = 0, yet P is still nonzero.
 
Four-acceleration A is the derivative of four-velocity with
respect to time.  We can define a quantity called the
four-force, under which definition

      F = mA.

This is generally easier to work with than three-dimensional
forces, especially when handling electromagnetism.  Electro-
magnetism involves more complicated quantities, like four-
dimensional tensors.  I should mention, though, that the electric
potential and magnetic vector potential combine quite nicely
(in the proper units) to form a four-vector, which transforms
by the very same rules as any other four-vector, as long as
you're careful to take into account *where* and *when* the
potential is being evaluated in each frame.

                     V.  Remarks 

From what I've said in this article it is possible to do
an extraordinary variety of calculations in relativistic
kinematics with a minimum of mathematical pain and a greatly
reduced amount of conceptual confusion.  Many things become
easier if you do as much of the work as possible in terms of
quantities with simple Lorentz transformation properties,
or what physicists call "manifest covariance."

Also, some of the simplicity of this notation carries
over into *general* relativity, where the metric g is
no longer a constant diagonal matrix, but codifies the
gravitational field!  Not to use covariant quantities in
such a situation would be patently insane.

Note that is ridiculous, with this formalism, to call
the time component of P the "mass" when what I've been
calling m shows up in so many nice places.

I hope this was comprehensible.

-- 
Matt McIrvin, grad student, Dept. of Physics, Harvard University
mcirvin@husc.harvard.edu      mumble mumble mumble mumble mumble